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

For the Hilbert type multiple integral inequality $$ \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m,\rho }, \Vert y \Vert _{n, \rho }\bigr) f(x)g(y) \,\mathrm{d} x \,\mathrm{d} y \leq M \Vert f \Vert _{p, \alpha } \Vert g \Vert _{q, \beta } $$ ∫ R + n ∫ R + m K ( ∥ x ∥ m , ρ , ∥ y ∥ n , ρ ) f ( x ) g ( y ) d x d y ≤ M ∥ f ∥ p , α ∥ g ∥ q , β with a nonhomogeneous kernel $K(\|x\|_{m, \rho }, \|y\|_{n, \rho })=G(\|x\|^{\lambda _{1}}_{m, \rho }/ \|y\|^{\lambda _{2}}_{n, \rho })$ K ( ∥ x ∥ m , ρ , ∥ y ∥ n , ρ ) = G ( ∥ x ∥ m , ρ λ 1 / ∥ y ∥ n , ρ λ 2 ) ($\lambda _{1}\lambda _{2}> 0$ λ 1 λ 2 > 0 ), in this paper, by using the weight function method, necessary and sufficient conditions that parameters p, q, $\lambda _{1}$ λ 1 , $\lambda _{2}$ λ 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.

THE STRUCTURAL FEATURES OF HILBERT-TYPE LOCAL FRACTIONAL INTEGRAL INEQUALITIES WITH ABSTRACT HOMOGENEOUS KERNEL AND ITS APPLICATIONS

In this paper, by using the theory of local fractional calculus and some techniques of real analysis, the structural characteristics of Hilbert-type local fractional integral inequalities with abstract homogeneous kernel are studied. At the same time, the necessary and sufficient conditions for these inequalities to take the best constant factor are discussed. As an application, some best constant factor inequalities with specific kernels are obtained.

A New Hilbert-Type Inequality with the Homogeneous Kernel of Degree - 2 and with the Integral

By using the weight functions and by means of Hadamard's inequality, we present a new Hilbert-type inequality with the integral in whole plane, a best constant factor and a homogeneous kernel of degree-2.

A New Hilbert-Type Integral Inequality with the Homogeneous Kernel of Real Degree Form and the Integral in Whole Plane

In this paper,we build a new Hilbert's inequality with the homogeneous kernel of real order and the integral in whole plane. The equivalent inequality is considered. The best constant factor is calculated using ψ function.

A New Hilbert-Type Integral Inequality with the Homogeneous Kernel of Real Degree Form and the Integral in Whole Plane

In this paper,we build a new Hilbert's inequality with the homogeneous kernel of real order and the integral in whole plane. The equivalent inequality is considered. The best constant factor is calculated using ψ function.

A New Hilbert-Type Integral Inequality with the Homogeneous Kernel of Real Degree Form and the Integral in Whole Plane

In this paper, we build a new Hilbert's inequality with the homogeneous kernel of real order and the integral in whole plane. The equivalent inequality is considered. The best constant factor is calculated using ψ function.

A More Accurate Half-Discrete Hilbert Inequality with a Nonhomogeneous Kernel

By using the way of weight functions and the idea of introducing parameters, a more accurate half-discrete Hilbert inequality with a nonhomogeneous kernel and a best constant factor is presented. We also consider its best extension with the parameters, the equivalent forms, and the operator expressions as well as some reverses.

A Hilbert-Type Integral Inequality with the Homogeneous Kernel of Degree-3 and a Best Constant Factor

By establishing the weight function, we present a new Hilbert-type inequality with the integral in whole plane and with a best constant factor, and its kernel is a homogeneous form of degree-3, and also we put forward its equivalent form.