On boundedness of the generalized B-potential integral operators in the Lorentz spaces


Guliyev V. S. , Serbetci A., Ekincioglu İ.

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS, vol.18, no.12, pp.885-895, 2007 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 18 Issue: 12
  • Publication Date: 2007
  • Doi Number: 10.1080/10652460701510980
  • Journal Name: INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS
  • Journal Indexes: Science Citation Index Expanded, Scopus
  • Page Numbers: pp.885-895
  • Keywords: Laplace-Bessel differential operator, O'Neil type inequality, generalized B-potential integrals, B-fractional integral, Lorentz spaces

Abstract

In this paper, we study the convolution operator (B-convolution), and the generalized B-potential integral and fractional integral (B-fractional integral) with rough kernel, associated with the Laplace-Bessel differential operator Delta(B) = Sigma(k)(i=1) B-i + Sigma(n)(j=k+1)(partial derivative(2)/partial derivative x(j)(2)), B-i = (partial derivative(2)/partial derivative x(i)(2)) + (gamma(i)/x(i))(partial derivative/partial derivative x(i)). We get O'Neil type inequality for the B-convolution. By using O'Neil type inequality we obtain a pointwise rearrangement estimate of the generalized B-potential integral. We prove the boundedness of the generalized B-potential integral operator in the Lorentz spaces, and the proof is based on the pointwise estimate of the rearrangement of this operator.