Degenerate operators on the half-line

We study elliptic and parabolic problems governed by the singular elliptic operators $$ y^{\alpha}\left(D_{yy}+\frac{c}{y}D_y\right)-V(y),\qquad\alpha \in\mathbb R $$ in $\mathbb R_+$, where $V$ is a potential having non-negative real part.


Introduction
In this paper we study solvability and regularity of elliptic and parabolic problems associated to the degenerate operators L = y α D yy + c y D y − V and D t − L in the half-line R + .
Here c, α are real numbers and V ∈ L 1 loc (R + , y c−α ) is a potential having non-negative real part. The operator B = D yy + c y D y is a Bessel operator and satisfies the scaling property I −1 s BI s = s 2−α B, I s u(y) = u(sy).
We study L in the weighted spaces L p m := L p (R + , y m dy), m ∈ R, and we characterize all m such that L generates a C 0 -semigroup. When V ≥ 0 we also prove that the generated semigroup is analytic and we show that it has maximal regularity, which means that both D t v and Lv have the same regularity as (D t − L)v. In the case V (y) = y α we finally characterize the domain of L.
We observe that the results already available for B, see [11,Section 3] and also [10,9,8,14,7] for the N -d version of B, imply the corresponding ones for y α B in L p m by a change of variables, as described in Section 3. The change of variables varies the underlying measure and explains why need the full scale of L p m spaces.
More effort is needed to add the potential term. We consider first B − V in L 2 (R + ; y c dy). We use form methods to construct an analytic semigroup, we prove kernel bounds for complex times via Davies-Gaffney estimates and provide a core. Then, with the methods of Section 3, we deduce similar results for y α B − V in L 2 (R + ; y c−α dy). Next we prove that the semigroup can be extended to L p m under sharp conditions on p and m. Finally we prove that for every ǫ > 0 the family of operators e z(y α B−V ) : z ∈ Σ π 2 −ǫ , 0 ≤ V ∈ L 1 loc R + , y c−α is R-bounded in L p m , which implies the maximal regularity of the semigroup when V ≥ 0. As a motivation for our investigation, we point out that, in the special case V (y) = y α , all the results above play a crucial role in [13] in the investigation of the degenerate operators Let us suppose, for simplicity, b = 0, α 1 = α 2 := α. Assuming that y α (∆ x u + B y u) = f and taking the Fourier transform F u orû with respect to x we obtain y α |ξ| 2û (ξ, y) = −y α |ξ| 2 (y α |ξ| 2 − y α B y ) −1f (ξ, y). Therefore y α ∆ x L −1 = F −1 y α |ξ| 2 (y α |ξ| 2 − y α B y ) −1 ) F and the boundedness of y α ∆ x L −1 is equivalent to that of the multiplier ξ ∈ R N → y α |ξ| 2 (y α |ξ| 2 − y α By) −1 .
For this reason we prove in Section 8 that certain multiplier associated to y α B − V satisfy a vector valued Mikhlin theorem. These results rely on square function estimates which we deduce from kernel bounds and the following equality, which allows to treat λ or |ξ| 2 as spectral parameters simultaneously λ − y α B + |ξ| 2 y α −1 = |ξ| 2 − B + λ y α −1 1 y α .
We restrict ourselves to α < 2 and consider y α B with Neumann boundary condition at 0, namely lim y→0 y c D y u(y) = 0. This is equivalent to require y α−1 D y u ∈ L p m , see [12,Proposition 5.11]. The restriction α < 2 is not really essential since one can deduce from it the case α > 2, which requires a boundary condition at ∞, using the change of variables described in Section 3.
Besides this, our strategy can be easily adapted to different boundary conditions and to more general operators y α D yy + c y D y − b y 2 −V . We do this (in much more generality) in [13,Sections 7,8].
The paper is organized as follows. In Section 2 we briefly recall the harmonic analysis background needed in the paper, as square function estimates, R-boundedness and a vector valued multiplier theorem.
