Maximal regularity for elliptic operators with second-order discontinuous coefficients

We prove maximal regularity for parabolic problems associated to the second-order elliptic operator L=Δ+(a-1)∑i,j=1Nxixj|x|2Dij+cx|x|2·∇-b|x|-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} L =\Delta +(a-1)\sum _{i,j=1}^N\frac{x_ix_j}{|x|^2}D_{ij}+c\frac{x}{|x|^2}\cdot \nabla -b|x|^{-2} \end{aligned}$$\end{document}with a>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0$$\end{document} and b,c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b,\ c$$\end{document} real coefficients.


Introduction
In this paper, we consider second-order elliptic operators of the form with a > 0 and b, c constant real coefficients. The leading coefficients are uniformly elliptic but discontinuous at 0, if a = 1, and singularities in the lower order terms appear when b or c is different from 0. The operator commutes with dilations, in the sense that I −1 s L I s = s 2 L, if I s u(x) = u(sx). When c = 0 and a = 1, L reduces to a Schrödinger operator with inverse square potential. Operators of this form have been widely investigated in previous works. In particular generation properties of analytic semigroups in L p spaces endowed with the Lebesgue measure, sharp kernel estimates and Rellich-type inequalities have been proved (see [3,[10][11][12][13][14][15][16]). Here, we prove that the following parabolic problem associated with L ∂ t u(t) − Lu(t) = f (t), t > 0, u(0) = 0 (2) has maximal L q regularity, that is for each f ∈ L q (0, ∞; X ) there exists u ∈ W 1,q (0, ∞; X ) ∩ L q (0, ∞; D(L)) satisfying (2). Here, X is the underlying function space where L acts and D(L) is the domain of L in X .
The functional analytic approach we use for proving maximal regularity is widely described in [9] and in the new books [6,7]. The whole theory relies on a deep interplay between harmonic analysis and structure theory of Banach spaces but largely simplifies when the underlying Banach spaces are L p spaces, by using classical square function estimates. This last approach has been employed extensively in [4], showing that uniformly parabolic operators have maximal regularity, under very general boundary conditions. Here, we show that the same happens for a class of degenerate second-order operators.
We deduce maximal regularity from the R-boundedness of the generated semigroup in closed sectors of the right half plane, see [4,Chapter 4]. This last is deduced through an extrapolation result in [1] which involves a family of Muckenhoupt weighted estimates. We show that L has maximal regularity in L p when it generates a semigroup in L p , that is when the necessary and sufficient conditions of Theorems 2.1 and 2.2 are satisfied. We consider L p spaces with respect to radial power weights |x| m not just for the sake of generality but because our proof relies on weighted estimates: we are unable to obtain the result just fixing the Lebesgue measure or the symmetrizing measure but we have to work simultaneously in different homogeneous spaces. Our result is known for Schrödinger operators with inverse square potentials, due to a recent result of Bui, see [2]. We adopt his strategy in Sect. 4.1 but new complications arise due to the lack of symmetry of L in L p (R N ). At a first sight one could think that the symmetrizing measure (see Sect. 3) plays the same role as the Lebesgue measure for Schrödinger operators. This is not true, however, and there are situations where the symmetrizing measure is not doubling and the whole machinery of harmonic analysis in homogeneous spaces breaks down. Let us explain these points by considering Proposition 4.9. If μ m is the Lebesgue measure, maximal regularity follows if L generates in L 2 , which is not always the case since L is not symmetric. On the other hand, if μ m is the symmetrizing measure |x| γ dx, one needs N + γ > 0 in order to work with a doubling measure and this condition would impose extra conditions on the coefficients of L, not needed for the generation of a semigroup. The generality of the weights |x| m allows to play with an extra parameter m, prove maximal regularity when |x| m dx is doubling and L generates both in L 2 (dμ m ) and in L p (dμ m ) and then recover the general situation by similarity transformations. When |x| m is doubling, that is when m + N > 0, R boundedness follows from domination of the semigroup by maximal functions. Both the Euclidean maximal function M f and the weighted maximal function M μ m f are used and the proof distinguish between m ≥ 0 and −N < m < 0, where |x| m ∈ A p . spaces from the previous ones. Section 4 is devoted to the main maximal regularity result.
Notation. We use for R N \{0}. The unit sphere { x = 1} in R N is denoted by S N −1 . We denote by C + := {z; Re z > 0} and for δ ≥ 0, δ = {z ∈ C : |Arg z| < δ}. We adopt standard notation for L p and Sobolev spaces and the Lebesgue measure is understood when no measure is explicitly written.

