Higher order elliptic equations in weighted Banach spaces

We consider higher order linear, uniformly elliptic equations with non-smooth coefficients in Banach-Sobolev spaces generated by weighted general Banach Function Space (BFS). Supposing boundedness of the Hardy-Littlewood Maximal and Calderon-Zygmund singular operators in BFSs we obtain local solvability in the Sobolev-BFS and establish interior Schauder type a priori estimates for the. elliptic operator. These results will be used in order to obtain Fredholmness of the operator under consideration in weighted BFSs with suitable weight. In addition, we analyze some examples of weighted BFS that verify our assumptions and in which the corresponding Schauder type estimates and Fredholmness of the operator hold true.


Introduction
The question of solvability in the small and Schauder type estimates play an essential role in the theory of the Fredholmness of boundary value problems for elliptic equations in appropriate BFS.There is a vast number of papers and monographs dedicated to the Fredholmness of Partial Differential Equations (PDEs) in the frame of classical function classes as the Hölder and Sobolev spaces.The appearance of new function spaces and their intensive study had a very big impact on the regularity and existence theory for PDEs.The so-called non standard function spaces turn to be interesting not only from theoretical point of view but also for the applied mathematics and mathematical physics.Among the most studied spaces we can find the Morrey L p,λ spaces, the weighted Lebesgue spaces and various generalizations, the variable Lebesgue spaces L p(•) , the grand Lebesgue spaces L p) , the Orlicz spaces L Φ and many others (see [1,4,5,7,15,16,[19][20][21][22] and the references therein).
In the present paper we are interested on elliptic differential operators of higher order in weighted general BFS X w (Ω) with Muckenhoupt type weight w in a bounded domain Ω ⊂ R n with a smooth enough boundary.In order to obtain the desired a priori estimates we need that the Hardy-Littlewood maximal and the Calderón-Zygmund singular operators are bounded in X w (Ω).Then under certain conditions on the coefficients we obtain solvability in the small, that is local solvability, in Sobolev spaces W m Xw (Ω) generated by weighted BFS X w (Ω).More over, we obtain interior Schauder type inequalities in X w (Ω) for the solutions of the linear uniformly elliptic equations under consideration.Such estimates play an exceptional role in the establishing of the Fredholmness of the corresponding elliptic operators.At the end, some examples of BFS are given.

Definitions and auxiliary results
The question of existence and regularity of the solutions of linear elliptic equations is strongly related to the question of continuity of certain integral operators in the corresponding function spaces.For this goal we recall the definitions of these operators and introduce the function spaces that we are going to use.
Let f ∈ L 1 (R n ) and Mf be the Hardy-Littlewood maximal operator (2.1) and Kf be the Calderón-Zygmund integral operator For each γ ∈ (0, n) we consider the Riesz potential Simple calculations show that we can estimate the Riesz potential via the maximal function (see [6]) Following [2], we define the BFS and give some of its properties.
. ., for any constant a ∈ R and for any Lebesgue measurable set E ⊂ R n the following properties hold: We can extend the Definition 2.1 to BFSs X(Ω) builded on a bounded domain Ω with |Ω| > 0 and regular boundary ∂Ω by taking f ∈ F (Ω) and extending it as zero out of Ω.
It follows immediately, by (P 5) that X(Ω) ⊂ L 1 (Ω).The associated space X ′ (Ω), endowed with the associated norm • X ′ , consists of all g ∈ F (Ω), such that (cf.[2]) At this point we can extend the Hölder inequality in the case of BFSs.Proposition 2.2.Let X(Ω) and X ′ (Ω) be associated BFSs.If f ∈ X(Ω) and g ∈ X ′ (Ω), then f g ∈ L 1 (Ω) and Let w be a positive weight, that is a Lebesgue measurable function on Ω for which 0 < w(x) < ∞.Then we can define X w (Ω) as the space of all f ∈ F (Ω) for which In what follows we assume that w ∈ X(Ω) and w −1 ∈ X ′ (Ω).The definition of general Sobolev BFS follows naturally from the definition of the classical Sobolev spaces.By W m Xw (Ω) we denote the Sobolev space of all functions f ∈ X w (Ω) differentiable in distributional sense up to order m, that is We denote by • W m Xw (Ω) the space of W m Xw (Ω)-functions compactly supported in Ω, endowed with the same norm (2.6).
In the case when Ω ≡ B r (0) we simplify the notion, writing X w (r) and W m Xw (r) instead of X w (B r ) and W m Xw (B r ).In our further considerations, we are going to use the following norm which is equivalent of (2.6).
To be able to adapt the classical techniques from the L p -theory to nonstandard function spaces we assume that that space X w (Ω) and the weight function w possess the following properties.
Property 1.Let X w (Ω) be a weighted BFS, suppose that w is such that the following inclusions hold: (A) The operators (2.1) and (2.2) are bounded in X w (Ω), i.e.M, K ∈ [X w (Ω)] and the estimates hold with constants independent of f. (B) There exists p 0 ∈ (1, +∞) such that (2.9) Property 2. For any bounded domain Ω with ∂Ω ∈ C m , the Sobolev-Banach space W m Xw (Ω) has the extension property.This means that for each domain Ω ′ such that Ω ⋐ Ω ′ , there exists a linear bounded operator, called extension operator, such that Property 3.For a given space X w (Ω) we suppose that one of the following conditions hold.
Concerning the Condition 2 we can construct the following example.Let w be a positive weight belonging to L 1+0 (Ω) = ∪ δ>0 L 1+δ (Ω), and consider the space There exists α > 1 such that w ∈ L α (Ω) and if we denote Applying the Hölder inequality we obtain In addition, the classical weighted Lebesgue spaces L p w (Ω), p ∈ (1, ∞), w ∈ A p (Ω) verify the Properties 1, 2, and 3 (cf.[8] . It is easy to see that the (E) holds for any weighted space X w (Ω), just extending the functions as zero out of Ω, while for the Sobolev spaces it is not so obvious.This property is well-known in the case of classical Sobolev spaces W p,m (Ω), p ∈ (1, +∞) (see [3]) and weighted Sobolev spaces W p,m w (Ω) with a Muckenhoupt weight (see [8,10,11]).In the our case, we assert that the extension property holds for all k = 0, . . ., m, Concerning the Conditions 1 and 2 they are not mutually exclusive and we suppose that for a given BFS at least one of them holds true.

