Bergman projection induced by kernel with integral representation

Bounded Bergman projections $P_ω:L^p_ω(v)\to L^p_ω(v)$, induced by reproducing kernels admitting the representation $\frac{1}{(1-\overline{z}ζ)^γ}\int_0^1\frac{dν(r)}{1-r\overline{z}ζ},$ and the corresponding (1,1)-inequality are characterized in terms of Bekollé-Bonami-type conditions. The two-weight inequality for the maximal Bergman projection $P^+_ω:L^p_ω(u)\to L^p_ω(v)$ in terms of Sawyer-testing conditions is also discussed.

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