text:
\section*{INTRODUCTION. MOTIVATION AND ACTUALITY}

The estimates for monotone operators on cones of nonnegative functions play an important role in different branches of analysis, such as the theory of function spaces, the approximation theory, and applications to the theory of partial differential equations.

Let us describe shortly the general setting of the problem. Let $\mu$ be a nonnegative continuous Borel measure on ${\mathbb R}_+=(0,\infty); L_0=L_0({\mathbb R}_+)$ be the set of all Borel-measurable real valued functions, $L_0^+=\left\lbrace f \in L_0 : f\geq 0 \right\rbrace;$
$$\dot{L}_0=\left\lbrace f \in L_0: |f|< \infty \quad \mu-\text{almost everywhere} \right\rbrace, \quad  \dot{L}_0^+=\left\lbrace f \in \dot{L}_0: f\geq 0 \right\rbrace.$$
Let $\Omega \subset \dot{L}_0^+$  be a cone of functions, so that $g \in \Omega, \alpha\geq 0 \Rightarrow \alpha g \in \Omega$. Let $A:\dot{L}_0^+ \rightarrow L_0^+ $  be a monotone operator; $H_{\Omega}(A)$ be the norm of restriction of operator $A$ on $\Omega$
\begin{equation}\label{eq2.3.1}
H_{\Omega}(A)=\sup_{g \in \Omega}\left[ \left(\int\limits_{0}^{\infty} (Ag)^{q}d\gamma \right)^{1/q} \left(\int\limits_{0}^{\infty} g^{p}d\beta \right)^{-1/p} \right],
\end{equation}
with $0<p,q\leq\infty;\, \beta, \gamma$ being nonnegative continuous Borel measures on ${\mathbb R}_+.$

Typical examples of cones are the following:
\begin{equation}\label{eq2.3.2}
\Omega=\Omega_{(k)}= \left\lbrace g: 0\leq g(t)<\infty;\, g(t)t^{-k}\downarrow on \quad  {\mathbb R}_{+}\right\rbrace ,  k \in {\mathbb R};
\end{equation}
\begin{equation}\label{eq2.3.3}
\Omega=\Omega^{(m)}= \left\lbrace g: 0\leq g(t)<\infty;\, g(t)t^{-m}\uparrow on \quad {\mathbb R}_{+}\right\rbrace ,  m \in {\mathbb R};
\end{equation}

\begin{equation}\label{eq2.3.4}
\Omega=\Omega_{(k)}^{(m)}=\Omega_{k}\cap \Omega^{m}, m<k;
\end{equation}

 We see that $\Omega_{(0)}$ is the cone of nonnegative decreasing functions, $\Omega^{(0)}$  is the cone of nonnegative increasing functions, $\Omega_{(1)}^{(0)}$ is the cone of quasi-concave functions.

The typical operators connected with this problem are the following: the identity operator $A=I$ ; Hardy type operators  $A=H, A=\tilde{H} ;$ convolutions with nonnegative kernels and so on. For example, the corresponding Hardy- and Copson-type  operators are
\begin{equation}\label{eq2.3.5}
(Hg)(t)=\int\limits_{(0, t]} gd\mu; \quad (\tilde{H}g)(t)=\int\limits_{[t, \infty)}gd\mu.
\end{equation}

\textit{The general problem is to find necessary and sufficient conditions for the finiteness: $H_{\Omega}(A)<\infty$, and to establish order-sharp estimates for  $H_{\Omega}(A)$.}

Note that this problem is important for applications. We illustrate it by some examples from different branches of analysis.

\subsection*{Integration theory}\label{subsec2.1.1}
Let $\ddot{L}_0=\ddot{L}_0({\mathbb R}^n)$  be the subspace of all functions $f: {\mathbb R}^n \rightarrow {\mathbb R}$  measurable with respect to the $n$-dimensional Lebesgue measure $\mu_n$ , they are finite  $\mu_n$-almost everywhere, and such that for $f \in \ddot{L}_0$  the distribution function $\lambda_f$ is not identical to infinity, where
\begin{equation}\label{eq2.3.6}
\lambda_f(y)=\mu_n\left\lbrace x \in {\mathbb R}^n: |f(x)|>y\right\rbrace, y \in {\mathbb R}_+.
\end{equation}
Then $\lambda_f \in \Omega_{(0)}, \lambda_f(y)\rightarrow 0 \,(y\rightarrow +\infty).$ Let $f^{\ast} \in \Omega_{(0)}$
be the \textit{decreasing rearrangement} of the function $f,$ i.e. $f^{\ast}$  is a left-continuous inverse function for the positive decreasing function $\lambda_f,$ namely

\begin{equation}\label{eq2.3.7}
f^{\ast}(t)=\inf \left\lbrace y \in {\mathbb R}_+: \lambda_f(y)< t\right\rbrace, t \in {\mathbb R}_+.
\end{equation}
For  $f^{\ast} \in L_1(0,t), t \in {\mathbb R}_+,$ we define the \textit{elementary maximal function} $f^{\ast\ast}$:
\begin{equation}
f^{\ast\ast}(t)=t^{-1}\int\limits_{0}^{t}f^{\ast}d\tau.
\label{eq2.3.8}
\end{equation}
It is easy to see that  $f^{\ast\ast}\downarrow, tf^{\ast\ast}\uparrow$ , so that $f^{\ast\ast} \in \Omega_{(0)}^{(-1)}.$ These definitions, and the relations between $f^{\ast}$ and $f^{\ast\ast}$  are considered in details in the books \cite{BL}, \cite{BS}, and \cite{KPS}.

It is well known that functions $f$ and $f^{\ast}$  are equimeasurable: they have equal distribution functions.
 Therefore, integral properties of $f$ are determined by   $f^{\ast} \in \Omega_{(0)}$. For example,

$$\int_{{\mathbb R}^n} |f|^p d\mu_n=\int_{{\mathbb R}_+}(f^{\ast})^pd\mu_1, p>0.$$

Let  $Mf$ be the Hardy- Littlewood maximal function for  $f \in  L_1^{loc}({\mathbb R}^n)$:
\begin{equation}\label{eq2.3.9}
(Mf)(x) =\sup \left\lbrace \mu_n(Q)^{-1}\int_{Q} |f| d\mu_n : x \in Q\right\rbrace, x \in {\mathbb R}^n.
\end{equation}

Here, the supremum is taken over all cubes $Q$  with sides parallel to the axes containing the given point $x$. It is well known that integral properties of the maximal function play important role in problems of functional series theory, Fourier analysis, approximation theory and so on (see \cite{BS}). Integral properties of $Mf$ are determined by $ f^{\ast\ast} \in \Omega_{(0)}^{(-1)}$. For example,

$$\int_{{\mathbb R}^n} |Mf|^pd\mu_n \cong\int_{{\mathbb R}_+}(f^{\ast\ast})^pd\mu_1, p>0.$$

The cause is related to the following fundamental two-sided estimate:  $(Mf)^{*} \cong f^{**}$, see \\
 \cite[Ch. 2]{BS}. Therefore, when we use decreasing rearrangements in study of integral properties of functions (maximal functions), the problems are reduced to the corresponding integral properties on some cones of monotone functions.

\subsection*{Embedding theory of function spaces}\label{subsec2.1.2}
Here we consider some examples appearing in the embedding theory of function spaces.

\begin{center}
	\textbf{Weighted Lorentz spaces with general weights}
\end{center}

Let us recall two main variants of weighted Lorentz spaces with general weights $0<v,w \in \dot{L}_0^+$:
\begin{equation}\label{eq2.3.10}
\Lambda_{p, v}=\left\lbrace f: ||f||_\Lambda=\left( \int\limits_{0}^{\infty} (f^{\ast})^p vdt\right)^{1/p}<\infty \right\rbrace,
\end{equation}

\begin{equation}\label{eq2.3.11}
\Gamma_{q, w}=\left\lbrace f: ||f||_\Gamma=\left( \int\limits_{0}^{\infty} (f^{\ast\ast})^q wdt\right)^{1/q}<\infty \right\rbrace.
\end{equation}

Here $0<p,q <\infty.$  Classical Lorentz spaces correspond to power weights; about general properties and recent developments in the theory of these spaces see \cite{BL}, \cite{BS}, \cite{CPSS}, \cite{CRS}, \cite{FiR}-\cite{GPS}, \cite{LiK}.

Obviously, $\Gamma_{p, v}\subset \Lambda_{p, v}$ because $f^{\ast} \leq f^{\ast\ast}$. From the above definitions, it follows easily that

\begin{equation}\label{eq2.3.12}
\Lambda_{p, v}\subset \Lambda_{q, w} \Leftrightarrow G_{\Omega_{(0)}}(p,q)<\infty, 0<p,q <\infty,
\end{equation}

where for the cone $\Omega$ we define
\begin{equation}
 G_{\Omega}(p,q)=\sup_{g \in \Omega} \left[ \left( \int\limits_{0}^{\infty}g^qwdt\right)^{1/q}\left( \int\limits_{0}^{\infty}g^pvdt\right)^{-1/p}\right].
 \label{eq2.3.13}
\end{equation}

By the same reasons

\begin{equation}\label{eq2.3.14}
\Gamma_{p, v}\subset \Gamma_{q, w} \Leftrightarrow G_{\Omega_{(0)}^{(-1)}}(p,q)<\infty, 0<p,q <\infty.
\end{equation}

Further, (\ref{eq2.3.8}), (\ref{eq2.3.10}), and (\ref{eq2.3.11}) imply
\begin{equation}\label{eq2.3.15}
\Lambda_{p, v}\subset \Gamma_{q, w} \Leftrightarrow H_{\Omega_{(0)}}(p,q)<\infty, 0<p,q <\infty,
\end{equation}

where
\begin{equation}\label{eq2.3.16}
H_{\Omega_{(0)}}(p,q)=\sup_{g \in \Omega_{(0)}} \left[ \left( \int\limits_{0}^{\infty}\left(\int\limits_{0}^{t}gd\tau\right)^q t^{-q}wdt\right)^{1/q}\left( \int\limits_{0}^{\infty}g^pvdt\right)^{-1/p}\right].
	\end{equation}

	Therefore, the integral properties of the identity operator, and Hardy type operators on the cone of decreasing functions  play a significant role in embedding problems for Lorentz spaces.

	\begin{center}
		\textbf{Besov spaces with generalized smoothness}
	\end{center}

Let $f \in L_p({\mathbb R}^n), 1\leq p\leq \infty.$ Consider the modulus of continuity of order $k$ in  $L_p({\mathbb R}^n),$

\begin{equation}\label{eq2.3.17}
\omega_p^k(f;t)=\sup \left\lbrace ||\Delta_h^k f||_{L_p} : h \in {\mathbb R}^n,\, |h|\leq t\right\rbrace, t \in {\mathbb R}_+.
	\end{equation}

Introduce Besov spaces with generalized smoothness:

\begin{equation}\label{eq2.3.18}
B_{p\theta}^{v(\cdot)}({\mathbb R}^n)=\left\lbrace f \in L_p: ||f||_B=||f||_{L_p}+||f||_b<\infty\right\rbrace,
\end{equation}

\begin{equation}\label{eq2.3.19}
||f||_b=\left( \int\limits_{0}^{\infty}\omega_p^k(f;t)^{\theta}v(t)dt\right)^{{1}/{\theta}},
\end{equation}
where $0<\theta<\infty.$ Classical Besov space  $B_{p\theta}^{\alpha}({\mathbb R}^n)$ corresponds to the power weight $v(t)=t^{-\alpha\theta-1}$, $0<\alpha<k,$ see S.M. Nikolskii \cite[Ch. 4]{Nik}. Different variants of Besov spaces with generalized smoothness were considered in the papers of M.Z. Berkolaiko, A.S. Dzhafarov, A. Gogatishvili, M.L. Goldman, D. Haroske, G.A. Kalyabin, H.-G. Leopold, P.I. Lizorkin, S. Moura, Yu.V. Netrusov, B. Opic, P. Oswald, W. Sickel, and many others ; see surveys and references in \cite{FL},  \cite{G1}, \cite{KL}.

