on Hardy inequalities

Overview

The Hardy inequality, first established by G. H. Hardy in the 1920s, provides a fundamental bound relating the $L^p$ norm of a function (or sequence) to the $L^p$ norm of its average. Originally motivated by attempts to simplify the proof of Hilbert’s inequality, it has since become a cornerstone of functional analysis and partial differential equations, particularly in the study of Sobolev spaces and boundary behavior. In this post, we explore the classical 1D forms, modern refinements involving logarithmic weights, and generalizations to multi-dimensional domains.

🏷️ Classical 1D Hardy Inequalities

The Hardy inequality exists in two primary forms: discrete and continuous. Both provide a sharp bound on the “averaging” operator.

1D Continuous Hardy Inequality

Let $p>1$ and $f\in L^p(0,\infty )$ be a non-negative function. Define $F(x)={\int }_0^{x}f(t)dt$. Then

$${\int }_0^{\infty }{\left(\frac{F(x)}{x}\right)}^{p}dx\le {\left(\frac{p}{p-1}\right)}^p{\int }_0^{\infty }f(x{)}^{p}dx.$$

The constant $C_p={\left(\frac{p}{p-1}\right)}^p$ is sharp and is not achieved for any non-zero $f\in L^p$.

Proof: Continuous Case (Hardy's Original)

We use integration by parts for the integral on a finite interval $[ϵ,X]$.

$${\int }_{ϵ}^X{\left(\frac{F(x)}{x}\right)}^{p}dx=-\frac{1}{p-1}{\int }_{ϵ}^{X}F(x{)}^p\frac{d}{dx}(x^{1-p})dx$$

Integrating by parts:

$$={\left[-\frac{x^{1-p}}{p-1}F(x{)}^p\right]}_{ϵ}^X+\frac{p}{p-1}{\int }_{ϵ}^X{x}^{1-p}F(x{)}^{p-1}f(x)dx$$

Dropping the negative term at $X$ and estimating the boundary term at $ϵ$:

$$\le \frac{{ϵ}^{1-p}}{p-1}F(ϵ{)}^p+\frac{p}{p-1}{\int }_{ϵ}^X\frac{F(x{)}^{p-1}}{x^{p-1}}f(x)dx$$

By Hölder’s inequality with $p^{\prime }$ where $1/p+1/p^{\prime }=1$:

$${\int }_{ϵ}^X\frac{F(x{)}^{p-1}}{x^{p-1}}f(x)dx\le {\left({\int }_{ϵ}^{X}f(x{)}^{p}dx\right)}^{1/p}{\left({\int }_{ϵ}^X{\left(\frac{F(x)}{x}\right)}^{p}dx\right)}^{1/p^{\prime }}$$

Let $W={\int }_{ϵ}^X(F(x)/x{)}^{p}dx$. As $ϵ\to 0$, $F(ϵ{)}^p=o({ϵ}^{p-1})$, so the boundary term vanishes. We obtain:

$$W\le \frac{p}{p-1}\Vert f{\Vert }_p{W}^{1/p^{\prime }}⟹W^{1/p}\le \frac{p}{p-1}\Vert f{\Vert }_p$$

Raising to the power $p$ and letting $X\to \infty$ yields the result [@hardyNotesSomePoints1925].

Discrete Hardy Inequality

For a non-negative sequence $\{a_n\}$, the discrete form is:

$$\sum _{n=1}^{\infty }{\left(\frac{1}{n}\sum _{k=1}^n{a}_k\right)}^p\le {\left(\frac{p}{p-1}\right)}^p\sum _{n=1}^{\infty }{a}_n^p.$$

This was famously proved by Elliott using an adaptation of Hardy’s integral proof [@kufnerPrehistoryHardyInequality2006].

