Fourier multipliers of generalized Lipschitz spaces (Q1079768)

Let G be a locally compact abelian group with a two-way infinite sequence \(\{G_ n\}^{\infty}_{n=-\infty}\) of compact open subgroups such that \(G_{n+1}\subsetneqq G_ n\) for every integer n, \(\cup^{\infty}_{n=-\infty}G_ n=G\), and \(\cap^{\infty}_{n=- \infty}G_ n=\{0\}\). For \(-\infty <\alpha <\infty\), \(1\leq p\leq \infty\) and \(0<q\leq \infty\), the author defines the generalized Lipschitz space Lip(\(\alpha\),p;q;G) in terms of the distributions on G. The space Lip(\(\alpha\),p;q;G) is then characterized as an interpolation space. Using this characterization, the author shows that every Fourier multiplier from \(Lip(\alpha_ 1,p_ 1;q_ 1;G)\) to \(Lip(\alpha_ 2,p_ 2;q_ 2;G)\) is given by a distribution \(\sigma\) on \(\hat G \)(= the dual group of G) such that the inverse Fourier transform \({\check \sigma}\in Lip(\alpha_ 2-\alpha_ 1-1+1/p_ 1,p_ 2;\infty;G)\), where \(-\infty <\alpha_ 1, \alpha_ 2<\infty\), \(1\leq p_ 1, p_ 2<\infty\), and \(0<q_ 1, q_ 2\leq \infty\). He also characterizes the space of all Fourier multipliers from \(Lip(\alpha_ 1,p_ 1;q_ 1;G)\) to \(Lip(\alpha_ 2,p_ 2;q_ 2;G)\), where \(-\infty <\alpha_ 1, \alpha_ 2<\infty\), \(1\leq p_ 1, p_ 2<\infty\), and \(0<q_ 1\leq q_ 2\leq \infty.\) Similar results have been obtained by \textit{H. Q. Bui} [Hiroshima Math. J. 11, 81-96 (1981; Zbl 0473.46020)] for generalized Lipschitz spaces defined on the n-dimensional Euclidean space.