Let us note that the embedding problems for Besov spaces may be reduced to the estimates of some positive operators on the cone  $\Omega_{(k)}^{(0)}$. Indeed, it is well known that

$$\Omega_{(k)}^{(0)}\approx\Omega\equiv \left\lbrace h(t)=\omega_p^k(f;t); f \in L_p({\mathbb R}^n)\right\rbrace.$$

Here the equivalence means that functions from these cones are pointwise comparable, so that there exists a constant  $c=c(k) \in [1,\infty)$:

$$h_1 \in \Omega \Rightarrow \exists h_2 \in \Omega_{(k)}^{(0)};\quad  h_1  \in \Omega_{(k)}^{(0)} \Rightarrow \exists h_2 \in \Omega: c^{-1}\leq h_1 h_2^{-1}\leq c.$$

Therefore, we can replace $\Omega$  by $\Omega_{(k)}^{(0)}$ in the considerations below.
Let us present some examples. For $p=q$

 \begin{equation}
 B_{p\theta}^{v(\cdot)}({\mathbb R}^n) \subset B_{q\tau}^{w(\cdot)}({\mathbb R}^n) \Leftrightarrow G_{\Omega_{(k)}^{(0)}} (\theta,\tau)<\infty,
 \label{eq2.3.20}
 \end{equation}

Let $1\leq p < q\leq \infty,\, k>n(\frac{1}{p}-\frac{1}{q}).$ Then,

 \begin{equation}\label{eq2.1.21}
B_{p\theta}^{v(\cdot)}({\mathbb R}^n) \subset L_q({\mathbb R}^n) \Leftrightarrow \tilde{G}_{\Omega_{(k)}^{(0)}}<\infty,
\end{equation}

where

 \begin{equation}\label{eq2.1.22}
\tilde{G}_{\Omega_{(k)}^{(0)}}=\sup_{g \in \Omega_{(k)}^{(0)}} \left[ \left( \int\limits_{0}^{\infty}\left[ g(t)t^{-\alpha}\right] ^{q^{\ast}} \frac{dt}{t}\right)^{{1}/{q^{\ast}}}\left(\int\limits_{0}^{\infty}g(t)^{\theta}vdt\right)^{-{1}/{\theta}}\right],
\end{equation}

\begin{equation}\label{eq2.1.23}
q^{\ast}=q,\, 1\leq q<\infty; \quad q^{\ast}=1,\, q=\infty;\, \alpha= n\left( \frac{1}{p}-\frac{1}{q}\right).
	\end{equation}

	Last conclusion is based on the following sharp estimate (see \cite{G1}, \cite{He}, \cite{U1}, and \cite{U2}):
	for  $1\leq p<q\leq\infty, c=c\left( p, q, k, n\right) \in {\mathbb R}_+,$

	\begin{equation}\label{eq2.1.24}
	||f||_{L_q}\leq c\left\lbrace \left( \int\limits_{0}^{\infty}\left[ \omega^k_p(f;t)t^{-\alpha}\right] ^{q^{\ast}} \frac{dt}{t}\right)^{{1}/{q^{\ast}}}+||f||_{L_p}\right\rbrace.
		\end{equation}

	\begin{remark} 	The embedding problems (\ref{eq2.3.20}) for $1\leq p, q\leq \infty$, (\ref{eq2.1.21}) were studied in \cite{G1}-\cite{G6}, where the criteria for embeddings in explicit form were established. They stimulated our interest to the estimates for monotone operators on different cones of functions with monotonicity conditions.
\end{remark}

	\subsection*{Interpolation Theory}

	Let  $\left\lbrace A_0, A_1 \right\rbrace $ be a so-called Banach pare (see \cite{KPS}), so that we can add the elements from $A_0$ and $A_1.$ For  $a \in A_0 + A_1$ the famous Peetre's  $K$-functional is defined as
$$K(t, a)\equiv K(t, a; A_0, A_1)=\inf_{a_0+a_1=a} \left\lbrace ||a_0||_{A_0}+t||a_1||_{A_1}\right\rbrace.$$

Here the infimum is taken over all representations $a_0+a_1=a, a_0 \in A_0, a_1 \in A_1.$

It is obvious that $K(\cdot , a) \in \Omega_{(1)}^{(0)}.$ For
$v \in L_0^+$ we define the intermediate space

$$(A_0, A_1)_{v, q}=\left\lbrace a \in A_0+A_1: ||a||_{v, q}=\left( \int_{0}^{\infty}K(t, a)^qv(t)dt\right)^{1/q} <\infty\right\rbrace,$$

These spaces play a significant role in real method of interpolation, see books \cite{BL}, \cite{BS}, \cite{KPS}, \cite{Tri} and references there. The classical case considered by J. Peetre corresponds to $v(t)=t^{-\theta q-1}, 0<\theta <1.$ Many authors (see some names below) studied general spaces.

When we study the relations between different intermediate spaces, we have to invoke the results on the estimates on the cone  $\Omega_{(1)}^{(0)}.$

The question arises about relationship between the estimates that are valid on the cones with properties of monotonicity and, for example, the estimates on the set of all functions from $\dot{L}_0^+.$ In Section 2 we demonstrate  that such difference can be essential.

These actual problems were actively studied recently by many researchers. In this context, we mention here some names of researchers in alphabetical order:

M. Arino, S. Astashkin, C. and G. Bennett, E. Berezhnoi, J. Bergh, O. Besov, S. Bloom, N. Bokayev, Y. Brudnyi, V. Burenkov, M. Carro, A. Cianchi,  A. Gogatishvili, M. Goldman, K.-G. Grosse-Erdmann, D. Haroske, H. Heinig, S. Janson, G. Kalyabin, A. Kaminska, G. Karadzhov, R. Kerman, V. Kolyada, N. Krugljak, A. Kufner, L. Maligranda, J. Martin,   M. Milman, B. Muckenhoupt, Yu. Netrusov, J. Neves, E. Nursultanov, V. Ovchinnikov, R. Oinarov, K. Oskolkov, B. Opic, L.-E. Persson, L. Pick, D. Prokhorov, E. Sawyer, R. Sharpley, V. Sickel, G. Sinnamon, J. Soria, V. Stepanov,  S. Tikhonov, H. Triebel, P. Ulyanov,...

The paper is organized as follows. In Section 1 we present the survey of general results concerning exact calculations of the norms for monotone operators on the cones of functions with monotonicity conditions in general ideal spaces. These results modify the approach developed in paper \cite{BG} by V. Burenkov and M. Goldman. Section 2 contains applications of these approaches for calculations of the norms for different concrete operators such as integral operators on the cones, the dilation operator and embedding operators in weighted Lorentz spaces with general weights. In Section 3 we present (without proves) the order-sharp estimates for Hardy-type operators on the cones in wider set of parameters than in Section 2.

\section{CALCULATIONS OF THE NORMS OF MONOTONE OPERATOR ON THE CONES}\label{sec1}

\subsection{Some general theorems on the cones of functions with monotonicity properties}\label{sbsec1.1}

Let $(M, \varSigma_{M}, \beta),\, (N, \varSigma_{N}, \gamma)$  be measure spaces with nonnegative $\sigma$-additive full measures $\beta, \gamma;\,$ $S(M, \varSigma_{M}, \beta),\, S(N, \varSigma_{N}, \gamma)$ be the spaces of real-valued measurable functions. First, we recall the concept of the ideal space (shortly: IS) $X \subset S(M, \varSigma_{M}, \beta)$ with the  (quasi)norm $\|\cdot\|_X$ (see \cite{BaG}).

The space $X$ is called an IS if the following conditions are satisfied:

\begin{align*}
(B1)\quad \|f\|_X=0 & \Leftrightarrow f=0 \quad (\beta -a.e.), \quad  \|\alpha f\|_X= \alpha \|f\|_X, \, \alpha \geq 0; \\
& \exists C \in [1, \infty): \|f+g\|_X \leq C\left(\|f\|_X+ \|g\|_X\right) ;\\
(B2)\quad 0\leq f\leq g & \quad (\beta -a.e.)\Rightarrow \|f\|_X \leq \|g\|_X;\\
(B3)\quad 0\leq f_m \uparrow f & \quad (\beta -a.e.)\Rightarrow \|f_m\|_X \uparrow \|f\|_X;\\
(B4)\quad \|f\|_X<\infty & \Rightarrow   |f|< \infty \quad (\beta -a.e.).
\end{align*}

The space $X$ is a normed Banach space if $C=1$ in the triangle inequality,  and it is a quasi-normed quasi-Banach space if $C>1$.

We say that the (quasi)norm  in the ideal space  $X \subset S(M, \varSigma_{M}, \beta)$   is \textit{order-continuous} if

\begin{equation}\label{eq1.1.1}
\left\lbrace x_{m} \in X, m \in {\mathbb N};\, 0\leq x_{m}\downarrow 0 \quad \beta-\text{a.e.}\right\rbrace \Rightarrow \|x_{m}\|_{X}\downarrow 0.
\end{equation}

An ideal space $X \subset S(M, \varSigma_{M}, \beta)$ is $l_{p}$-concave for $p \in {\mathbb R}_{+}$ if

\begin{equation}\label{eq1.1.2}
\left( \sum_{m} \|x_{m}\|_{X}^{p}\right)^{1/p} \leq \left\| \left( \sum_{m}|x_{m}|^{p}\right)^{1/p}\right\| _{X}.
\end{equation}

An ideal space $Y \subset S(N, \varSigma_{N}, \gamma)$ is $l_{q}$-convex for $q \in {\mathbb R}_{+}$ if
\begin{equation}\label{eq1.1.3}
\left\|  \left( \sum_{m}|y_{m}|^{q}\right)^{1/q}\right\|_{Y}\leq \left( \sum\limits_{m}||y_{m}||_{Y}^{q}\right)^{1/q}.
\end{equation}

It means that the convergence of the series in the right-hand side of (\ref{eq1.1.2}) or (\ref{eq1.1.3}) implies the convergence of the series in the left-hand side, and the corresponding inequalities hold.

Note that any normed ideal space is $l_{1}$-convex, and $l_{q}$-convexity for $0<q<1$  implies the triangle inequality in the form

\begin{equation}\label{eq1.1.4}
\|f+g\|_{Y}\leq \left( \|f\|_{Y}^{q}+ \|g\|_{Y}^{q}\right)^{1/q} \leq 2^{1/q -1}\left( \|f\|_{Y}+\|g\|_{Y}\right).
\end{equation}

Also, note that  $Y=L_{q}(N, \gamma), \, 0<q<\infty$, is $l_{\rho}$-convex for any $\rho \in (0, q]$  (see Lemma \ref{lm1.3.7} below), and it is  $l_{p}$-concave for any $p \in [q, \infty).$

Let $D \subset X$ be a cone of nonnegative functions such that
$$f, g \in D;\, c_1, c_2 \geq 0 \Rightarrow c_1 f+c_2 g \in D.$$

An operator $T: D\rightarrow Y$    is called   $l_{r}$-convex for $0<r<\infty$ if for any $f_{m} \in D, m \in {\mathbb Z}$ such that  $\left(\sum\limits_{m}f_{m}^{r}\right)^{1/r} \in D,$

\begin{equation}\label{eq1.1.5}
\left| T\left[  \left( \sum\limits_{m}f_{m}^{r}\right)^{1/r}\right]\right|  \leq \left( \sum\limits_{m}|Tf_{m}|^{r}\right)^{1/r}
\end{equation}
$\gamma$-almost everywhere on  $N$;  and for all  $f \in D; \alpha\geq 0\Rightarrow T[\alpha f]=\alpha T[f].$

Note that $l_{1}$-convexity of operator $T$ coincides with its countable sublinearity:

$$
\left| T\left[\left(\sum\limits_{m}f_{m}\right)\right]\right|  \leq \left( \sum\limits_{m}|Tf_{m}|\right).
$$

An operator $T$ is called \textit{ monotone} if
\begin{equation}\label{eq1.1.6}
\left\lbrace f, g \in D;\, 0\leq f \leq g \quad \beta-\text{a.e.}\right\rbrace \Rightarrow \left\lbrace  0\leq Tf \leq Tg \quad \gamma-\text{a.e.}\right\rbrace.
\end{equation}

An example of   $l_{r}$-convex monotone operator gives the operator

$$
T\left[f\right] = \left( L[f^{r}]\right)^{1/r},
$$

where  $L$ is a countable sublinear monotone operator. Moreover, this formula gives the correspondence between  $l_{r}$- convex and countable sublinear operators.

We will consider the case   $M=J:=(a,b),\, -\infty \leq a , b \leq \infty,$ with nonnegative continuous Borel measure  $\beta,$ and restrictions of operators on the following cones of nonnegative decreasing left-continuous functions on  $J:=(a,b):$

\begin{align}\label{eq1.1.7}
\Omega &= \left\lbrace g \in X:\, 0\leq g \downarrow ;\, g(t)=g(t-0),\, t \in (a,b)\right\rbrace,\nonumber\\
\dot{\Omega}&= \left\lbrace g \in \Omega:\, \lim\limits_{t \rightarrow b-0} g(t)=0\right\rbrace.
\end{align}

So, we define the norms of such restrictions

\begin{equation}\label{eq1.1.8}
\|T\|_{\Omega} =\sup \left\lbrace \|T[g]\|_{Y}:  g \in \Omega,\, \|g\|_{X}\leq 1\right\rbrace,
\end{equation}

\begin{equation}\label{eq1.1.9}
\|T\|_{\dot{\Omega}} =\sup \left\lbrace \|T[g]\|_{Y}:  g \in \dot{\Omega},\, \|g\|_{X}\leq 1\right\rbrace.
\end{equation}

Denote

\begin{equation}\label{eq1.1.10}
\dot{\Omega}_{0}:=\left\lbrace \chi_{(a, t]}: a<t<b\right\rbrace,\,  \Omega_{0}:=\dot{\Omega}_{0}\bigcup \chi_{(a, b)};
\end{equation}

\begin{equation}\label{eq1.1.11}
F(x, t) =T\left[\chi_{(a, t]}\right](x),\, a<t<b;\quad  F(x, b) =T\left[\chi_{(a, b)}\right] (x).
\end{equation}

\begin{theorem}\label{th1.1.1}
	Let $0<p \leq q\leq r<\infty;\, X \subset S(J, \beta)$ be an ideal    $l_{p}$-concave space with order-continuous (quasi)norm;
	\begin{equation}\label{eq1.1.12}
	||\chi_{(a, b)}||_{X}=\infty;
	\end{equation}
$Y \subset S(N, \gamma)$ be an ideal $l_{q}$-convex space , and $T: \dot{\Omega}\rightarrow Y$ be an $l_{r}$- convex positive operator. Then,

\begin{equation} \label{eq1.1.13}
||T||_{\dot{\Omega}} =||T||_{\dot{\Omega}_{0}}:=\sup\limits_{a<t<b} \left[ ||F(\cdot, t)||_{Y} ||\chi_{(a,t]}(\cdot)||_{X}^{-1}\right].
\end{equation}
\end{theorem}

\begin{remark}\label{rem1.1.2}
Note that in the case of nondegeneracy
$$||\chi_{(a, b)}||_{X}=\infty \Rightarrow \Omega=\dot{\Omega} \Rightarrow ||T||_{\Omega}=||T||_{\dot{\Omega}},$$
because

\begin{equation}\label{eq1.1.14}
\left\lbrace 0\leq g\downarrow,\, \lim\limits_{t\rightarrow b-0} g(t)>0\right\rbrace \Rightarrow g \notin X.
\end{equation}

\end{remark}

It means that $||T||_{\Omega}=||T||_{\dot{\Omega}}$ in case (\ref{eq1.1.12}).

