Conditions for the validity of a class of optimal Hilbert type multiple integral inequalities with nonhomogeneous kernels

Abstract

For the Hilbert type multiple integral inequality ∫R+n∫R+mK(∥x∥m,ρ,∥y∥n,ρ)f(x)g(y)dxdy≤M∥f∥p,α∥g∥q,β\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\int _{\\mathbb{R}_{+}^{n}} \\int _{\\mathbb{R}_{+}^{m}} K\\bigl( \\Vert x \\Vert _{m,\\rho }, \\Vert y \\Vert _{n, \\rho }\\bigr) f(x)g(y) \\,\\mathrm{d} x \\,\\mathrm{d} y \\leq M \\Vert f \\Vert _{p, \\alpha } \\Vert g \\Vert _{q, \\beta } $$\\end{document} with a nonhomogeneous kernel K(|x|_{m, rho }, |y|_{n, rho })=G(|x|^{lambda _{1}}_{m, rho }/ |y|^{lambda _{2}}_{n, rho }) (lambda _{1}lambda _{2}> 0), in this paper, by using the weight function method, necessary and sufficient conditions that parameters p, q, lambda _{1}, lambda _{2}, α, β, m, and n should satisfy to make the inequality hold for some constant M are established, and the expression formula of the best constant factor is also obtained. Finally, their applications in operator boundedness and operator norm are also considered, and the norms of several integral operators are discussed.