In Section 3, we exploit an elementary change of variables, in a functional analytic setting, to reduce our operators to the simpler case where α = 0. Section 4 is devoted to the study of the Bessel operator y α B. In Sections 5, 6 and 7 we perturb the Bessel operator by adding the potential V and we prove real and complex kernel estimates, generation results and maximal regularity for y α B − V . Finally in Section 8 we treat the case V (y) = y α and characterize the domain of y α B − y α .
Notation. For m ∈ R we consider the measure y m dy in R + and we write L p m for L p (R + , y m dy). Similarly W k,p m = {u ∈ L p m : ∂ α u ∈ L p m |α| ≤ k}. When we write V ∈ L q loc (R + , y m dy), we mean that V ∈ L q ([0, b], y m dy) for every b < ∞.

Harmonic analysis and maximal regularity
The study of maximal regularity of parabolic problems of the form u t = Au + f, u(0) = 0, where A is the generator of an analytic semigroup on a Banach space X, consists in proving estimates like . This can be interpreted as closedness of D t − A on the intersection of the respective domains or, equivalently, boundedness of the operator Nowadays this strategy is well established and relies on Mikhlin vector-valued multiplier theorems. Let us state the relevant definitions and main results we need, referring the reader to [5], [16] or [6].
Let S be a subset of B(X), the space of all bounded linear operators on a Banach space for every finite sum as above, where (x i ) ⊂ X, (S i ) ⊂ S and ε i : Ω → {−1, 1} are independent and symmetric random variables on a probability space Ω. The smallest constant C for which the above definition holds is the R-bound of S, denoted by R(S). It is well-known that this definition does not depend on 1 ≤ p < ∞ (however, the constant R(S) does) and that R-boundedness is equivalent to boundedness when X is an Hilbert space. When X is an L p space (with respect to any σ-finite measure), testing R-boundedness is equivalent to proving square functions estimates, see [6,Remark 2.9 ].
The best constant C for which the above square functions estimates hold satisfies κ −1 C ≤ R(S) ≤ κC for a suitable κ > 0 (depending only on p). The proposition above R-boundedness follows from domination.
Corollary 2.2 Let S, T ⊂ B(L p (Σ)), 1 < p < ∞ and assume that T is R bounded and that for every S ∈ S there exists T ∈ T such that |Sf | ≤ |T f | pointwise, for every f ∈ L p (Σ). Then S is R-bounded.
Let (A, D(A)) be a sectorial operator in a Banach space X; this means that ρ(−A) ⊃ Σ π−φ for some φ < π and that λ(λ + A) −1 is bounded in Σ π−φ . The infimum of all such φ is called the spectral angle of A and denoted by φ A . Note that −A generates an analytic semigroup if and only if φ A < π/2. The definition of R-sectorial operator is similar, substituting boundedness of λ(λ + A) −1 with R-boundedness in Σ π−φ . As above one denotes by φ R A the infimum of all φ for which this happens; since R-boundedness implies boundedness, we have φ A ≤ φ R A . The R-boundedness of the resolvent characterizes the regularity of the associated inhomogeneous parabolic problem, as we explain now.
An analytic semigroup (e −tA ) t≥0 on a Banach space X with generator −A has maximal regular- ). This means that the mild solution of the evolution equation is in fact a strong solution and has the best regularity one can expect. It is known that this property does not depend on 1 < q < ∞ and T > 0. A characterization of maximal regularity is available in UMD Banach spaces, through the R-boundedness of the resolvent in a suitable sector ω + Σ φ , with ω ∈ R and φ > π/2 or, equivalently, of the scaled semigroup e −(A+ω ′ )t in a sector around the positive axis. In the case of L p spaces it can be restated in the following form, see [6, Theorem 1.11] Theorem 2.3 Let (e −tA ) t≥0 be a bounded analytic semigroup in L p (Σ), 1 < p < ∞, with generator −A. Then T (·) has maximal regularity of type L q if and only if the set {λ(λ+ A) −1 , λ ∈ Σ π/2+φ } is R-bounded for some φ > 0. In an equivalent way, if and only if there are constants 0 < φ < π/2, C > 0 such that for every finite sequence or, equivalently, there are constants 0 < φ ′ < π/2, C ′ > 0 such that for every finite sequence .
