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

Valentina Casarino, and Silvia Secco. "$L^{p}-L^{q}$ boundedness of analytic families of fractional integrals." Studia Mathematica 184.2 (2008): 153-174. <http://eudml.org/doc/285188>.

@article{ValentinaCasarino2008,
abstract = {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, $f ∈ ^\{∞\}_\{c\}(ℝ²)$, 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 < ∞.},
author = {Valentina Casarino, Silvia Secco},
journal = {Studia Mathematica},
keywords = {fractional integration along curves; product kernels; strong endpoint bounds},
language = {eng},
number = {2},
pages = {153-174},
title = {$L^\{p\}-L^\{q\}$ boundedness of analytic families of fractional integrals},
url = {http://eudml.org/doc/285188},
volume = {184},
year = {2008},
}

TY - JOUR
AU - Valentina Casarino
AU - Silvia Secco
TI - $L^{p}-L^{q}$ boundedness of analytic families of fractional integrals
JO - Studia Mathematica
PY - 2008
VL - 184
IS - 2
SP - 153
EP - 174
AB - 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, $f ∈ ^{∞}_{c}(ℝ²)$, 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 < ∞.
LA - eng
KW - fractional integration along curves; product kernels; strong endpoint bounds
UR - http://eudml.org/doc/285188
ER -