Hardy's inequality

Hardy's inequality is an inequality in mathematics, named after G. H. Hardy. It states that if $a_{1},a_{2},a_{3},\dots$ is a sequence of non-negative real numbers, then for every real number p > 1 one has

\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 $a_{n}=0$ for all n.

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

\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.

Multidimensional version

In the multidimensional case, Hardy's inequality can be extended to $L^{p}$ -spaces, taking the form [2]

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

where $f\in C_{0}^{\infty }(R^{n})$ , and where the constant ${\frac {p}{n-p}}$ is known to be sharp.

Proof of the inequality

Integral version: a change of variables gives
$\left(\int _{0}^{\infty }\left({\frac {1}{x}}\int _{0}^{x}f(t)\,dt\right)^{p}\ dx\right)^{1/p}=\left(\int _{0}^{\infty }\left(\int _{0}^{1}f(sx)\,ds\right)^{p}\,dx\right)^{1/p}$ ,
which is less or equal than $\int _{0}^{1}\left(\int _{0}^{\infty }f(sx)^{p}\,dx\right)^{1/p}\,ds$ by Minkowski's integral inequality. Finally, by another change of variables, the last expression equals
$\int _{0}^{1}\left(\int _{0}^{\infty }f(x)^{p}\,dx\right)^{1/p}s^{-1/p}\,ds={\frac {p}{p-1}}\left(\int _{0}^{\infty }f(x)^{p}\,dx\right)^{1/p}$ .
Discrete version: assuming the right-hand side to be finite, we must have $a_{n}\to 0$ as $n\to \infty$ . Hence, for any positive integer j, there are only finitely many terms bigger than $2^{-j}$ . This allows us to construct a decreasing sequence $b_{1}\geq b_{2}\geq \cdots$ containing the same positive terms as the original sequence (but possibly no zero terms). Since $a_{1}+a_{2}+\cdots +a_{n}\leq b_{1}+b_{2}+\cdots +b_{n}$ for every n, it suffices to show the inequality for the new sequence. This follows directly from the integral form, defining $f(x)=b_{n}$ if $n-1<x<n$ and $f(x)=0$ otherwise. Indeed, one has
$\int _{0}^{\infty }f(x)^{p}\,dx=\sum _{n=1}^{\infty }b_{n}^{p}$
and, for $n-1<x<n$ , there holds
${\frac {1}{x}}\int _{0}^{x}f(t)\,dt={\frac {b_{1}+\dots +b_{n-1}+(x-n+1)b_{n}}{x}}\geq {\frac {b_{1}+\dots +b_{n}}{n}}$
(the last inequality is equivalent to $(n-x)(b_{1}+\dots +b_{n-1})\geq (n-1)(n-x)b_{n}$ , which is true as the new sequence is decreasing) and thus
$\sum _{n=1}^{\infty }\left({\frac {b_{1}+\dots +b_{n}}{n}}\right)^{p}\leq \int _{0}^{\infty }\left({\frac {1}{x}}\int _{0}^{x}f(t)\,dt\right)^{p}\,dx$ .

Notes

Hardy, G. H. (1920). "Note on a theorem of Hilbert". Mathematische Zeitschrift. 6 (3–4): 314–317. doi:10.1007/BF01199965.
Ruzhansky, Michael; Suragan, Durvudkhan (2019). Hardy Inequalities on Homogeneous Groups: 100 Years of Hardy Inequalities. Birkhäuser Basel. ISBN 978-3-030-02894-7.

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]

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[Hardy1920-1] Hardy, G. H. (1920). "Note on a theorem of Hilbert". Mathematische Zeitschrift. 6 (3–4): 314–317. doi:10.1007/BF01199965.

[2] Ruzhansky, Michael; Suragan, Durvudkhan (2019). Hardy Inequalities on Homogeneous Groups: 100 Years of Hardy Inequalities. Birkhäuser Basel. ISBN 978-3-030-02894-7.