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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A two-parameter “Bergman space” inequality
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by J. Michael Wilson PDF
Proc. Amer. Math. Soc. 125 (1997), 755-762 Request permission

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.
References
  • 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 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
  • Received by editor(s): February 7, 1995
  • Additional Notes: The author was supported by NSF grant DMS 9401498.
  • Communicated by: J. Marshall Ash
  • © Copyright 1997 American Mathematical Society
  • 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