Weighted theory of Toeplitz operators on the Bergman space

We study the weighted compactness and boundedness properties of Toeplitz operators on the Bergman space with respect to Bekolle-Bonami type weights. Let T-u denote the Toeplitz operator on the (unweighted) Bergman space of the unit ball in C-n with symbol u is an element of L-infinity. We characterize the compact Toeplitz operators on the weighted Bergman space A(sigma)(p) for all sigma in a subclass of the Bekolle-Bonami class B-p that includes radial weights and powers of the Jacobian of biholomorphic mappings. Concerning boundedness, we show that T-u extends boundedly on L-sigma(p) for p is an element of(1, infinity) and weights sigma in a u-adapted class of weights containing B-p, and we establish analogous weighted endpoint weak-type (1, 1) bounds for weights beyond B-1.