Finally we state a version of the operator-valued Mikhlin multiplier theorem in the N-dimensional case, see [5,Theorem 3.25] or [6,Theorem 4.6].
We end this section with the following lemma on radially symmetric multipliers.
The proof of the lemma now follows by Corollary (2.2) since from (1) one has for f ∈ L p (Σ)

Degenerate operators and similarity transformations
We investigate when the operators can be transformed one into the other by means of change of variables. Here α, c are unrestricted real coefficients. For β ∈ R, β = −1 let Observe that The following properties hold.
(i) For every m ∈ R, T β maps isometrically L p m onto L p m wherẽ m = m − β β + 1 .

The Bessel operator y α B n
In this section we consider for α < 2, c ∈ R the operator in the space L p m under Neumann boundary conditions. According to Proposition 3.2, we use the isometry All the results for y α B in L p m are then immediate consequence of those ofB in L p m already proved in [11,Section 3] (see also [10,9,8,14] for some analogous results for the N -d version of B).
and refer to [12] where these spaces are studied in detail in R N +1 + . The Neumann boundary condition is enclosed in the requirement y α−1 D y u ∈ L p m . This last is redundant when (m + 1)/p > 1 − α and equivalent to D y u(y) → 0 as y → 0, when (m + 1)/p < 1 − α, see [12,Proposition 4.3]. Consequently we write y α B n or, more pedantically y α B n m,p if necessary, for its realization in L p m .

Remark 4.1
The restriction α < 2 is not really essential since one can deduce from it the case α > 2, which requires boundary condition at ∞, using the change of variables described in Section 3 or directly from the equality 2B which is valid for any α = 2. However, here and in what follows, we keep to it in order to simplify the exposition.
Proof. We use the identity 2B n and apply [11,Proposition 3.3] in Under the hypothesis of Theorem 4.2, the domain of y α B n consists of all functions in the maximal domain satisfying a Neumann condition at 0, see [12,Proposition 4.6,4.7], that is (the condition lim y→0 y c D y u = 0 can be deleted in the range 0 < m+1 p ≤ c − 1). When c ≥ 1 the domain can also be described involving a Dirichlet, rather than Neumann, boundary condition We close this section by describing a core which does not depend on α, m, p and on the coefficients of the operator.
Proof. The proof immediately follows by observing that, by [11,Proposition 5.4], D is a core when α = 0, that is forB ñ m,p , and the isometry T − α 2 leaves invariant D since α < 2.
Remark 4. 4 We point out that, by the proof of [11,Proposition 5.4] or by [12,Remark 4.14], it The operator B n − V We start our investigation by adding a potential 0 ≤ V ∈ L 1 loc (R + , y c dy) to B n . Here we prove kernel bounds and construct a core.

Kernel bounds
For c + 1 > 0 and 0 ≤ V ∈ L 1 loc (R + , y c dy) we prove upper bounds for the heat kernel of B n − V , following the method used in [3,Sections 3,4].
Setting H 1 c = {u ∈ L 2 c , u ′ ∈ L 2 c }, we recall that from [11, Section 2] the operator B n c,2 is associated to the non-negative, symmetric and closed form in L 2 We consider the perturbed form a V in L 2 c defined by and define B n − V in L 2 c as the operator associated to the form a V The positivity of V implies that the norm induced by the form a V is stronger then the one induced by a: as an immediate consequence one deduces that a V is closed. By standard theory on sesquilinear forms we have the following result.
, then a V is a non-negative, symmetric and closed form in L 2 c . Its associated operator −B n + V is non-negative and self-adjoint and B n − V generates a contractive analytic semigroup e z(B n −V ) : z ∈ C + in L 2 c . Moreover: it is positive and L ∞ -contractive) and it is dominated by e tB n , that is t≥0 is a semigroup of integral operators and its heat kernel p V , taken with respect the measure ρ c dρ, satisfies Proof. The first claim follows from the property of a V . e t(B n −V ) is sub-markovian from [ ] since e t(B n −V ) is dominated by the positive integral operator e tB n whose kernel satisfies the stated estimate, see [11,Proposition 2.8] where, however, the kernel is written with respect to the Lebesgue measure.
