Sobolev inequality with non-uniformly degenerating gradient

. In this paper we prove the following weighted Sobolev inequality in a bounded domain Ω ⊂ R n , n ≥ 1, of a homogeneous space ( R n , ρ , wdx ) , under suitable compatibility conditions on the positive weight functions ( v , w , ω 1 , ω 2 , . . . , ω n ) and on the quasi-metric ρ , where q ≥ p > 1 and M S denotes the strong maximal operator. Some corollaries on non-uniformly degenerating gradient inequalities are derived.


Introduction
In this paper we aim to prove the following weighted Sobolev type inequality in a bounded domain Ω ⊂ R n , n ≥ 1, of a homogeneous space (R n , ρ, wdx) where q ≥ p > 1 and M S denotes the strong maximal operator. This can be done under suitable compatibility conditions on the positive weight functions (v, w, ω 1 , ω 2 , . . . , ω n ) and on the quasi-metric ρ. We say that (1.1) is a non-uniform weighted Sobolev inequality since the functions ω i ω −1 j , i, j = 1, . . . , n, are not assumed to be neither bounded nor bounded away from zero in any compact subset of Ω.
The study of the above mentioned qualitative properties of second order elliptic equations in absence of uniform ellipticity condition and in lack of uniform degeneration relies on Poincaré-Sobolev type weighted inequalities having non-uniformly degenerating gradient. Meanwhile, the theory has also been extended to more general contexts, such as that of Carnot-Carathéodory metrics associated with families of vector fields (see, e.g., [8][9][10]13]).
To deduce the inequality (1.1) one could first derive a suitable representation formula in terms of integral operators of potential type, and then use some continuity results for these operators in proper metric spaces, endowed with doubling measures (see, for example, [18]). In this paper, we show a new different approach to obtain the inequality (1.1). The arguments of our proofs are inspired by those of [31] (see, also the recent papers [29,32]), where the Euclidean metric was considered. However, the ideas of [31,32] cannot be simply adapted to homogeneous spaces and to non-uniformly degenerating gradients, since not all the homogeneous spaces posses the Besicovitch covering property (see, e.g., [35]). To overcome this difficulty, we use the "5B" covering lemma that holds in any homogeneous space, see e.g., [7,39]. We refer to [37], and to the references therein where the Euclidean metric and equal weights ω i , , i = 1, . . . , n, are considered.
In general, when dealing with multi-weighted Sobolev inequalities the task is to find sufficient (and hopefully necessary) conditions on the measures ω i (x)dx, i = 1, . . . , n, and v(x)dx which give where 1 ≤ p ≤ q < ∞ and the constant C does not depend on f and Ω. For equal weights ω i , i = 1, . . . , n, sharp sufficient conditions can be found in [4,15] and in the papers [20,32,34]. Though this subject has been extensively studied in the last years it is still far from its full characterization (see, [6,[14][15][16][17][18][19][20][21][22]). Some progresses in deriving sufficient conditions for the Sobolev-Poincaré type inequalities with Grushin type weights were made in the works [15,29]. In this article we give sufficient conditions for the inequality (1.2) to hold and we show some generalizations.
A useful result by Macías and Segovia (see [28]) asserts that, every quasi-metric space is metrizable, i.e. there exist a distance d and a positive number α > 0 such that ρ α is equivalent to d.
Now, let us denote by B(x, r) = {y ∈ R n : ρ(y, x) < r} the ρ-metric ball with center in x ∈ R n and radius r > 0, and let µ be a nonnegative Borel measure on R n satisfying the doubling condition. We say the measure µ is a doubling measure if there exists C 1 such that The quasi-metric space (R n , ρ) equipped with a doubling measure µ is called a homogeneous space and it is denoted by (R n , ρ, dµ) (see [7]). In Section 3 we will give an example of homogeneous space. In sequel, the notation Q n (x, r) ( or simply Q(x, r) ) denotes the n-dimensional Euclidean ball Q(x, r) = {y ∈ R n : |y − x| < r} centered in x and of radius r. For i = 1, . . . , n, we denote by i (B(x, r)) = sup |z i − y i | : z = (z 1 , . . . , z n ), y = (y 1 , . . . , y n ) ∈ B(x, r) and by d(Ω) = sup {ρ(x, y) : x, y ∈ Ω} the ρ-diameter of the domain Ω. We also let Σ be the collection of ρ-metric balls with center in Ω and radius less then d(Ω).
Given an integrable function f and a measurable set E ⊂ R n we denote by f (E) = E f (x)dx the weighted measure of E, while |E| denotes the Lebesgue measure of E. Denote by p the conjugate number of 1 < p < ∞ such that 1 p + 1 p = 1 A measurable function taking a.e. finite positive values is called a weight. A weight function f : R n → (0, ∞) belongs to the A p -Muckenhoupt weight class, 1 < p < ∞, with respect to the quasi-metric ρ, if for any ρ-metric ball B = B(x, r) ⊂ R n , one has while it belongs to the A 1 -class if where the constants C > 0 do not depend on x ∈ R n and r > 0.
A weight function f : R n → (0, ∞) belongs to the A ∞ -Muckenhoupt weight class A ∞ (dx) if there exist two constants C, δ > 0 such that for any ρ-metric ball B = B(x, r) and any measurable subset E ⊂ B it holds that Let g : R n → (0, ∞) be a weight function and µ be a doubling measure. We say g belongs to the A ∞ (µ) weight class if there exist two constants C, δ > 0 such that for any ρ quasi-metric ball B = B(x, r) and any measurable subset E ⊂ B one has For the main properties of the A p -Muckenhoupt's weight classes, we refer the reader, for instance, to [7]. It is well-known that A p ⊂ A ∞ for any fixed 1 ≤ p ≤ ∞ and moreover A ∞ = ∪ 1≤p<∞ A p . Furthermore, A p ⊂ A p−ε , for some ε > 0 depending on the constant C in the A p class definition.
In the statement and proof of Theorem 2.5 below, we use the strong maximal function M S w. For sake of completeness, let us recall its definition. Let R denote the collection of rectangles R in R n with sides parallel to the coordinate axes, we define the strong maximal function M S as In Theorem 2.6, we make use of the classical fractional maximal operator M ε defined as where the supremum is taken all over the Euclidean balls {Q} containing the point x.
In the proofs of our main results we avail ourselves of the so called "5B" covering lemma below. This lemma, unlike Besicovich covering property, is valid in any homogeneous space. . Let (X, ρ, µ) be a homogeneous space. Let B = {B α = B(x α , r α ) : α ∈ Γ} be a family of balls in X such that ∪ α∈Γ B α is bounded. Then there exists a sequence of disjoint balls {B(x i , r i )} i∈N ⊂ B such that for every α ∈ Γ there exists i satisfying r α ≤ 2r i and B α ⊂ B(x i , 5K 2 0 r i ).