Proof: Discrete Case via Step Functions

We can deduce the discrete version from the continuous one by defining a specific step function $f(x)$. Let $a_n$ be a non-negative sequence. Define:

$$f(x)=a_n\text{for }x\in [n-1,n).$$

Then ${\int }_0^{n}f(x)dx={\sum }_{k=1}^n{a}_k=A_n$. The continuous Hardy inequality gives:

$${\int }_0^{\infty }{\left(\frac{F(x)}{x}\right)}^{p}dx\le {\left(\frac{p}{p-1}\right)}^p{\int }_0^{\infty }f(x{)}^{p}dx={\left(\frac{p}{p-1}\right)}^p\sum _{n=1}^{\infty }{a}_n^p.$$

For $x\in [n,n+1)$, we have $F(x)={\int }_0^{x}f(t)dt\ge {\int }_0^{n}f(t)dt=A_n$. Thus:

$${\int }_n^{n+1}{\left(\frac{F(x)}{x}\right)}^{p}dx\ge A_n^p{\int }_n^{n+1}\frac{1}{x^p}dx.$$

While this direct step-function approach requires careful estimation of ${\int }_n^{n+1}{x}^{-p}dx$, the more general approach using ${\lambda }_n$ weights and scaling limits establishes the sharp constant [@kufnerPrehistoryHardyInequality2006].

☘️ Carleman’s Inequality as the $p\to \infty$ Limit

An interesting “endpoint” case of the Hardy inequality occurs as $p\to \infty$. If we consider the discrete inequality and take the limit, we recover Carleman’s inequality for geometric means.

Carleman's Inequality

Let $\{a_n\}$ be a sequence of non-negative real numbers such that $\sum a_n<\infty$. Then

$$\sum _{n=1}^{\infty }(a_1{a}_2\dots a_n{)}^{1/n}\le e\sum _{n=1}^{\infty }{a}_n,$$

where $e$ is the best possible constant. This arises from the Hardy constant ${\lim }_{p\to \infty }(1+\frac{1}{p-1}{)}^p=e$ (Masmoudi, 2011).

🏷️ Modern Refinements and Optimality

The classical constant $1/4$ (for $p=2$) can be refined by adding lower-order terms that capture the “lack of achievement” of the inequality.

Refined 1D Hardy (Brezis-Marcus)

For $u\in H^1(0,1)$ with $u(0)=0$, we have:

$${\int }_0^1\left(|u^{\prime }(t){|}^2-\frac{u(t{)}^2}{4t^2}\right)dt\ge {\int }_0^1\left(|u^{\prime }(t){|}^2+\frac{u(t{)}^2}{4t^2}\right)tdt.$$

Furthermore, using the logarithmic weight $X(t)=(1-\log t{)}^{-1}$:

$${\int }_0^1\left(|u^{\prime }(t){|}^2-\frac{u(t{)}^2}{4t^2}\right)dt\ge \frac{1}{4}{\int }_0^1\frac{u(t{)}^2}{t^2}{X}^2(t)dt.$$

Proof: Brezis-Marcus 1D Refinement

The key to the 1D refinement is a clever change of variables that “flattens” the singularity. Let $v(t)=t^{-1/2}u(t)$, which implies $u(t)=t^{1/2}v(t)$. Then

$$u^{\prime }(t)=t^{1/2}{v}^{\prime }(t)+\frac{1}{2}{t}^{-1/2}v(t).$$

Squaring both sides:

$$|u^{\prime }(t){|}^2=t|v^{\prime }(t){|}^2+\frac{1}{4t}v(t{)}^2+v(t)v^{\prime }(t)=t|v^{\prime }(t){|}^2+\frac{u(t{)}^2}{4t^2}+\frac{1}{2}(v(t{)}^2{)}^{\prime }.$$

Rearranging gives:

$$|u^{\prime }(t){|}^2-\frac{u(t{)}^2}{4t^2}=t|v^{\prime }(t){|}^2+\frac{1}{2}(v(t{)}^2{)}^{\prime }.$$

Integrating from $0$ to $1$:

$${\int }_0^1\left(|u^{\prime }(t){|}^2-\frac{u(t{)}^2}{4t^2}\right)dt={\int }_0^{1}t|v^{\prime }(t){|}^{2}dt+\frac{1}{2}v(1{)}^2\ge 0.$$

This identity not only proves the Hardy inequality but shows that the remainder is non-negative and relates to the derivative of the scaled function $v(t)$ [@brezisHardyInequalitiesRevisited1997].

