Hardy's inequality

Hardy's inequality is an inequality in mathematics, named after G. H. Hardy.

Its discrete version states that if ${\displaystyle a_{1},a_{2},a_{3},\dots }$ is a sequence of non-negative real numbers, then for every real number p > 1 one has

${\displaystyle \sum _{n=1}^{\infty }\left({\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right)^{p}\leq \left({\frac {p}{p-1}}\right)^{p}\sum _{n=1}^{\infty }a_{n}^{p}.}$

If the right-hand side is finite, equality holds if and only if ${\displaystyle a_{n}=0}$ for all n.

An integral version of Hardy's inequality states the following: if f is a measurable function with non-negative values, then

${\displaystyle \int _{0}^{\infty }\left({\frac {1}{x}}\int _{0}^{x}f(t)\,dt\right)^{p}\,dx\leq \left({\frac {p}{p-1}}\right)^{p}\int _{0}^{\infty }f(x)^{p}\,dx.}$

If the right-hand side is finite, equality holds if and only if f(x) = 0 almost everywhere.

Hardy's inequality was first published and proved (at least the discrete version with a worse constant) in 1920 in a note by Hardy. ^[1] The original formulation was in an integral form slightly different from the above.

Statements

General discrete Hardy inequality

The general weighted one dimensional version reads as follows: ^{[2] : §332}: if ${\displaystyle a_{n}\geq 0}$, ${\displaystyle \lambda _{n}>0}$ and ${\displaystyle p>1}$,

${\displaystyle \sum _{n=1}^{\infty }\lambda _{n}{\Bigl (}{\frac {\lambda _{1}a_{1}+\dotsb +\lambda _{n}a_{n}}{\lambda _{1}+\dotsb +\lambda _{n}}}{\Bigr )}^{p}\leq {\Bigl (}{\frac {p}{p-1}}{\Bigr )}^{p}\sum _{n=1}^{\infty }\lambda _{n}a_{n}^{p}.}$

General one-dimensional integral Hardy inequality

The general weighted one dimensional version reads as follows: ^{[2] : §330}

• If ${\displaystyle \alpha +{\tfrac {1}{p}}<1}$, then

${\displaystyle \int _{0}^{\infty }{\biggl (}y^{\alpha -1}\int _{0}^{y}x^{-\alpha }f(x)\,dx{\biggr )}^{p}\,dy\leq {\frac {1}{{\bigl (}1-\alpha -{\frac {1}{p}}{\bigr )}^{p}}}\int _{0}^{\infty }f(x)^{p}\,dx}$

• If ${\displaystyle \alpha +{\tfrac {1}{p}}>1}$, then

${\displaystyle \int _{0}^{\infty }{\biggl (}y^{\alpha -1}\int _{y}^{\infty }x^{-\alpha }f(x)\,dx{\biggr )}^{p}\,dy\leq {\frac {1}{{\bigl (}\alpha +{\frac {1}{p}}-1{\bigr )}^{p}}}\int _{0}^{\infty }f(x)^{p}\,dx.}$

Multidimensional Hardy inequalities with gradient

Multidimensional Hardy inequality around a point

In the multidimensional case, Hardy's inequality can be extended to ${\displaystyle L^{p}}$-spaces, taking the form ^[3]

${\displaystyle \left\|{\frac {f}{\cdot }}\right\|_{L^{p}(\mathbb {R} ^{n})}\leq {\frac {p}{n-p}}\|\nabla f\|_{L^{p}(\mathbb {R} ^{n})},2\leq n,1\leq p<n,}$

where ${\displaystyle f\in C_{c}^{\infty }(\mathbb {R} ^{n})}$, and where the constant ${\displaystyle {\frac {p}{n-p}}}$ is known to be sharp; by density it extends then to the Sobolev space ${\displaystyle W^{1,p}(\mathbb {R} ^{n})}$.