Definition 2.2.
Throughout this paper, we consider the quasi-metrics ρ satisfying the following "S"-condition |B # | ≤ C|B|, (2.4) for all the ρ quasi-metric balls B (see next section for their definition). Here B # is the smallest parallelepiped with edges parallel to coordinate axes containing the ρ quasi-metric ball B. Definition 2.3. Moreover, we assume that there exists a constant c > 0 such that for every B and every x, y ∈ B, t ∈ (0, 1) one has Now, we are ready to state our main results.
Theorem 2.4. Let q ≥ p ≥ 1, (R n , ρ, dx) be a homogeneous space and Ω ⊂ R N be a bounded domain. Assume that the ρ quasi-metric balls B ∈ Σ satisfy the S-condition (2.4) and (2.5). Let v ∈ A ∞ (dx) and ω 1−p i , i = 1, 2, . . . , n, be doubling functions on Σ. If for all Lipschitz continuous functions f : Ω → R vanishing on ∂Ω, and with a constant C 0 depending only p, q, n and on C, δ in (2.2).
Theorem 2.4 is an easy consequence of the next assertion.
for all Lipschitz continuous functions f : Ω → R vanishing on ∂Ω, and with a constant C 0 depending only on p, q, n and on C, δ in (2.3).
We remark that, in Theorem 2.5, the Sobolev type weight inequality (2.9) is proven with different weights for the partial derivatives. This is due to the fact that the weights and the metric must be in a balance with the geometry of the quasi-metric balls. Taking v ≡ ω i ≡ 1 in (2.9) we get the measure w(x)dx to be a doubling function on Σ, hence we obtain the inequality Moreover, let us mention that the doubling condition on the weights ω In the next Theorem 2.6 we give a better estimate. In order to do that, the sufficiency condition (2.8) needs to be suitably strengthened (see (2.11)). Theorem 2.6 below gives, locally, a finer inequality since Theorem 2.6. Let q ≥ p ≥ 1 and Ω ⊂ R N be a bounded domain. Let (R N , ρ, wdx) be a homogeneous space and assume that there exists a positive constant C 1 such that . . , n, be doubling functions on Σ. Assume that the ρ quasi-metric balls B ∈ Σ satisfy the Scondition (2.4) and (2.5). If for all Lipschitz continuous functions f : Ω → R vanishing on ∂Ω, and with a constant C 0 depending on p, q, n and on C, δ in (2.3).

