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The size of \((L^ 2,L^ p)\) multipliers. (English) Zbl 0795.43005

Let \(G\) be an infinite compact abelian group and \(\Gamma\) its dual. Then a complex-valued function \(\varphi\) on \(\Gamma\) is an \((L^ p, L^ q)\)- multiplier \((\varphi\in M(p,q))\) if the operator \(f\to (\varphi \widehat{f})\) maps \(L^ p\) into \(L^ q\). (\(\widehat{f}\) denotes the Fourier transform of \(f\).) The multiplier \(\varphi\) is \(L^ p\)-improving if \(\varphi\in M(2,p)\) for some \(p>2\). This paper continues the work of the author in characterizing \(L^ p\)-improving multipliers, this time in terms of the size of the function \(\varphi\). The main examples considered here are of Riesz-products and one-sided Riesz products. A characterization of those Riesz products that are \(L^ p\)-improving is given and is best possible as a necessary condition, but the conditions are not sufficient. This answers an open problem contained in a paper of C. C. Graham, the author and D. L. Ritter [J. Funct. Anal. 84, No. 2, 472-495 (1989; Zbl 0678.43001)]. Another possible characterization is discussed in terms of \(\Lambda(p)\) sets, and estimates of \(\Lambda(p)\) constants for sums of dissociate sets are sharpened.
Reviewer: G.V.Wood (Swansea)

MSC:

43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
42A45 Multipliers in one variable harmonic analysis
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.)

Citations:

Zbl 0678.43001
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