To extend the above heat kernel estimates to the half plane C + we need the following lemma. Then one has In particular the function Q c satisfies, for some constants C ≥ 1, the doubling condition , r , 1 and we may therefore assume r = 1. The local integrability of y c implies that Q c (y 0 , 1) is continuous as a function of y 0 and moreover Q c (y 0 , 1) → (0,1) y c dy > 0 as y 0 → 0. Therefore if y 0 ≤ 1 then On the other hand if y 0 > 1 then y ≃ y 0 for any y ∈ (y 0 , y 0 + 1) which implies The last two inequalities yields Q c (y 0 , 1) ≃ (y 0 ) c (y 0 ∧ 1) −c . The doubling condition follows from the previous estimates and the fact that for 0 < r < s one has Furthermore for every ǫ > 0 there exist k ǫ , C ǫ > 0 such that, for every z ∈ Σ π 2 −ǫ and y, ρ > 0, Proof. Using the previous lemma we rewrite Proposition 5.1 (ii) as Furthermore by [4,Theorem 3.3], e t(B n −V ) satisfies the Davies-Gaffney estimates for all t > 0, U 1 , U 2 open subsets of (0, +∞), r := d(U 1 , U 2 ) = min{|x − y| : x ∈ U 1 , y ∈ U 2 } and f i in L 2 (U i , y c dy). By [4, Corollary 4.4] and Lemma 5.2 we then obtain for z ∈ Σ π 2 −ǫ and y, ρ > 0 This is an equivalent form (after modifying the constant in the exponential) of the estimate in the statement, by [11,Lemma 10.2] with γ 1 = γ 2 = − c 2 .
Remark 5. 4 We remark that in [4], the authors work in an abstract metric measure space (M, d, µ) and assume that the heat kernel p associated with a semigroup e −zL , where L is a non-negative self-adjoint operator on L 2 (M, dµ), is continuous with respect to the space variables. In such a case, in fact,  y,s)) . This holds, outside a set of zero measure, when the measure µ is doubling, by the Lebesgue differentiation theorem.

A core for B n − V
We prove that under mild hypotheses the set is a core for B n − V in L 2 c . Note that this is true when V = 0, by Proposition 4.3. We need some elementary lemmas. Unless explicitly stated, we only assume that 0 ≤ V ∈ L 1 loc (R + , y c dy).
Proof. By Proposition 4.3, D is dense in D(B n ) with respect to the graph norm. Moreover, since V ∈ L 2 c locally, D ⊂ D(a V ). The claim follows from the density of D(B n ) in H 1 c .
Proof. Let u ∈ D(B n ). Then u ∈ D(a) and there exists f ∈ L 2 c such that for every v ∈ D(a V ), hence for every v ∈ H 1 c , by Lemma 5.5.
Lemma 5.7 Let u ∈ D(B n − V ) and η be a smooth function such that η = 1 for 0 ≤ y ≤ 1 and η = 0 for y ≥ 2. Then ηu ∈ D(B n − V ) and Proof. Let u ∈ D(B n − V ), then ηu ∈ D(a V ) and, setting f = (B n − V )u, for every v ∈ D(a V ).
Proof. Let η be a smooth function such that η = 1 for 0 ≤ y ≤ 1 and η = 0 for y ≥ 2. Setting η k (y) = η y k , by Lemma 5.7, u k = η k u ∈ D(B n − V ) and Then u k → u, η k (B n − V )u → (B n − V )u in L 2 c by dominated convergence and, since D y η k = 0 in [0, 1], Lemma 5.7 shows that functions with compact support are a core for B n − V . To show that D is a core we need more information on the behavior near y = 0 of functions in the domain of B n − V .
