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Porosity and the \(\ell ^p\)-conjecture for semigroups

Abstract

In this paper, we consider the size of the set \(\big \{(f, g)\in \ell ^p(S)\times \ell ^q(S): \exists \,x\in S,\, |f|*|g|(x)<\infty \big \}\), where \(p\in (1,+\infty )\), \(q\in (0,+\infty ]\), and \(S\) stands for a discrete semigroup. In particular, we prove that if \(S\) is an infinite discrete semigroup, \(p\in (1,+\infty )\), \(q\in (1,+\infty ]\) with \(1/p+1/q<1\), then the set \(\big \{(f, g)\in \ell ^p(S)\times \ell ^q(S): |f|*|g|\in \ell ^\infty (S)\big \}\) is a \(\sigma \)-\(c\)-lower porous set in \(\ell ^p(S)\times \ell ^q(S)\) for some \(c>0\). By means of this notion of porosity we also provide a strengthening of a famous result by Rajagopalan on the \(\ell ^p\)-conjecture.

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Correspondence to J. B. Seoane-Sepúlveda.

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Akbarbaglu, I., Maghsoudi, S. & Seoane-Sepúlveda, J.B. Porosity and the \(\ell ^p\)-conjecture for semigroups. RACSAM 110, 7–16 (2016). https://doi.org/10.1007/s13398-014-0215-1

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Keywords

  • Lebesgue space
  • \(\sigma \)-\(c\)-Lower porous set
  • Semigroup
  • Convolution

Mathematics Subject Classification

  • 43A15
  • 46E30
  • 54E52