JournalsrmiVol. 23, No. 3pp. 743–770

The fractional maximal operator and fractional integrals on variable LpL^p spaces

  • Claudia Capone

    Consiglio Nazionale delle Ricerche, Napoli, Italy
  • David Cruz-Uribe

    Trinity College, Hartford, USA
  • Alberto Fiorenza

    Università degli Studi di Napoli Federico II, Italy
The fractional maximal operator and fractional integrals on variable $L^p$ spaces cover
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Abstract

We prove that if the exponent function p()p(\cdot) satisfies log-Hölder continuity conditions locally and at infinity, then the fractional maximal operator MαM_\alpha, 0<α<n0 < \alpha < n, maps Lp()L^{p(\cdot)} to Lq()L^{q(\cdot)}, where 1p(x)1q(x)=αn\frac{1}{p(x)}-\frac{1}{q(x)}=\frac{\alpha}{n}. We also prove a weak-type inequality corresponding to the weak (1,n/(nα))(1,n/(n-\alpha)) inequality for MαM_\alpha. We build upon earlier work on the Hardy-Littlewood maximal operator by Cruz-Uribe, Fiorenza and Neugebauer [The maximal function on variable LpL^p spaces. Ann. Acad. Sci. Fenn. Math. 28 (2003), 223-238]. As a consequence of these results for MαM_\alpha, we show that the fractional integral operator IαI_\alpha satisfies the same norm inequalities. These in turn yield a generalization of the Sobolev embedding theorem to variable LpL^p spaces.

Cite this article

Claudia Capone, David Cruz-Uribe, Alberto Fiorenza, The fractional maximal operator and fractional integrals on variable LpL^p spaces. Rev. Mat. Iberoam. 23 (2007), no. 3, pp. 743–770

DOI 10.4171/RMI/511