We start by recalling some well-known facts about the modified Bessel functions I ν and K ν which constitute a basis of solutions of the modified Bessel equation We recall that for Re z > 0 one has where limiting values are taken for the definition of K ν when ν is an integer. The basic properties of these functions we need are collected in the following lemma, see e.g., [1, Sections 9.6 and 9.7].

Lemma 5.9
For ν > −1, I ν is increasing and K ν is decreasing (when restricted to the positive real half line). Moreover they satisfy the following properties if z ∈ Σ π/2−ε .
for suitable constants C ν,ǫ > 0 which may be different in lower an in the upper estimate.
The following estimates of the resolvent operator of B n − V is a consequence of the domination property stated in Proposition 5.1.

Proposition 5.10
Let c + 1 > 0 and λ > 0. Then, for every f ∈ L 2 c , is the integral kernel (taken with respect to the measure ρ c dρ) of the operator (λ − B n ) −1 .
We now prove local pointwise estimates for functions in the domain of B n − V .

Proposition 5.11
Let c + 1 > 0. Then there exists C > 0, independent of V , such that for every u ∈ D(B n − V ) and 0 < y < 1 Proof. Let u ∈ D(B n − V ) and f = u − (B n − V )u ∈ L 2 c so that u = (I − B n + V ) −1 f . Let us distinguish between the following cases and always take 0 < y < 1.
(i) If −1 < c < 1, Lemma 5.9 implies that for y ≤ 1 Then one has (ii) If c = 1, Lemma 5.9 gives for y ≤ 1 | log ρ|, y < ρ < 1, Then analogously (iii) Let now 1 < c. Then Lemma 5.9 implies that for y ≤ 1 If c < 3 one has If c = 3 then we get and finally if c > 3 We can now show that, under stronger assumptions, the potential term V can be seen as a perturbation of B n near 0, that is V u ∈ L 2 c for every u ∈ D(B n ) having compact support. In particular we prove that D is a core for B n − V . ∈ L 2 loc (R + , y c dy).
Proof. Let u ∈ C r . Then the hypotheses on V and Proposition 5.11 imply that V u ∈ L 2 c and Then by Lemma 5.6 u ∈ D(B n − V ) if and only if u ∈ D(B n ). This show the equality D(B n − V ) ∩ C r = D(B n ) ∩ C r . Using Proposition 5.11 again we also have V u L 2 c ≤ C 1 u − Bu L 2 c for any u ∈ D(B n )∩C r , which proves the equivalence of the graph norms. Finally, let u ∈ D(B n − V ). We have to prove that u can be approximated in the graph norm with functions belonging to D. Using Lemma 5.8 we may suppose, without any loss of generality, that supp u ⊆ (0, r). Then, by Proposition 4.3, there exist (u n ) ⊂ D such that u n → u in the graph norm · D(B n ) . We may also assume, after multiplying by a suitable cut-off function, that supp u n ⊆ (0, 2r) for every n. Then the previous point implies that V u n → V u in L 2 c , too.
6 The operator y α B n − V in L 2

c−α
We consider now for c ∈ R, α < 2, and 0 ≤ V ∈ L 1 loc (R + , y c−α dy) the operator As in Section 4 we use the isometry T − α 2 u(y) = 1 − α under which y α B − V becomes similar to one obtains that, when c > −1 + α, y α B n − V generates a contractive analytic semigroup We state the properties obtained so far, together with a density result which is a restating of Proposition 5.12 under the isometry T − α 2 . Proposition 6.1 Let c + 1 − α > 0 and 0 ≤ V ∈ L 1 loc (R + , y c−α ). Then the operator y α B n − V generates a contractive analytic semigroup in L 2 c−α . If, in addition, (i) c < 3 − α and V ∈ L 2 loc (R + , y c−α ) or Let aṼ be the form in L 2 c , defined in (3), associated toB n −Ṽ . In L 2 c−α we introduce the form a α,V which is the image of aṼ under the isometry T 0,− α 2 , that is To keep consistency of notation we often write a 0,V = a V . By construction, y α B n − V is the operator associated to the form a α,V in L 2 The next lemma, which follows from the considerations above, will be used later to relate the resolvents of y α B n − y α and B n − y −α . Lemma 6.3 Let a α,y α and a y −α be the sesquilinear forms associated respectively to the operator y α B n − y α in L 2 c−α and B n − y −α in L 2 c . Then on the common form domain Note that the above operators act in different Hilbert spaces; in particular their domains are different. However, the form domains coincide.
