NECESSARY AND SUFFICIENT CONDITIONS AND OPTIMAL CONSTANT FACTORS FOR THE VALIDITY OF MULTIPLE INTEGRAL HALF-DISCRETE HILBERT TYPE INEQUALITIES WITH A CLASS OF QUASI-HOMOGENEOUS KERNELS

The problem of equivalent parameters and the best constant factor for the existence of quasi-homogeneous half-discrete Hilbert type inequality $ \int_{R_ + ^m} {\sum\limits_{n = 1}^\infty G } \left( {{n^{{\lambda _1}}}/\left\| x \right\|_{m, \rho }^{{\lambda _2}}} \right){a_n}f(x){\rm{d}}x \le M{\left\| {\tilde a} \right\|_{p, \alpha }}{\left\| f \right\|_{q, \beta }} $ is discussed, and their applications in the study of operator boundedness and norm are also considered.