Now, let us consider the case
\begin{equation}\label{eq1.1.15}
||\chi_{(a, b)}||_{X}<\infty.
\end{equation}

 Then $\Omega_{0}\subset \Omega$  and we have the following statement.

\begin{theorem}\label{th1.1.3}
	Let $0<p\leq q\leq r<\infty;$ $X \subset S(J, \beta)$   be an ideal $l_{p}$- concave space with order-continuous (quasi)norm, and condition (\ref{eq1.1.15}) be satisfied. Let $Y \subset S(J, \gamma)$  be an ideal $l_{q}$ - convex  space, and  $T:\Omega \rightarrow Y$ be an $l_{r}$- convex positive operator. Then,

	\begin{equation}\label{eq1.1.16}
	||T||_{\dot{\Omega}} =||T||_{\dot{\Omega}_{0}}:=\sup\limits_{a<t<b} \left[ ||F(\cdot, t)||_{Y} ||\chi_{(a,t]}(\cdot)||_{X}^{-1}\right],
		\end{equation}

	\begin{equation}\label{eq1.1.17}
||T||_{\Omega}=||T||_{\Omega_{0}}:=\max\left\lbrace ||T||_{\dot{\Omega}_{0}},  ||F(\cdot, b)||_{Y} ||\chi_{(a,b)}(\cdot)||_{X}^{-1}\right\rbrace.
			\end{equation}
\end{theorem}

\subsection{Corollaries}

\begin{corollary}\label{cl1.3.5}
Let $0<p \leq \min \left\lbrace q,r\right\rbrace <\infty;\, X \subset S(J, \beta)$   be an ideal  $l_p$-concave space with order-continuous (quasi)norm, and condition (\ref{eq1.1.12}) be satisfied. Let $Y=L_{q}(N, \gamma),$   and $T$  be an $l_r$-convex positive operator. Then  equality   (\ref{eq1.1.13}) holds.
\end{corollary}

\begin{corollary}\label{cl1.3.6}
	Let $0<p \leq \min\left\lbrace q,r\right\rbrace <\infty;\, X \subset S(J, \beta)$   be an ideal  $l_p$-concave space with order-continuous (quasi)norm, and condition (\ref{eq1.1.15}) be satisfie. Let $Y=L_{q}(N, \gamma),$   and $T$  be an $l_r$-convex positive operator. Then equalities   (\ref{eq1.1.16}), (\ref{eq1.1.17}) hold.
\end{corollary}

 To prove these corollaries we need a lemma concerning properties of convexity for Lebesgue spaces.

\begin{lemma}\label{lm1.3.7}
	Let  $0<q<\infty.$ Then $Y=L_{q}(N, \gamma)$    is an ideal  $l_{\rho}$-convex  space for any $\rho \in (0,q].$
\end{lemma}

\begin{proof}[Proof of Corollaries \ref{cl1.3.5} and \ref{cl1.3.6}]
Denote  $\rho=\min\left\lbrace q, r\right\rbrace.$    Then we have $p\leq\rho \leq r.$   According to Lemma \ref{lm1.3.7} we see that  $Y=L_{q}(N, \gamma)$    is   $l_{\rho}$-convex, and we can apply Theorem \ref{th1.1.1} with $\rho$  instead of  $q.$   Thus,  equality (\ref{eq1.1.13}) holds for $Y=L_{q}(N, \gamma).$    Proof of Corollary \ref{cl1.3.6} is the same. We only apply Theorem \ref{th1.1.3} instead of Theorem \ref{th1.1.1}.

\end{proof}

\subsection{Generalization of monotonicity condition}\label{sbsec1.3}

Analogous results are valid on the cones of functions with the monotonicity property with respect to a given positive function $k \in C(J).$ Define
\begin{equation}\label{eq1.4.1}
\Omega_{k}\equiv \Omega(X,k)\!=\left\lbrace g\!\in X\!: g \geq 0,\, g(t)/k(t)\downarrow;\, g(t)=g(t-0),\, t \in (0,b)\right\rbrace ,
\end{equation}
\begin{equation}\label{eq1.4.2}
\dot{\Omega}_{k}\equiv \dot{\Omega}(X,k)=\left\lbrace g \in \Omega_{k}:\,  g(t)/k(t)\rightarrow 0, t \rightarrow b-0\right\rbrace
\end{equation}
(in this notation  $\Omega_{1}=\Omega,\,\dot{\Omega}_{1}=\dot{\Omega},$  see (\ref{eq1.1.7})). Denote

 \begin{equation}\label{eq1.4.3}
\begin{aligned}
& \dot{\Omega}_{k,0}\equiv \dot{\Omega}_{0}(X,k)=\left\lbrace k\chi_{(a,t]}: a<t<b\right\rbrace,\\
&{\Omega}_{k,0}\equiv {\Omega}_{0}(X,k)=\dot{\Omega}_{k,0}\cup \left\lbrace k\chi_{(a,b)}\right\rbrace.
\end{aligned}
\end{equation}

\begin{theorem}\label{th1.4.1}
Let the conditions of Theorem \ref{th1.1.1} hold with the replacement of  $\Omega$     by  $\Omega_{k}$    and condition (\ref{eq1.1.12}) being replaced by
 \begin{equation}\label{eq1.4.4}
|| k\chi_{(a,b)}||_{X}=\infty.
\end{equation}
Then
 \begin{equation}\label{eq1.4.5}
||T||_{\dot{\Omega}_{k}}=||T||_{\dot{\Omega}_{k,0}}:=\sup_{a<t<b}\left[ ||F_{k}(\cdot, t)||_{Y} ||k(\cdot)\chi_{(a,t]}(\cdot)||^{-1}_{X}\right],
\end{equation}
where we denote
 \begin{equation}\label{eq1.4.6}
F_{k}(x, t)=T[k\chi_{(a,t]}](x),\,a<t<b;\, F_{k}(x, b)=T[k\chi_{(a,b)}](x).
\end{equation}
\end{theorem}

\begin{remark}\label{rm1.4.2}
	Note that
	\begin{equation}\label{eq1.4.7}
	||k\chi_{(a,b)}||_{X}=\infty\Rightarrow {\Omega}_{k}=\dot{\Omega}_{k}\Rightarrow ||T||_{{\Omega}_{k}}=||T||_{\dot{\Omega}_{k}}
	\end{equation}
	because
	$\left\lbrace 0\leq g \downarrow,\, \lim\limits_{t\rightarrow b-0}[g(t)/k(t)]>0\right\rbrace$ leads to $ g \notin X.$
\end{remark}

Now we consider the case
\begin{equation}\label{eq1.4.8}
||k\chi_{(a,b)}||_{X}<\infty.
\end{equation}
In this case  ${\Omega}_{k, 0}\subset {\Omega}_{k},$    and we have the following statement.

\begin{theorem}\label{th1.4.3}
	Let the conditions of Theorem \ref{th1.1.3} hold with the replacement of    $\Omega$   by   ${\Omega}_{k}$   and condition (\ref{eq1.1.15}) being replaced by (\ref{eq1.4.8}). Then

	\begin{equation}\label{eq1.4.9}
	||T||_{\dot{\Omega}_{k}}=||T||_{\dot{\Omega}_{k,0}}:=\sup_{a<t<b}\left[ \left\|F_{k}(\cdot, t)\right\|_{Y} \left\| k(\cdot)\chi_{(a,t]}(\cdot)\right\|^{-1}_{X}\right],
	\end{equation}
		\begin{equation}\label{eq1.4.10}
	||T||_{{\Omega}_{k}}=||T||_{{\Omega}_{k,0}}:=\max\left\lbrace \left\| T\right\|_{\dot{\Omega}_{k,0}}, \left\|F_{k}(\cdot, b)\right\|_{Y}||k(\cdot)\chi_{(a,b)}(\cdot)||^{-1}_{X}\right\rbrace.
	\end{equation}
\end{theorem}
\begin{proof}[Proof of Theorem \ref{th1.4.1}]
	Formally, this theorem is more general than Theorem \ref{th1.1.1}, but we can easily reduce it to that one.

	We consider the $l_{r}$-convex positive operator $T: \dot{\Omega}(X, k)\rightarrow Y.$ Let us define
	\begin{equation}\label{eq1.4.11}
X_{k}:=\left\lbrace f \in S(J, \beta):\, fk \in X\right\rbrace =\left\lbrace f=g/k:\, g \in X\right\rbrace, ||f||_{X_{k}}= ||kf||_{X}.
\end{equation}
	Then we have the equivalence:
		\begin{equation}\label{eq1.4.12}
	g \in \dot{\Omega}(X, k)\Leftrightarrow f=g/k \in \dot{\Omega}(X_{k}, 1);\, ||f||_{X_{k}}= ||kf||_{X},
	\end{equation}
	see (\ref{eq1.4.1}), (\ref{eq1.4.2}), (\ref{eq1.4.11}). Therefore,
	\begin{equation*}
	||T||_{\dot{\Omega}(X,k)}\!=\!\sup\left\lbrace \frac{||T[g]||_{Y}}{||g||_{X}}\!: g \!\in\!\dot{\Omega}(X, k)\right\rbrace\!=\!
	\sup\left\lbrace  \frac{||T[kf]||_{Y}}{||f||_{X_{k}}}\!: 	f\!\in\!\dot{\Omega}(X_{k}, 1)\right\rbrace.
	\end{equation*}

	Note that $X_{k}$   together with $X$    is an ideal  $l_{p}$-concave space with order-continuous (quasi)norm, and the operator
	$$T_{k}: \dot{\Omega}(X_{k}, 1)\rightarrow Y; \, T_{k}[f]:=T[kf], \, f \in \dot{\Omega}(X_{k}, 1),$$
	is  $l_{r}$-convex together with the operator $T.$ Thus, we apply Theorem \ref{th1.1.1} and obtain that
		\begin{equation*}
	||T||_{\dot{\Omega}(X,k)}=\sup_{a<t<b}\left\lbrace \left\|T[k\chi_{(a,t]}]\right\|_{Y}\left\|k\chi_{(a,t]}\right\|^{-1}_{X}\right\rbrace.
	\end{equation*}
\end{proof}

\begin{proof}[Proof of Theorem \ref{th1.4.3}]
It is essentially the same as one for Theorem  \ref{th1.4.1}. We only replace $\dot{\Omega}(X,k)$  by ${\Omega}(X,k),$ $\dot{\Omega}(X_k, 1)$  by  ${\Omega}(X_k, 1),$ and apply Theorem \ref{th1.1.3} instead of Theorem \ref{th1.1.1}, taking into account that (\ref{eq1.1.15}) has now the form \eqref{eq1.4.8}.
\end{proof}

\section{Applications}\label{sec1.5}

\subsection{Calculation of the norm of an integral operator on the cone of functions with the monotonicity property}\label{subsec1.5.1}

Let $K=K(x, \tau)$ be a nonnegative measurable function of variables $(x, \tau) \in N\otimes J,$ where $(N; \gamma)$  and  $(J; \mu)$ are measure spaces with the nonnegative     $\sigma$-finite    $\sigma$-additive measure $\gamma$  and the nonnegative continuous Borel measure    $\mu$ on  $J=(a,b).$
\begin{equation}\label{eq1.5.1}
T_{r\mu}[f](x)=\left( \int_{(a, b)}K(x, \tau)|f(\tau)|^{r}d\mu(\tau)\right)^{1/r},\quad r \in (0, \infty).
\end{equation}
It is  an $l_{r}$-convex monotone operator. For $r=1$ its restriction on the set of nonnegative  $\mu$ - measurable functions coincides with the restriction of the linear integral operator

\begin{equation}\label{eq1.5.2}
T[f](x)=\int_{(a, b)}K(x, \tau)f(\tau)d\mu(\tau).
\end{equation}

All the results of Section 1 are applicable here. In particular, for the restrictions of  $T_{r\mu}$ on the cone    $\Omega_{k}$   application  of Theorems \ref{th1.4.1} and \ref{th1.4.3} gives the following results.

