We obtain a family of improved Hardy, Leray, and Poincaré inequalities for \(2 \le p < n\) , incorporating explicit multidimensional remainder terms. These inequalities feature additive contributions of the form \(\sum _{i=1}^n x_i^{-p}\) with an explicit constant that depends only on p, independent of the dimension n. We establish that the structural factor \(\big (\tfrac{p-1}{p}\big )^p\) appearing in the remainder term of the Hardy inequality is sharp. The results extend known cases beyond \(L^2\) , 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.
