Hardy inequality

for series

If $ p > 1 $, $ a _ {n} \geq 0 $ and $ A _ {n} = a _ {1} + \dots + a _ {n} $, $ n = 1, 2 \dots $ then

$$ \sum _ {n = 1 } ^ \infty \left ( \frac{A _ {n} }{n} \right ) ^ {p} < \ \left ( \frac{p}{p - 1 } \right ) ^ {p} \sum _ {n = 1 } ^ \infty a _ {n} ^ {p} , $$

except when all the $ a _ {n} $ are zero. The constant $ ( p/( p - 1)) ^ {p} $ in this inequality is best possible.

The Hardy inequalities for integrals are:

$$ \int\limits _ { 0 } ^ \infty x ^ {-} p \left | \int\limits _ { 0 } ^ { x } f ( t) dt \right | ^ {p} dx < \left ( \frac{p}{p - 1 } \right ) ^ {p} \int\limits _ { 0 } ^ \infty | f ( x) | ^ {p} dx,\ \ p > 1 , $$

and

$$ \int\limits _ { 0 } ^ \infty \left | \int\limits _ { x } ^ \infty f ( t) dt \right | ^ {p} dx < p ^ {p} \int\limits _ { 0 } ^ \infty | xf ( x) | ^ {p} dx,\ \ p > 1. $$

The inequalities are valid for all functions for which the right-hand sides are finite, except when $ f $ vanishes almost-everywhere on $ ( 0, + \infty ) $. (In this case the inequalities turn into equalities.) The constants $ ( p/( p - 1)) ^ {p} $ and $ p ^ {p} $ are best possible.

The integral Hardy inequalities can be generalized to arbitrary intervals:

$$ \int\limits _ { a } ^ { b } \left | x ^ {\alpha - 1 } \int\limits _ { a } ^ { x } f ( t) dt \right | ^ {p} dx \leq c \int\limits _ { a } ^ { b } | x ^ \alpha f ( x) | ^ {p} dx,\ \ \alpha < 1 - { \frac{1}{p} } , $$

$$ \int\limits _ { a } ^ { b } \left | x ^ {\alpha - 1 } \int\limits _ { x } ^ { b } f ( t) dt \right | ^ {p} dx \leq c \int\limits _ { a } ^ { b } | x ^ \alpha f ( x) | ^ {p} dx,\ \alpha > 1 - { \frac{1}{p} } , $$

where $ 0 \leq a < b \leq + \infty $, $ 1 < p < + \infty $, and where the $ c $' s are certain constants.

Generalized Hardy inequalities are inequalities of the form

$$ \tag{1 } \int\limits _ { a } ^ { b } \left | \phi ( x) \int\limits _ { a } ^ { x } f ( t) dt \right | ^ {p} \ dx \leq c \int\limits _ { a } ^ { b } | \psi ( x) f ( x) | ^ {p} dx, $$

$$ \tag{2 } \int\limits _ { a } ^ { b } \left | \phi ( x) \int\limits _ { x } ^ { b } f ( t) dt \right | ^ {p} dx \leq c \int\limits _ { a } ^ { b } | \psi ( x) f ( x) | ^ {p} dx. $$

If $ a = 0 $ and $ b = + \infty $, inequality (1) holds if and only if

$$ \sup _ {x > 0 } \left [ \int\limits _ { x } ^ \infty | \phi ( t) | ^ {p} dt \right ] ^ {1/p} \left [ \int\limits _ { 0 } ^ { x } | \psi ( t) | ^ {- p ^ \prime } dt \right ] ^ {1/p ^ \prime } < + \infty , $$

and (2) holds if and only if

$$ \sup _ {x > 0 } \left [ \int\limits _ { 0 } ^ { x } | \phi ( t) | ^ {p} \ dt \right ] ^ {1/p} \left [ \int\limits _ { x } ^ \infty | \psi ( t) | ^ {- p ^ \prime } \ dt \right ] ^ {1/p ^ \prime } < + \infty , $$

$$ { \frac{1}{p} } + { \frac{1}{p ^ \prime } } = 1. $$

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