JournalszaaVol. 22 , No. 4DOI 10.4171/zaa/1178

On Sobolev Theorem for Riesz-Type Potentials in Lebesgue Spaces with Variable Exponent

  • Vakhtang Kokilashvili

    Georgian Acadademy of Sciences, Tbilisi, Georgia
  • Stefan Samko

    University of Algarve, Faro, Portugal
On Sobolev Theorem for Riesz-Type Potentials in Lebesgue Spaces with Variable Exponent cover

Abstract

The Riesz potential operator of variable order α(x)\alpha(x) is shown to be bounded from the Lebesgue space Lp()(Rn)L^{p(\cdot)}({\Bbb R}^n) with variable exponent p(x)p(x) into the weighted space Lρq()(Rn)L^{q(\cdot)}_\rho({\Bbb R}^n), where ρ(x)=(1+x)γ\rho(x) = (1 + 'x')^{-\gamma} with some γ>0\gamma > 0 and 1q(x)=1p(x)α(x)n{1 \over q(x)} = {1 \over p(x)} - {\alpha(x) \over n} when pp is not necessarily constant at infinity. It is assumed that the exponent p(x)p(x) satisfies the logarithmic continuity condition both locally and at infinity and 1<p()p(x)P<1 < p(\infty) \le p(x) \le P < \infty (xRn)(x \in {\Bbb R}^n).