We consider a double analytic family of fractional integrals along the curve , introduced for α = 2 by L. Grafakos in 1993 and defined by , where ψ is a bump function on ℝ supported near the origin, , z,γ ∈ ℂ, Re γ ≥ 0, α ∈ ℝ, α ≥ 2. We determine the set of all (1/p,1/q,Re z) such that maps to boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel is a product kernel on ℝ², adapted to the curve ; as a consequence, we show that the operator , θ, ϱ ∈ ℝ, is bounded on for 1 < p < ∞.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm184-2-5, author = {Valentina Casarino and Silvia Secco}, title = {$L^{p}-L^{q}$ boundedness of analytic families of fractional integrals}, journal = {Studia Mathematica}, volume = {187}, year = {2008}, pages = {153-174}, zbl = {1134.42004}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm184-2-5} }
Valentina Casarino; Silvia Secco. $L^{p}-L^{q}$ boundedness of analytic families of fractional integrals. Studia Mathematica, Tome 187 (2008) pp. 153-174. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm184-2-5/