A semi-discrete Hilbert-type inequality involving partial sums and multidimensional variable upper limit integral functions and its applications

By using the properties of Gamma function and the construction theorem for the semi-discrete Hilbert-type inequality with quasi-homogeneous kernel, we discuss the construction conditions for a semi-discrete Hilbert-type inequality involving partial sums and multidimensional variable upper limit integral functions. The necessary and sufficient conditions for the validity of this inequality and the estimation formula for the optimal constant factor are obtained, extending existing results from a new perspective, Generalize the results discussed previously for single-variable functions to the case of multivariable functions, and present the relationships between the parameters in a concise and clear manner. Finally, the obtained results are utilized to discuss the boundedness of the relevant operators and the computation of operator norms.