An example of homogeneous space
Let ω : R n → (0, ∞) be a positive measurable function, such that σ(x) = 1 is in the Muckenhoupt A 2 -weight class all over the n-dimensional Euclidean balls. This condition used in proofs of the corollaries below. Observe that this gives that also ω is in the Muckenhoupt's A 2 -class all over the n-dimensional Euclidean balls. For and assume that h x (0) = 0, lim t→+∞ h x (t) = +∞ for a fixed x ∈ R n . Then we may consider with a constant K 0 ≥ 1 independent of z 1 , z 2 , z 3 ∈ R N , (see, e.g., [1,15]). Therefore, the above defined quasi-metric space (R N , ρ) endowed with the Lebesgue measure is a homogeneous space.
In general, the balls of a homogeneous space are not convex, therefore the conditions (2.4), (2.5) may be failed. The condition (2.4) means that the Lebesgue measure of a metric ball comparable with Lebesgue measure of its circumscribed parallelepiped. Also as we have noted the balls of a metric space are not convex the line segment connecting any two points of a ball may get out of that ball. The meaning of condition (2.5) is that, although the points on a line segment get out of the ball its points are contained on the comparable ball. It easily seen that the balls of metric (3.1) are convex and conditions (2.4), (2.5) are satisfied for that.

Applications
In this section, we give two examples of applications of Theorem 2.4. To this aim, let ρ be the quasi-metric defined in (3.1). It is not difficult to see that the ball B(z 0 , R) with center in z 0 = (x 0 , y 0 ) ∈ R N and radius R > 0 of this quasi-metric is given by Let ω as in the beginning of Section 3 and f : (x, y) ∈ R n × R m → f (x, y) ∈ R be a Lipschitz continuous function. The degenerated gradient of f is given by For m ≥ 2 we can prove the following result: n+m−2 , t = n n+m−2 and let ω ∈ A 2 -Muckenhoupt class function on R n . Then, for any function f , Lipschitz continuous in the ball B(z 0 , R) ⊂ R N , vanishing on ∂B(z 0 , R). The positive constant C 0 in (4.2) depends on n, m and on the constants in the A 2 -condition from (2.1).
For m = 1, we have: n−1 , and let ω −1 be a classical A 1+ 1 n -Muckenhoupt class function on R n . Then, for any x ∈ R n and r > 0. Then for all f ∈ Lip 0 (B z 0 R ) C 0 depends on the A ∞ , A 2 conditions for v, ω and n, q.
The given above corollaries generalize the two-weight Sobolev inequalities to the case of non-uniformly degenerate gradient ∇ ω f . Therefore, those inequalities are of the well-known inequalities type by Chanillo-Wheeden, Fabes-Kenig-Serapioni with ω ≡ 1. Such inequalities may be applied to the study of equations with Grushin type operator ∂ x i ω(x)∂ x i + ∂ 2 y j or its generalizations ∂ x i ω(x)w∂ x i + ∂ y j w∂ y j when w(x, y) is a function of two variables x, y obligated to satisfy some conditions.
Note that, the condition (4.4) is a balance condition of Chanillo-Wheeden type [4] for the case of non-uniformly degenerate gradient inequality of the Sobolev type. Note again, the function v depends only the variable x while the function f is dependent of two variables z = (x, y).

