We consider a double analytic family of fractional integrals S(z)(gamma,alpha) along the curve t -> | t|^(alpha), introduced in the particular case alpha = 2 by L. Grafakos in 1993, depending on two complex parameters z, alpha and defined in a suitable way . We determine the characteristic set of this family, that is, set of all (1/p, 1/q, Re z) such that S(z)(gamma,alpha) maps L(p)(R(2)) to L(q)(R(2)) boundedly, and we prove that is does not depend on alpha. Our proof is based on product-type kernel arguments. More precisely, we prove that at the height Re z=-1 the convolution kernel is a product kernel on R^2, adapted to the curve t -> |t|^(alpha); as a consequence, we show that the corresponding operator is bounded on L(p)(R(2)) for 1 < p < infinity.

L^p-L^q boundedness of analytic families of fractional integrals

CASARINO, VALENTINA;
2008

Abstract

We consider a double analytic family of fractional integrals S(z)(gamma,alpha) along the curve t -> | t|^(alpha), introduced in the particular case alpha = 2 by L. Grafakos in 1993, depending on two complex parameters z, alpha and defined in a suitable way . We determine the characteristic set of this family, that is, set of all (1/p, 1/q, Re z) such that S(z)(gamma,alpha) maps L(p)(R(2)) to L(q)(R(2)) boundedly, and we prove that is does not depend on alpha. Our proof is based on product-type kernel arguments. More precisely, we prove that at the height Re z=-1 the convolution kernel is a product kernel on R^2, adapted to the curve t -> |t|^(alpha); as a consequence, we show that the corresponding operator is bounded on L(p)(R(2)) for 1 < p < infinity.
2008
File in questo prodotto:
Non ci sono file associati a questo prodotto.
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/141594
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 4
  • ???jsp.display-item.citation.isi??? 4
social impact