Here we investigate properties of y α B − V , α < 2, in L p m when 0 < m+1 p < c + 1 − α. We introduce the family of integral operators S β Here κ is a positive constant but we omit the dependence on it. The following result has been proved for α = 0 in [11].
We can now prove our main results for the operator y α B − V .
Theorem 7.2 Let 0 ≤ V ∈ L 1 loc (R + , y c−α dy). For any p ∈ (1, ∞) such that 0 < m+1 p < c + 1 − α, the semigroup e z(y α B n −V ) initially defined on L 2 c−α extends to a bounded analytic semigroup on L p m of angle π/2 which consists of integral operators. Moreover the generated semigroup has maximal regularity and the following properties hold.
(i) For every ǫ > 0 there exist C = C(ǫ, α) > 0 (independent of V ) such that (ii) For every ǫ > 0 the families of operators Proof. By Proposition 5.3 and (6), (i) holds for any f ∈ L 2 c−α . The boundedness of e z(y α B n −V ) in L p m follows from the previous lemma and (i) extends to L p m . The semigroup law is inherited from L 2 c−α via a density argument and we have only to prove the strong continuity at 0. Using the isometry T − α 2 , we may suppose that by the strong continuity of e z(B n −V ) in L 2 c . By density and uniform boundedness of the family (e z(B n −V ) ) z∈Σ π 2 −ǫ this holds for every f ∈ L p m , g ∈ L p ′ m . The semigroup is then weakly continuous, hence strongly continuous.
In our investigation of degenerate Nd problems, see [13], we need also a weaker version of the result above for potentials having non-negative real part. We formulate it in the next proposition.
loc (R + , y c−α dy) be a potential having non-negative real part. Then for any 1 < p < ∞ such that 0 < m+1 p < c + 1 − α, y α B n − V generates a C 0 -semigroup on L p m . The generated semigroup consists of integral operators and the following estimates hold In particular the families of operators Proof. Using the isometry T 0,− α 2 we may assume that α = 0. Let us treat first the symmetric case in L 2 c . The generation results can be proved as in Proposition 5.1 (where we assumed V ≥ 0). If a is the form associated with B n , then B n − V is associated to a V := a(u, v) + V u, v L 2 c and, by the standard theory on sesquilinear forms, B n − V generates a C 0 -semigroup on L 2 c . The domination properties follow from [15,Theorem 2.21]. Let u, v ∈ D (a V ) = D(a) ∩ L 2 (R + , |V |y c dy) such that uv ≥ 0. Since e tB n is positive one has Re a(u, v) ≥ a(|u|, |v|). Moreover Re V uv y c dy ≥ Re a(|u|, |v|) which by [15,Theorem 2.21] again implies the stated domination of the generated semigroups (one easily verifies that D (a V ) is an ideal of D(a) since this last is an ideal in itself, by the positivity of e tB n , see [15,Proposition 2.20]). The extrapolation on L p m follows as in Theorem 7.2. The domination of the resolvent is a straightforward consequence of that of the semigroup. The Rboundedness of the semigroup follows by domination from the R-boundedness of (e tB n ) t≥0 proved in Theorem 7.2. The R-boundedness of the resolvent follows as in Theorem 7.2.
8 The operator y α B n − y α We end the paper by thoroughly investigating the special case V (y) = y α , keeping α < 2. We prove, in particular, that the domain of y α B − V is D(y α B) ∩ D(V ), under slightly more restrictive hypotheses than those of Theorem 7.2.