Statement of the problem
We consider the m-order linear differential operator with m being even number.
The operator L(x, D) is uniformly elliptic that is, there exist positive constants λ and Λ, such that We say that the operator L satisfies in x 0 ∈ Ω the (P x0 )-property if there exists a ball B r (x 0 ) ⋐ Ω and functions Let x 0 ∈ Ω be a point in which (3.2) and (P x0 ) hold.Consider the tangential operator where under the value a α (x 0 ) we understand the value of g α (x 0 ) defined by (P x0 ).
The fundamental solution J x0 of the equation L x0 ϕ = 0 is called a parametrix for the equation Lϕ = 0, having a singularity at the point x 0 .By the properties of the fundamental solution, for any multiindex β we have These properties ensure that the m-order derivatives of J x0 are Calderón-Zygmund kernels (see [18,19] and the references therein for more details).It is well know, from the classical theory, that for any function ϕ ∈ C m 0 (Ω) the following representation holds Introducing the operators we can rewrite (3.5) in the form Our goal is to show that T x0 +S x0 L is identity operator in W m Xw (Ω).Let us calculate the m-order derivatives of S x0 ϕ (cf.[3,12,21]) with |β| = m.Hence K β ϕ are Calderón-Zygmund integrals satisfying (2.8).Calculating the lower order derivatives we observe that they have weak singularity and can be treated as Riesz potential (2.3).
Lemma 3.1 (Main Lemma).Let the Property 1 and condition (P x0 ) hold at some point , where σ(r) → 0 as r → 0 and it depends on the coefficients of L, but not on ϕ.
Proof.Following [4] we assume that x 0 = 0 and simplify the notation writing S 0 , L 0 and T 0 .Let n ≥ 3 be an odd number, in case of even dimension it can be introduced a fictitious new variable and extend all functions as constants along the new variable.Take an arbitrary ϕ ∈ W m Xw (r) with a compact support in B r , then Since a α (0) − a α L ∞ (r) → 0 as r → 0, we have In order to estimate the Sobolev-Banach norm of (3.10) we need to calculate the derivatives up to order m.Let β be a multi-index and for |β| < m we have By (3.4) we have where R γ is defined by (2.3) with γ = m − |β|.Applying (2.4) and Property (1) we obtain Making use of (3.11) we get for r small enough and σ(r) vanishing function as r → 0. Consider now the case |β| = m.By (2.9) it follows that W m Xw (r) ⊂ W 1,m (r) and therefore, it holds (cf.where c depends on known quantities but not on r and ψ.By the properties of the kernel it follows that D β J 0 is singular for each |β| = m and therefore by (2.8) and from (3.13) we have Hence the following estimate holds ≤ σ(r) ϕ W m Xw ;r (r) .Taking into account (3.12) we obtain r) , σ(r) → 0 as r → 0.

