$$L^p$$ Hardy-Type Inequalities with Multidimensional Remainder Terms

Abstract We obtain a family of improved Hardy, Leray, and Poincaré inequalities for $$2 \le p &lt; n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> , incorporating explicit multidimensional remainder terms. These inequalities feature additive contributions of the form $$\sum _{i=1}^n x_i^{-p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:msubsup> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> with an explicit constant that depends only on p , independent of the dimension n . We establish that the structural factor $$\big (\tfrac{p-1}{p}\big )^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:mstyle> <mml:mfrac> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>p</mml:mi> </mml:mfrac> </mml:mstyle> <mml:msup> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:math> appearing in the remainder term of the Hardy inequality is sharp. The results extend known cases beyond $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , 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.