View source for Hardy-Littlewood theorem

The Hardy–Littlewood theorem in the theory of functions of a complex variable: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h0463701.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h0463702.png" /> and if the power series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h0463703.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h0463704.png" /></td> </tr></table>

then the partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h0463705.png" /> satisfy the asymptotic equality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h0463706.png" /></td> </tr></table>

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.C. Titchmarsh,   "The theory of functions" , Oxford Univ. Press  (1979)</TD></TR></table>

''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 [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h0463707.png" /> be a non-negative summable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h0463708.png" />, and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h0463709.png" /></td> </tr></table>

Then:

1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637011.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637012.png" /></td> </tr></table>

2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637013.png" />, then for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637014.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637015.png" /></td> </tr></table>

3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637016.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637017.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637018.png" /> depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637019.png" />. Here

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637020.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637021.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637022.png" />-periodic function that is summable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637023.png" />, and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637024.png" /></td> </tr></table>

Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637026.png" /> is constructed for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637027.png" />. From the theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637028.png" /> one obtains integral inequalities for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637029.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,   J.E. Littlewood,   "A maximal theorem with function-theoretic applications"  ''Acta. Math.'' , '''54'''  (1930)  pp. 81–116</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR></table>

''A.A. Konyushkov''

====Comments====
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637030.png" /> is called the Hardy–Littlewood maximal function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637031.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.M. Stein,   G. Weiss,   "Fourier analysis on Euclidean spaces" , Princeton Univ. Press  (1971)</TD></TR></table>