Hardy-Littlewood theorem

The Hardy–Littlewood theorem in the theory of functions of a complex variable: If $ a _ {k} \geq 0 $, $ k = 0, 1 \dots $ and if the power series

$$ f ( z) = \ \sum _ {k = 0 } ^ \infty a _ {k} z ^ {k} $$

with radius of convergence 1 satisfies on the real axis the asymptotic equality

$$ f ( x) = \ \sum _ {k = 0 } ^ \infty a _ {k} x ^ {k} \sim \ \frac{1}{1 - x } ,\ \ x \uparrow 1, $$

then the partial sums $ s _ {n} $ satisfy the asymptotic equality

$$ s _ {n} = \ \sum _ {k = 0 } ^ { n } a _ {n} \sim n,\ \ n \rightarrow \infty . $$

This theorem was established by G.H. Hardy and J.E. Littlewood [1] and is one of the Tauberian theorems.

References

[1]	G.H. Hardy, J.E. Littlewood, "Tauberian theorems concerning power series and Dirichlet's series whose coefficients are positive" Proc. London. Math. Soc. (2) , 13 (1914) pp. 174–191
[2]	E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979)

E.D. Solomentsev

The Hardy–Littlewood theorem on a non-negative summable function. A theorem on integral properties of a certain function connected with the given one. It was established by G.H. Hardy and J.E. Littlewood [1]. Let $ f $ be a non-negative summable function on $ [ a, b] $, and let

$$ \theta ( x) = \ \theta _ {f} ( x) = \ \sup _ {\begin{array}{c} \xi \in [ a, b] \\ \xi \neq x \end{array} } \ \frac{1}{x - \xi } \int\limits _ \xi ^ { x } f ( t) dt. $$

Then:

1) If $ f \in L _ {p} ( a, b) $, $ 1 < p < \infty $, then

$$ \int\limits _ { a } ^ { b } \theta ^ {p} ( x) \ dx \leq 2 \left ( \frac{p}{p - 1 } \right ) ^ {p} \int\limits _ { a } ^ { b } f ^ { p } ( x) dx. $$

2) If $ f \in L _ {1} ( a, b) $, then for all $ \alpha \in ( 0, 1) $,

$$ \int\limits _ { a } ^ { b } \theta ^ \alpha ( x) dx \leq \ \frac{2 ( b - a) ^ {1 - \alpha } }{1 - \alpha } \int\limits _ { a } ^ { b } f ( x) dx. $$

3) If $ f \mathop{\rm ln} ^ {+} f \in L _ {1} ( a, b) $, then

$$ \int\limits _ { a } ^ { b } \theta ( x) dx \leq 4 \int\limits _ { a } ^ { b } f ( x) \mathop{\rm ln} ^ {+} f ( x) dx + A, $$

where $ A $ depends only on $ b - a $. Here

$$ \mathop{\rm ln} ^ {+} u = \ \left \{ \begin{array}{ll} 0 & \textrm{ if } u < 1, \\ \mathop{\rm ln} u & \textrm{ if } u \geq 1. \\ \end{array} \right .$$

Let $ f $ be a $ 2 \pi $- periodic function that is summable on $ [- \pi , \pi ] $, and let

$$ M ( x) = \ M _ {f} ( x) = \ \sup _ {0 < | t | \leq \pi } \ { \frac{1}{t} } \int\limits _ { x } ^ { {x } + t } | f ( u) | du. $$

Then $ M _ {f} ( x) \leq \theta _ {| f | } ( x) $, where $ \theta _ {| f | } ( x) $ is constructed for $ [- 2 \pi , 2 \pi ] $. From the theorem for $ \theta $ one obtains integral inequalities for $ M $.

References

[1]	G.H. Hardy, J.E. Littlewood, "A maximal theorem with function-theoretic applications" Acta. Math. , 54 (1930) pp. 81–116
[2]	A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988)

A.A. Konyushkov

Comments

The function $ M _ {f} $ is called the Hardy–Littlewood maximal function for $ f $.

References

[a1]	E.M. Stein, G. Weiss, "Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1971)