Toeplitz operators and Carleson measures in strongly pseudoconvex domains

We study mapping properties of Toeplitz operators associated to a finite positive Borel measure on a bounded strongly pseudoconvex domain D⋐Cn. In particular, we give sharp conditions on the measure ensuring that the associated Toeplitz operator maps the Bergman space Ap(D) into Ar(D) with r>p, generalizing and making more precise results by Čučković and McNeal. To do so, we give a geometric characterization of Carleson measures and of vanishing Carleson measures of weighted Bergman spaces in terms of the intrinsic Kobayashi geometry of the domain, generalizing to this setting results obtained by Kaptanoğlu for the unit ball.