\begin{theorem}\label{th1.5.1}
Let  $0<p\leq q\leq r<\infty;$ $X \subset S(J, \beta)$ be an ideal $l_p$--concave space with order-continuous (quasi) norm;  $Y \subset S(N, \gamma)$   be an $l_q$-convex ideal space, and
\begin{equation}\label{eq1.5.3}
\|k\chi_{(a,b)}\|_{X}=\infty.
\end{equation}
Then,
\begin{equation}\label{eq1.5.4}
\|T_{r\mu}\|_{\Omega_k}=\|T_{r\mu}\|_{\dot{\Omega}_{k,0}}:=\sup_{a<t<b}\left\lbrace  \|T_{r\mu}[k\chi_{(a,t]}]\|_{Y}\|k\chi_{(a,t]}\|_{X}^{-1}\right\rbrace .
\end{equation}
Here
\begin{equation}\label{eq1.5.5}
T_{r\mu}[k\chi_{(a,t]}](x)= \int_{(a, t]}K(x, \tau)k(\tau)^{r}d\mu(\tau),\quad x \in N.
\end{equation}
In the case
\begin{equation}\label{eq1.5.6}
\|k\chi_{(a,b)}\|_{X}<\infty,
\end{equation}
  we have $\|T_{r\mu}\|_{\dot{\Omega}_{k}}=\|T_{r\mu}\|_{\dot{\Omega}_{k,0}}$ (see \eqref{eq1.5.4}),

  \begin{equation}\label{eq1.5.7}
  \|T_{r\mu}\|_{\Omega_k}=\max \left\lbrace \|T_{r\mu}\|_{\dot{\Omega}_{k,0}};\quad  \|T_{r\mu}[k\chi_{(a,b)}]\|_{Y}\|k\chi_{(a,b)}\|_{X}^{-1}\right\rbrace .
  \end{equation}
\end{theorem}

\begin{remark}\label{rm1.5.2}
For the restrictions on the cone $\Omega=\Omega_1$ of nonnegative decreasing left-continuous functions we have to set $k(\tau)=1$ in \eqref{eq1.5.3}-- \eqref{eq1.5.7}.
\end{remark}

\begin{remark}\label{rm1.5.3}
In the case  $Y=L_q(N, \gamma)$ the results of Theorem \ref{th1.5.1} hold if  $0<p\leq \min \left\lbrace q,r\right\rbrace <\infty$, see Corollaries \ref{cl1.3.5} and \ref{cl1.3.6}.
\end{remark}

\begin{remark}\label{rm1.5.4}
	As a particular case of operator \eqref{eq1.5.1} we will consider later the case where  $(N, \gamma)=(J, \gamma)$ with the nonnegative continuous  Borel measure $\gamma$ on $J=(a,b)$, and $T_{r\mu}$ coincides with the generalized Hardy-type operator
	\begin{equation}\label{eq1.5.8}
	A_{r\mu}[f](x)=\left( \int_{(a, x]}|f(\tau)|^{r}d\mu(\tau)\right)^{1/r},\quad x \in (a, b).
	\end{equation}
	Then,
	\begin{align}
	A_{r\mu}[k\chi_{(a,t]}](x)=\left( \int_{(a, x]}k(\tau)^{r}d\mu(\tau)\right)^{1/r},\quad x \leq t;\nonumber\\
	A_{r\mu}[k\chi_{(a,t]}](x)=\left( \int_{(a, t]}k(\tau)^{r}d\mu(\tau)\right)^{1/r},\quad x > t.\label{eq1.5.9}
	\end{align}
	and we have the equalities: in case \eqref{eq1.5.3}

\begin{equation}\label{eq1.5.10}
\|A_{r\mu}\|_{\Omega_k}=\|A_{r\mu}\|_{\dot{\Omega}_{k,0}}:=\sup_{a<t<b}\left\lbrace \left\|A_{r\mu}[k\chi_{(a,t]}]\right\|_{Y}\left\|k\chi_{(a,t]}\right\|_{X}^{-1}\right\rbrace;
\end{equation}
in case \eqref{eq1.5.6} formula \eqref{eq1.5.10} remains true for  $\|A_{r\mu}\|_{\dot{\Omega}_{k}}$, but

 \begin{equation}\label{eq1.5.11}
\|A_{r\mu}\|_{\Omega_k}=\max \left\lbrace \|A_{r\mu}\|_{\dot{\Omega}_{k,0}};\quad  \left\| A_{r\mu}[k\chi_{(a,b)}]\right\|_{Y}\left\|k\chi_{(a,b)}\right\|_{X}^{-1}\right\rbrace .
\end{equation}

\end{remark}
\subsection{Calculation of the associated norm over the cone of functions with the monotonicity property}\label{subsec1.5.2}

Let us describe the results concerning calculation of associate norms mentioned in {\bf Introduction}. For   $0<p\leq \infty$ and for nonnegative Lebesgue-measurable functions $f, g \in \dot{L}^{+}_{0}({\mathbb R}_{+})$   let us define
\begin{align*}
C_{p}(f, g)=\left(\, \int_{{\mathbb R}_+}fgdt\right) \left(\int_{{\mathbb R}_+}g^{p}dt\right)^{-1/p};\\
A(f, p)=\sup_{g \in  \dot{L}^{+}_{0}} C_{p}(f, g);\quad B(f, p)=\sup_{g \in  \Omega_k} C_{p}(f, g).
\end{align*}
These quantities present the associate norms on the sets $\dot{L}^{+}_{0}({\mathbb R}_{+})$ and $\Omega_k$, respectively. Here,  $k$ is a given positive continuous function on $[0,\infty)$. Classical results are the following:
 \begin{equation}\label{eq1.5.12}
A(f, p)=\begin{cases}\infty,& 0<p<1, \, f\neq 0;\\
\|f\|_{L_{p'}}, & 1\leq p\leq\infty,\frac{1}{p}+\frac{1}{p'}=1.\end{cases}
\end{equation}
The first assertion illustrates nonexistence of bounded linear non-zero functionals on $L_p({\mathbb R}_+)$  for $0<p<1$. The second one shows the calculation of the associate norm for the norm in $L_p({\mathbb R}_+)$, $1\leq p\leq\infty$.

The results on the cone are essentially different. Namely,
let $0<p\leq 1$. We define
\begin{equation*}
D_{p}(f, k, t)=\left( \int\limits_{(0,t]}fk d\tau\right) \left(\int\limits_{(0,t]}k^{p}d\tau\right)^{-1/p}.
\end{equation*}
\begin{theorem}\label{th1.5.5}
	In  the notation of this Section
	\begin{equation}\label{eq1.5.13}
	B_k(f, p)=\sup \left\lbrace  D_{p}(f, k, t):\, 0<t<\infty\right\rbrace.
	\end{equation}
\end{theorem}

\begin{proof}
	These results follow immediately from Theorems \ref{th1.4.1} and \ref{th1.4.3}. Indeed, here $X=L_p({\mathbb R}_{+}, \beta)$, $0<p\leq 1$, $\beta$  is the Lebesgue measure on ${\mathbb R}_{+}$; $Y=L_1({\mathbb R}_{+}, \gamma)$, $d\gamma(t)=g(t)dt$; $T=I$ is the identity operator which is $l_1$-convex, and  Theorems \ref{th1.4.1} and \ref{th1.4.3} are applicable here.

	If $\int\limits_{(0, \infty)}k^pd\tau=\infty$, then \eqref{eq1.5.13} follows from Theorem \ref{th1.4.1}. In the case  $\int\limits_{(0, \infty)}k^pd\tau<\infty$  we apply Theorem \ref{th1.4.3}, so that \eqref{eq1.4.10} implies
	\begin{equation}\label{eq1.5.14}
	B_k(f, p)=\max \left\lbrace \sup \left\lbrace  D_{p}(f, k, t):\, 0<t<\infty\right\rbrace; D_{p}(f, k, \infty)\right\rbrace.
	\end{equation}

	In particular, the condition $\int\limits_{(0, \infty)}fk\,d\tau<\infty$ is necessary for the finiteness of $B_k(f, p)$. Nevertheless, let us note that the right--hand sides of \eqref{eq1.5.13} and \eqref{eq1.5.14} coincide because $D_{p}(f, k, t)$ is  continuous on ${\mathbb R}_+$.
\end{proof}

\begin{remark}
	The results on the cone $\Omega_1$ and $\dot{\Omega}_1$ of decreasing functions we obtain by setting here $k(\tau)\equiv 1$.
\end{remark}

\subsection{Calculation of the norm of the dilation operator 	and triangle inequality in Lorentz space}\label{subsec1.5.3}

Let $L_0=L_0({\mathbb R}^n)$  be the space of all Lebesgue-measurable functions $f: {\mathbb R}^n\rightarrow {\mathbb R}$; $\ddot{L}_0=\ddot{L}_0({\mathbb R}^n)$  be the subspace of functions having the decreasing rearrangements $f^{\ast}$ which is not identical to infinity (see \eqref{eq2.3.7}).

Note that $f^{\ast}$ is a nonnegative decreasing left-continuous function on  ${\mathbb R}_+$.  Now, let  $0<p<\infty$, $v$  be a positive Lebesgue--measurable function on  ${\mathbb R}_+$.
Introduce a weighted Lorentz space:
\begin{equation}\label{eq1.5.17}
\Lambda_{p, v}=\left\lbrace f \in \ddot{L}_0:\, \|f\|_{\Lambda_{p, v}}=\left( \int_{0}^{\infty}(f^{\ast})^{p}v d\tau\right)^{1/p}<\infty\right\rbrace.
\end{equation}

For  $v(\tau)\equiv 1$, we have $\Lambda_{p, v}=L_p({\mathbb R}^n)$. The criterion for nontriviality is
\begin{equation}\label{eq1.5.18}
\Lambda_{p, v}\neq{0} \Leftrightarrow \exists t \in {\mathbb R}_+:\, \int\limits_{(0, t)}vd\tau<\infty.
\end{equation}

Now we consider the dilation operator $\sigma_m,\, m \in {\mathbb R}_+$,
\begin{equation}\label{eq1.5.19}
\sigma_m(f)(x)=f(m^{-1}x), x \in {\mathbb R}^n.
\end{equation}
Then,
\begin{equation}\label{eq1.5.20}
\sigma_m(f)^{\ast}=\sigma_{m^n}(f^{\ast}).
\end{equation}

Indeed, by \eqref{eq2.3.6}
\begin{equation}\label{eq1.5.21}
\begin{aligned}
\lambda_{\sigma_m(f)}(y)&=\mu_n \left\lbrace x \in {\mathbb R}^n:\, |f(m^{-1}x)|>y\right\rbrace \\
&=m^n\mu_n \left\lbrace x \in {\mathbb R}^n:\, |f(x)|>y\right\rbrace=m^n\lambda_{f}(y),
\end{aligned}
\end{equation}
so that
\begin{equation*}
\begin{aligned}
\left[ \sigma_m(f)\right] ^{\ast}(\tau)&=\inf \left\lbrace y>0:\, \lambda_{\sigma_m(f)}(y)< \tau\right\rbrace \\
&=\inf \left\lbrace y>0:\, \lambda_{f}(y)< m^{-n}\tau\right\rbrace=f^{\ast}(m^{-n}\tau)=\sigma_{m^n}(f^{\ast})(\tau).
\end{aligned}
\end{equation*}

We define
\begin{equation}\label{eq1.5.22}
V(\tau)=\int\limits_{(0, \tau]}v(\rho)d\rho,\, \tau \in {\mathbb R}_+,\quad V(\infty)=\int_{{\mathbb R}_+}v(\rho)d\rho=\lim\limits_{\tau \rightarrow \infty}V(\tau).
\end{equation}

\begin{theorem}\label{th1.5.6}
	In the notation of this Section let condition \eqref{eq1.5.18} be satisfied. For $E=\Lambda_{p, v}$   the following equality for the norm of   $\sigma_{m}: E\rightarrow E$   holds:
	\begin{equation}\label{eq1.5.23}
	\|\sigma_{m}\|=\sup_{\tau \in {\mathbb R}_{ +}} \left[ V(m^n\tau)V(\tau)^{-1}\right]^{1/p};
	\end{equation}
\end{theorem}

\begin{proof}
	We have
		\begin{align*}
		&\|\sigma_{m}(f)\|_{\Lambda_{p, v}}=\left( \int_{{\mathbb R}_{+}}\left[\sigma_{m}(f)^{\ast}\right]^{p}vd\tau\right)^{1/p}= \left( \int_{{\mathbb R}_{+}}\left[\sigma_{m^n}(f^{\ast})\right]^{p}vd\tau\right)^{1/p}\\
		&= \left( \int_{{\mathbb R}_{+}}\left[f^{\ast}(m^{-n}\tau)\right]^{p}v(\tau)d\tau\right)^{1/p}=m^{n}\left( \int_{{\mathbb R}_{+}}\left[f^{\ast}(\rho)\right]^{p}v(m^{n}\rho)d\rho\right)^{1/p},
		\end{align*}
		Therefore,
		\begin{align*}
		\|\sigma_{m}\|=m^{n}\sup_{f \in \Lambda_{p, v}}\left[\left(\int_{{\mathbb R}_{+}}\left[f^{\ast}(\rho)\right]^{p}v(m^{n}\rho)d\rho\right)^{1/p} \left( \int_{{\mathbb R}_{+}}\left[f^{\ast}(\rho)\right]^{p}v(\rho)d\rho\right)^{-1/p}\right].
		\end{align*}
	Note that in this context

	\begin{equation*}
	\Omega\equiv \left\lbrace g:\, 0\leq g \downarrow,\, g(t)=g(t-0);\, \int_{{\mathbb R}_{+}}g^pvd\rho<\infty\right\rbrace,
	\end{equation*}
	\begin{equation*}
	\dot{\Omega}\equiv \left\lbrace g \in \Omega:\, g(t)\rightarrow 0\,(t\rightarrow \infty)\right\rbrace.
	\end{equation*}

	\begin{equation}\label{eq1.5.24}
	V(\infty)=\infty,\, f \in \Lambda_{p, v}\Leftrightarrow g=f^{\ast} \in \dot{\Omega};
	\end{equation}
		\begin{equation}\label{eq1.5.25}
	V(\infty)<\infty,\, f \in \Lambda_{p, v}\Leftrightarrow g=f^{\ast} \in \Omega.
	\end{equation}
	Therefore,  $V(\infty)=\infty$ or $V(\infty)<\infty$  imply (respectively)
		\begin{align*}
\|\sigma_{m}\|=m^{n}\sup_{g \in \dot{\Omega}}\left[\left(\int_{{\mathbb R}_{+}}g(\rho)^{p}v(m^{n}\rho)d\rho\right)^{1/p} \left( \int_{{\mathbb R}_{+}}g(\rho)^{p}v(\rho)d\rho\right)^{-1/p}\right];
	\end{align*}
	\begin{align*}
	\|\sigma_{m}\|=m^{n}\sup_{g \in {\Omega}}\left[\left(\int_{{\mathbb R}_{+}}g(\rho)^{p}v(m^{n}\rho)d\rho\right)^{1/p} \left( \int_{{\mathbb R}_{+}}g(\rho)^{p}v(\rho)d\rho\right)^{-1/p}\right];
	\end{align*}