Generation results in L p (R N )
In this section, we recall, without proofs, the main results concerning generation and domain characterization proved in [13,15]. Kernel estimates are, instead, proved in [3,11,16].
If 1 < p < ∞, we define the maximal operator L p,max through the domain The operator L p,min is defined as the closure, in L p (R N ) of (L , C ∞ c (R N \{0}) (the closure exists since this operator is contained in the closed operator L p,max ) and it is clear that L p,min ⊂ L p,max .
Let us employ spherical coordinates on R N \{0}. For every x ∈ R N \{0} we write x = r ω, where r := |x|, ω := x |x| ∈ S N −1 . If u ∈ C 2 (R N ), D r u, D rr u are the radial derivatives of u and ∇ τ u is the tangential component of its gradient. They are defined through the formulas Denoting by 0 the Laplace-Beltrami operator on S N −1 , the operator L takes the form The equation Lu = 0 has radial solutions |x| −s 1 , |x| −s 2 where s 1 , s 2 are the roots of the indicial equation f (s) = −as 2 + (N − 1 + c − a)s + b = 0 given by where Introducing the parameter we may write The above numbers are real if and only if D ≥ 0. In the critical case D = 0, we often write s 0 for s 1 = s 2 . When D < 0 the equation u − Lu = f cannot have positive distributional solutions for certain positive f , see [15].
Assuming D ≥ 0 we have shown in [13,15] that there exists an intermediate operator L p,min ⊂ L p,int ⊂ L p,max which generates a semigroup in L p (R N ) if and only if N p ∈ (s 1 , s 2 + 2). From now on, we assume that D ≥ 0 and 1 < p < ∞ throughout this paper. When N p ∈ (s 1 , s 2 + 2), then σ (L) = C for every L p,min ⊂ L ⊂ L p,max .
generates a bounded positive analytic semigroup of angle π/2 on L p . Moreover, When N p ∈ (s 0 , s 0 + 2), then σ (L) = C for every L p,min ⊂ L ⊂ L p,max .
We refer the reader to [15,Theorem 3.29] for a detailed discussion of the inclusion D(L p,int ) ⊂ W 2, p (R N ). As a consequence of the previous results, L generates a semigroup in some L p (R N ), 1 < p < ∞, if and only if (s 1 , s 2 + 2) ∩ [0, N ] = ∅. We remark that, in general, the generated semigroup is not contractive. If a = 1, in fact, it is contractive if and only if s 1 ≤ (N − 2)/ p ≤ s 2 , see [13,Proposition 4.2].
The formal adjoint of L is given by where Let us compute the numbers s * 1 , s * 2 , D * defined as in (5), (6) and relative to L * . We have Observe that N p > s 1 is equivalent to N p < s * 2 + 2 and N p < s 2 is equivalent to The coefficients of the formal adjoint L * of L, taken with respect to the Lebesgue measure and defined in (8), then satisfy In the following propositions we write the above results in a different way and we also clarify when L p,int coincides with L p,min or L p,max . To shorten the notation we write We now characterize L p,int coincides with L p,min or L p,max .
Therefore, the following properties are equivalent: Next we show that, under the assumptions of Theorems 2.1 and 2.2, the generated semigroup e zL consists of integral operators, see [3,11].
(i) p L is analytic on C + , for every fixed x, y ∈ , it is continuous on C + × × and p L |y| −γ is symmetric on × for every fixed z ∈ C + . Furthermore, one has where s 1 is defined in (5) and s * 1 in (10). (iii) For every 1 < p, q < ∞ satisfying the condition of Proposition 2.4, the semigroups generated by L, respectively, in L p R N and in L q R N are consistent.
Proof. The existence of the heat kernel as well as its regularity is proved in [3, Theorems 4.3 and 4.5, Corollary 4.6] and in [16] (note that by [16,Lemma 5.4] the semigroups in L p R N and in L 2 R N , |x| γ , as well as their generators, are consistent). The decomposition (13) [16,Corollary 4.15]. The consistency of e zL in all the spaces L p R N follows after observing that the kernel p L is independent of p.
In particular, when s 1 , s * 1 ≤ 0 the heat kernel of the generated semigroup satisfies Gaussian estimates.

