Distorted Hankel integral operators
A. AleksandrovV. Peller
47B3547B1046E35Hankel operatoraveraging projectionBesov spacesSchatten classes
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 $2\beta(\beta+1)^{-1} < p < 2\beta(\beta-1)^{-1}$.
Indiana University Mathematics Journal
2004
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10.1512/iumj.2004.53.2525
10.1512/iumj.2004.53.2525
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Indiana Univ. Math. J. 53 (2004) 925 - 940
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