	In both cases we come to formula  \eqref{eq1.5.23}   applying Corollaries \ref{cl1.3.5} and \ref{cl1.3.6}, respectively. In the last case we take into account that
\begin{equation*}
\sup_{0<\tau \leq \infty}	\left[ V(m^n\tau)V(\tau)^{-1}\right]^{1/p}=\sup_{\tau \in {\mathbb R}_{+}}\left[ V(m^n\tau)V(\tau)^{-1}\right]^{1/p},
\end{equation*}
because the function $V$  is positive and continuous on ${\mathbb R}_{+}$.
\end{proof}

\begin{remark}\label{rm1.5.7}
	Note that
	\begin{equation*}
\|\sigma_{m}\|=\sup_{\tau \in {\mathbb R}_{+}}\left[ V(m^n\tau)V(\tau)^{-1}\right]^{1/p}<\infty \Leftrightarrow V \in \Delta_2.
	\end{equation*}
	It is the well-known  $\Delta_2$- condition for the function $V$.
\end{remark}
Let us discuss now the triangle inequality for a Lorentz space $\Lambda_{p,v}$.

\begin{theorem}\label{th1.5.8}
In the notation of this Section, let  $\|\sigma_{2^{1/n}}\|<\infty$. Then\\ $f, g$ $\in \Lambda_{p,v}$ implies $f+g \in \Lambda_{p,v}$, and
	\begin{equation}\label{eq1.5.26}
\|f+g\|_{\Lambda_{p,v}}\leq \|\sigma_{2^{1/n}}\|\left(  \|f\|_{\Lambda_{p,v}}+\|g\|_{\Lambda_{p,v}}\right),\, 1\leq p<\infty;
\end{equation}
\begin{align}\label{eq1.5.27}
&\|f+g\|_{\Lambda_{p,v}}\leq \|\sigma_{2^{1/n}}\|\left(  \|f\|_{\Lambda_{p,v}}^{p}+\|g\|_{\Lambda_{p,v}}^{p}\right)^{1/p}\nonumber \\
&\leq 2^{1/p -1}\|\sigma_{2^{1/n}}\|\left(  \|f\|_{\Lambda_{p,v}}+\|g\|_{\Lambda_{p,v}}\right),\, 0< p<1.
\end{align}
\end{theorem}

\begin{proof}
	For $f, g \in \ddot{L}_0$  there is the well-known  inequality (see \cite{BS})
	\begin{equation*}
	(f+g)^{\ast}(\tau_1+\tau_2)\leq f^{\ast}(\tau_1)+g^{\ast}(\tau_2),\, 0<\tau_1, \tau_2 <\infty.
	\end{equation*}
	Thus,
		\begin{equation*}
	(f+g)^{\ast}(\tau)\leq f^{\ast}(\tau/2)+g^{\ast}(\tau/2)=\sigma_{2}(f^{\ast})(\tau)+\sigma_{2}(g^{\ast})(\tau),\, 0<\tau <\infty,
	\end{equation*}
	so that in view of \eqref{eq1.5.20}
	\begin{equation*}
	(f+g)^{\ast}\leq [\sigma_{2^{1/n}}(f)]^{\ast}+[\sigma_{2^{1/n}}(g)]^{\ast}.
	\end{equation*}
		\begin{equation*}
	\|f+g\|_{\Lambda_{p,v}}\!=\!\left( \int\limits_{0}^{\infty}\![(f+g)^{\ast}]^{p}vd\tau\right)^{1/p}\!\leq\!\left( \int\limits_{0}^{\infty}\! [(\sigma_{2^{1/n}}(f))^{\ast}+(\sigma_{2^{1/n}}(g))^{\ast}]^{p}vd\tau\right)^{1/p}\!.
	\end{equation*}
	Now, we apply the triangle inequality for Lebesgue spaces in the form
		\begin{equation}\label{eq1.5.28}
	\|h_1+h_2\|_{L_{p,v}}\leq \|h_1\|_{L_{p,v}}+\|h_2\|_{L_{p,v}},\, 1\leq p<\infty;
	 \end{equation}
	 \begin{equation}\label{eq1.5.29}
	 \begin{aligned}
	 &\|h_1+h_2\|_{L_{p,v}}\leq \left( \|h_1\|_{L_{p,v}}^{p}+ \|h_2\|_{L_{p,v}}^{p}\right)^{1/p}\\
	&\leq 2^{1/p-1}\left( \|h_1\|_{L_{p,v}}+ \|h_2\|_{L_{p,v}}\right),\, 0<p<1.
	\end{aligned}
	\end{equation}
	Here,  $h_1=(\sigma_{2^{1/n}}(f))^{\ast}$, $h_2=(\sigma_{2^{1/n}}(g))^{\ast}$. Therefore, for $1\leq p<\infty$,
	\begin{equation*}\begin{aligned}
	\|f+g\|_{\Lambda_{p,v}}&\leq \|(\sigma_{2^{1/n}}(f))^{\ast}\|_{L_{p,v}}+ \|(\sigma_{2^{1/n}}(g))^{\ast}\|_{L_{p,v}}\\
	&\leq\|(\sigma_{2^{1/n}}\|\left(\|f\|_{\Lambda_{p,v}}+\|g\|_{\Lambda_{p,v}}\right).
		\end{aligned}
		\end{equation*}
		Analogously, we obtain \eqref{eq1.5.27}  applying \eqref{eq1.5.29}.
\end{proof}

\begin{remark}\label{rm1.5.9}
Let us recall the equivalence
\begin{equation}\label{eq1.5.30}
\|\sigma_{2^{1/n}}\|=\sup_{\tau \in {\mathbb R}_{+}} \left[V(2\tau)V(\tau)^{-1} \right]^{1/p}<\infty \Leftrightarrow V \in \Delta_2.
\end{equation}
Thus, this condition guaranties that $\Lambda_{p,v}$   is  a linear (quasi)normed space. It is known that condition \eqref{eq1.5.30} is necessary for such linearity, see \cite{CPSS,CRS}.
\end{remark}

\subsection{Calculation of the norms of embedding operators for Lorentz spaces}

We preserve the notation of Section 2 and consider the conditions for the embedding of Lorentz spaces.
\begin{equation*}
\Lambda_{p,v}\subset  \Lambda_{q,w},\, 0<p\leq q.
\end{equation*}
Here $v,w$ are positive Lebesgue-measurable functions on  $(0,\infty)$.

\begin{theorem}\label{th1.5.10}
	For $0<p\leq q < \infty$  the following equivalence takes place:
\begin{equation}\label{eq1.5.31}
\Lambda_{p,v}\subset  \Lambda_{q,w} \Leftrightarrow \, A_{pq}:=\sup_{t \in {\mathbb R}_+} \left[ W(t)^{1/q}V(t)^{-1/p}\right] <\infty,
\end{equation}
	where $W(t)=\int\limits_{(0, t]}wd\tau$.  Moreover, $\|J\|=A_{pq}$   for the embedding operator $J:\Lambda_{p,v}\rightarrow  \Lambda_{q,w}$.
\end{theorem}

\begin{proof}
	We have
	\begin{equation}\label{eq1.5.32}
	\|J\|=\sup_{f \in \Lambda_{p,v}} \left[ \left( \int_{{\mathbb R}_+}(f^{\ast})^{q}wd\tau\right) ^{1/q}\left( \int_{{\mathbb R}_+}(f^{\ast})^{p}vd\tau\right)^{-1/p}\right].
	\end{equation}
	Now, the equivalences \eqref{eq1.5.24} or \eqref{eq1.5.25} admit us to come to the cones $\dot{\Omega}$  or $\Omega$. Note that the identity  operator is $l_r$-convex for any $r \in {\mathbb R}_+$, and for $0<p\leq q <\infty$ we can apply Corollaries \ref{cl1.3.5}  and \ref{cl1.3.6}, respectively. In both cases we have
	\begin{equation*}
	\|J\|=\sup_{t \in {\mathbb R}_+} \left[ W(t)^{1/q}V(t)^{-1/p}\right].
	\end{equation*}
	As before we take into account that
	\begin{equation*}
\sup_{t \in (0, \infty]} \left[ W(t)^{1/q}V(t)^{-1/p}\right]=\sup_{t \in {\mathbb R}_+} \left[ W(t)^{1/q}V(t)^{-1/p}\right].
	\end{equation*}
	Now let us consider the other variant of Lorentz spaces. For the decreasing rearrangement $f^{\ast}$ we introduce the mean-value $f_{r,\mu}^{\ast\ast}$ with respect to the nonnegative continuous Borel measure $\mu$ on ${\mathbb R}_+$:
	\begin{equation*}
	f_{r,\mu}^{\ast\ast}(\tau)=\left( \frac{1}{M(\tau)}\int\limits_{(0, \tau]}(f^{\ast})^rd\mu\right)^{1/r};\quad M(\tau)=\int\limits_{(0, \tau]}d\mu,\, \tau \in {\mathbb R}_+.
	\end{equation*}
	For $0<q,r<\infty$  the generalized $\Gamma$- Lorentz space is determined as follows
	\begin{equation*}
	\Gamma_{qr}(\gamma,\mu)=\left\lbrace f \in \ddot{L}_{0}:\, \|f\|_{\Gamma_{qr}(\gamma,\mu)}=\left(\int_{0}^{\infty} \left( f_{r,\mu}^{\ast\ast}\right)^{q}d\gamma\right)^{1/q}<\infty\right\rbrace,
	\end{equation*}
	where $\gamma$ is a nonnegative continuous Borel measure.
	Classical variant of this space (see \cite{CPSS, CRS}) we obtain in the case: $r=1,\,\mu$ is  a Lebesgue measure, $d\gamma(\tau)=w(\tau)d\tau$.  Some variants of generalization were introduced in  \cite{FiR}-\cite{GPS}. Here we consider the case $0<p\leq \min\left\lbrace q,r\right\rbrace$.
\end{proof}
 \begin{theorem}\label{th1.5.11}
 	Let  $0<p\leq \min\left\lbrace q,r\right\rbrace$. Then the following equivalences take place.
 	\begin{enumerate}
 		\item If $V(\infty)=\infty$, then
 		\begin{equation}\label{eq1.5.33}
 	\Lambda_{p,v}\subset \Gamma_{qr}(\gamma,\mu)\Leftrightarrow \dot{J}_{pqr}:= \sup_{t \in {\mathbb R}_{ +}}\left[ W_{qr}(t)^{1/q}V(t)^{-1/p}\right] <\infty,
 		\end{equation}
 		and $\|J\|=\dot{J}_{pqr}$  for the embedding operator $J: \Lambda_{p,v}\rightarrow \Gamma_{qr}(\gamma,\mu)$, where
 		\begin{equation}\label{eq1.5.34}
 		 W_{qr}(t)=\int_{{\mathbb R}_+}\left( \frac{M(\min\left\lbrace t, \tau\right\rbrace )}{M(\tau)}\right) ^{q/r}d\gamma(\tau).
 		\end{equation}

 		\item If $V(\infty)<\infty$, then
 			\begin{equation}\label{eq1.5.35}
 		\Lambda_{p,v}\subset \Gamma_{qr}(\gamma,\mu)\Leftrightarrow J_{pqr}:= \sup_{t \in (0, \infty]}\left[ W_{qr}(t)^{1/q}V(t)^{-1/p}\right] <\infty,
 		\end{equation}
 		and $\|J\|=J_{pqr}$.
 	\end{enumerate}
 \end{theorem}

\begin{proof}
	\begin{enumerate}
		\item We define
			\begin{equation}\label{eq1.5.36}
	X=L_{p,v}({\mathbb R}_+),\, Y=L_{q}({\mathbb R}_+, \tilde{\gamma});\, d \tilde{\gamma}(\tau)=M(\tau)^{-q/r}d\gamma(\tau),\, \tau \in {\mathbb R}_+.
		\end{equation}
		We have by definitions
			\begin{equation*}
		\|f\|_{\Gamma_{qr}(\gamma,\mu)}=\left( \int_{{\mathbb R}_+}\left(f_{r\mu}^{**}\right) ^qd\gamma\right)^{1/q}=
		\left( \int_{{\mathbb R}_+}\left(\int\limits_{(0, \tau]}(f^{*})^{r}d\mu\right)^{q/r}d\tilde{\gamma}(\tau)\right)^{1/q},
		\end{equation*}
		\begin{equation*}
		\|J\|=\sup_{f \in \Lambda_{p,v}}\left[ \left( \int_{{\mathbb R}_+}\left(\int\limits_{(0, \tau]}(f^{*})^{r}d\mu\right)^{q/r}d\tilde{\gamma}(\tau)\right)^{1/q} \left( \int_{{\mathbb R}_+}(f^{*})^{p}vd\tau\right)^{-1/p}\right].
		\end{equation*}
		Now we take into account equivalences \eqref{eq1.5.24} and \eqref{eq1.5.25} and come to the cones $\dot{\Omega}$  or $\Omega$.

