## 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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