A collection of sharp dilation invariant inequalities for differentiable functions

TL;DR

This paper derives optimal constants for various dilation-invariant integral inequalities involving derivatives, including new and known results, enhancing understanding of inequalities in analysis.

Contribution

It provides the best constants for several dilation-invariant inequalities, some of which are new contributions to the field.

Findings

Optimal constants for quadratic form of the gradient

New weighted G{ a}rding inequality for biharmonic operator

Sharp Hardy's inequality with remainder term

Abstract

We find best constants in several dilation invariant integral inequalities involving derivatives of functions. Some of these inequalities are new and some were known without best constants. The contents: 1. Estimate for a quadratic form of the gradient, 2. Weighted G{\aa}rding inequality for the biharmonic operator, 3.Dilation invariant Hardy's inequality with remainder term, 4. Generalized Hardy-Sobolev inequality with sharp constant, 5. Hardy's inequality with sharp Sobolev remainder term.