		\item    If $V(\infty)=\infty$  we have $\Omega=\dot{\Omega}$  and
		\begin{equation*}
		\|J\|=\sup_{g \in \dot{\Omega}}\left[ \left( \int_{{\mathbb R}_+}\left(\int\limits_{(0, \tau]}g^{r}d\mu\right)^{q/r}d\tilde{\gamma}(\tau)\right)^{1/q} \left( \int_{{\mathbb R}_+}g^{p}vd\tau\right)^{-1/p}\right].
		\end{equation*}

		In the case  $g(\tau)=\chi_{(0,t]}(\tau)$   we have
			\begin{equation*}
		\int\limits_{(0, \tau]}g^{r}d\mu=M(\min\left\lbrace t,\tau\right\rbrace );\, \int_{{\mathbb R}_+}g^{p}vd\tau=V(t).
			\end{equation*}
			Thus, condition \eqref{eq1.1.13} is satisfied, and application of Corollary \ref{cl1.3.5} implies
			\begin{equation*}
			\|J\|=\sup_{t \in {\mathbb R}_+}\left[ \left( \int_{{\mathbb R}_+}M(\min\left\lbrace t,\tau\right\rbrace )^{q/r}d\tilde{\gamma}(\tau)\right)^{1/q}V(t)^{-1/p}\right].
			\end{equation*}
			Therefore, equality \eqref{eq1.5.33} holds.

			If $V(\infty)<\infty$,  then
			\begin{equation*}
			\|J\|=\sup_{g \in \Omega}\left[ \left( \int_{{\mathbb R}_+}\left(\int\limits_{(0, \tau]}g^{r}d\mu\right)^{q/r}d\tilde{\gamma}(\tau)\right)^{1/q} \left( \int_{{\mathbb R}_+}g^{p}vd\tau\right)^{-1/p}\right].
			\end{equation*}
			In this case condition \eqref{eq1.1.15} is satisfied, and application of Corollary \ref{cl1.3.6} implies
			\begin{equation*}
			\|J\|=\sup_{t \in (0, \infty]}\left[ \left( \int_{{\mathbb R}_+}M(\min\left\lbrace t,\tau\right\rbrace )^{q/r}d\tilde{\gamma}(\tau)\right)^{1/q}V(t)^{-1/p}\right].
			\end{equation*}
			Therefore, equality \eqref{eq1.5.35} holds.
	\end{enumerate}
\end{proof}
\begin{remark}\label{rm1.5.12}
	We obtain \eqref{eq1.5.33}  and \eqref{eq1.5.35} applying results for general operators (Corollaries \ref{cl1.3.5} and \ref{cl1.3.6}). Now, let us note that $M(\min\left\lbrace t,\tau\right\rbrace )$ increases by $t$  for any $\tau \in {\mathbb R}_+$,  $\lim\limits_{t \rightarrow +\infty}M(\min\left\lbrace t,\tau\right\rbrace)=M(\tau)$.    Therefore, by Levy's theorem
	\begin{equation}\label{eq1.5.37}
	W_{qr}(t)\uparrow W_{qr}(\infty)=\int_{{\mathbb R}_{+}}d\gamma.
	\end{equation}
	Therefore, in the case  $0<V(\infty)<\infty$    we have
		\begin{equation}\label{eq1.5.38}
\dot{J}_{pqr}\geq \lim_{t \rightarrow +\infty}\left[ 	W_{qr}(t)^{1/q}V(t)^{-1/p}\right] =\left[  W_{qr}(\infty)V(\infty)^{-1/p}\right].
	\end{equation}
	It means that in the considered case $J_{pqr}=\dot{J}_{pqr}$.
\end{remark}
\begin{corollary}\label{cl1.5.13}
	In Theorem \ref{th1.5.11} let $p=q\leq r,\, d\gamma(\tau)=v(\tau)d\tau.$ Then the equality $\Lambda_{p,v}=\Gamma_{pr}(\gamma,\mu)$ is equivalent to the following  condition:
	\begin{equation*}
	\|J\|=\sup_{t \in {\mathbb R}_+}\left[ \left( \int_{{\mathbb R}_+}M(\min\left\lbrace t,\tau\right\rbrace)^{p/r}M(\tau)^{-p/r}v(\tau)d\tau
	\right)^{1/p}V(t)^{-1/p}\right]<\infty.
	\end{equation*}
	Moreover,
		\begin{equation}\label{eq1.5.39}
\|f\|_{\Lambda_{p,v}}\leq \|f\|_{\Gamma_{pr}(\gamma,\mu)}\leq
	\|J\|\|f\|_{\Lambda_{p,v}}.
	\end{equation}
\end{corollary}
 Indeed, the right-hand-side inequality follows by Theorem \ref{th1.5.11}. For the left-hand-side one it is enough to note that
 	\begin{equation*}
 f_{r\mu}^{**}(\tau)=\left(\frac{1}{M(\tau)} \int\limits_{(0, \tau]}(f^{*})^{r}d\mu \right)^{1/r}\geq f^{*}(\tau).
 \end{equation*}
 \begin{remark}\label{rm1.5.14}
 	In this Section we considered  the applications to the theory of Lorentz spaces of the results obtained in  Section 1 for operators on the cones with monotonicity properties. They are connected with some restrictions on parameters such as $0<p\leq q\leq r$ in Theorems \ref{th1.1.1}, \ref{th1.1.3}, \ref{th1.4.1}, \ref{th1.4.3}, or $0<p\leq q$ in Theorem \ref{th1.5.10}, or $0<p\leq \min\left\lbrace q,r\right\rbrace $  in Corollaries \ref{cl1.3.5}, \ref{cl1.3.6}, and  Theorem \ref{th1.5.11}. The technic of consideration connected with such restrictions admit us to calculate \textit{exactly} the norms of related operators. In the next Section we formulate a result  about order-sharp estimates for Hardy-type operators on the general cone $ \Omega_{k}$ without such restrictions on parameters.
 \end{remark}

\section{ESTIMATES FOR HARDY-TYPE OPERATORS}

Let $b \in (0, \infty], \ p, q, r \in (0, \infty)$ be fixed, $\beta, \gamma, \mu$ be nonnegative continuous Borel measures on  $(0, b); \beta \in N_{p}(k)$ that is
$$0<\omega_{kp}(t):=\left(\int\limits_{(0,t)}k^qd\beta \right) \in C(0,b);\quad \omega_{kp}(+0)=0,\, \omega_{kp}(b-0)=\infty.$$
 Consider
	\begin{equation}
\label{eq3.1}
H_{ \Omega_{k}} \equiv H_{ \Omega_{k}}   \left( {A_{r\mu}}  \right)=
\sup\limits_{f\in \Omega_{k }}  \left[\left( \int\limits_{(0, b)} \left( A_{r\mu}f\right)^{q} d \gamma \right)^{ 1 / q}\left(\int\limits_{(0, b)}  {f^{ p} d \beta }  \right)^{- 1 / p}\right].
	\end{equation}
	Here,

	\begin{equation}
	\label{eq3.2}
	\left(  {A_{r \mu}   f}  \right) \left(  { t}
	\right)=\left( \int\limits_{( { 0, t}]} { f^{r} d \mu
	}\right)^{{1}/{r}}, \, t \in (0, b),
	\end{equation}
	is the Hardy-type operator.
	We need some notation to formulate the results. Define
	\begin{equation}
	\label{eq3.3}
	\Psi_{r}(\tau)\equiv 	\Psi_{r}(k, \mu; \tau)=\left( \int\limits_{( {0, \tau } ]}
	k^{r} d \mu\right)^{{1}/{r}} ,  \quad \tau \in (0, b),
	\end{equation}
		\begin{equation}
	\label{eq3.4}
\mathfrak{J}_{p q r}\equiv 	\mathfrak{J}_{p q r}(k, \beta, \gamma, \mu)=\sup_{t \in (0, b)}\left[\left( \int\limits_{\left( {0, b}  \right)}
	\Psi_{r}(\min \left\lbrace t, \tau \right\rbrace )^{q}d\gamma(\tau)\right)^{{1}/{q}}\omega_{k p}(t)^{-1}\right].
	\end{equation}
	\begin{equation}
	\label{eq3.5}
	V_{ pr}  \left(  { t}  \right)\equiv 	V_{ pr}  \left(  {k, \beta, \mu; t}  \right)=\mathop {\sup} \limits_{\tau \in
	({0, t }  ]} \left[  \Psi_{r} \left(\tau \right) \omega _{k p} \left(  { \tau }\right)^{-1} \right],\quad p \leq r;
	\end{equation}
	and for $p>r, \sigma=\frac{pr}{p-r},$
	\begin{equation}
	\label{eq3.6}
	V_{ pr}  \left(  { t}  \right)=\left\lbrace  {\int\limits_{({0, t }]} {\Psi_{r}  \left({\tau }\right)}^{ \sigma} \left(- d \left[ {\omega _{ k p} \left({ \tau }  \right)}^{ -\sigma}\right]\right)  } +\left[\Psi_{r}(t) \omega _{ k p}(t)^{-1}\right]^{\sigma}  \right\rbrace^{ \frac{1}{\sigma}};
	\end{equation}
	\begin{equation}
	\label{eq3.7}
	W_{ q}  \left(  { t}  \right)=	W_{ q}  \left(  {\gamma; t}  \right)=\left(\ {\int\limits_{\left(
			{ t, b }  \right)} {d \gamma} }  \right)^{ 1  / q}, \, t \in (0, b);
	\end{equation}

	\begin{equation}
	\label{eq3.8}
	E_{ p q r} \equiv E_{ p q}\left( k, r, \beta, \gamma, \mu\right) =\mathop {\sup} \limits_{t \in (0, b)}  \left[\left(\int\limits_{ (0, t]} {\Psi_{r}^{ q} d \gamma}	\right)^{ 1 / q}\omega _{k p}  \left( t\right)^{-1}\right],\quad p \leq  q;
	\end{equation}
	and for $p > q, s=\frac{pq}{p-q},$
	\begin{equation}
	\label{eq3.9}
	E_{ p q r}  = \left\lbrace \int\limits_{(0, b) }  \left(\int\limits_{ (0,t]} \Psi_{r}^{ q}d \gamma   \right)^{{ s}\mathord{\left/ {\vphantom {{ s}  { q}}} \right.
					\kern-\nulldelimiterspace} { q}}\left(  {- d \left[
				\omega _{k p}(t)^{-s}   \right]}
			\right)  \right\rbrace ^{ { 1}  \mathord{\left/ {\vphantom {{ 1}  { s}}}
			\right. \kern-\nulldelimiterspace} { s}};
	\end{equation}
	\begin{equation}
	\label{eq3.10}
	F_{ p q r}\equiv 	F_{ p q} (k, r, \beta, \gamma, \mu) =\mathop {\sup} \limits_{t\in
	(0, b)}  \left[
	{V_{ pr}  \left(  { t}  \right)W_{ q} \left(  { t}  \right)}
	\right],\quad p \leq  q,
	\end{equation}
	\begin{equation}
	\label{eq3.11}
	F_{ p q r} =\left\lbrace {\int\limits_{(0, b) }}  V_{ p r}(t)^{ s}\left( -d \left[W_{ q}(t)^{ s}\right]  \right) \right\rbrace^{ 1 / s},\quad p > q, s=\frac{pq}{p-q}.
	\end{equation}

	\begin{theorem}\label{th3.1} Let the above notation and conditions hold, $\beta \in N_p(k)$. Then the following assertions take place:
		\begin{equation}
		\label{eq3.12}
		 H_{ \Omega _{ k}}=\mathfrak{J}_{p q r},\, p\leq \min \left\lbrace q; r\right\rbrace;
		\end{equation}
			\begin{equation}
			\label{eq3.13}
			c ^{ - 1}\left(  { (E_{ p q r })^{r} +(F_{ p q r})^{r} }  \right)^{{1}/{r}}\leq H_{ \Omega _{ k}}(A_{r \mu})  \leq c\left( (E_{ p q r})^{r} +(F_{p q r})^{r}\right)^{{1}/{r}},
			\end{equation}
		for $p> \min \left\lbrace q; r\right\rbrace$	with $c	=c\left(p, q, r \right)\in \left[ { 1, \infty} \right).$
		\end{theorem}
	\begin{remark}\label{rm3.1} Note that the convergence of integral in (\ref{eq3.11}) and integration by part imply the following equality
		\begin{equation}\label{eq3.11'}
		F_{ p q r} =\left\lbrace {\int\limits_{(0, b) }}  W_{ q}(t)^{ s}\left( d \left[V_{ pr}(t)^{ s}\right]  \right) \right\rbrace^{ 1 / s}.
	\end{equation}
\end{remark}
	\begin{remark} Obviously, (\ref{eq3.13}) implies estimate
		\begin{equation}
	\label{eq3.13'}
	c_{1}^{ - 1}\left(  { E_{ p q r } +F_{ p q r} }  \right)\leq H_{ \Omega _{ k}} \leq c_{1}\left( E_{ p q r} +F_{p q r}\right),
	\end{equation}
	with some constant $c_{1}	=c_{1}\left(p, q, r \right)\in \left[ { 1, \infty} \right).$
\end{remark}

\section*{Acknowledgments}
This work is supported by the Ministry of Science and Higher Education of the Russian Federation: agreement no. 075-03-2020-223/3 (FSSF-2020-0018).

