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Weighted Berezin transformations with application to Toeplitz operators of Schatten class on parabolic Bergman spaces. (English) Zbl 1198.47046

Let \(L^{(\alpha)}=\partial_t+(-\partial_{x_1}^2-\dots-\partial_{x_n}^2)^\alpha\), \(0<\alpha\leq 1\), and \((x,t)\in{\mathbb R}_+^{n+1}={\mathbb R}^n\times(0,\infty)\). The parabolic Bergman space \(b_\alpha^2\) is the set of \(L^2\)-functions that are \(L^{(\alpha)}\)-harmonic on \({\mathbb R}_+^{n+1}\). The orthogonal projection from \(L^2\) onto \(b_\alpha^2\) is represented as an integral operator with a kernel \(R_\alpha\). For a positive Radon measure \(\mu\) on \({\mathbb R}_+^{n+1}\), let \(T_\mu\) denote the Toeplitz operator defined, as usual, by \((T_\mu u)(X)=\int R_\alpha(X,Y)u(Y)\,d\mu(Y)\) for \(u\in b_\alpha^2\). By \(\widehat{\mu}^{(\alpha)}\) denote the averaging function \(\mu(Q^{(\alpha)}(X))/V(Q^{(\alpha)}(X))\), where \(V\) is the Lebesgue measure on \({\mathbb R}_+^{n+1}\) and \(Q^{(\alpha)}(X)\) is an \(\alpha\)-parabolic Carleson box. Let \(V^*\) denote the invariant measure on \({\mathbb R}_+^{n+1}\) with respect to \(\alpha\)-parabolic similarities. Suppose that \(\psi:[0,\infty)\to[0,\infty)\) is a convex and strictly increasing function such that \(\psi(0)=0\) and \(\psi(s)\to\infty\) as \(s\to\infty\) and consider the corresponding Orlicz space \(L^\psi(V^*)\) and the Schatten-Orlicz space \(S^\psi(b_\alpha^2)\).
The main result of the paper says that, if \(\mu\) is a Radon measure satisfying \(\int(1+t+|x|^{2\alpha})^{-\delta}\,d\mu(x,t)<\infty\) for some \(\delta\in{\mathbb R}\), then \(T_\mu\in S^\psi(b_\alpha^2)\) if and only if \(\widehat{\mu}^{(\alpha)}\in L^\psi(V^*)\).

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
35K05 Heat equation
26D10 Inequalities involving derivatives and differential and integral operators
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
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