Equivalent Conditions of a Hilbert-Type Multiple Integral Inequality Holding. (15th April 2020) - Liao, Jianquan; Hong, Yong; Yang, Bicheng
Abstract : Let ∑ i = 1 n 1 / p i = 1 p i > 1, in this paper, by using the method of weight functions and technique of real analysis; it is proved that the equivalent parameter condition for the validity of multiple integral Hilbert-type inequality ∫ R + n K x 1, ⋯, x n ∏ i = 1 n f i x i   d x 1 ⋯ d x n ≤ M ∏ i = 1 n f i p i, α i with homogeneous kernel K x 1, ⋯, x n of order λ is ∑ i = 1 n α i / p i = λ + n − 1, and the calculation formula of its optimal constant factor is obtained. The basic theory and method of constructing a Hilbert-type multiple integral inequality with the homogeneous kernel and optimal constant factor are solved.