Generation results in L p (R N , |x| m dx)
In this section, we consider the operator L (keeping the assumption (7), is important since this measure symmetrizes the operator and will be discussed at the end of this section. The weighted case is deduced by the unweighted one by the use of the following multiplication operator.
whereL is the operator defined as in (1) with parameters b, c replaced, respectively, byb Moreover, the discriminantD and the parameterγ ,s 1,2 ofL defined as in (5), (6) and (7) are given bỹ Then, recalling (4) one has which is the first required claim. The second assertion follows from the definitions (5), (6) sincẽ and Vol. 21 (2021) Maximal regularity for elliptic operators 3621 as one easily verifies (with the usual logarithmic correction when D = 0). The operator L m, p,min is the closure, in L p (R N , dμ m ) of (L , C ∞ c ( )). We also define where ∇u, D 2 u consist of weak derivatives of u in (not in R N ). For a fixed ball B centered at the origin, In what follows we extend the result of the previous section by showing that L generates a semigroup in L p ( , dμ m ) for any 1 < p < ∞ satisfying s 1 < N +m p < s 2 + 2. All the results for L in L p R N , dμ m are immediate consequence of those of L in L p (R N , dx), using the isometry T −m/ p .
generates a bounded positive analytic semigroup of angle π/2 on L p R N , dμ m . Moreover, When N +m p ∈ (s 0 , s 0 + 2), then σ (L) = C for every L m, p,min ⊂ L ⊂ L m, p,max .
Let us denote by I N +m the closed interval with endpoints 0 and N + m (note that N + m < 0 is allowed). As a consequence of the previous Theorems, L generates a semigroup in some L p (R N , dμ m ), 1 < p < ∞, if and only if (s 1 , s 2 +2)∩ I N +m = ∅.
In the following Proposition we compute the adjoint L * m of L with respect to the measure μ m = |x| m dx. and The parameters s * m 1,2 , γ * m , D * m defined as in (5), (6) and relative to L * m are Proof. Note that if 1 < p < ∞ and k ∈ R, then the adjoint operator of the isometry (1− p) . In In what follows when N + m > 0 we adopt the notation Proposition 3.5. The following properties are equivalent. In particular: (a) When N + m < 0 then (i) is equivalent to s 1 , s * m 1 < 0 and

Furthermore, p = 2 satisfies the above assumptions if and only if |γ − m|
The coincidence of L m, p,int with L m, p,min or L p,max can be obtained similarly from Proposition 2.5 but we do not state it here.
Let us compute the kernel of the generated semigroup Proposition 3.6. Let 1 < p < ∞ satisfies the condition of Proposition 3.5. Then, for with p m (z, x, y) = |y| −m p L (z, x, y) and p L defined in (13). Moreover, the following properties hold (i) For every ε > 0, there exist C ε > 0 and κ ε > 0 such that for z ∈ C + satisfying | arg z| ≤ π 2 − ε, and (x, where s 1 is defined in (5), s * 1 in (10) and s * m (iii) If m, n ∈ R and 1 < p, q < ∞ satisfy the generation conditions of Proposition 2.4 in both L p R N , dμ m and L q R N , dμ n . Then, the semigroups generated by L in L q R N , dμ m and in L q R N , dμ n are consistent.
Proof. Let us consider the isometry T − m p : L p ( , dx) → L p ( , |x| m dx). Then, by the definition, we have for every u ∈ L p ( , |x| m dx) e zL u = T − m p e zL Tm p u whereL is the operator on L p R N defined as in (1) with parametersb,c defined by (15). Let p(z, x, y) the heat kernels of e zL z∈C + on L p R N , dμ m taken with respect to the Lebesgue measure. The last relation between the semigroups translate into the analogous equality for the heat kernels: Recalling (13) and setting x = r ω, y = ρη, r, ρ > 0, |ω| = |η| = 1, the last relation yields where we used the fact that, from (16), we haveD = D,s 1,2 = s 1,2 + m p ,γ = γ −2 m p . The other claims follow directly from Theorem 2.6. The estimate for m ≥ 0 follows after observing that, in this case, Let us end this section with some comments on the special case m = γ , see (7). Writing we can see that L is associated to the symmetric form which is coercive if and only if D ≥ 0. Then, L is self-adjoint in L 2 (dμ γ ) and the generation conditions in L p (dμ γ ) of Proposition 3.5 take the form We refer to [3,11,16] for this approach.

