Absolute convergence of Fourier series on totally disconnected groups

Abstract

LetG 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_af: 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.