Similarly, if ${\displaystyle p>n\geq 2}$, then one has for every ${\displaystyle f\in C_{c}^{\infty }(\mathbb {R} ^{n})}$

${\displaystyle {\Big (}1-{\frac {n}{p}}{\Big )}^{p}\int _{\mathbb {R} ^{n}}{\frac {\vert f(x)-f(0)\vert ^{p}}{|x|^{p}}}dx\leq \int _{\mathbb {R} ^{n}}\vert \nabla f\vert ^{p}.}$

Multidimensional Hardy inequality near the boundary

If ${\displaystyle \Omega \subsetneq \mathbb {R} ^{n}}$ is an nonempty convex open set, then for every ${\displaystyle f\in W^{1,p}(\Omega )}$,

${\displaystyle {\Big (}1-{\frac {1}{p}}{\Big )}^{p}\int _{\Omega }{\frac {\vert f(x)\vert ^{p}}{\operatorname {dist} (x,\partial \Omega )^{p}}}\,dx\leq \int _{\Omega }\vert \nabla f\vert ^{p},}$

and the constant cannot be improved. ^[4]

Fractional Hardy inequality

If ${\displaystyle 1\leq p<\infty }$ and ${\displaystyle 0<\lambda <\infty }$, ${\displaystyle \lambda \neq 1}$, there exists a constant ${\displaystyle C}$ such that for every ${\displaystyle f:(0,\infty )\to \mathbb {R} }$ satisfying ${\displaystyle \int _{0}^{\infty }\vert f(x)\vert ^{p}/x^{\lambda }\,dx<\infty }$, one has ^{[5] : Lemma 2}

${\displaystyle \int _{0}^{\infty }{\frac {\vert f(x)\vert ^{p}}{x^{\lambda }}}\,dx\leq C\int _{0}^{\infty }\int _{0}^{\infty }{\frac {\vert f(x)-f(y)\vert ^{p}}{\vert x-y\vert ^{1+\lambda }}}\,dx\,dy.}$

Proof of the inequality

Integral version (integration by parts and Hölder)

Hardy’s original proof ^{[1] [2] : §327 (ii)} begins with an integration by parts to get

${\displaystyle {\begin{aligned}\int _{0}^{\infty }\left({\frac {1}{x}}\int _{0}^{x}f(t)\,dt\right)^{p}dx&=\int _{0}^{\infty }\left(\int _{0}^{x}f(t)\,dt\right)^{p}{\frac {1}{x^{p}}}dx\\[0.2em]&={\frac {p}{p-1}}\int _{0}^{\infty }\left(\int _{0}^{x}f(t)\,dt\right)^{p-1}{\frac {f(x)}{x^{p-1}}}dx\\[0.2em]&={\frac {p}{p-1}}\int _{0}^{\infty }\left({\frac {1}{x}}\int _{0}^{x}f(t)\,dt\right)^{p-1}f(x)dx\end{aligned}}}$

Then, by Hölder's inequality,

${\displaystyle \int _{0}^{\infty }\left({\frac {1}{x}}\int _{0}^{x}f(t)\,dt\right)^{p}dx\leq {\frac {p}{p-1}}\left(\int _{0}^{\infty }\left({\frac {1}{x}}\int _{0}^{x}f(t)\,dt\right)^{p}dx\right)^{1-{\frac {1}{p}}}\left(\int _{0}^{\infty }f(x)^{p}\,dx\right)^{\frac {1}{p}},}$

and the conclusion follows.

