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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
MathSciNet review: 1415376
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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.


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Additional Information

J. Michael Wilson
Affiliation: Department of Mathematics, University of Vermont, Burlington, Vermont 05405

DOI: http://dx.doi.org/10.1090/S0002-9939-97-04039-2
PII: S 0002-9939(97)04039-2
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