Maximal regularity
An analytic semigroup T (·) on a Banach space X with generator B has maximal regularity of type L q ( D(B)). 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. In recent years this concept has thoroughly been studied and applied in various directions, see [4,9] for applications to uniformly parabolic problems under general boundary conditions. A characterization of maximal regularity is available in UMD Banach spaces, through the R-boundedness of the resolvent in sector larger than the right half plane or, equivalently, of the semigroup in a sector around the positive axis, can be largely simplified when X = L p (with respect to any measure), through the classical tool of square functions estimates. This is what we use and recall here after, see [9, Theorem 1.11, Remark 2.9]

Muckenhoupt weighted estimates
Let (S, d, ν) be a space of homogeneous type, that is a metric space endowed with a Borel measure ν which is doubling on balls. When X = L p (S, d, ν) the square function estimate in Theorem 4.1 can be reduced to a family of Muckenhoupt weighted estimates of the type see Theorem 4.4 as follows. With this in mind, we recall preliminarily, the definition and the essential properties about Muckenhoupt weights. For the proof of the following results as well as for further details, we refer the reader to [1, Chapter 2 and 5] and [18,Chapter 1]. Let w be a weight, i.e., a nonnegative locally integrable function defined on S; we use the notation Let M denote the uncentered maximal operator over balls in S defined by where the supremum is taken over all balls of S containing x. We recall that M is bounded on L p (w) if and only if w ∈ A p , see for example [5,Theorem 7.3]. We say that w ∈ A p , 1 < p < ∞, if there exists a constant C such that for every ball B ⊆ S one has For p = 1, we say that w ∈ A 1 if there is a constant C such hat M ν w ≤ C w a.e.. The weight w is in the reverse Hölder class of order q, w ∈ R H q , 1 < q ≤ ∞, if there is a constant C such that for every ball B ⊆ S with the usual modification for q = ∞. The best constants appearing in the previous inequalities are referred, respectively, as the A p and the R H q constants of w.
We sum up in the following propositions the properties we need about these classes of weights.
A proof of the following result is in [18,Corollary 14] or [5,Chapter 7]. Lemma 4.3. Let w ∈ A p ∩ R H r , 1 < r, p < ∞. Then, there exists a constant C > 1 such that for any ball B and any measurable subset E ⊂ B, We now state an extrapolation result originally due to Rubio de Francia, adapted as in [1,Theorem 4.9], which allows to reduce the square function estimate in Theorem 4.1 to a family of Muckenhoupt weighted estimates. Only weights and pairs of functions appear and no operator is involved. In what follows we consider families F = {( f, g) : is the set of all nonnegative, measurable functions defined on S.  Let (S, d, ν) be a space of homogeneous type and let F ⊆ L 0 Then, for all p 0 < q < q 0 and ( f, g) ∈ F we have

All the constant C above may vary from line to line but depend only on the A p and R H q constants of w.
Combining Theorems 4.1 and 4.4 we derive the following characterization of maximal regularity in terms of boundedness over L p (w) spaces. (S, d, ν) be a space of homogeneous type and let T (·) be a bounded analytic semigroup in L p (S, ν) defined in a sector δ , δ > 0. Let 0 < p 0 < q 0 ≤ ∞. Suppose that p 0 < 2 < q 0 and that there exists p with p 0 ≤ p ≤ q 0 (and p < ∞ if q 0 = ∞), such that for f ∈ L p (S, ν),

