Hardy inequality
From Encyclopedia of Mathematics
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for series
If , and , then
except when all the are zero. The constant in this inequality is best possible.
The Hardy inequalities for integrals are:
and
The inequalities are valid for all functions for which the right-hand sides are finite, except when vanishes almost-everywhere on . (In this case the inequalities turn into equalities.) The constants and are best possible.
The integral Hardy inequalities can be generalized to arbitrary intervals:
where , , and where the 's are certain constants.
Generalized Hardy inequalities are inequalities of the form
(1) |
(2) |
If and , inequality (1) holds if and only if
and (2) holds if and only if
References
[1] | G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934) |
[2] | S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian) |
[3] | B. Muckenhoupt, "Hardy's inequality with weights" Studia Math. , 44 (1972) pp. 31–38 |
How to Cite This Entry:
Hardy inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hardy_inequality&oldid=13888
Hardy inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hardy_inequality&oldid=13888
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article