A two-parameter “Bergman space” inequality

Author:
J. Michael Wilson

Journal:
Proc. Amer. Math. Soc. **125** (1997), 755-762

MSC (1991):
Primary 42B25, 42B30, 42C10

DOI:
https://doi.org/10.1090/S0002-9939-97-04039-2

MathSciNet review:
1415376

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Abstract | References | Similar Articles | Additional Information

Abstract: For $f\in L^{1}([0,1]\times [0,1])$, define $\lambda _{R} \equiv \langle f,h_{(R)}\rangle$, where $h_{(R)}(x,y)=h_{(I)}(x)\cdot h_{(J)}(y)$ is a tensor product of one-parameter Haar functions. Let $1<p\leq q<\infty$ and $q\geq 2$. We prove a sufficient condition, which is close to necessary, on double sequences of weights $\{\mu _{R}\}_{R}$ and non-negative $v\in L^{1}([0,1]\times [0,1])$, which ensures that the inequality \begin{equation*} \left (\sum _{R}\vert {\lambda _{R}}\vert ^{q}\mu _{R}\right )^{1/q}\leq \left (\int _{[0,1]\times [0,1]}\vert {f}\vert ^{p} v dx\right )^{1/p}\end{equation*} holds for all $f\in L^{1}([0,1]\times [0,1])$. We extend our result to an inequality concerning two-parameter wavelet families.

- R. L. Wheeden, J. M. Wilson, “Weighted norm estimates for gradients of half-space extensions,”
*Indiana University Math. Journal***44**(1995), 917-969. - J. Michael Wilson,
*Some two-parameter square function inequalities*, Indiana Univ. Math. J.**40**(1991), no. 2, 419–442. MR**1119184**, DOI https://doi.org/10.1512/iumj.1991.40.40022

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Additional Information

**J. Michael Wilson**

Affiliation:
Department of Mathematics, University of Vermont, Burlington, Vermont 05405

Keywords:
Haar functions,
Littlewood-Paley theory,
wavelets,
Bergman space,
weighted norm inequality

Received by editor(s):
February 7, 1995

Additional Notes:
The author was supported by NSF grant DMS 9401498.

Communicated by:
J. Marshall Ash

Article copyright:
© Copyright 1997
American Mathematical Society