Local existence of strong solutions
In the next section we are going to obtain some local results concerning solvability in weighted Sobolev-Banach space builded upon X w .Proof.
Because of the equivalence of the norms (2.6) and (2.7) the Lemma 3.1 holds also in W m Xw;r (r).
For any u ∈ W m Xw;r (r) we consider the equation Lu = f.By (4.5) we can rewrite it as follows f = Lu = (L 0 − L 0 T 0 )u = L 0 (Id − T 0 )u, where the identity operator here acts in W m Xw;r (r).Applying S 0 we obtain
Taking r small enough such that T 0 [W m Xw ;r (r)] < 1, we obtain that the operator Id − T 0 is boundedly invertible in W m Xw ;r (r) and by Lemma 4.1 the function u = (Id − T 0 ) −1 S 0 f, is a solution of the equation Lu = f in W m Xw ;r (r).

Interior Schauder type estimates
Our goal now is to obtain local interior Schauder type estimates for the solutions of Lϕ = f in W m Xw (Ω).For this purpose we need some auxiliary lemma.Let ω ∈ C ∞ 0 ([0, 1] be such that for any 0 < r 1 < r 2 ≤ 1.The norm of ξ is bounded, as it is proved in [5] with a constant independent of r 1 and r 2 . Lemma 5.1.Let the conditions of Theorem 4.3 be fulfilled in B r2 ⋐ Ω.Then for any 0 < r 1 < r 2 as in (5.1) and u ∈ W m Xw ;r2 (Ω) the following estimate holds with a constant independent of r 1 , r 2 , and u.
Proof.Take ϕ = ξu ∈ W m Xw ;r2 (r 2 ) with a compact support in B r2 .Then by Corollary 4.2 we have (5.2) and by Lemma 3.1 there exists r > 0 such small that holds for all r 2 ∈ (0, r).Then by (5.2) we obtain where (5.4) Calculating the higher order derivatives we obtain with a constant c independent of ϕ.The integral operator is of Calderón-Zygmund type and, by Property 1, the following estimate holds (5.5) For the lower order derivatives of (5.4) we have the following expression In this case the kernel has a weak singularity and the integral operator is a Riesz type integral (2.3) that we can estimate as with a constant independent of r 2 and ϕ.Unifying (5.5) and (5.6) we obtain . On the other hand, it is easy to see that Lϕ can be represented in the form (5.8) where M (u; ξ) is a linear combination of derivatives D α u, the order of which does not exceed (m − 1), multiplied by the derivatives of ξ of order at most m.Precisely Following [5] we obtain analogously (5.9) Unifying (5.9) and (5.7), we obtain (5.10) Then making use of (5.3) we obtain the desired estimate In order to establish interior Schauder's estimate we need the following result.
In order to simplify the calculus, we begin assuming n = 1, m = 2 and then we extend the result via induction to more general situation.Let Ω = (a, b) be an interval of length b − a = ε for a fixed ε and suppose that u ∈ C 2 0 (Ω ′ ), then by [14,Theorem 7.27] we have (5.11) By density arguments it follows that (5.11) holds true also for any u ∈ Xw (Ω ′ ).We can estimate the norm of u ′ w by (5.11): Then for an arbitrary interval Ω = (a, b) we construct covering with disjoint intervals I l with length ε, such that Ω = l I l .Since (5.12) holds for all I l , the Hölder inequality for p > 1 gives where we have used that for each Let the Condition 1 holds.Then for each p ∈ [1, p 1 ] we have For any p ≤ min{p 0 , p 1 }, the Property 1 implies X w (Ω) ⊂ L p (Ω) and hence (5.14) If x ∈ R n then (5.14) holds for any partial derivative D i u for i = 1, . . ., n. Moreover it holds also for the higher order derivatives, that is for all s = 1, . . ., m − 1.Consider now the case when Condition 2 holds, hence there exists As before, it is sufficient to consider the case of n = 1, m = 2 and b − a = ε.Taking into account (5.11) and [14,Lemma 7.27] (see also [1,21]) we have Taking an arbitrary interval Ω = (a, b) we construct a covering with a family of disjoint intervals of length ε, that is, Ω = l I l , where Ω I l = ∅, I i I j = ∅, i = j.Since (5.15) holds for all I l , summing up these estimates with respect to l we obtain Extending this to functions of n variables and taking ε = w −1 X ′ (Ω) ǫ, we obtain (5.16) . ., m − 1, for any p ≥ 2p 2 and a constant c depending on known quantities and on the norms w −1 X ′ (Ω) and w Xw (Ω) .