Integral version (scaling and Minkowski)

A change of variables gives

${\displaystyle \left(\int _{0}^{\infty }\left({\frac {1}{x}}\int _{0}^{x}f(t)\,dt\right)^{p}\ dx\right)^{1/p}=\left(\int _{0}^{\infty }\left(\int _{0}^{1}f(sx)\,ds\right)^{p}\,dx\right)^{1/p},}$

which is less or equal than ${\displaystyle \int _{0}^{1}\left(\int _{0}^{\infty }f(sx)^{p}\,dx\right)^{1/p}\,ds}$ by Minkowski's integral inequality. Finally, by another change of variables, the last expression equals

${\displaystyle \int _{0}^{1}\left(\int _{0}^{\infty }f(x)^{p}\,dx\right)^{1/p}s^{-1/p}\,ds={\frac {p}{p-1}}\left(\int _{0}^{\infty }f(x)^{p}\,dx\right)^{1/p}.}$

Discrete version: from the continuous version

Assuming the right-hand side to be finite, we must have ${\displaystyle a_{n}\to 0}$ as ${\displaystyle n\to \infty }$. Hence, for any positive integer j, there are only finitely many terms bigger than ${\displaystyle 2^{-j}}$. This allows us to construct a decreasing sequence ${\displaystyle b_{1}\geq b_{2}\geq \dotsb }$ containing the same positive terms as the original sequence (but possibly no zero terms). Since ${\displaystyle a_{1}+a_{2}+\dotsb +a_{n}\leq b_{1}+b_{2}+\dotsb +b_{n}}$ for every n, it suffices to show the inequality for the new sequence. This follows directly from the integral form, defining ${\displaystyle f(x)=b_{n}}$ if ${\displaystyle n-1<x<n}$ and ${\displaystyle f(x)=0}$ otherwise. Indeed, one has

${\displaystyle \int _{0}^{\infty }f(x)^{p}\,dx=\sum _{n=1}^{\infty }b_{n}^{p}}$

and, for ${\displaystyle n-1<x<n}$, there holds

${\displaystyle {\frac {1}{x}}\int _{0}^{x}f(t)\,dt={\frac {b_{1}+\dots +b_{n-1}+(x-n+1)b_{n}}{x}}\geq {\frac {b_{1}+\dots +b_{n}}{n}}}$

(the last inequality is equivalent to ${\displaystyle (n-x)(b_{1}+\dots +b_{n-1})\geq (n-1)(n-x)b_{n}}$, which is true as the new sequence is decreasing) and thus

${\displaystyle \sum _{n=1}^{\infty }\left({\frac {b_{1}+\dots +b_{n}}{n}}\right)^{p}\leq \int _{0}^{\infty }\left({\frac {1}{x}}\int _{0}^{x}f(t)\,dt\right)^{p}\,dx}$.

Discrete version: Direct proof

Let ${\displaystyle p>1}$ and let ${\displaystyle b_{1},\dots ,b_{n}}$ be positive real numbers. Set ${\displaystyle S_{k}=\sum _{i=1}^{k}b_{i}}$. First we prove the inequality

${\displaystyle \sum _{n=1}^{N}{\frac {S_{n}^{p}}{n^{p}}}\leq {\frac {p}{p-1}}\sum _{n=1}^{N}{\frac {b_{n}S_{n}^{p-1}}{n^{p-1}}},}$

*

Let ${\displaystyle T_{n}={\frac {S_{n}}{n}}}$ and let ${\displaystyle \Delta _{n}}$ be the difference between the ${\displaystyle n}$-th terms in the right-hand side and left-hand side of * , that is, ${\displaystyle \Delta _{n}:=T_{n}^{p}-{\frac {p}{p-1}}b_{n}T_{n}^{p-1}}$. We have:

${\displaystyle \Delta _{n}=T_{n}^{p}-{\frac {p}{p-1}}b_{n}T_{n}^{p-1}=T_{n}^{p}-{\frac {p}{p-1}}(nT_{n}-(n-1)T_{n-1})T_{n}^{p-1}}$

or

${\displaystyle \Delta _{n}=T_{n}^{p}\left(1-{\frac {np}{p-1}}\right)+{\frac {p(n-1)}{p-1}}T_{n-1}T_{n}^{p}.}$

According to Young's inequality we have:

