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Abstract. Let B be a locally integrable matrix function, W a matrix A p weight with 1 < p < ∞, and T be any of the Riesz transforms. We will characterize the boundedness of the commu-in terms of the membership of B in a natural matrix weighted BMO space. To do this, we will characterize the boundedness of dyadic paraproducts on L p (W ) via a new matrix weighted Carleson embedding theorem. Finally, we will use some of the ideas from these proofs to (among other things) obtain quantitative weighted norm inequalities for these operators and also use them to prove sharp L 2 bounds for the Christ/Goldberg matrix weighted maximal function associated with matrix A 2 weights.

Abstract. We show that Zorboska's criterion for compactness of Toeplitz operators with BMO 1 symbols on the Bergman space of the unit disc holds, by a different proof, for the Segal-Bargmann space of Gaussian square-integrable entire functions on C n . We establish some basic properties of BMO p for p ≥ 1 and complete the characterization of bounded and compact Toeplitz operators with BMO 1 symbols. Via the Bargmann isometry and results of Lo and Englis, we also give a compactness criterion for the Gabor-Daubechies "windowed Fourier localization operators" on L 2 (R n , dv) when the symbol is in a BMO 1 Sobolev-type space. Finally, we discuss examples of the compactness criterion and counterexamples to the unrestricted application of this criterion for the compactness of Toeplitz operators.

In this paper we study the compactness of operators on the Bergman space of the unit ball and on very generally weighted Bargmann-Fock spaces in terms of the behavior of their Berezin transforms and the norms of the operators acting on reproducing kernels. In particular, in the Bergman space setting we show how a vanishing Berezin transform combined with certain (integral) growth conditions on an operator T are sufficient to imply that the operator is compact. In the weighted Bargmann-Fock space setting we show that the reproducing kernel thesis for compactness holds for operators satisfying similar growth conditions. The main results extend the results of Xia and Zheng to the case of the Bergman space when 1 < p < ∞, and in the weighted Bargmann-Fock space setting, our results provide new, more general conditions that imply the work of Xia and Zheng via a more familiar approach that can also handle the 1 < p < ∞ case.2000 Mathematics Subject Classification. 32A36, 32A, 47B05, 47B35.