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

#### 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.
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2008
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11577/141594`
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