Introducing the notion A
, for all k = 0, . . ., m, we rewrite (5.16) where C 1 is a constant, independent of F. It is obvious that dF (x) dx = f (x), a.e. in Ω. Moreover Considering functions of n variables, fixing all variables except one and applying the above procedure we can extend this result in the n-dimensional case.
Let us note that in the definition of the Morrey spaces L p,λ (Ω) the supremum is taken over all intersections E = B ∩ Ω where B are balls in R n .This gives the continuous embedding M p,λ (Ω) ⊂ L p,λ (Ω) for all λ ∈ (0, 1).
It is easy to see that M p,λ (Ω) is nonseparable, moreover, it is a rearrangement invariant space (cf.[6]).
Let T be a quasi-linear operator of (p, p; q, q)-compatible weak type, (see [2] for the definition).Then we have the following results (cf.[2,6]).Lemma 6.2.The Boyd indices of the Marcinkiewicz space M p,λ (Ω) are Theorem 6.3.Let X be a rearrangement invariant space on an nonatomic complete σ-finite space with infinite measure.Then any linear (quasilinear) operator T of (p, p; q, q)-compatible weak type is bounded in X; i.e., T ∈ X if and only if the Boyd indices α X and β X satisfy the inequality The Theorem 6.3 (cf.[2]) implies the validity of the Property1 for M q,λ (Ω) for any q ∈ (1, ∞), λ ∈ (0, 1).The validity of Property 2 can be proved analogously as in the case of the Morrey spaces.Hence, by Theorem 4.3 we have the following result.Corollary 6.4.Under the conditions of Theorem 6.3 there exists u ∈ W m M q,λ (Ω), a solution of Lu = f, for all f ∈ M q,λ (Ω), q ∈ (1, ∞), λ ∈ (0, 1).
6.2.Grand Lebesgue spaces.The Grand Lebesgue spaces L q) (Ω), q ∈ (1, ∞), are non separable Banach spaces endowed by the norm The Property 1 holds for L q) (Ω) as it is proved in [17] and the following continuous embedding is valid Consider the Grand Sobolev spaces W m L q) (Ω) builded on the spaces L q) (Ω).Corollary 6.6.Suppose that the conditions of Theorem 5.3 hold true and x 0 ∈ Ω verifies the property (P x0 ).Then there exists a solution u ∈ W m L q) (B r (x 0 )) of the equation Lu = f , for r small enough and for all f ∈ L q) (B r (x 0 )).Corollary 6.7.Under the conditions of Theorem 5.3 the following a priori estimate holds We denote this function set as LH 0 (Ω).It follows immediately (see [9]) that if p(•) ∈ LH 0 (Ω) than it is uniformly continuous and p(•) ∈ L ∞ (Ω).
where the supremum is taken over all cubes Q ⊂ R n with sides parallel to coordinate axes and p ′ (•) is the conjugate function, 1 p(x) + 1 p ′ (x) = 1 for all x ∈ R. The properties of the weighted Variable Lebesgue spaces (see [9,17]) ensure the validity of the Property 1.For a bounded domain Ω we have the embedding L p+ (Ω) ⊂ L p(•) (Ω) ⊂ L p− (Ω) with a measurable function p(•) verifying (2.10).In the case of Variable Lebesgue spaces without any weight, the estimate (6.3) is obtained in

Corollary 4 . 2 .Theorem 4 . 3 .
Let ϕ ∈ W m Xw;r (B r (x 0 )).Under the conditions of Lemma 4.1 it holds (1) ϕ = T x0 ϕ + S x0 Lϕ, a.e. in B r (x 0 ); (2) if for some f ∈ X w (B r (x 0 )) it holds ϕ = T x0 ϕ + S x0 f , then ϕ is a solution of the equation Lϕ = f.The previous lemma allows us to proof the following local existence result.Let (3.2), (P x0 ), and the Properties 1 and 2 hold.Then the equation Lu = f admits a solution in W mXw (B r (x 0 )), for all f ∈ X w (Ω), and r > 0 small enough.Proof.By the Corollary 4.2 we can give the proof in the space W m Xw;r (r), taking x 0 = 0 ∈ Ω.Since L 0 S 0 = Id in X w (r), and keeping in mind (3.7) we have(4.5)

Corollary 6 . 9 .
Let the conditions of Theorem 4.3 hold and suppose that w ∈ LH 0 (Ω) ∩ A p(•) (R n ), p − > 1, then there exists a solution u ∈ W p(•),m w (B r (x 0 )) of the equation Lu = f for r small enough and for all f ∈ L p(•) w (Ω).The validity of the Property 2 in this case follows by Lemma 6.1 and [9].Theorem 6.10.Let Ω ⊂ R n be a bounded domain with C k smooth boundary and p(•) ∈ LH 0 (Ω) such that 1 < p − ≤ p + < +∞.Then for every k ≥ 1 there exists a bounded linear extension operatorθ k ∈ W p(•),k (Ω); W p(•),k (R n ) .Applying the Theorem 5.3 to the case of weighted Variable Lebesgue spaces we are able to obtain a local a priori estimate for the solution.Corollary 6.11.Let Ω 0 ⋐ Ω.Under the conditions of Theorem 5.3 and Corollary 6.9 the following a priori estimate holds (6.3) u W p(•),m (Ω0) ≤ c( Lu L p(•) (Ω) + u L p(•) (Ω) ).