$L^{p}-L^{q}$ boundedness of analytic families of fractional integrals

Valentina Casarino; Silvia Secco

L^{p} - L^{q}

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

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Résumé

We consider a double analytic family of fractional integrals $S_{z}^{γ, α}$ along the curve ${t \mapsto | t |}^{α}$ , introduced for α = 2 by L. Grafakos in 1993 and defined by $(S_{z}^{γ, α} f) (x ₁, x ₂) : = 1 / Γ (z + 1 / 2) {\int \int | u - 1 |}^{z} ψ (u - 1) {f (x ₁ - t, x ₂ - u | t |}^{α} {) d u | t |}^{γ} d t / t$ , where ψ is a bump function on ℝ supported near the origin, $f \in_{c}^{\infty} (ℝ ²)$ , z,γ ∈ ℂ, Re γ ≥ 0, α ∈ ℝ, α ≥ 2. We determine the set of all (1/p,1/q,Re z) such that $S_{z}^{γ, α}$ maps $L^{p} (ℝ ²)$ to $L^{q} (ℝ ²)$ boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel $K_{- 1 + i θ}^{i ϱ, α}$ is a product kernel on ℝ², adapted to the curve ${t \mapsto | t |}^{α}$ ; as a consequence, we show that the operator $S_{- 1 + i θ}^{i ϱ, α}$ , θ, ϱ ∈ ℝ, is bounded on $L^{p} (ℝ ²)$ for 1 < p < ∞.

Publié le : 2008-01-01

Zbl 1134.42004

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},
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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/