Abstract
Abstract In this paper, we establish equivalent parameter conditions for the validity of multiple integral half-discrete Hilbert-type inequalities with the nonhomogeneous kernel G ( n λ 1 ∥ x ∥ m , ρ λ 2 ) G\left({n}^{{\lambda }_{1}}\parallel x{\parallel }_{m,\rho }^{{\lambda }_{2}}\hspace{-0.16em}) ( λ 1 λ 2 > 0 {\lambda }_{1}{\lambda }_{2}\gt 0 ) and obtain best constant factors of the inequalities in specific cases. In addition, we also discuss their applications in operator theory.
Highlights
Introduction and preliminary knowledgeSuppose that 1 p + 1 q = 1 (p > 1), a∼ {am} ∈ lp, b∼ {bn}
(ii) For c = 0, assuming the best constant factor of (5) is M0, we can see from the previous proof that
(i) T1 is a bounded operator from lpα to Lpβ(1−p) ( +m) and T2 is a bounded operator from Lqβ ( +m) to lqα(1−q) if and only if l ≥ 0; (ii)
Summary
The classical Hilbert series inequality was obtained in 1925 [1]:. where the constant factor π is the best. The classical Hilbert series inequality was obtained in 1925 [1]:. If f (x) ∈ Lp(0, +∞), g(y) ∈ Lq(0, +∞), the corresponding Hilbert integral inequality was obtained in 1934 [2]:. Later on the equivalent conditions for validity of multiple integral halfdiscrete Hilbert-type inequality with generalized homogeneous kernel were discussed [4]. In [5], the parameter conditions for the optimal constant factor of half-discrete Hilbert-type inequality with homogeneous kernel in one dimension were established. To further discuss the multiple integral half-discrete Hilbert-type inequality, we need to introduce the following notations: Suppose that m ∈. We will discuss the equivalent parameter conditions under which the multiple integral half-discrete Hilbert-type inequality. What conditions do the parameters α, β, λ1, λ2, p, q meet if there is a constant M > 0 such that (4) holds?
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