$L^p$-$L^q$ boundedness of analytic families of fractional integrals
Volume 184 / 2008
Abstract
We consider a double analytic family of fractional integrals $S^{\gamma,\alpha}_{z}$ along the curve $t\mapsto |t|^{\alpha}$, introduced for $\alpha =2$ by L. Grafakos in 1993 and defined by $$ (S^{\gamma,\alpha}_{z}f)(x_1,x_2):= \frac{1}{{\mit\Gamma}({z+1\over2})}\int\int |u-1|^{z}\psi(u-1) f(x_1-t,x_2- u|t|^{\alpha}) \,du\, |t|^{\gamma}\,\frac{dt}{t}, $$ where $\psi$ is a bump function on $\mathbb R$ supported near the origin, $f\in{\cal C}^{\infty}_{\rm c} (\mathbb R^2)$, $z,\gamma\in\mathbb C$, $\mathop{\rm Re}\nolimits \gamma \ge 0$, $\alpha\in\mathbb R$, $\alpha\ge 2$.
We determine the set of all (${{1}/{p}}, {{1}/{q}},\mathop{\rm Re}\nolimits z$) such that $S^{\gamma,\alpha}_{z}$ maps $L^p(\mathbb R^2)$ to $L^q (\mathbb R^2) $ boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel $K^{i\varrho,\alpha}_{-1+i\theta}$ is a product kernel on $\mathbb R^2$, adapted to the curve $t\mapsto |t|^{\alpha}$; as a consequence, we show that the operator $S^{i\varrho,\alpha}_{-1+i\theta}$, $\theta, \varrho \in \mathbb R$, is bounded on $L^p(\mathbb R^2)$ for $1< p< \infty$.