Proofs of the main results
Let us start proving Theorem 2.5.

Proof of Theorem 2.5
Assume that f is not equal to zero almost everywhere in Ω, otherwise the result of Theorem 2.5 is trivial. For α > 0 set Ω α = x ∈ Ω : | f (x)| > α . Since f is continuous the set Ω α is open. Let a fixed α be such that the set Ω 3α is nonempty. Choose a countable covering of Ω α made up of connected components Ω α,j ⊂ Ω α , j ∈ N. Denote the parts of Ω 3α and Ω 2α contained in Ω α,j by Ω 3α,j and Ω 2α,j , respectively (note that the sets Ω 3α,j and Ω 2α,j need not to be connected).
For the reader's convenience, let us recall that the weight function w of the homogeneous space (R n , ρ, wdx) satisfies the doubling condition on the ρ quasi-metric balls. Let b ∈ Ω 3α,j be a fixed point. Let us show that there exists a ρ-quasi metric ball B = B (b, r(b)) such that where γ is a small positive number that will be chosen later on. To this aim, let γ > 0 and define the function which is continuous and negative for sufficiently small t > 0 since b is an interior point of Ω 3α,j . From the doubling property of w on the ρ-quasimetric balls it follows that there exists a positive real number τ such that w (B (b, d(Ω)) \ Ω) ≥ τw (B (b, d(Ω))) .
(5.4) By (5.1) and since v ∈ A ∞ (wdx) we have again therefore, eventually reducing γ Thus, by (5.4) we get In case 2), we have two possibilities: 2a) If 2a) takes place, let us show that where M S w denotes the strong maximal function of w, B * * is the ρ-metric ball B * * = cB * and (B * ) # denotes the smallest rectangular with edges parallel to coordinate axes containing B * . To prove inequality (5.8), we follow an idea of [31], formula (3.7). DenoteÂ = B * \ Ω 2α,j and Z = B * ∩ Ω 3α,j . Let the points x ∈Â and y ∈ Z be arbitrary fixed. Since the quasimetric balls are not assumed to be convex, the line segment xy = {x + t(y − x) : t ∈ [0, 1]} connecting x, y may get out of the ball B * as t varies in (0, 1). But, due to hypothesis (2.5) it will stay in the congruent ball B * * = cB * .
Also, the line segment xy intersects the surfaces {z ∈ Ω α,j : | f (z )| = α} and {z ∈ Ω j α : Here, t 2 corresponds to the value of t for which xy meets for the first time the surface ∂Ω 2α,j after leaving ∂Ω α,j while t 1 corresponds to the value of t when xy intersects the surface ∂Ω α,j .
Having this in mind and using (5.1), (5.6) it follows that Whence, By Fubini's theorem, Let us now make the change of variable z = x + t(y − x) in the interior integral to pass from y to z. Since dy = t −n dz, one has For t ∈ (0, 1) and z ∈ G it follows |x s − z s | < tl s (B * ), s = 1, 2, . . . , n, therefore applying Fubini's formula again, we get where M S is the strong maximal operator. Therefore, In the case 2b) we can argue as in case 2a) by puttingÂ = B * \ Ω α,j and Z = Ω 2α,j ∩ B * . Thus, we have In this case the line segment xy intersects the surfaces {z ∈ Ω α,j : | f (z )| = α} and {z ∈ Ω j α : | f (z )| = 2α} in points that can be expressed as z = x + t 1 (y − x) and z = x + t 2 (y − x) where t 1 , t 2 ∈ [0, 1], with t 2 > t 1 depend on x, y. Here, t 2 corresponds to the value of t for which xy meets for the first time the surface ∂Ω 2α,j after leaving ∂Ω α,j while t 1 corresponds to the value of t when xy intersects the surface ∂Ω α,j .
In this case, in place of (5.11), we get the following inequality where G = B * * ∩ Ω α,j \ Ω 2α,j . Therefore, Now, since Ω 3α,j ⊂ Ω 2α,j , combining (5.5), (5.12), and (5.13) we have (5.14) Summing up over j = 1, 2, . . . , we obtain Writing (5.15) for the system of balls B * m , we get Summing up over m = 1, 2, . . . , we get (5.20) Using Hölder's inequality, this implies By the property (5.17) of the covering {B m } and by the doubling assumption on σ i M S w on the ρ quasi-metric balls, we get where κ i = σ i M S w. Note that in (5.22) we have used that the condition (2.4) and (2.8) and the doubling assumption on the measures yield c p m κ i (B m ) ≤ C 3 A p (λ(B m )) p /q . Now, by (5.1) and since p /q ≥ 1, Observe that the same inequality can be obtained also for integrals over the sets Ω α \ Ω 2α . Thus, by (5.21), we get Therefore, from (5.25) and Hölder's inequality, we get Now, by Fubini's theorem, On the other hand, Minkowski's inequality gives Using the last inequalities and choosing This completes the proof of Theorem 2.5

