Two Weight Bump Conditions for Matrix Weights

Abstract

In this paper we extend the theory of two weight, A p bump conditions to the setting of matrix weights. We prove two matrix weight inequalities for fractional maximal operators, fractional and singular integrals, sparse operators and averaging operators. As applications we prove quantitative, one weight estimates, in terms of the matrix A p constant, for singular integrals, and prove a Poincaré inequality related to those that appear in the study of degenerate elliptic PDEs.2010 Mathematics Subject Classification. Primary 42B20, 42B25, 42B35.

Variation of Calderón--Zygmund Operators with Matrix Weight

“…In Section 3, we first extend the pointwise domination of variation in Section 2 to the vector-valued setting. Then we obtain a version of domination of V ρ (T n , * f ) by convex body valued sparse operators introduced in [30], and by following some idea from [11], present the proof of Theorem 1.5.…”

Sharp $$A_{1}$$ Weighted Estimates for Vector-Valued Operators

“…In our next result we show that the A 1 type conditions constants control the corresponding A ∞ constants. We include in the statement the case of the A q constant that was already established in [3] for the sake of completeness.…”

“…Very recently D. Cruz-Uribe, J. Isralowitz and K. Moen [3] extended (1.1) to every 1 < p < ∞, providing the following estimate…”

Two Matrix Weighted Inequalities for Commutators with Fractional Integral Operators

“…We refer the reader to Section 5.2 in [CUIM18] for the standard Orlicz space related definitions used in the statement of Proposition 1.5.…”

“…Our proof is a combination and modification of the arguments in [IPT,CUIM18,Li06]. For additional information on Orlicz spaces, see e.g., [BL12].…”