redpajama_set_name: RedPajamaArXiv


Interview about the possible impact of South African Finance Minister Nene's purported request to be withdrawn from National Treasury. The discussion reviews the first round of effects following Minister Nene's testimony at the Commission of Inquiry into State Capture where he admitted to oversight of not declaring his meetings with the Gupta Family. The interview also evaluates the possible impact if President Cyril Ramaphosa accepted Minister Nene's purported resignation and if he didn't. The recommended continuity until the May 2019 national elections and dealing with the wider implications of Minister Nene's oversight as a solution to the challenge.


text:
Book review - Puff Piece
Puff Piece
John Safran
Penguin/Hamish Hamilton 353pp
In journalism a 'puff piece' is a weightless story praising a person or business and ignoring, or even covering up, any flaws they may possess. While John Safran's latest book certainly focuses on a brand name – the words Philip Morris appear on almost every page – the result is anything but an enhancement of the cigarette manufacturer's reputation.
As he makes this clear from the outset, the company's history of deceit is long and harmful, the latest example being its response to the European Parliament's ban in May 2020 of menthol cigarettes.
Philip Morris agreed to stop the product's sale but announced a substitute: 'It's not a menthol cigarette, they assure everyone. It's a menthol HeatStick. But a HeatStick is just tobacco rolled in paper with a filter at one end. You know, like a cigarette.'
This non-cigarette, now being marketed well beyond the menthol ban phase, is activated by a device called an IQOS (pronounced 'eye kose') that heats the tobacco to a high temperature to release what the company denies is smoke, without actually burning, enabling the not-smoker to draw nicotine and tobacco, plus cancer-causing tar, into the lungs.
The lack of combustion is the basis of the company's claim that they are not producing a cigarette.
'When you're finished,' Safran says, 'you flick the HeatStick into the bin or onto the ground. Like you'd flick a cigarette butt into the bin or onto the ground.'
The book's tone is darkly humorous and unrelentingly angry with a company that has killed millions around the world while buying its way out of and around every legal attempt to stop it. The IQOS is its latest play, coming at a time when anti-smoking pressures are peaking and, as luck would have it, another 'healthy' smoking device is sweeping the world, or at least the West.
Enter the e-cigarette, a battery-operated device that heats a liquid to produce a vapour that users inhale. Hence the term 'vaping'. The fluid in these devices can be bought in a myriad of flavours. What can't be bought, as of 1 October this year, is liquid nicotine, which has been banned for importation or sale in Australia without a prescription relating to smoking cessation.
In its latest cynical, massively funded strategy Philip Morris is riding the slipstream of the vaping movement by seeking to portray the IQOS as a parallel healthy alternative to smoking. As Safran reveals, they have even helped fund the vaping movement's lobbying activities and legal actions.
The pitch is that the device doesn't generate smoke, but aerosol, and while your lungs are indeed drawing in nicotine and tobacco, this aerosol means 'the levels of harmful chemicals are significantly reduced compared to cigarette smoke' (the company claims).
It's 'sleek, blue and metallic-looking, roughly the length of a cigarette… like a pen an astronaut would have… it slides into a matching charging device… and the whole package fits into your hand. If I saw a woman holding one on a red carpet I'd assume it's some sort of compact or tampon case that I'm unfamiliar with because I don't read Vogue.'
This is a whip-smart book, laugh out loud funny despite being bleakly infuriating.
'I began this book assuming people would be enraged by Philip Morris… The word 'corporations' once took top spot… But, using my Twitter feed and left-leaning news outlets as a barometer, the word 'whiteness' has substituted in… Philip Morris, the lucky buggers, are the beneficiaries of a cultural shift. Judging eyes have drifted away from the corporations, the very same moment they're sending the IQOS into the world.'
Some 21,000 Australians die prematurely each year from smoking cigarettes, often from cancer, and as Safran says, people hate the product as a result. They should also hate the tobacco companies, especially Philip Morris because of its subterfuge claim that HeatSticks do not produce smoke.
As Safran comes to realise, he was blinded all along by the smoke screen (which might have been a better title): 'I realise the precise truth is that what makes cigarettes so harmful is the tar in the smoke they produce… I've been wild-goose chasing smoke this whole time… I should have been chasing tar… the deal maker or breaker is not 'Does the HeatStick produce smoke?' It's 'Does the HeatStick produce tar?'
The answer is 'yes', as Philip Morris, a.k.a. the 'Marlboro cowboys', admit but try to conceal when releasing the product's ingredient list to the American FDA: 'all but one constituent is given their ordinary name… Carbon monoxide is carbon monoxide, mercury is mercury. Only tar is scrubbed out, reworded as 'NFDPM' – Nicotine Free Dry Particulate Matter.'
The company admits, without shame, that the HeatStick produces 78 per cent of the tar produced by a cigarette. This, having lost a case arguing that 'low tar' cigarettes are a healthier option for smokers.
Will the claim that this unattractively named device is preferable to cigarettes enable the device to be sold here, as it is in Europe, Japan and elsewhere?
One hopes not, Your Honour.
Written by Robin Osborne

redpajama_set_name: RedPajamaCommonCrawl


The integrated business comprising of several companies and firms is managed by Wadia brothers, under the leader ship Of Ashok Wadia. Shri Ashok Wadia is assisted by his two brothers and a strong inter-disciplinary professional team with experience in Infrastructure and real estate development, Banking, Investment and Project Management, Architecture, Facilitation and leisure development, wholesale distribution and retail marketing.
Super headed by Shri Ashok Wadia, aged about 50 years is a visionary business legendary that through his enterprising acumen created gigantic supply chain system in liquor trade both at retail as also at wholesale level, real estate, logistics management and development of real estate joint ventures as Strategic Business Units (SBUs).
Shri Ashok Wadia, aged around Fifty years is a visionary business person having graduated into the traditional family business of liquor trading from his early twenties. In about two decades Shri Ashok Wadia has created a sprawling multi-business behemoth singularly or in strategic partnership with leading business entities in the areas of commercial and residential real estate development, infrastructure management, logistics, leisure and tourism and acquisition of land banks in the prime areas without resorting to any external funding.
Shri Ashish Wadia has been actively involved in the retail and wholesale liquor trading, bid management and development of efficient supply chain system related to liquor trading in the NCR region.
THE Group is capitalizing on its inherent strength of large chunks of land Bank, diversified experience and expertise in trading, real estate development, Investment and Project Management. Several Land Banks, already owned by the Group, in the prime vicinities are at different stages of development. These initiatives will be the future building blocks of the nation, value creation for the economy and the prospective owners.


My oldest child – Matter – turns 15 this weekend. It's no April Fool's prank that our incorporation date is the first of April, 15 years ago. We're enthusiastically celebrating this milestone as an agency later this year, but it's appropriate to pause briefly to reflect on our people and the smart, hard and creative work that drove our growth. We're nearly 150 professionals strong – all of whom support some of the best brands in their respective categories. I'm proud of our path and who we are at 15 years of age.
This being shared, I'm joined by our staff in modestly reflecting on where we've been – while remaining focused primarily on where we're headed. And, I'm excited. Not just about the agency in which we work, but also the category where we apply our skills. The ever-evolving communications market is vibrant and filled with opportunity for brand-oriented companies to better establish themselves, and for professionals who want to be creative in their approach.
Emotional brand marketing will have a heightened importance. Smart and savvy brand managers will seek ways to further connect their beloved products and services to the hopes, dreams and directions of their audiences – in a personal way that drives further brand loyalty. But, this will need to be carefully balanced with heightening concerns about privacy. As marketers become more digitally savvy about tailoring messaging to hyper-targeted audiences, there will be a greater drive to fully disclose HOW the brand knows what the audience wants, and that the information wasn't harvested through data the customer didn't know the brand possessed. This act of balance between privacy and emotional appeal will separate the brands that do well from the ones that fail.
We will see increased attention given to the quality of content. That goes for still and moving images, and copy in all forms and formats. Is it as good (engaging) as it needs to be? That question will be asked by marketing professionals as they further assess not just the avenues their content travels, but also the content itself. And, agency professionals will be held to a higher standard with expectations even greater than today.
I'm certain customer engagement will continue to be a priority – and I'm further certain the channels for connecting with customers will be countless. I suspect, however, that some of the current and popular platforms will expire, fade away or just die on the vine – trend data shows that's already afoot with younger generations. Acknowledging that all things change over time, staying educated on new technologies – opportunities to interact with customers, really – will continue to be key to a brand's success.
Search marketing will be a BIG – bigger than it is today! – percentage of marketing budget for consumer-facing entities and those who market products and services in technology and B-to-B. It will further drive the creation of content, but it will be the backbone of many programs as opposed to being merely a supporting tactic. We're working here at Matter to move search to the beginning of the planning discussions, and I expect that's where it will be across all client accounts shortly.
I think a brand's genuine and organic commitment to charitable work – as a real extension of the business, not a line about CSR in the annual report – will be even more important. The challenges of those less fortunate are being elevated in the public dialogue – driven by a global "do-better" generation – and businesses will need to have real initiatives that demonstrate how they are paying attention and working to solve serious social issues at home and around the world. Companies that ignore this will be left behind by a generation of customers who prioritize helping others.
Much has changed in our market since we started our business 15 years ago, and much will change in the 15 years upcoming. I'm looking forward to all of it!


To be a leader in the Pharmaceutical Industry by providing enhanced services, relationship and profitabitability.
To provide quality services that exceeds of our esteemed customers.
To build long term relationship with our customers and clients.


text:
How do Hibs replace Martin Boyle? Because they need to
Looking at bare statistics, Martin Boyle scored 65 goals in 265 appearances over the course of his seven years at Hibs.
By Moira Gordon
21st Jan 2022, 3:02pm
Martin Boyle has been a talisman for Hibs.
Considering that return, it would be fair to assume that finding someone else who can come in and score around 10 goals a season from a forward position might not be impossible, especially now that Hibs have taken in £3million from his sale and don't need to rifle through the bargain bins
But such a limited field of view does not get close to illustrating Martin Boyle's real worth to Hibs in recent years or explain just how difficult he will be to replace, especially in a January transfer window which is notoriously difficult to negotiate – where available players have usually been sidelined and are not ready to hit the ground running and the quality targets tend to be under contract, with owners unwilling to part unless sizable sums are spent.
The fact is, replacing like for like will be an almost impossibly big ask.
Boyle has impressed for Australia.
Because Boyle was far more than those 65 goals. He was an on-field leader, who could take games and turn them on their head in the blink of an eye. A man with startling pace and a gallus willingness to run at defences, who, on his day, was unplayable, as Rangers discovered in this season's League Cup semi-final.
He scored three that day. But on other occasions he drew a blank. In those matches it was what he did for the team that counted.
A man who could inject fear into opposition defences, who could push them to the byeline and whip in a cross or cut inside and nip through gaps, or play one-twos, his pace also meant that in a straight sprint, few could match him let alone contain him.
But, it was only in the last season-and-a-half that he emerged as a reliable and rampant goalscorer, though. A player who could cut through defences, he finally added incisive finishing to his repertoire. Before the arrival of Jack Ross as manager, the assists – as valuable as they were – had always outnumbered his goals.
One of Boyle's finest moments for Hibs was his hat-trick in the recent 3-1 win over Rangers.
When Hibs returned from the Championship exile in 2017/18, he reintroduced himself to the top flight with 13 assists and Hibs pushed up the table.
Last term, he was mercurial, and while he still weighed in with a dozen assists, for the first time since his first full campaign at Hibs back in 2015/16, he finished with more goals than assists, netting 15. As part of a deadly frontline, he was a massive factor in Hibs highest league finish in 16 years as they booked a return to Europe and also got to one final and another semi final in the cups.
That ability to drive the team forward, taking the pressure off the defence and gaining territory with a quick counterattack or a injection of pace to help bust through a stodgy rearguard, offered those around him an on-field calmness born of the belief that he is always capable of producing a killer moment, be that an inviting cross or a breakaway goal. And, that is something that will be hard to replace,
The upcoming days will be vital for Hibs. With a new manager and an initial upturn in fortunes, they have high hopes of moving up the league standings. They received kudos for the early signings made in the transfer window and there was renewed positivity but that might not last if they can't find a way to cover the loss of Boyle. Or enough of his key attributes.
Hibs manager Shaun Maloney may bring in more signings.
A player with a bit of mischief and an almost Peter Pan love of the game, finding an identikit version will be a daunting and not necessarily fruitful search.
Of course they can find players who can offer different elements, but few come ready made to get a stadium buzzing with excitement and expectation as they switch on the afterburners and home in on goal.
Then you add in the pace, the direct running, the assists and the goals. He provided five and 14 of them respectively this season before jumping on the plane to Saudi Arabia.
And, scant statistics cannot quantify his mental and psychological contribution to a dressing room that has had it's downs as well as ups during his team in Edinburgh. His effervescence helped to lift spirits and bolster the team's bouncebackability, and his positive personality resonated with fans who felt inclined to respond in kind.
Without him, against Cove Rangers on Thursday night, the performance and the atmosphere was flat.
While the team carved out openings as he watched from the stand, they did not do it with the kind of pace and verve that led to the away keeper Kyle Gourlay being peppered with shots.
One-paced, they lacked his cockiness and his nuisance value.
Manager Shan Maloney admitted they need to recruit wisely and quickly.
"People with Martin's attributes make a difference," he said. "It's hard to find and that's why he's been one of the best in his position.
"But we can't not replace that. At times we saw we have some good play but when you have a player who can go by an opponent it changes really long possession into a good chance."
A popular figure, in a social media post the Australian international said: "It's not goodbye, it's see you later." And, many will hope they will see him in the green and whale again.
Few if any begrudge him his major payday, but they will be slightly worried about how they manage to fill the void in the meantime.
Martin Boyle opens up on Hibs exit and insists 'there's no bridges burned'
Martin BoyleCove RangersEuropeJack Ross


