The size of Lp-improving measures

Abstract

A measure μ on the locally compact abelian group G is “ L p -improving” if for some 1⩽p<r<∞, μ∗L p(G)⊆L r(G) . L-space properties of such μ are discussed: if μ is L p -improving and non-negative, and f is a bounded Borel function, then fμ is L p -improving. The non-negativity of μ and boundedness of f are essential, as is shown in several ways. In particular, there exists on each infinite compact abelian group an L p -improving μ such that |μ| is not L p -improving. The distribution function of every L p -improving measure on the circle group is shown to satisfy a Lipschitz condition, but that is not sufficient for a measure to be L p -improving, except in restricted cases. The Gelfand transform on Γ‖Γ of an L p -improving measure is shown to be “small” in a quantitative sense; from that the strong continuity of L p -improving measures follows. The spectrum of an L p -improving measure as an operator on L p is shown to be precisely the closure of μ(Γ) . Related results and open problems are included.