Lᵖ Hardy-Type Inequalities with Multidimensional Remainder Terms

Abstract We obtain a family of improved Hardy, Leray, and Poincaré inequalities for 2 p 2 ≤ p n, incorporating explicit multidimensional remainder terms. These inequalities feature additive contributions of the form ₈=₁ⁿ xᵢ^-p ∑ i = 1 n x i - p with an explicit constant that depends only on p, independent of the dimension n. We establish that the structural factor (p-1p) ᵖ (p - 1 p) p appearing in the remainder term of the Hardy inequality is sharp. The results extend known cases beyond L² L 2, unify different inequalities under a common framework, and yield stronger remainder estimates. They also provide a foundation for further generalizations to weighted and singular settings.