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We obtain new explicit formulas for the Bergman kernel function on two families of Hartogs domains. To do so, we first compute the Bergman kernels on the slices of these Hartogs domains with some coordinates fixed, evaluate these kernel functions at certain points off the diagonal, and then apply a first order differential operator to them. We find, for example, explicit formulas for the kernel function onand on {(z 1 , z 2 , w) ∈ C 3 : |z 1 | 2 + |z 2 | 2 + |w| 2 < 1 + |z 2 w| 2 and |w| < 1}.We use our formulas to determine the boundary behavior of the kernel function of these domains on the diagonal. AMS Classification Number: 32A05, 32A07, 32A25, 32A36, 32A40.

Abstract. We obtain L p regularity for the Bergman projection on some Reinhardt domains. We start with a bounded initial domain Ω with some symmetry properties and generate successor domains in higher dimensions. We prove: If the Bergman kernel on Ω satisfies appropriate estimates, then the Bergman projection on the successor is L p bounded. For example, the Bergman projection on successors of strictly pseudoconvex initial domains is bounded on L p for 1 < p < ∞. The successor domains need not have smooth boundary nor be strictly pseudoconvex.

We establish a weighted inequality for the Bergman projection with matrix weights for a class of pseudoconvex domains. We extend a result of Aleman-Constantin and obtain the following estimate for the weighted norm of P : P L 2 (Ω,W) ≤ C(B 2 (W)) 2. Here B 2 (W) is the Bekollé-Bonami constant for the matrix weight W and C is a constant that is independent of the weight W but depends upon the dimension and the domain.

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