The analysis of holomorphic sections of high powers L of holomorphic ample line bundles L → M over compact Kähler manifolds has been widely applied in complex geometry and mathematical physics. Any polarized Kähler metric g with respect to the ample line bundle L corresponds to the Ricci curvature of a hermitian metric h on L. Any orthonormal basis {SN 0 , ..., S dN} of H(M,L ) induces a holomo...

We investigate Bergman type operators on the complex unit ball, which are singular integral induced by modified kernel. consider $L^p$-$L^q$ boundedness and compactness of operators. The results can be viewed as Hardy–Littlewood–Sobolev (HLS) theorem in case ball. also give some sharp norm estimates fact gives upper bounds optimal constants HLS inequality Moreover, a trace formula is given.