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

• Autorzy: Valentina Casarino , Silvia Secco
• Języki publikacji: EN

Abstrakty

EN

We consider a double analytic family of fractional integrals $S^{γ,α}_{z}$ along the curve $t ↦ |t|^{α}$, introduced for α = 2 by L. Grafakos in 1993 and defined by
$(S^{γ,α}_{z}f)(x₁,x₂): = 1/Γ(z+1/2) ∫∫ |u-1|^{z}ψ(u-1)f(x₁-t,x₂-u|t|^{α})du|t|^{γ}dt/t$,
where ψ is a bump function on ℝ supported near the origin, $f ∈ 𝓒^{∞}_{c}(ℝ²)$, z,γ ∈ ℂ, Re γ ≥ 0, α ∈ ℝ, α ≥ 2.
We determine the set of all (1/p,1/q,Re z) such that $S^{γ,α}_{z}$ maps $L^{p}(ℝ²)$ to $L^{q}(ℝ²)$ boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel $K^{iϱ,α}_{-1+iθ}$ is a product kernel on ℝ², adapted to the curve $t ↦ |t|^{α}$; as a consequence, we show that the operator $S^{iϱ,α}_{-1+iθ}$, θ, ϱ ∈ ℝ, is bounded on $L^{p}(ℝ²)$ for 1 < p < ∞.

Kategorie tematyczne

• 47B38: Operators on function spaces (general)
• 42B20: Singular and oscillatory integrals (Calder\'on-Zygmund, etc.)
• 44A35: Convolution

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2008
• Tom: 184
• Numer: 2
• Strony: 153-174

Daty

wydano

2008

Twórcy

autor

Valentina Casarino

• Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

autor

Silvia Secco

• Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Identyfikatory

DOI

10.4064/sm184-2-5