Theorem 4.5. Let
The following three lemmas will be crucial in the proof of maximal regularity. Lemma 4.6. Let w ∈ A p , p ≥ 1, and let ν w be the measure wdν. Denote by M ν w and M ν the maximal function defined by ν w and ν. Then, (S, d, ν w ) is a space of homogeneous type and which, taking the supremum over B, yields the required claim. The case p = 1 follows similarly.
Lemma 4.7. Let p be a nonnegative, locally integrable function on R N and consider the measure ν = p dx. Let M ν be the uncentered maximal operator relative to ν, defined as in (22). If 0 ≤ φ ∈ L 1 R N , ν is radial and decreasing then If p is homogeneous of degree k, i.e., p(t x) = p(x)t k for all x ∈ R N and t > 0, then Proof. Let us suppose preliminarily that φ is a simple function and let us write, for some a 1 , . . . , a k > 0 and balls B 1 , . . . , B k centered at 0, In the general case the first required claim follows since φ can be approximated by a sequence of simple functions which increase to it monotonically. To prove the second claim it is enough to observe that, under the homogeneity assumptions on p, one has Lemma 4.8. Let m ∈ R such that N + m > 0 and let dμ m = |x| m dx. For every k ∈ R let us consider the radial weight w(x) = |x| k . The following properties hold.
Proof. To prove (i), we start by considering balls of center x 0 and radius 1. Fix R > 1 and assume first that |x 0 | ≤ R. Then, both |x| k and |x| − k p−1 are integrable in B(x 0 , 1) with respect to the measure μ m and for some positive constant C depending on R. On the other hand, when |x 0 | > R, then In what follows we write A p (μ m ), R H p (μ m ), M μ m to denote, respectively, the class of Muckenhoupt weights, the reverse Hölder class and the maximal function over balls taken with respect to the measure μ m . When m = 0 we write A p , R H p , M.
We fix 0 < δ < π/2 and we recall that for z ∈ δ , z = ωt, t = |z|, Proposition 3.6 yields the pointwise estimate When m ≥ 0 we also have We prove the R-boundedness of the family (T (z)) z∈ δ using the extrapolation result of Theorem 4.4. We follow the proof in [2, Theorem 2.9] but new complications arise because the operator is non-symmetric and the measure μ m is not the Lebesgue one. In particular we have to distinguish between the cases m ≥ 0 and −N < m < 0 and both the maximal functions with respect to the Lebesgue measure and the weighted one appear.
For the reader's convenience, in what follows we write for z ∈ δ , B = B(0, √ t), t = |z|, and Proposition 4.9. Let N + m > 0, 1 < p < ∞ and let us suppose that the generation conditions of Proposition 3.5 are satisfied, that is Then, for every weight there exist C > 0 depending on δ and the A p (μ m ) and R p (μ m ) constants of w such that for every z ∈ δ one has Finally, if |γ − m| < 2 1 + √ D then T (·) has maximal regularity on We split the proof in four lemmas according to (27). Proof. Assume first that m ≥ 0. Then, using (26) and Lemma 4.7 with p(y) = |y| m we get The claim then follows since M μ m is bounded on L p (w). When −N < m < 0 we use (25) (and Lemma 4.7 with respect to the Lebesgue measure) to get Lemma 4.12. The estimate of Proposition 4.9 holds for (T 3 (z)) z∈ δ .
Proof. Using (25) we get Let us fix r such that N +m s * m 1 < r < p < N +m s 1 . Applying Hölder's inequality we obtain To prove the boundedness of T 4 (z) we apply a duality argument. With this aim, let p, m satisfy the assumptions in (28) and let T * m (z) be the adjoint of T (z) taken with respect to the measure μ m . The conditions in (28) are equivalent to If L * m is the adjoint operator defined by Proposition 3.4, then the last conditions assures that L * m generates a bounded analytic semigroup in L p R N , μ m which coincides with T * m (z).
Proof. We apply a duality argument. Let g ∈ L p (R N , wμ m ); since T 4 (z) = T * m 3 (z) we obtain Using Hölder's inequality we then yield ˆR Using Lemma 4.12 with L and p replaced, respectively, by L * m and p we get ˆR which concludes the proof.
We can finally prove Proposition 4.9.

Maximal regularity without restrictions on m
Let us start with the case of the Lebesgue measure, that is when m = 0.
Theorem 4.14. Let 1 < p < ∞ and let us suppose that the generation conditions of Proposition 2.4 are satisfied, that is Then, T (·) has maximal regularity on L p (R N ).
Proof. Let s 1 < N p < s 2 + 2. If |γ | < 2 1 + √ D , the required claim follows by using Proposition 4.9 with m = 0 and w = 1. Let us suppose now |γ | ≥ 2 1 + √ D ; in this case, from Proposition 3.5, L does not generate a semigroup in L 2 R N and therefore p = 2. Let m ∈ R and let us consider the isometry