Two-sided weighted Fourier inequalities

Liflyand, Elijah; Tikhonov, Sergey

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

Fourier transform estimates for $\parallel \widehat{f}{\parallel }_{L_{q,\tilde{w}}}$ via ${\parallel f\parallel }_{L_{p,w}}$ from above and from below are studied. For $p=q$, equivalence results, i.e.,

$$C_1{\parallel f\parallel }_{L_{p,w}}\le \parallel \widehat{f}{\parallel }_{L_{p,\tilde{w}}}\le C_2{\parallel f\parallel }_{L_{p,w}},\tilde{w}\left(x\right)=w(1/x)x^{p-2},1\le p<\infty ,$$

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.

Classification : 42A38, 26D15, 46E30

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