Proof of Theorem 2.6
To prove Theorem 2.6 we may argue following along the lines the proof of Theorem 2.5 until formula (5.10). Then, from hypothesis (2.10), for t ∈ (0, 1) and z = x + t(y − x) ∈ G, using the condition (2.10) it follows therefore applying Fubini's formula again, Now, by the definition of the fractional order Hardy-Littlewood maximal operator over Euclidean balls and since B(x, 2K 0 r(B * )t) z it follows By (5.29), one has Arguing further as in Theorem 2.5 we obtain estimate (5.28) withĀ in place of A. The proof of Theorem 2.6 is then complete.

Proof of Theorem 2.4
Theorem 2.4 is a corollary of Theorem 2.5 for w ≡ 1.

Proof of Corollary 4.1
The result follows from Theorem 2.4. It is enough to choose (x, y) ∈ R N , with N = n + m, v(x, y) = ω(x) t , t = n n+m−2 , and ω 1 = · · · = ω n = ω(x), ω i ≡ 1, i = n + 1, n + 2, . . . , n + m in the statement of Theorem 2.4. Observe that the A ∞ -condition on the ρ-quasimetric balls on ω(x) n n+m−2 as well as the A 2 -condition on the ρ-quasimetric balls for ω are satisfied, in view of (3.1) and (4.1). Indeed, it is well-known that the A p condition for some p ≥ 1 implies the A ∞ condition. Therefore, in order to show that ω t belongs to A ∞ let us show that it belongs to A p , for some p ≥ 1. To this aim, observe that, by our assumptions, σ ∈ A 2 hence σ(Q) The last inequality implies ω t ∈ A p with p = 1 + n n+m−2 . For what concerns hypothesis (2.6), by the definition of the quasimetric ρ given in Section 3, in this case it can be derived by the following inequality Cr|B(z, r)| −( 1 2 − 1 q ) 1 |Q(x, r)| Q(x,r) ω t ds 1 q 1 |Q(x, r)| Q(x,r) σ(s)ds 1 2 ≤ A, (5.31) where B(z, r) is a ρ-quasimetric ball of center z and radius r, 0 < r < R, while Q(x, r) is the projection of B(z, r) on R n . Thus, in order to satisfy condition (2.6) we need to estimate the left hand side of (5.31) from above. To this aim, observe that r|B(z, r)| −( 1 2 − 1 q ) 1 |Q(x, r)| Q(x,r) ω t ds where we used the fact that q = 2(n+m) n+m−2 gives 1 − (n + m) 1 2 − 1 q = 0 and Hölder's inequality. Now, since t q = 1 2 − m 2 1 2 − 1 q = n 2(n+m) , taking into account assumption ω ∈ A 2 we get

Proof of Corollary 4.2
To prove this result, one can follow along the lines the proof of Corollary 4.1, for t = n n−1 , with suitable modifications.

Proof of Corollary 4.3
The proof of Corollary 4.3 is obtained from Theorem 2.4 similarly to that of Corollary 4.1, so we leave the proof to the Reader.