Fourier transform of anisotropic Hardy spaces
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- by Marcin Bownik and Li-An Daniel Wang PDF
- Proc. Amer. Math. Soc. 141 (2013), 2299-2308 Request permission
Abstract:
We show that if $f$ is in an anisotropic Hardy space $H_A^p$, $0 < p \leq 1$, with respect to a dilation matrix $A$, then its Fourier transform $\hat {f}$ satisfies the pointwise estimate \[ |\hat f(\xi )| \le C ||f||_{H^p_A} \rho _*(\xi )^{\frac {1}{p}-1}.\] Here, $\rho _*$ is a quasi-norm associated with the transposed matrix $A^*$. This leads to necessary conditions for functions $m$ to be multipliers on $H_A^p$, as well as further pointwise characterizations on $\hat {f}$ and a generalization of the Hardy-Littlewood inequality on the integrability of $\hat {f}$. This last result is strengthened through the use of rearrangement functions.References
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Additional Information
- Marcin Bownik
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403–1222
- MR Author ID: 629092
- Email: mbownik@uoregon.edu
- Li-An Daniel Wang
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403–1222
- Address at time of publication: Department of Mathematics, Trinity College, Hartford, Connecticut 06106
- MR Author ID: 1015472
- Email: lwang3@uoregon.edu, daniel.wang@trincoll.edu
- Received by editor(s): October 7, 2011
- Published electronically: February 14, 2013
- Communicated by: Michael T. Lacey
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 2299-2308
- MSC (2010): Primary 42B30
- DOI: https://doi.org/10.1090/S0002-9939-2013-11623-0
- MathSciNet review: 3043011