Lᵖ Hardy-Type Inequalities with Multidimensional Remainder Terms

We obtain a family of improved Hardy, Leray, and Poincaré inequalities for 2 ≤ p < n 2 ≤ p < n , incorporating explicit multidimensional remainder terms. These inequalities feature additive contributions of the form ∑ ᵢ₌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-1/p )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.