On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights

Abstract

Abstract. We prove generalized Fefferman-Stein type theorems on sharp functions with Ap weights in spaces of homogeneous type with either finite or infinite underlying measure. We then apply these results to establish mixednorm weighted Lp-estimates for elliptic and parabolic equations/systems with (partially) BMO coefficients in regular or irregular domains. IntroductionThe objective of this paper is two-fold. The first is to present a few generalized versions of the Fefferman-Stein theorem on sharp functions. One of our main theorems in this direction proveswhere (X , µ) is a space of homogeneous type, w is a Muckenhoupt weight, f # dy is the sharp function of f using a dyadic filtration of partitions of X , and L p,q (X , w dµ) is a mixed norm. A space X of homogeneous type is equipped with a quasi-distance and a doubling measure µ. A brief description of spaces of homogeneous type is given in Section 2. For more discussions, see [45,44,11]. If X is the product of two spaces (X 1 , µ 1 ) and (X 2 , µ 2 ) of homogeneous type, w = w 1 (x ′ )w 2 (x ′′ ), and µ(x) = µ 1 (x ′ )µ 2 (x ′′ ), where x ′ ∈ X 1 and x ′′ ∈ X 2 , then the L p,q (X , w dµ) norm is defined asSee (2.13) for a precise definition of the mixed norm. If X is the Euclidean space R d , µ is the Lebesgue measure on R d , w ≡ 1, and p = q, the inequality (1.1) is the celebrated Fefferman-Stein theorem on sharp functions. See, for instance, [26,49], where the measure of the underlying space is clearly infinite. Here we deal with both the cases of a finite measure µ(X ) < ∞ and an infinite measure µ(X ) = ∞. We also present a different form of the Fefferman-Stein theorem. See Theorems 2.3 and 2.4 and Corollaries 2.6 and 2.7.The second objective, which is in fact a motivation of writing this paper, is to apply the generalized versions of the Fefferman-Stein theorem to establishing 2010 Mathematics Subject Classification. 35R05, 42B37, 35B45, 35K25, 35J48. Key words and phrases. sharp/maximal functions, elliptic and parabolic equations, weighted Sobolev spaces, measurable coefficients.H.

Weighted L-estimates for stationary Stokes system with partially BMO coefficients

“…Indeed, suppose that both quantities (the flatness and the mean oscillations) are bounded by a positive number ρ. Then, in elliptic and parabolic cases with partially BMO coefficients on Reifenberg flat domains (see, for instance, [11]), there exists a unique solution to a given equation in a Sobolev space if ρ is sufficiently small. It is important that the size of ρ is determined only by parameters such as the dimension, the ellipticity constant, and q if solutions are to be found in a L q -based Sobolev space.…”

“…The regularity assumptions in this paper on coefficients and domains have also been considered in recent papers, [12,11] for instance, on elliptic and parabolic equations/systems. In particular, as far as coefficients are concerned, the assumption allowing coefficients to be merely measurable in one direction cannot be relaxed in view of the counterexamples about the unique solvability of elliptic equations in Sobolev spaces (see [28,15]) when the coefficients are only measurable functions of two variables.…”

“…In particular, as far as coefficients are concerned, the assumption allowing coefficients to be merely measurable in one direction cannot be relaxed in view of the counterexamples about the unique solvability of elliptic equations in Sobolev spaces (see [28,15]) when the coefficients are only measurable functions of two variables. See [11] and the references therein for a comprehensive study on elliptic and parabolic equations/systems in Sobolev space with Muckenhoupt weights.…”

“…For the proofs of the main results for Sobolev spaces with weights, we make use of the mean oscillation estimate approach presented, for instance, in [13,12,11]. As explained in [13], mean oscillation estimates are well suited to the perturbation argument for coefficients having small mean oscillations.…”

“…As explained in [13], mean oscillation estimates are well suited to the perturbation argument for coefficients having small mean oscillations. Moreover, as shown in [11], they are in an appropriate form to deal with L q -estimates with Muckenhoupt weights via the Hardy-Littlewood maximal function theorem and the FeffermanStein theorem on sharp functions, which are valid for L q spaces with Muckenhoupt weights. The mean oscillation estimates rely on Theorem 2.4, which is for the unweighted case.…”