# Hardy-Littlewood theorem

The Hardy–Littlewood theorem in the theory of functions of a complex variable: If , and if the power series

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

then the partial sums satisfy the asymptotic equality

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

## Contents

#### 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 be a non-negative summable function on , and let

Then:

1) If , , then

2) If , then for all ,

3) If , then

where depends only on . Here

Let be a -periodic function that is summable on , and let

Then , where is constructed for . From the theorem for one obtains integral inequalities for .

#### 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 is called the Hardy–Littlewood maximal function for .

#### References

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

**How to Cite This Entry:**

Hardy-Littlewood theorem.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Hardy-Littlewood_theorem&oldid=22551