Two-sided weighted Fourier inequalities
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 2, pp. 341-362.

Fourier transform estimates for $\parallel \stackrel{^}{f}{\phantom{\rule{0.166667em}{0ex}}\parallel }_{{L}_{q,\stackrel{˜}{w}}}$ via ${\parallel f\parallel }_{{L}_{p,w}}$ from above and from below are studied. For $p=q$, equivalence results, i.e.,

 ${C}_{1}{\parallel f\parallel }_{{L}_{p,w}}\le \parallel \stackrel{^}{f}{\phantom{\rule{0.166667em}{0ex}}\parallel }_{{L}_{p,\stackrel{˜}{w}}}\le {C}_{2}{\parallel f\parallel }_{{L}_{p,w}},\phantom{\rule{1em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\stackrel{˜}{w}\left(x\right)=w\left(1/x\right){x}^{p-2},\phantom{\rule{1em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}1\le p<\infty ,$

are shown to be valid for functions from certain classes under the Muckenhoupt conditions: $w\in {A}_{p}$ or $w\in {A}_{2p}$. Sharpness of these conditions is proved.

Publié le :
Classification : 42A38,  26D15,  46E30
@article{ASNSP_2012_5_11_2_341_0,
author = {Liflyand, Elijah and Tikhonov, Sergey},
title = {Two-sided weighted Fourier inequalities},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {341--362},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 11},
number = {2},
year = {2012},
zbl = {1278.42006},
mrnumber = {3011994},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2012_5_11_2_341_0/}
}
Liflyand, Elijah; Tikhonov, Sergey. Two-sided weighted Fourier inequalities. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 2, pp. 341-362. http://www.numdam.org/item/ASNSP_2012_5_11_2_341_0/

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