The boundedness of the generalized anisotropic potentials with rough kernels in the Lorentz spaces

GULİYEV V. S., ŞERBETÇİ A., EKİNCİOĞLU İ.

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS, vol.22, no.12, pp.919-935, 2011 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 22 Issue: 12
  • Publication Date: 2011
  • Doi Number: 10.1080/10652469.2010.548334
  • Journal Name: INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.919-935
  • Keywords: Laplace-Bessel differential operator, generalized anisotropic potential integral, rough anisotropic fractional integral, Lorentz spaces, SINGULAR-INTEGRALS, SOBOLEV THEOREM, OPERATORS
  • Ankara University Affiliated: Yes

Abstract

In this paper, we study the generalized anisotropic potential integral K(alpha,gamma) circle times f and anisotropic fractional integral I(Omega,alpha,gamma) f with rough kernels, associated with the Laplace-Bessel differential operator Delta(B). We prove that the operator f -> K(alpha,gamma) circle times f is bounded from the Lorentz spaces L(p,r,gamma) (R(k)(n),(+)) to L(q,s,gamma) (R(k)(n),(+)) for 1 <= p < q <= infinity, 1 <= r <= s <= infinity. As a result of this, we get the necessary and sufficient conditions for the boundedness of I(Omega,alpha,gamma) from the Lorentz spaces L(p,s,gamma) (R(k)(n),(+)) to L(q,r,gamma) (R(k)(n),(+)), 1 < p < q < infinity, 1 <= r <= s <= 8 and from L(1,r,gamma) (R(k)(n),(+)) to L(q,infinity,gamma) (R(k)(n),(+)) = WL(q,gamma) (R(k)(n),(+)), 1 < q < infinity, 1 <= r <= 8. Furthermore, for the limiting case p = Q/alpha, we give an analogue of Adams' theorem on the exponential integrability of I(Omega,alpha,gamma) in L(Q/alpha,gamma) (R(k)(n),(+)).