WEIGHTED A PRIORI ESTIMATES FOR SOLUTION OF ( − Δ ) m u = f WITH HOMOGENEOUS DIRICHLET CONDITIONS

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Introduction
We will use the standard notation for Sobolev spaces and for derivatives, namely, if For u ∈ W k,p (Ω), its norm is given by We consider the homogeneous problem where ∂ ∂ ν is the normal derivative.
In the classic paper [1], the authors obtained a priori estimates for solutions of (1.1) for a smooth domain Ω given by A key tool to prove those estimates was the Calderón-Zygmund theory for singular integral operators.
On the other hand, after the pioneering work of Muckenhoupt [7] , a lot of work on continuity in weighted norms has been developed.In particular, weighted estimates for a wide class of singular integral operators have been obtained for weights in the class of Muckenhoupt A p .
Therefore, it is a natural question whether analogous weighted a priori estimates can be proved for the derivatives of solutions of elliptic equations.
For the Laplace equation (m = 1), it was proved in [5] that for a weight ω belonging to the The goal of this paper is to extend the results of [5] for powers of the Laplacian operator with homogeneous Dirichlet boundary conditions, i.e. it is to prove that for ω ∈ A p , where the constant C depends on Ω, m, n and the weight ω.
The main ideas for the proof of these estimates are similar to those given in [5].However, non trivial technical modifications are needed because, for m ≥ 2, the Green function is not positive in general and therefore, we cannot apply the maximum principle.

Preliminaries
In what follows we consider the problem (1.1) in a bounded domain Ω with ∂ Ω ∈ C 6m+4 for n = 2 and ∂ Ω ∈ C 5m+2 for n > 2 (the regularity on the boundary is necessary in order to use the results of the Green function given in [6]).
Anal.Theory Appl., Vol. 26, No.4 (2010) 341 The solution of (1.1) is given by where G m (x, y) is the Green function of the operator (−Δ) m in Ω which can be written as where Γ(x − y) is a fundamental solution and h(x, y) satisfies where K j (y, P) are the Poisson kernels and dS denotes the surface measure on ∂ Ω.
We recall that any fundamental solution associated to (1.1) is smooth away from the origin and it is homogeneous of degree 2m − n if n is odd or if 2m < n and the logarithmic function appears if n is even and 2m ≥ n.However, in both cases, under our assumption on the boundary domain, we have the known estimates of the Green function G m (x, y) and the Poisson kernels In what follows the letter C will denote a generic constant not necessarily the same at each occurrence.
3 The Estimates for the Derivatives of u In this section we state pointwise estimates for the first 2m − 1 derivatives of the function u and a weak estimate for the 2m derivative.These estimates are needed for proving the main result of this work.
Proposition 3.1.Given two measurable functions f and g in Ω, for |α| = 2m we have that Proof.We write where Then, using (2.7) we have Anal.Theory Appl., Vol. 26, No.4 (2010) In order to estimate the term II in (3.1), we first observe that for (x, y) ∈ D 2 , we have that and therefore, by the same arguments used before we have that and the Proposition is proved.
In order to see how to estimate in Ω \D, we consider separately the function Proof.In view of (2.3) we must find estimates for D α x (
From the general properties of the fundamental solution Γ(x − y) we have that for |α| + j ≥ 2m − n + 1, and for 0 ≤ j ≤ m − 1, by (2.8) we have that for y ∈ Ω and P ∈ ∂ Ω.
Then by (3.3), (3.4) and the fact that if |x−y| ≤ d(x) then d(y) < 2d(x), we have for In order to see that each integral is finite we write ∂ Ω = F 1 ∪ F 2 , with where P 0 ∈ ∂ Ω is that |y − P 0 | = d(y).And now, the convergence of these integrals follows in a standard way.
It follows from the previous Proposition that for each x ∈ Ω and |α| ≥ 2m − n + 1 we have that D α x h(x, y) is bounded uniformly in a neighborhood of x and so is On the other hand, although D α x Γ is a singular kernel for |α| = 2m, taking β such that |β | = 2m − 1, we have where c is a bounded function and K is a Calderón -Zygmund operator given by We will also make use of the maximal operator K f (x) = sup Proof.Using the representation formula for u, by (3.5), (3.6) and (3.7) we have By the results given above, for I, II and III we have pointwise estimates, and obtain ( in the same way as in [5]) that However, for IV we have just a weak estimate.Indeed, from Proposition ?? we have and the Theorem is proved.

Main Result
We can now state and prove our main result.First we recall the definition of the A p class for 1 < p < ∞.A non-negative locally integrable function ω belongs to A p if there exists a constant For any weight ω, L p ω (Ω) is the space of measurable functions f defined in Ω such that and W k,p ω (Ω) is the space of functions such that and u a weak solution of (1.1), then there exists a constant C depending only on n, m, ω and Ω such that Proof.Since M is a bounded operator in L p ω (Ω), by Lemma 3.1 it follows that Therefore, it only remains to estimate kD α x uk L p ω (Ω) for |α| = 2m.
Let ω ∈ A p and g(x) := (D α x u(x)) p−1 ω(x).By Theorem ?? we see that Since K and M are bounded operators in L p ω (Ω), applying the Hölder inequality, it follows that where In the same way, we obtain that For the last term in (4.1), taking into account that ω −q/p ∈ A q , we have Then, by (4.2), (4.3), (4.4) and (4.5)we have By the definition of g(x), Then we obtain and the Theorem is proved for u ∈ W 2m,p ω (Ω).
Finally, we will show that the weak solutio u of (1.1) belongs to W 2m,p ω (Ω).
We have (−Δ) m u = f , with f ∈ L p ω (Ω), then there exists a sequence It is easy to see, from Lemma 3.1 that u k ∈ W 2m−1,p ω (Ω), and obviously u k ∈ W 2m,p ω,loc (Ω).Moreover, for all compact sets K ⊂ Ω, we have where C(K) is a constant depending on the measure of K. Indeed, taking (Ω), satisfies (1.1) with f = g k ∈ L p ω (Ω), and we can use (4.6).Then, it follows from the dominated convergence theorem that u k ∈ W 2m,p ω (Ω) and applying (4.6), we have the boundary operators defined in [1].
Indeed, we define l 1 > max j (2m − m j ) and l 0 = max j (2m − m j ).If the coefficients a α ∈ we have that the Green function G m and the Poisson kernels K j for 0 ≤ j ≤ m − 1 exist whenever l 1 > 2(l 0 + 1) for n = 2 and l 1 > 3 2 l 0 for n ≥ 3.
Our results show that, at least in the case p > m + 1, the estimate remains valid when

ε>0Theorem 3 . 3 .
|K ε f (x)|.Here and in what follows we consider f defined in R n extending the original f by zero.Now we are in conditions to give the following estimate: Given g a measurable function and |α| = 2m.Then there exists a constant C depending only on n, m and Ω such that for any x ∈ Ω, Ω) and by the classical trace theorems in Sobolev spaces and the definition of ω ∈ A p , it follows that v satisfies the homogeneous boundary conditions and by uniqueness of the solution, the Theorem is proved.R. G. Durán et al: : Weighted a priori estimates for solution of (−Δ) m u = f Remark 4.2.The result of Theorem 4.1 is also valid for u a weak solution of ⎧