We investigate Bergman type operators on the complex unit ball, which are singular integral operators induced by the modified Bergman kernel. We consider the $L^p$-$L^q$ boundedness and compactness of Bergman type operators. The results of boundedness can be viewed as the Hardy–Littlewood–Sobolev (HLS) type theorem in the case unit ball. We also give some sharp norm estimates of Bergman type operators which in fact gives the upper bounds of the optimal constants in the HLS type inequality on the unit ball. Moreover, a trace formula is given.
The first author was partially supported by PCPSF (2020M680172, 2020T130016). The second author was partially supported by NSFC (12026250, 11722102) and STIP (21JC1400800).
The first author would like to thank Professor Genkai Zhang for his helpful discussions and warm hospitality when the author visited Chalmers University of Technology. The first author gratefully acknowledges the many suggestions of Professor Zipeng Wang. The authors would like to thank Professor H. Kaptanoğlu devoutly for sending us their recent work.
"The $L^p$-$L^q$ Boundedness and Compactness of Bergman Type Operators." Taiwanese J. Math. 26 (4) 713 - 740, August, 2022. https://doi.org/10.11650/tjm/220101