Muckenhoupt type weights and Berezin formulas for Bergman spaces

Abstract

By means of Muckenhoupt type conditions, we characterize the weights ω on C such that the Bergman projection of Fα2,ℓ=H(C)∩L2(C,e−α2|z|2ℓ), α>0, ℓ>1, is bounded on Lp(C,e−αp2|z|2ℓω(z)), for 1<p<∞. We also obtain explicit representation integral formulas for functions in the weighted Bergman spaces Ap(ω)=H(C)∩Lp(ω). Finally, we check the validity of the so called Sarason conjecture about the boundedness of products of certain Toeplitz operators on the spaces Fαp,ℓ=H(C)∩Lp(C,e−αp2|z|2ℓ).