${\displaystyle T_{n-1}T_{n}^{p-1}\leq {\frac {T_{n-1}^{p}}{p}}+(p-1){\frac {T_{n}^{p}}{p}},}$

from which it follows that:

${\displaystyle \Delta _{n}\leq {\frac {n-1}{p-1}}T_{n-1}^{p}-{\frac {n}{p-1}}T_{n}^{p}.}$

By telescoping we have:

${\displaystyle {\begin{aligned}\sum _{n=1}^{N}\Delta _{n}&\leq 0-{\frac {1}{p-1}}T_{1}^{p}+{\frac {1}{p-1}}T_{1}^{p}-{\frac {2}{p-1}}T_{2}^{p}+{\frac {2}{p-1}}T_{2}^{p}-\\&\qquad -{\frac {3}{p-1}}T_{3}^{p}+\dotsb +{\frac {N-1}{p-1}}T_{N-1}^{p}-{\frac {N}{p-1}}T_{N}^{p}\\&\qquad =-{\frac {N}{p-1}}T_{N}^{p}<0\end{aligned}}}$

proving * . Applying Hölder's inequality to the right-hand side of * we have:

${\displaystyle \sum _{n=1}^{N}{\frac {S_{n}^{p}}{n^{p}}}\leq {\frac {p}{p-1}}\sum _{n=1}^{N}{\frac {b_{n}S_{n}^{p-1}}{n^{p-1}}}\leq {\frac {p}{p-1}}\left(\sum _{n=1}^{N}b_{n}^{p}\right)^{1/p}\left(\sum _{n=1}^{N}{\frac {S_{n}^{p}}{n^{p}}}\right)^{(p-1)/p}}$

from which we immediately obtain:

${\displaystyle \sum _{n=1}^{N}{\frac {S_{n}^{p}}{n^{p}}}\leq \left({\frac {p}{p-1}}\right)^{p}\sum _{n=1}^{N}b_{n}^{p}.}$

Letting ${\displaystyle N\rightarrow \infty }$ we obtain Hardy's inequality.

See also

• Carleman's inequality

Notes

1. Hardy, G. H. (1920). "Note on a theorem of Hilbert". Mathematische Zeitschrift. 6 (3–4): 314–317. doi:10.1007/BF01199965. S2CID   122571449.
2. Hardy, G. H.; Littlewood, J.E.; Pólya, G. (1952). Inequalities (Second ed.). Cambridge, UK.
3. Ruzhansky, Michael; Suragan, Durvudkhan (2019). Hardy Inequalities on Homogeneous Groups: 100 Years of Hardy Inequalities. Birkhäuser Basel. ISBN   978-3-030-02894-7.
4. Marcus, Moshe; Mizel, Victor J.; Pinchover, Yehuda (1998). "On the best constant for Hardy's inequality in $\mathbb {R}^n$". Transactions of the American Mathematical Society. 350 (8): 3237–3255. doi: 10.1090/S0002-9947-98-02122-9 .
5. Mironescu, Petru (2018). "The role of the Hardy type inequalities in the theory of function spaces" (PDF). Revue roumaine de mathématiques pures et appliquées. 63 (4): 447–525.

References

• Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952). Inequalities (2nd ed.). Cambridge University Press. ISBN   0-521-35880-9.

• Kufner, Alois; Persson, Lars-Erik (2003). Weighted inequalities of Hardy type. World Scientific Publishing. ISBN   981-238-195-3.

• Masmoudi, Nader (2011), "About the Hardy Inequality", in Dierk Schleicher; Malte Lackmann (eds.), An Invitation to Mathematics, Springer Berlin Heidelberg, ISBN   978-3-642-19533-4 .
• Ruzhansky, Michael; Suragan, Durvudkhan (2019). Hardy Inequalities on Homogeneous Groups: 100 Years of Hardy Inequalities. Birkhäuser Basel. ISBN   978-3-030-02895-4.

External links

• "Hardy inequality", Encyclopedia of Mathematics , EMS Press, 2001 [1994]