Hardy's inequality

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

If the right-hand side is finite, equality holds if and only if for all n.

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

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.

General one-dimensional version

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The general weighted one dimensional version reads as follows:[2]: §329 

  • If  , then
 
  • If  , then
 

Multidimensional versions

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Multidimensional Hardy inequality around a point

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In the multidimensional case, Hardy's inequality can be extended to  -spaces, taking the form [3]

 

where  , and where the constant   is known to be sharp; by density it extends then to the Sobolev space  .

Similarly, if  , then one has for every  

 

Multidimensional Hardy inequality near the boundary

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If   is an nonempty convex open set, then for every  ,

 

and the constant cannot be improved.[4]

Fractional Hardy inequality

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If   and  ,  , there exists a constant   such that for every   satisfying  , one has[5]: Lemma 2 

 

Proof of the inequality

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Integral version

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A change of variables gives

 

which is less or equal than   by Minkowski's integral inequality. Finally, by another change of variables, the last expression equals

 

Discrete version: from the continuous version

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Assuming the right-hand side to be finite, we must have   as  . Hence, for any positive integer j, there are only finitely many terms bigger than  . This allows us to construct a decreasing sequence   containing the same positive terms as the original sequence (but possibly no zero terms). Since   for every n, it suffices to show the inequality for the new sequence. This follows directly from the integral form, defining   if   and   otherwise. Indeed, one has

 

and, for  , there holds

 

(the last inequality is equivalent to  , which is true as the new sequence is decreasing) and thus

 .

Discrete version: Direct proof

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Let   and let   be positive real numbers. Set  . First we prove the inequality

  (*)

Let   and let   be the difference between the  -th terms in the right-hand side and left-hand side of *, that is,  . We have:

 

or

 

According to Young's inequality we have:

 

from which it follows that:

 

By telescoping we have:

 

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

 

from which we immediately obtain:

 

Letting   we obtain Hardy's inequality.

See also

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Notes

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  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.{{cite book}}: CS1 maint: location missing publisher (link)
  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

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  • 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.
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