The Regularity Problem for Uniformly Elliptic Operators in Weighted Spaces

Abstract

This paper studies the regularity problem for block uniformly elliptic operators in divergence form with complex bounded measurable coefficients. We consider the case where the boundary data belongs to Lebesgue spaces with weights in the Muckenhoupt classes. Our results generalize those of S. Mayboroda (and those of P. Auscher and S. Stahlhut employing the first order method) who considered the unweighted case. To obtain our main results we use the weighted Hardy space theory associated with elliptic operators recently developed by the last two named authors. One of the novel contributions of this paper is the use of an “inhomogeneous” vertical square function which is shown to be controlled by the gradient of the function to which is applied in weighted Lebesgue spaces.

The regularity problem for degenerate elliptic operators in weighted spaces

“…When w ≡ 1 (that is, we are working with the class of uniformly -or non-degenerate-elliptic operators) and v ≡ 1 then, clearly, r w = 1, W w v max{r w , q − (L w )}, q + (L w ) = (q − (L w ), q + (L w )) = Ø, and our result gives (1.10) in the range max 1, nq − (Lw) n+q − (Lw) , q + (L w ) , hence we fully recover [29,Theorem 4.1]. If we still assume that w ≡ 1 and we let v ∈ A ∞ (w) = A ∞ (dx), then our assumption (1.9) agrees with that in [11,Theorem 1.10] and the range of p's here is slightly worse than the one in that result (the lower end-point in [11,Theorem 1.10] has been pushed down using an extra technical argument that we have chosen not to follow here).…”

“…This amounted to control non-tangentially the full gradient of the solution given by the Poisson semigroup in terms of the gradient of the boundary datum. In turn, using the weighted Hardy space theory developed in [27,28,30], the solvability of the regularity problem in the block case for data in Lebesgue spaces with Muckenhoupt weights has been recently studied in [11].…”

Boundary value problems and Hardy spaces for elliptic systems with block structure

“…Now, we can take a Ẇ1,2convergent atomic decomposition u = i λ i m i as in Proposition 8. 31. By the solution of the Kato problem we have L 2 -convergence of…”

“…We point out that Step V in the proof of [74,Thm. 4.1] has a flaw that can be fixed (personal communication of S. Hofmann) or treated differently, see the argument in [31].…”