IUMJ

Title: Distorted Hankel integral operators

Authors: A. B. Aleksandrov and Vladimir V. Peller

Issue: Volume 53 (2004), Issue 4, 925-940

Abstract:

For $\alpha$, $\beta>0$ and for a locally integrable function (or, more generally, a distribution) $\varphi$ on $(0,\infty)$, we study the integral operators $\mathfrak{G}^{\alpha,\beta}_{\varphi}$ on $L^2(\mathbb{R}_{+})$ defined by \[(\mathfrak{G}^{\alpha,\beta}_{\varphi}f)(x)=\int_{\mathbb{R}_{+}}\varphi(x^{\alpha}+y^{\beta})f(y)\,\mathrm{d}y.\] We describe the bounded and compact operators $\mathfrak{G}^{\alpha,\beta}_{\varphi}$ and the operators $\mathfrak{G}^{\alpha,\beta}_{\varphi}$ of Schatten-von Neumann class $\mathbf{S}_p$. The main results of the paper are given in Section 5, where we study continuity properties of the averaging projection $\mathcal{Q}_{\alpha,\beta}$ onto the operators of the form $\mathfrak{G}^{\alpha,\beta}_{\varphi}$. In particular, we show that if $\alpha\le\beta$ and $\beta>1$, then $\mathfrak{G}^{\alpha,\beta}_{\varphi}$ is bounded on $\mathbf{S}_p$ if and only if $2\beta(\beta+1)^{-1} < p < 2\beta(\beta-1)^{-1}$.