Fourier transforms of functions in Herz spaces on certain groups
The main result of the authors is a generalization of the Hausdorff-Young inequality in the context of certain locally compact abelian groups. A l.c.a.g. G is given with a two-sided sequence \(\{G_ n\}^{\infty}_{n=-\infty}\) of compact open subgroups, with \(G_{n+1}\subsetneqq G_ n\) for all n. The Haar measures \(\mu\) on G and \(\lambda\) on the dual group \(\Gamma\) are chosen so that \(\mu (G_ 0)=1=\lambda (\Gamma_ 0)\); \(\Gamma_ n\) is the annihilator of \(G_ n\), and set \(m_ n=\mu (G_ n)^{-1}=\lambda (\Gamma_ n).\) Let \(\xi_{\Lambda}\) denote the characteristic function of a set \(\Lambda\). A measurable function f on G is said to belong to the Herz space K(\(\alpha\),p,q;G) if \[ \| f\|_{K(\alpha,p,q;G)}=(\sum^{\infty}_{j=-\infty}m_{j+1}^{- \alpha q} \| fx_{G_ j\setminus G_{j+1}}\|^ q_ p)^{1/q}<\infty \quad if\quad q<\infty, \] or \[ \| f\|_{K(\alpha,p,\infty,G)}=\sup_{j\in {\mathbb{Z}}}m^{- \alpha}_{j+1} \| f\xi_{G_ j\quad \setminus G_{j+1}}\|_ p<\infty. \] Similarly define \(\| g\|_{K(\alpha,p,q;\Gamma)}\) if g is a measurable function on \(\Gamma\), except that in the above definition \(\alpha\) is replaced by -\(\alpha\). Recall that the Lorentz norms are given by \[ \| f\|^*_{p,q}=(\int^{\infty}_{0}t^{p/q-1}f^*(t)^ q dt)^{1/q},\quad if\quad 0<p,q<\infty,\quad and\quad =\sup_{t>0}t^{1/p}f^*(t),\quad if\quad 0<p<\infty,\quad q=\infty. \] \(f^*\) is the non-decreasing rearrangement of f. Theorem. Let \(1<p\leq 2\), \(0<\alpha <1/p'\), \((1/p+1/p'=1)\) and \(0<q\leq \infty\). Then we have \[ K(\alpha,p,q;G) \hookrightarrow L^{p/(p-\alpha p-1),q}(\Gamma) \hookrightarrow K(-\alpha,p',q;\Gamma). \] (Here \(L^{r,s}(\Gamma)\) is the set of functions f on \(\Gamma\) such that \(\| f\|^*_{r,s}<\infty\), and ''\(\hookrightarrow ''\) means ''is continuously embedded in''.)
scientific article; zbMATH DE number 3948965