Weighted Alpert Wavelets
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(2021) 27:1
Weighted Alpert Wavelets Rob Rahm1 · Eric T. Sawyer2 · Brett D. Wick3 Received: 14 June 2019 / Revised: 15 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper we construct a wavelet basis in L 2 (Rn ; μ) possessing vanishing moments of a fixed order for a general locally finite positive Borel measure μ. The approach is based on a clever construction of Alpert in the case of the Lebesgue measure that is appropriately modified to handle the general measures considered here. We then use this new wavelet basis to study a two-weight inequality for a general Calderón–Zygmund operator on R and conjecture that under suitable natural conditions, including a weaker energy condition, the operator is bounded from L 2 (R; σ ) to L 2 (R; ω) if certain stronger testing conditions hold on polynomials. An example is provided showing that this conjecture is logically different from existing results in the literature. Keywords Calderon-Zygmund operators · Two-Weight inequality · Alpert basis
1 Introduction and Statement of Main Results The use of weighted Haar wavelet expansions has its roots in connection with the T b theorem in [2,3], and came to fruition in treating the two weight norm inequality for the Hilbert transform in [9,13], μ the two part paper [5,6,8] . The key features of the weighted Haar expansion h I I ∈D are threefold (below, D is a dyadic lattice; explained below):
Communicated by Stephan Dahlke.
B
Rob Rahm [email protected] Eric T. Sawyer [email protected]
1
Texas A&M University, College Station, TX, USA
2
McMaster University, Hamilton, Canada
3
Washington University, St. Louis, USA 0123456789().: V,-vol
1
Page 2 of 41
Journal of Fourier Analysis and Applications
(2021) 27:1
μ (1) The Haar functions h I I ∈D form an orthonormal basis of L 2 (μ): f =
μ
f , hI
I ∈D
μ
h I both pointwise μ -a.e. and in L 2 (μ) ,
L 2 (μ)
μ μ where h J , h I L 2 (μ) = δ IJ , (2) Telescoping identities hold:
1K
μ
f , hI
I ∈D : K I ⊂L
L 2 (μ)
μ
μ
μ
hI = EK f − EL f ,
K L,
(3) Moment vanishing conditions hold:
μ
h I (x) dμ (x) = 0,
I ∈ D.
In the setting of Lebesgue measure, Alpert [1] introduced new wavelets with more vanishing moments in (3), while retaining orthonormality (1) and telescoping (2). The expense of imposing these extra moment conditions is that one requires additional functions in order to obtain the expansion. The purpose of this note is to extend existence of Alpert wavelets to arbitrary locally finite positive Borel measures in Euclidean space Rn , and to investigate degeneracy and uniqueness in the one-dimensional case as well. To state the main result in this paper requires some notation. Let μ be a locally finite positive Borel measure on Rn , and fix k ∈ N. Let P n denote the collection of cubes with sides parallel to the coordinate axes. If Q is such a cube, its “dyadic children” or just its “children”—denoted C (Q)—are the 2n subcubes whose side lengths are half the side length of Q. So
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