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Measures and convolution products with small transforms. (English) Zbl 0207.43702
Let \(G\) be an LCA group and \(S\) a closed subset of the dual \(\Gamma\). Let \(\mu_i\) be a finite complex-valued regular measure on \(G\) such that \(\hat \mu_i = \hat f_i\) a. e. on \(\Gamma \sim S\) for some \(f\in L^{p_i}(G)\), \(1\le p_i\le 2\) and \(i = 1,2\). Using a result of the author which stems from multiplier theory [Proc. Am. Math. Soc. 29, 511–515 (1971; Zbl 0216.14804); erratum 48, 515 (1975; Zbl 0298.43006)], we show the following:
(i) \(\mu_1 * \mu_2\) is absolutely continuous if \(S\cap (\gamma - S)\) has finite Haar measure for a dense subset of \(\gamma\in\Gamma\);
(ii) The support of the Fourier-Stieltjes transform of the singular part of \(\mu_i\) is contained in \(S\) if \(S\) has the property that whenever the support of the transform of a measure is contained in \(S\) so is the support of the transform of its singular part.
(i) and (ii) are generalizations of theorems due respectively to I. Glicksberg [Ill. J. Math. 9, 418–427 (1965; Zbl 0188.20401)] and R. Doss [Pac. J. Math. 26, 257–263 (1968; Zbl 0165.48803)].
MSC:
43A05 Measures on groups and semigroups, etc.
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
43A70 Analysis on specific locally compact and other abelian groups
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References:
[1] Doss, R.: On measures with small transforms. Pacific J. Math.26, 257-263 (1968). · Zbl 0165.48803
[2] ?: On the Fourier-Stieltjes transforms of singular or absolutely continuous measures. Math. Z.97, 77-84 (1967). · Zbl 0162.45202 · doi:10.1007/BF01111125
[3] Glicksberg, I.: Fourier-Stieltjes transforms with small supports. Illinois J. Math.9, 418-427 (1965). · Zbl 0188.20401
[4] Pigno, L.: Restrictions ofL p transforms. Proc. Amer. Math. Soc.29, 511-515 (1971). · Zbl 0216.14804
[5] Weiss, G.: Complex methods in harmonic analysis. Amer. Math. Monthly77, 465-474 (1970). · Zbl 0198.18401 · doi:10.2307/2317379
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