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Startup Class Technologies Bets Big on the Future of Online Learning...
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Digital Learning in Higher Ed
Startup Class Technologies Bets Big on the Future of Online Learning (and Zoom)
By Jeffrey R. Young Sep 24, 2021
23WaysArt / Shutterstock
It may not seem too surprising that one of the best-funded edtech startups in the past year of pandemic has been a company that piggybacks on the success of Zoom to add tools for running online classes. But the sheer size of its fundraising may raise some eyebrows.
Class has raised more than $165 million from a mix of sources including GSV Ventures, Owl Ventures and Reach Capital since it was founded nearly a year ago. Last month EdSurge sat down with its founder and CEO, Michael Chasen, to find out what he's seen so far and where the company hopes to go next.
Chasen is a familiar figure in edtech: he co-founded Blackboard, one of the largest providers of learning management systems to colleges and schools, and served as its CEO for years. As he watched his own kids adjust to online schooling during the pandemic, he felt that Zoom lacked features to let teachers handle standard classroom activities such as taking attendance or giving quizzes.
He knew that Zoom had a development kit, or SDK, that let other software integrate on top of the video platform, so he decided to build those features into what became Class.
"Now you can use Zoom, but take attendance, hand out assignments, give tests or quizzes, proctor those exams, and talk one-on-one with the students," he says. "We let you replicate the physical class in an online environment."
The plan when the company started was to start with higher education and K-12 and later expand into the corporate learning market. But Chasen said Class got so many inbound requests from the corporate side that they have done more there already than originally thought.
In corporate training, he said, "they moved these classes online, and they found that the employees are more engaged with the live teacher. If you tell your employees, 'You could take this management course, it's self-paced, it's on your own,' Half of them get around to it. Half of them don't don't really care. If you tell them, 'It's at seven o'clock on Wednesday night, there's a teacher there,' everybody shows up. And they're more engaged. And now with Zoom, you can really have a live class [remotely]."
Most colleges were already doing at least some online education even before the pandemic hit. But Chasen says that higher education also offered many online classes asynchronously in the past, meaning that students could go through them on demand rather than showing up at a set time. But he said colleges are now also moving to have more live sessions in online courses, and they're looking for tools to make that happen.
Class already has a well-funded rival to offer a next-generation online classroom, a startup called Engageli that has raised more than $47 million in the past year. That company's tool was built from the ground up, while Class is an add-on to Zoom, which means that institutions who want to use Class have to also purchase a license to Zoom if they haven't already done so.
Chasen argues that standing on the shoulder of a fast-growing video platform means he can offer a more robust and secure experience. "Zoom has multi-billion dollars of video and audio architecture behind them for streaming these classes or meetings live. I could never even build that," he said. "I was able to focus all of our development on really adding the teaching and learning tools to Zoom. I didn't have to worry about the audio video transcribing or anything like that."
But if Zoom is already built, why does Class need all the investment money?
"Zoom is actually a very expensive platform to develop on," Chasen explains. Because it's a downloadable app, his team had to create separate versions of Class for Windows, Chrome, Mac OS and several mobile operating systems. That means his development costs are nearly five times as much as if he built a piece of software for the web. Currently, he estimates there are between 80 to 100 people at Class working on "development and consulting services."
While Zoom has grown in education since the start of the pandemic, there are still many schools and colleges that have already adopted a rival video platform, such as Microsoft Teams or Google Classroom.
What did Chasen learn from his experience as the longtime CEO of Blackboard?
He said his biggest advantage is that he knows so many figures in higher ed and K-12 based on his previous work, which made it easier to form advisory groups and get feedback as he developed Class.
At Blackboard, Chasen had a reputation as something of a business shark, buying up competitors and suing rivals. And many professors and college leaders criticized the company during that period for not feeling like a partner.
Chasen says he has learned from that experience as well.
"When I started Blackboard I was a lot younger and I didn't have a lot of experience," he said. "I don't think we were working as closely as we should have with institutions to be getting that feedback and getting input along the way." In contrast, he says one of the first things he did at Class was create advisory boards to get community input.
At the moment, Chasen sees plenty of schools eager to go back to in-person. But he said that many districts have started or expanded virtual academies to give options to those students that do better online or need the online option.
He said that he sees K-12 as more of a "long-term opportunity," since schools were doing little online education before the pandemic. Now, many see it as something to keep in the mix of options in the future.
Jeffrey R. Young (@jryoung) is the managing editor of EdSurge and producer and host of the EdSurge Podcast. He can be reached at jeff [at] edsurge [dot] com
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text:
Virginia Surtees
Virginia Surtees biography of Jane Welsh Carlyle will be reviewed by John Bayley.
Browning's Last Duchess
Virginia Surtees, 9 October 1986
Entries for September 1869 from the diary of the Hon. Mrs George Howard (Rosalind, youngest daughter of the second Lord Stanley of Alderley) are published here for the first time, with the permission of the Hon. Simon Howard. They cover the visit of the 57-year-old Robert Browning to Naworth Castle, the Cumberland home of the George Howards. Browning had recently published his great poem The Ring and the Book, with its dedication to his dead wife. He was mentally tired, and uneasy over his son Pen's performance at Oxford, but had agreed to go to Scotland in August with Sarianna his sister and Pen, and join forces at North Berwick with his friends the Storys. Before starting, he had told John Forster that Pen and the Storys had persuaded him to this course, but that he meant to stay only a month.
Entries for September 1869 from the diary of the Hon. Mrs George Howard (Rosalind, youngest daughter of the second Lord Stanley of Alderley) are published here for the first time, with the...
Pieces about Virginia Surtees in the LRB
Burnished and braced
Alethea Hayter, 12 July 1990
'Relaxation is my bane, Lady Morpeth. All my habits and tastes lean that way and in consequence I am going to wage war upon them all. I dread a languid yellow old age, hot, perfumed and...
Wives, Queens, Distant Princesses
John Bayley, 23 October 1986
Suppose Mr and Mrs Coleridge to be young SDP yuppies today, who have asked us to dinner. What impression of each should we get? Of an amiable but very silly young man who talked too much and put...
Super-Real
Peter Campbell, 18 March 1982
You know a Pre-Raphaelite picture when you see one, but definitions come hard. The paintings are likely to be detailed, but Rossetti's are soft and generalised; they often take subjects from...


An unsuccessful candidate for the Lynwood City Council has been charged with two misdemeanor election code violations for failing to register a campaign committee and using a fictitious address on a mass mailer, the district attorney's office said Tuesday.
Iris Pygatt, 51, was expected to surrender to authorities and be arraigned on the two charges today in Compton Municipal Court, said Deputy Dist. Atty. Herb Lapin.
Lapin said his office would recommend that the court release Pygatt, a Lynwood real estate agent, on her own recognizance after booking and arraignment. Pygatt declined to comment on the charges.
Pygatt, who finished sixth in a field of 12 candidates in the November election, is charged with failing to register her campaign committee with the secretary of state and sending out a mass mailer with a fictitious address, Lapin said.
Both are violations of the state election code, Lapin said. The maximum penalty for each offense is a $500 fine and six months in jail.
Lapin said there were complaints filed after the election that a four-page mailer listed a bogus committee called the Democratic Coalition of Lynwood, with an address of 1719 Century Blvd.
There was also a complaint that Pygatt had substituted her picture and name in the mass mailer for a third candidate who was running with two incumbents as a slate, giving the impression that the Democratic Coalition was supporting her and the two incumbents, Lapin said.
Lapin said he would not reveal who the complainants were because it is not department policy. However, he said, an investigation conducted by the district attorney's office showed that the Democratic Coalition did not exist, had not been filed with the secretary of state and that the address did not exist either in Lynwood or Los Angeles.
The complaint that Pygatt had substituted her picture and name for a third candidate was not part of the district attorney's charges. "We pursued what we could legally pursue with adequate evidence," Lapin said.
During the election for three council seats, Councilmen E. L. Morris and Louis Thompson were running as a slate with Louis J. Heine. Morris and Thompson were reelected, running third and second, respectively; Heine finished fourth. Evelyn M. Wells came in first.
Meanwhile, a hearing on a motion to dismiss a lawsuit filed by two other unsuccessful candidates in the Lynwood City Council election has been scheduled for Wednesday in Los Angeles Superior Court.
Attorney Henry Barbosa said he will ask the court to dismiss a complaint that Morris, Thompson and Wells won the election because there were numerous irregularities, including ballot tampering during the election.
"I will ask for dismissal because the claims have no merit," Barbosa said, adding that the suit only has "generalized accusations" with no specifics.
The lawsuit was filed Dec. 4 by Carlos Manlapaz and Benito R. Miranda. Miranda finished eighth in the race and Manlapaz finished 10th.


text:
Q: Why can't I end an active socket in node.js without causing a segmentation fault? I'm trying to stop some http servers in node.js but I don't want to wait for all keep-alive connections to timeout before the servers can fully close. So, I've been trying to keep track of all active sockets and then when the server needs to be stopped I want to call .end() on them to let the client know that I want to close the socket. However, when I do this I get a segmentation fault.
I have used a segmentation-fault handler to get a better stack trace but I can't figure out what could be going wrong.
Here is the code to keep track of the connections:
connectionNum = 1
activeHttpConnections = {}

trackConnection = (map, socket) ->
    # save active socket
    id = connectionNum
    map[id.toString()] = socket

    # remove socket once it closes
    socket.on 'close', ->
        map[id.toString()] = null
        delete map[id.toString()]

    # reset connection number eventually
    if connectionNum > 1000000
        connectionNum = 1
    else
        connectionNum++

When the server is started I call:
httpServer.on 'connection', (socket) -> trackConnection activeHttpConnections, socket

And when I stop the servers I call:
# close active connection so that servers can stop
for key, socket of activeHttpConnections
    socket.end()

When .end() is called I get the following stacktrace:
PID 29943 received SIGSEGV for address: 0x58
/home/ryan/git/openhim-core-js/node_modules/segfault-handler/build/Release/segfault-handler.node(+0x1185)[0x7f50a037e185]
/lib/x86_64-linux-gnu/libpthread.so.0(+0x10340)[0x7f50a2384340]
node(_ZN4node10StreamWrap8ShutdownERKN2v820FunctionCallbackInfoINS1_5ValueEEE+0x166)[0x9d1846]
node(_ZN2v88internal25FunctionCallbackArguments4CallEPFvRKNS_20FunctionCallbackInfoINS_5ValueEEEE+0x9b)[0x616d9b]
node[0x6356b6]
[0x270b80a060a2]

Any idea what I could be doing wrong, or what this stacktrace could point to?

redpajama_set_name: RedPajamaStackExchange


text:
package io.cattle.platform.docker.api.container;

import io.cattle.platform.api.utils.ApiUtils;
import io.cattle.platform.core.dao.InstanceDao;
import io.cattle.platform.core.model.Instance;
import io.cattle.platform.docker.api.ContainerProxyActionHandler;
import io.cattle.platform.docker.api.model.ServiceProxy;
import io.github.ibuildthecloud.gdapi.request.ApiRequest;
import io.github.ibuildthecloud.gdapi.request.resource.impl.AbstractNoOpResourceManager;
import io.github.ibuildthecloud.gdapi.util.ResponseCodes;

import java.util.List;

import javax.inject.Inject;

import org.apache.commons.lang3.StringUtils;

public class ServiceProxyManager extends AbstractNoOpResourceManager {

    @Inject
    InstanceDao instanceDao;
    @Inject
    ContainerProxyActionHandler actionHandler;

    @Override
    protected Object createInternal(String type, ApiRequest request) {
        ServiceProxy proxy = request.proxyRequestObject(ServiceProxy.class);
        String service = proxy.getService();
        String[] parts = StringUtils.split(service, ".", 2);
        List<? extends Instance> instances = null;

        if (parts.length == 2) {
            instances = instanceDao.findInstanceByServiceName(ApiUtils.getPolicy().getAccountId(), parts[1], parts[0]);
        } else {
            instances = instanceDao.findInstanceByServiceName(ApiUtils.getPolicy().getAccountId(), parts[0]);
        }

        if (instances.size() == 0) {
            request.setResponseCode(ResponseCodes.NOT_FOUND);
            return null;
        }

        return actionHandler.perform(null, instances.get(0), request);
    }

    @Override
    public Class<?>[] getTypeClasses() {
        return new Class<?>[] { ServiceProxy.class };
    }

}

redpajama_set_name: RedPajamaGithub


Apparently we're not all as excited as we used to be about the New Years sales. Black Friday and pre-Christmas bargains have taken the gloss off of what was THE time to get some deals.
Don't buy something when there's a WHOLE RACK OF IT! There will be a good reason - neckline doesn't sit right, it has funny washing instructions, the colours hard to wear... You've been warned!
Buy for your summer holidays! OK, so it requires a little imagination but you can pick up some stunning dresses, bikinis and beachwear in the winter sales. The stores want it gone and out of their stock rooms so consider what you're holiday suitcase is missing.
Don't buy ANYTHING if you're first thought when you see it is only 'Wow, that's cheap!' You're consciousness has zoned in on the price not the object - try and buy the things you really love. It's still money.
Buy one thing and then pass one on. This is the keeping a good wardrobe and a clutter free home. For every item you bring through the door find an item at you're no longer wearing or using, thank it and send it with love to your nearest charity shop or better still hold a shwopping event in January. Generally a dull month so a good time to clear your wardrobe of the items you no longer love and get your friends round too with maybe a glass of grape juice.