On Lebesgue integrability of Fourier transforms in amalgam spaces
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 this paper is to establish an extension of Proposition 1.3 in [\textit{W. O. Bray}, ibid. 20, No. 6, 1234--1256 (2014; Zbl 1306.42011)] in the setting of the spaces $\left( L^{q},\ell ^{p}\right) ^{\alpha }\left(\mathbb{R}^{d}\right) $. \par The following result (Proposition 3.3) is a generalization of Proposition 1.3. \par Theorem 3.3. Assume that: \par (i) $\mu $ belongs $\mathcal{K}_{\sigma ,A}\left(\mathbb{R}^{d}\right) $ for some positive real numbers $\sigma $ and $A$; \par (ii) $1\leq \alpha \leq p\leq 2$ with $1<p$ and $\theta $ is a positive real number such that $\frac{1}{p^{\prime }}<\frac{1}{\alpha ^{^{\prime }}}+\frac{\theta }{d}$; \par (iii) $f$ \ belongs to $\left( L^{1},\ell ^{p}\right) ^{\alpha }\left(\mathbb{R}^{d}\right) $ and there is a positive real number $E$ such that \[ \left\Vert f-M_{\mu }^{t}f\right\Vert _{1,p,\alpha }\leq Et^{\theta },\text{ }t\in \left( 0,\infty \right), \] where \[M_{\mu }^{t}f(x)=\int\limits_{\mathbb{R}^{d}}f(x-ty)d\mu \left( y\right) ,\text{ }x\in\mathbb{R}^{d}. \] Then, for any real numbers $a$ and $\beta $ satisfying $\frac{1}{p^{\prime }} \leq \frac{1}{\beta }$ and $\frac{1}{p^{\prime }}\leq \frac{1}{\beta }\left( 1+\frac{a}{d}\right) <\frac{1}{\alpha ^{^{\prime }}}+\frac{\theta }{d}$, we have \[ \int\limits_{\mathbb{R}^{d}}\left\vert x\right\vert ^{\alpha }\left\vert \hat{f}\left( x\right) \right\vert ^{\beta }dx<\infty \] and therefore \[ \int\limits_{\mathbb{R}^{d}}\left( 1+\left\vert x\right\vert \right) ^{\alpha }\left\vert \hat{f} \left( x\right) \right\vert ^{\beta }dx<\infty . \]
scientific article; zbMATH DE number 7005628
scientific article