Hilbert-Type Inequalities in Real Analysis
Summary
Hilbert-type inequalities occupy a central position in real analysis by providing precise bounds for bilinear forms involving integrals or series. Originating from the classical Hilbert integral inequality, these results assert that the double integral or double sum of two non-negative functions, weighted by a kernel, is bounded by a constant multiple of the product of their Lp-norms. Over the past century this framework has been extended to cover discrete, continuous, and mixed (half-discrete) settings, as well as kernels that are homogeneous, non-homogeneous or variable. The determination of the best possible constant factor often requires detailed estimates involving special functions such as the beta or gamma functions. Contemporary work has expanded the theory to include variable upper limits, multi-parameter weight functions and reverse inequalities, with applications ranging from operator theory to estimates in partial differential equations and harmonic analysis.
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Hilbert-Type Inequalities in Real Analysis publication trend
The graph below shows the total number of articles in hilbert-type inequalities in real analysis across all publications each year (not limited to Nature Index journals).
Technical terms
Hilbert-type inequality: An inequality bounding a double integral or sum of products of functions by a constant times the product of their norms.
Kernel: The function K(x,y) appearing in the integrand or summand that controls the interaction between variables.
Homogeneous kernel: A kernel satisfying K(tx, ty) = t−λK(x,y) for some λ, ensuring scale-invariance properties.
Weight function: A non-negative function introduced into integrals or sums to generalise classical inequalities and adjust the distribution of mass.
Best possible constant factor: The smallest value C such that the inequality holds for all admissible functions; typically determined via extremal functions or special-function estimates.
References
- A Weighted Generalization of Hardy–Hilbert-Type Inequality Involving Two Partial Sums. Mathematics (2023).
- Condition for the Construction of a Hilbert-Type Integral Inequality Involving Upper Limit Functions. Symmetry (2024).
- An Equivalent Form Related to a Hilbert-Type Integral Inequality. Axioms (2023).
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