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
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- 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
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