Entropy bump conditions for fractional maximal and integral operators

Abstract

Abstract:We investigate weighted inequalities for fractional maximal operators and fractional integral operators. We work within the innovative framework of "entropy bounds" introduced by Treil-Volberg. Using techniques developed by Lacey and the second author, we are able to efficiently prove the weighted inequalities.

Off-Diagonal Two Weight Bumps for Fractional Sparse Operators

“…This was extended to the general p = q case by Lacey-Spencer in [9]. These operators were also studied in the off-diagonal p ≤ q, α > 0 setting in [16,18]. Our theorem here is "sharp" in the sense that when σ and w are A ∞ , we recover the sharp results of, for example [2,3,7].…”

“…The topic of this article is two-weight bump conditions for sparse operators in the "offdiagonal" setting (i.e q > p). We continue the line of investigation concerning entropy bumps that began with Treil-Volberg in [21] and was continued in [9,18,19] and the line of investigation concerning "direct comparison bumps" introduced in [19] and continued and improved in [10]. We will be concerned with sparse operators of the form ( 0 ≤ α < d):…”

$A_p$ weights and Quantitative Estimates in the Schrödinger Setting

“…The following was proven by one of us and Scott Spencer [34]. Below, for a weight w we define ρ w (Q) := 1 w(Q) ˆQ(M(wQ))(x)dx.…”

Off-diagonal sharp two-weight estimates for sparse operators

“…• Orlicz norm bumps, studied among others in [CUMP12, Ler13, CUM13a, CUM13b, CU17], with early history in [Neu83, Pér94a, Pér94b]; • testing conditions, pioneered in [Saw82,Saw88], with a recent culmination in [LSSUT14,Lac14]; and • entropy bumps, recently introduced in [TV16] and studied in [LS15,RS16]. However, an in-depth discussion of any of these topics would take us too far afield.…”