Quadratic sparse domination and Weighted Estimates for non-integral Square Functions

Abstract

We prove a quadratic sparse domination result for general non-integral square functions S. That is, we prove an estimate of the formwhere q * 0 is the Hölder conjugate of q 0 /2, M is the underlying doubling space and S is a sparse collection of cubes on M . Our result will cover both square functions associated with divergence form elliptic operators and those associated with the Laplace-Beltrami operator. This sparse domination allows us to derive optimal norm estimates in the weighted space L p (w).2010 Mathematics Subject Classification. 42B20, 42B37.

Operator-free sparse domination

“…• Non-integral operators falling outside the scope of Calderón-Zygmund theory [8] and the associated square functions [2]. • Rough homogeneous singular integrals [34].…”

“…We refer to [25,Chapter 7] for the definition of (Rademacher) type r ∈ [1,2] with constant τ r,X used in the following theorem. Let us note here that any Banach space has type 1 with constant τ 1,X = 1.…”

“…Theorem 6.4. Let X be a Banach space with type r ∈ [1,2], take q ∈ (0, ∞) and let α > 0. Let f : R n+1 + → X be strongly measurable.…”

“…During the last five years a number of sparse domination principles (that is, general results establishing sparse domination for a given class of operators) have appeared e.g. in the works [2,5,6,8,12,13,33,34,36,38].…”

Multi-scale sparse domination

“…The original Banach space domination result was refined and streamlined to a pointwise result [28,72,59,70], but it is the concept of sparse domination in terms of bilinear (or multilinear) forms [15,29] that has allowed to extend the subject to many operators in harmonic analysis beyond the scope of Caldern-Zygmund theory. Among other examples, one may find the bilinear Hilbert transform [29], singular integrals with limited regularity assumptions [26,12,71], Bochner-Riesz operators [13,61], spherical maximal functions [60], singular Radon transforms [25,80,50], pseudo-differential operators [10], maximally modulated singular integrals [34,8], non-integral square functions [6], and variational operators [32,31,14], as well as results in a discrete setting (see for instance [58,30,2]).…”