Special Toeplitz operators on strongly pseudoconvex domains. (English) Zbl 1124.32004

Let \(\Omega\) be a smoothly bounded domain in \(\mathbb C^N\). Denote by \(\delta(z)\) the distance from \(z\in\Omega\) to the boundary of \(\Omega\). By a distance-symbol Toeplitz operator the authors mean a Toeplitz operator, built from the Bergman kernel \(B\), with symbol equal to a positive power of \(\delta\). Thus a distance-symbol Toeplitz operator has the form \(T_{\delta^\eta}(g)(z)= \int_\Omega B(z, w) \delta(w)^\eta g(w)\, dV(w)\) for some power \(\eta>0\), where \(B(z, w)\) denotes the Bergman kernel associated to \(\Omega\) and \(dV\) is the volume measure on \(\Omega\). These operators are naturally expected to have better “smoothing” behavior, as \(\eta\) increases.
In this paper the authors study how much a distance-symbol Toeplitz operator, depending on the power \(\eta\), improves \(L^p\)-integrability. Their results are obtained on strongly pseudoconvex domains and depend on the dimension. In case where \(N=1\) and \(\eta\) is relatively small, the authors show that a special case of their results is sharp; the sharpness remains open in general. They wonder whether there are similar \(L^p\)-improving estimates which are independent of the dimension. They mention that their proofs can be carried out, with minimal changes, on other classes of domains where good estimates on the Bergman kernel are known, e.g. finite type domains in \(\mathbb C^2\) and convex domains of finite type in \(\mathbb C^N\). They also mention that further results can be obtained, with help of somewhat more complicated functional analysis, on Banach spaces other than Lebesgue spaces, e.g. Hölder spaces and Sobolev spaces .


32A36 Bergman spaces of functions in several complex variables
32T15 Strongly pseudoconvex domains
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators


Bergman kernel
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