This logarithmic correction is optimal; any power $X^{2-ϵ}$ would fail to provide a lower bound for some sequence of functions [@brezisHardyInequalitiesRevisited1997].

🏷️ Multi-dimensional Extensions

In ${\mathbb{R}}^n$, the distance to the boundary $\delta (x)=\text{dist}(x,\partial \Omega )$ replaces the radial coordinate $t$.

Hardy Inequality on Domains

Let $\Omega \subset {\mathbb{R}}^n$ be a smooth bounded domain. Then there exists a constant $\mu (\Omega )$ such that:

$${\int }_{\Omega }|\nabla u{|}^{2}dx\ge \mu (\Omega ){\int }_{\Omega }\frac{u^2}{\delta (x{)}^2}dx,\forall u\in H_0^1(\Omega ).$$

For convex domains, $\mu (\Omega )=1/4$. For non-convex domains, $\mu (\Omega )$ can be strictly less than $1/4$.

🌵 The Role of Convexity

If $\Omega$ is convex, Brezis and Marcus showed that there exists a “remainder” term related to the $L^2$ norm:

$${\int }_{\Omega }|\nabla u{|}^{2}dx\ge \frac{1}{4}{\int }_{\Omega }\frac{u^2}{{\delta }^2}dx+{\lambda }^{\ast }(\Omega ){\int }_{\Omega }{u}^{2}dx.$$

They proved that for convex domains, ${\lambda }^{\ast }(\Omega )\ge \frac{1}{4{\text{ diam}}^2(\Omega )}$. This indicates that the singularity at the boundary is robust enough to allow for a global $L^2$ improvement [@brezisHardyInequalitiesRevisited1997].

🏷️ Applications to PDEs

Hardy inequalities are indispensable tools for establishing the well-posedness of boundary value problems with singular coefficients.

Boundary Traces and Sobolev Spaces

In the theory of Sobolev spaces $W^{1,p}(\Omega )$, the existence of the trace operator $\gamma :W^{1,p}(\Omega )\to W^{1-1/p,p}(\partial \Omega )$ relies on estimates of the form:

$${\Vert \frac{f(x)-\gamma f(\pi (x))}{\delta (x)}\Vert }_{L^p(\Omega )}\le C\Vert \nabla f{\Vert }_{L^p(\Omega )},$$

which is a direct application of the Hardy inequality in the direction normal to the boundary (Masmoudi, 2011).

Degenerate Density in Euler Equations

When modeling compressible fluids near a vacuum boundary (e.g., a bubble of gas in space), the density $\rho$ often vanishes as $d(x)$ near the boundary. The resulting singular terms in the Euler equations require weighted Hardy inequalities to control the kinetic energy and ensure the stability of the solution.

🔗 See Also

• on interpolation theorems --- The Hardy inequality can be viewed as an endpoint case of the Hardy-Littlewood-Sobolev fractional integration, where the Riesz potential’s kernel aligns with the boundary singularity.
• on Grönwall’s inequality --- Provides the integral stability framework used to extend local well-posedness to global solutions in systems where Hardy-type control is established.
• on Moser iteration --- Leverages Sobolev embeddings (often initialized with Hardy estimates) to boot-strap $L^p$ bounds into $L^{\infty }$ regularity for elliptic and parabolic operators.

📚 References

🐻  Masmoudi, N. 2011. About the Hardy Inequality. An Invitation to Mathematics: From Competitions to Research, 165–180.