As explained in the Introduction this case plays a crucial role in [13] in the investigation of the degenerate operators in the spaces L p R N +1 + , y m dxdy . In particular we prove in Propositions 8.3 and 8.4 that the multipliers satisfy the hypothesis of Theorem 2.4.
We start with the following Lemma.
Proof. Under the assumptions y α B n − µy α and B n − λy −α generate a semigroup on L 2 c−α and L 2 c , respectively, see Theorem 7.2. Since Re λ > 0, µ > 0 both resolvents are well defined but map to different spaces.
By Lemma 6.3 they are defined on the common domain In order to prove that the equality u = (λ − y α B n + µy α ) −1 f holds, we have to show that u ∈ F and that for every v ∈ F , u satisfies the weak equality (µuv + D y uD y v + λy −α uv)y c dy, which is the same as (8).
In the next results we relate the resolvent of y α B n − y α with that of B n − 1 y α . We shall assume both the conditions 0 < m+1 p < c+1−α and −α < m+1 p < c+1−α (that is α − < m+1 p < c+1−α). The first guarantees that y α B n is a generator in L p m and the second that B n is a generator in L p m+αp .
In the next propositions we prove the boundedness of the multipliers N λ and M λ . We start with M λ , used in [13] to characterize the domain of L = y α (∆ x + B y ).
Using Lemma 2.5 it suffices to show that the family The map T f = f /y α is an isometry of L p m onto L p m+αp and by Corollary 8.2, The family and e t(B n − λ y α ) : t ≥ 0, λ ∈ C + is R-bounded in L p m+αp , by Theorem 7.3. The R-boundedness of the derivatives follows either by the resolvent equation or by differentiating the last equation under the integral and using [6, Corollary 2.14]. In fact, if h(µ, t) = µe −µt , then Next we deal with N λ which is crucial in [13] for the proof that L = y α (∆ x + B y ) generates an analytic semigroup.
In order to characterize the domain of y α B n − y α , we denote by D(y α ) = {u ∈ L p m : y α u ∈ L p m } the domain of the potential V = y α in L p m . Recalling that Theorem 4.2 assures that D(y α B n ) = W 2,p N (α, m), we consider, for 0 < m+1 p < c + 1 − α, the Banach space W 2,p N (α, m) ∩ D(y α ) = u ∈ W 2,p loc (R + ) : u, y α u, y α D yy u, y α 2 D y u, y α−1 D y u ∈ L p m endowed with norm y α Bu L p m + y α u L p m + u L p m .
Proof. The generation properties as well as the maximal regularity follows from Theorem 7.2. Without any loss of generality we may assume that µ = 1. We prove preliminarily that D is dense in W 2,p N (α, m) ∩ D(y α ) = D(y α B n ) ∩ D(y α ). Let u ∈ W 2,p N (α, m) ∩ D(y α ); up to using a standard cut-off argument we may suppose that supp u ⊆ [0, b] for some b > 0. Using Remark 4.4, let (u n ) n∈N ⊆ D such that supp u n ⊆ [0, b] and u n → u in W 2,p N (α, m). Then by [12, Proposition 3.2 (ii)] y α (u n − u) L p m ≤ C y α+1 (D y u n − D y u) L p m ≤ Cb 2 y α−1 D y (u n − u) L p m which tends to 0 as n → ∞. This proves the density of D.
Let us now characterize the domain. By definition D(y α B n − y α ) = (1 − y α B n + y α ) −1 (L p m ). Let u = (1 − y α B n + y α ) −1 f with f ∈ L p m . Using Corollary 8.2 (ii) we obtain which proves the inclusion D(y α B n − y α ) ⊆ D(y α B n ) ∩ D(y α ). To prove the reverse property we observe that, since the graph norm of y α B n − y α is clearly weaker than the norm of D(y α B n ) ∩ D(y α ), inequality (13) again shows that they are equivalent on D(y α B n − y α ), in particular on D which is dense in D(y α B n ) ∩ D(y α ), by the previous step. Therefore D(y α B n − y α ) = D(y α B n ) ∩ D(y α ) and in particular D is a core.