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The size of \(L^ p\)-improving measures. (English) Zbl 0678.43001

A measure \(\mu\) on a locally compact abelian group G is \(L^ p\)- improving if there is \(r>p\) with \(\mu *L^ p(G)\subseteq L^ r(G)\). (If such an r exists for one p with \(p>1\), then an r exists for any other p with \(1<p<\infty.)\) The present paper continues the study of such measures in three different ways. First, the relationship with absolute continuity is considered. If G is compact and \(\mu\) is non-negative, then for each bounded Borel function f the measure \(f\mu\) is also \(L^ p\)- improving. On the other hand, there exist \(L^ p\)-improving measures \(\mu\) on G for which \(| \mu |\) is not \(L^ p\)-improving. Also, for any \(L^ p\)-improving \(\mu\), there is a probability measure \(\nu\) equivalent to \(\mu\) but for which \(\nu\) is not \(L^ p\)-improving.
The second collection of results relates only to the case in which G is the circle group. The measure \(\mu\) is said to belong to the class Lip(\(\alpha)\) if the distribution function of \(\mu\) satisfies a Lipschitz condition of order \(\alpha\). If, for some \(p<2\), \(\mu *L^ p(G)\subseteq L^ 2(G)\), then \(\mu\) belongs to Lip(1/p-1/2). On the other hand, there are measures in every Lip(\(\alpha)\) class for \(0<\alpha <1\) which are not \(L^ p\)-improving. Moreover, if \(f\in L^ 1(G)\) is such that \(| \hat f(n)|\) decreases monotonically in both directions then f (considered as a measure) is \(L^ p\)-improving if and only if it belongs to some Lipschitz class.
The remaining results are principally concerned with the Fourier- Stieltjes transform \({\hat \mu}\) of an \(L^ p\)-improving measure \(\mu\) on a compact abelian group G. For example, lim sup\(| {\hat \mu}| \leq (2-2/p)^{1/2}\| \mu \|\). If \(sp_ p(\mu)\) denotes the spectrum of \(\mu\) as an operator on \(L^ p(G)\) and \(\Gamma\) the dual group of G, then for each p with \(1<p<\infty\), \(sp_ p(\mu)=cl {\hat \mu}(\Gamma)\) (and this conclusion also holds if \(\mu\) is absolutely continuous with respect to an \(L^ p\)-improving measure). The paper concludes with a list of open questions.
Reviewer: J.S.Pym

MSC:

43A05 Measures on groups and semigroups, etc.
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
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