Toeplitz operators with distributional symbols on Bergman spaces

Abstract

AbstractWe study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces $A^p(\mathbb{D})$, 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space $\smash{W_\nu^{-m,\infty}(\mathbb{D})}$ of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.