Proof of Hardy inequality in $\mathbb{R}^n$

This is in Evans' PDE book, page 296 in the second edition. The following discussions on this proof may also be useful,

  1. Hardy's inequality
  2. A technical step in proving Hardy's inequality

However, it doesn't seem like the constant 4 is explicitly computed in Evans, and moreover it may be improved if the domain of $u$ is not convex. In the dimension 1 variant, it is optimal, and you can refer to Computing the best constant in classical Hardy's inequality. In higher dimensions, you may want to look at this paper On the best constant for Hardy's inequality in $\mathbb R^n$ by Marcus, Mizel and Pinchover, and its references.

Tags:

Inequality

Sobolev Spaces

Functional Analysis

Lp Spaces

Lebesgue Integral

Related

Find the probability that the white ball labelled $1$ is drawn before all the black balls. Finding a certain entry in a matrix Show that there is a continuous real function $h$ on $[0,1]$ such that $\lim \sup |\frac{h(x+t)-h(x)}{t}| = \infty$ for all $x \in [0,1)$ Property of set exclusion set. What is the name of this result about isosceles triangles? Deciding which statements about matrix A is true where $A^3+A^2-3A+I=0$ Solution to one-dimensional Wave Equation with Method of Characteristics Let $\lim\limits_{n\to\infty}a_{n+1}-a_{n}=\alpha$. Show that $\lim\limits_{n\to\infty} \frac{a_n}{n}=\alpha$. Closed form of integral over fractional part $\int_0^1 \left\{\frac{1}{2}\left(x+\frac{1}{x}\right)\right\}\,dx$ How to integrate $ \int \frac{\sin {3x} + \cos{3x}}{ \sin^3 x + \cos^3 x } dx\ $ Constructive proof of the Cauchy Schwarz inequality Improved intermediate value theorem

Recent Posts