On Lebesgue integrability of Fourier transforms in amalgam spaces
type: Item
instance of: scholarly article
title: On Lebesgue integrability of Fourier transforms in amalgam spaces
zbMATH Open document ID: 1417.42010
author: Sekou Coulibaly
author: Ibrahim Fofana
published in: The Journal of Fourier Analysis and Applications
publication date: 2019-01-23T00:00:00Z
review text: Let $d$ be a positive integer. The space $\mathbb{R}^{d}$ is endowed with the Lebesgue measure. Let us set for any positive real number $r$ \begin{align*} I_{k}^{r} & = \prod\limits_{j=1}^{d}[k_{j}r,\left( k_{j}+1\right) r),\text{ } k=\left( k_{1},k_{2},\dots,k_{d}\right) \in\mathbb{Z}^{d}. \\ J_{x}^{r} & = \prod\limits_{j=1}^{d}(x_{j}-\frac{r}{2},x_{j}+\frac{r}{2}), \text{ }x=\left( x_{1},x_{2},\dots,x_{d}\right) \in\mathbb{R}^{d}. \end{align*} Let $q$, $\alpha $ and $p$ be elements of $\left[ 0,\infty \right] $. We define for any element $f\in L_{\mathrm{loc}}^{q}\left( \mathbb{R}^{d}\right) $ and any real number $r>0$, \begin{align*} _{r}\left\Vert f\right\Vert _{q,p} & = \left[ \sum\limits_{k\in\mathbb{Z}^{d}}\left( \left\Vert f\chi _{I_{k}^{r}}\right\Vert _{q}\right) ^{p}\right] ^{\frac{1}{p}}\text{ if }p<\infty , \\ _{r}\left\Vert f\right\Vert _{q,p} & = \sup_{x\in\mathbb{R}^{d}}\left\Vert f\chi _{J_{k}^{r}}\right\Vert _{q}\text{ \ if }p=\infty s \end{align*} where $\chi _{A}$ denotes the characteristic function of the subset $A$ of $\mathbb{R}^{d}$. We also define \begin{align*} \left( L^{q},\ell ^{p}\right) \left(\mathbb{R}^{d}\right) & = \left\{ f\in L_{\mathrm{loc}}^{q}\left(\mathbb{R}^{d}\right) :\left( _{1}\left\Vert f\right\Vert _{q,p}\right) <\infty \right\} \\ \left( L^{q},\ell ^{p}\right) ^{\alpha }\left(\mathbb{R}^{d}\right) & = \left\{ f\in L_{\mathrm{loc}}^{q}\left(\mathbb{R}^{d}\right) :\left\Vert f\right\Vert _{q,p,\alpha }=\sup_{r>0}r^{d\left( \frac{1}{\alpha }-\frac{1}{q}\right) }\left( _{r}\left\Vert f\right\Vert _{q,p}\right) <\infty \right\}. \end{align*} Let $\sigma $ and $A$ be positive real numbers. We say that $\mu $ belongs to $\mathcal{K}_{\sigma ,A}\left(\mathbb{R}^{d}\right) $ when $\int\limits_{\mathbb{R}^{d}}d\mu \left( x\right) =1$ and \[ A\min \left( 1,\left\vert x\right\vert ^{2\sigma }\right) \leq \left\vert 1-\left( 2\pi \right) ^{\frac{d}{2}}\hat{\mu}\left( x\right) \right\vert , \text{ }x\in\mathbb{R}^{d}. \] The aim of thi…
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 7005628
zbMATH Keywords: Fourier transform
zbMATH Keywords: amalgam spaces
zbMATH Keywords: modulus of continuity
MaRDI profile type: Publication
full work available at URL: s00041-017-9577-z
OpenAlex ID: W2769662178
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DOI: 10.1007/S00041-017-9577-Z
description: scientific article; zbMATH DE number 7005628