MathSciDoc: An Archive for Mathematician ∫

Let$G$denote a totally disconnected locally compact metric abelian group with translation invariant metric$d$and character group$Γ$_{$G$}. The Lipschitz spaces are defined by $$Lip\left( {\alpha ;p} \right) = \left\{ {f \in L^p \left( G \right):\left\| {\tau _a f - f} \right\|_p = O\left( {d\left( {a,0} \right)^\alpha } \right),a \to 0} \right\},$$ where$τ$_{$a$}$f$:$x$→$f$($x-a$) and α∈(0,1). For a suitable choice of metric it is shown that Lip (α;$p$)⊂$L$^{$r$}($Γ$_{$G$}), where α>1/$p$+1/$r$−1≧0 and 1≦$p$≦2. In the case$G$is compact the corresponding result holds for α>1/$r$−1/2 and$p$>2. In addition for$G$non-discrete the above result is shown to be sharp, in the sense that the range of values of α cannot be extended. The results include classical theorems of S. N. Bernstein, O. Szász and E. C. Titchmarsh.

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