Sharp Multidimensional Hardy-type Inequalities with Singular Kernels: Theory, Approximation Methods and Applications
O. A. Olaiju
Department of Mathematics and Statistics, Federal University of Technology, Ilaro, Nigeria.
O. P. Durojaye
Department of Mathematics and Statistics, Federal University of Technology, Ilaro, Nigeria.
E. O. Fatunmbi *
Department of Mathematics and Statistics, Federal University of Technology, Ilaro, Nigeria.
S. A. Adegbenro
Department of Mechanical Engineering, Federal University of Technology, Ilaro, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
This study examines multidimensional Hardy-type integral inequalities associated with weighted integral operators and singular kernels. Measurable functions on multidimensional domains are considered under convexity, weighted Lebesgue-space, and kernel-integrability assumptions. A class of generalised Hardy operators is formulated, and sufficient boundedness conditions are established using Jensen’s inequality, Hölder’s inequality, Minkowski’s integral inequality, and Fubini’s theorem. Particular attention is given to singular power-law kernels and their relationship with fractional-type integral operators. A finite-grid quadrature framework is also developed to approximate multidimensional Hardy operators. The singularity near the diagonal is treated through kernel regularisation to support stable computation. Numerical tests with exponential, Gaussian, and regularised power-law kernels compare the computed left- and right-hand sides of the proposed inequality and show that the reported test cases satisfy the inequality. Grid-refinement results further show decreasing approximation errors as the resolution increases. The framework is discussed in relation to elliptic partial differential equations, fractional integral operators, Sobolev-type estimates, harmonic analysis, operator theory, and computational methods for partial differential equations. Overall, the paper combines analytical estimates with numerical approximation to provide a structured treatment of Hardy-type inequalities with singular kernels in multidimensional settings, while recognising that sharp constants for the general case and broader high-dimensional validation remain unresolved.
Keywords: Hardy inequalities, singular kernels, weighted Lebesgue spaces, convex functions, numerical quadrature, multidimensional operators.