Sharp Matrix Weighted Strong Type Inequalities for the Dyadic Square Function
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Sharp Matrix Weighted Strong Type Inequalities for the Dyadic Square Function Joshua Isralowitz1 Received: 4 May 2019 / Accepted: 2 December 2019 / © Springer Nature B.V. 2020
Abstract In this paper we refine the recent sparse domination of the integrated p = 2 matrix weighted dyadic square function by T. Hytonen, S. Petermichl, and A. Volberg to prove a pointwise sparse domination of general matrix weighted dyadic square functions. We then use this to prove sharp two matrix weighted strong type inequalities for matrix weighted dyadic square functions when 1 < p ≤ 2. Keywords Matrix weight · Square function · Weighted norm inequalities Mathematics Subject Classification (2010) 42B20 · 42B25
1 Introduction Let be U an a.e. positive definite n × n matrix valued function on Rd (that is, a matrix weight), and for a measurable Cn valued function f on Rd define 1 p p p1 p fL (U ) := U (x)f(x) dx Rd
where |e| is the standard Euclidean norm of a vector e in Cn . We will say that a pair of matrix weights U, V is matrix Ap if p 1 p −1 [U, V ]Ap := sup − − V p (y)U p (x)p dy dx < ∞ I ⊆Rd I is a cube
I
I
where −I refers to the unweighted average and A is the standard matrix norm of an n × n matrix A. Clearly this is a condition that reduces to the classical Muckenhoupt two weight Ap condition in the scalar setting (when n = 1). If U = V then we will say U is a matrix Ap weight if [U ]Ap := [U, U ]Ap < ∞. While it is known that most “classical” operators
Joshua Isralowitz
[email protected] 1
University at Albany, SUNY, Albany, NY, 12159, USA
J. Isralowitz
from harmonic analysis (such as the maximal function, Calder´on-Zygmund operators, paraproducts, martingale transforms, square functions, etc.) are bounded on Lp (U ) for matrix Ap weights U , it is difficult to determine the sharp dependence of such operators on [U ]Ap . In fact, the only two such operators where sharp one weighted matrix weighted norm inequalities for p = 2 are known are for the dyadic square function, which was recently proved in [6] and the maximal function, which was proved in [7] by slightly modifying the ideas in [1]. Furthermore, among these two operators, sharp one matrix weighted Ap bounds for 1 < p < ∞ are only known for the maximal function, which were proved in [8] by slightly modifying the ideas in [4]. The purpose of this paper is to prove sharp strong type matrix weighted norm inequalities for the dyadic square function in the range 1 < p ≤ 2, providing the first sharp p = 2 estimates for a singular operator in the matrix weighted setting. Let D be a dyadic grid and let {hkJ } for k = 1, . . . , 2d − 1 and J ∈ D be any Haar system on Rd , meaning that {hkJ } is an orthonormal system of L2 (Rd ) with hkJ supported on J, J hkJ (x) dx = 0, and each hkJ is constant on dyadic subcubes of J . Also, for a function f : Rd → Cn let k fJ := f(x)hkJ (x) dx J
and define the matrix weighted dyadic square function SU f = SU,p f by ⎞ 12 ⎛ 1 2 p k 1 (x) (x)f J J ⎟ ⎜ U SU,p f(x) := ⎝ ⎠ . |J |
(1.
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