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On the sweeping out property for convolution operators of discrete measures. (English) Zbl 1225.42013

The paper deals with the validity of the sweeping out property for the sequence of convolution operators \(S_{\mu_n}\) defined on \(L^1(\mathbb T)\) by \((S_{\mu_n} f)(x)=(f\star \mu_n)(x)\), where \(\mu_n\) is a sequence of bounded discrete measures on the torus \(\mathbb T\) satisfying
\[ \mu_n(0)=0, \quad \mu_n((-\delta,\delta))\to 1\quad \text{as }n\to \infty, \]
for any \(0<\delta\leq 1/2\). The sweeping out condition consists in showing the following strong divergence property for the previous operators: the existence of a set \(E\subset\mathbb T\) such that
\[ \limsup_{n\to \infty} S_{\mu_n}\mathbb I_E(x)=1, \qquad \liminf_{n\to \infty} S_{\mu_n}\mathbb I_E(x)=0 \]
almost everywhere on \(\mathbb T\).
The proof is based on some technical lemmas and it is obtained by applying a general result proved by the same author in [Sb. Math. 200, No. 4, 521-548 (2009); translation from Mat. Sb. 200, No. 4, 53–82 (2009; Zbl 1167.42006)].

MSC:

42B25 Maximal functions, Littlewood-Paley theory
11K36 Well-distributed sequences and other variations
37Axx Ergodic theory

Citations:

Zbl 1167.42006
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References:

[1] Mustafa Akcoglu, Alexandra Bellow, Roger L. Jones, Viktor Losert, Karin Reinhold-Larsson, and Máté Wierdl, The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters, Ergodic Theory Dynam. Systems 16 (1996), no. 2, 207 – 253. · Zbl 0851.47004
[2] Mustafa A. Akcoglu, Dzung M. Ha, and Roger L. Jones, Sweeping out properties of operator sequences, Canad. J. Math. 49 (1997), no. 1, 3 – 23. · Zbl 0870.28007
[3] A. Bellow, Two problems, Proc. Oberwolfach Conference on Measure Theory (June 1987), Springer Lecture Notes in Math., 945, 1987.
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[5] G. A. Karagulyan, Riemann sums and maximal functions in \Bbb R\(^{n}\), Mat. Sb. 200 (2009), no. 4, 53 – 82 (Russian, with Russian summary); English transl., Sb. Math. 200 (2009), no. 3-4, 521 – 548. · Zbl 1167.42006
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[8] Walter Rudin, An arithmetic property of Riemann sums, Proc. Amer. Math. Soc. 15 (1964), 321 – 324. · Zbl 0132.03601
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