Lp-Lq boundedness of analytic families of fractional integrals
Valentina Casarino ; Silvia Secco
Studia Mathematica, Tome 187 (2008), p. 153-174 / Harvested from The Polish Digital Mathematics Library

We consider a double analytic family of fractional integrals Szγ,α along the curve t|t|α, introduced for α = 2 by L. Grafakos in 1993 and defined by (Szγ,αf)(x,x):=1/Γ(z+1/2)|u-1|zψ(u-1)f(x-t,x-u|t|α)du|t|γdt/t, where ψ is a bump function on ℝ supported near the origin, fc(²), z,γ ∈ ℂ, Re γ ≥ 0, α ∈ ℝ, α ≥ 2. We determine the set of all (1/p,1/q,Re z) such that Szγ,α maps Lp(²) to Lq(²) boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel K-1+iθiϱ,α is a product kernel on ℝ², adapted to the curve t|t|α; as a consequence, we show that the operator S-1+iθiϱ,α, θ, ϱ ∈ ℝ, is bounded on Lp(²) for 1 < p < ∞.

Publié le : 2008-01-01
EUDML-ID : urn:eudml:doc:285188
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     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}
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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/