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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
+
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<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>
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 +
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
 
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>
+
$$
 +
f ( x)  = \
 +
\sum _ {k = 0 } ^  \infty 
 +
a _ {k} x  ^ {k}  \sim \
  
then the partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h0463705.png" /> satisfy the asymptotic equality
+
\frac{1}{1 - x }
 +
,\ \
 +
x \uparrow 1,
 +
$$
  
<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>
+
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 [[#References|[1]]] and is one of the [[Tauberian theorems|Tauberian theorems]].
 
This theorem was established by G.H. Hardy and J.E. Littlewood [[#References|[1]]] and is one of the [[Tauberian theorems|Tauberian theorems]].
Line 18: Line 50:
 
''E.D. Solomentsev''
 
''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
+
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 $  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 }
  
<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>
+
\int\limits _  \xi  ^ { x }
 +
f ( t)  dt.
 +
$$
  
 
Then:
 
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
+
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  \
  
<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>
+
\frac{2 ( b - a) ^ {1 - \alpha } }{1 - \alpha }
  
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" />,
+
\int\limits _ { a } ^ { b }  f ( x) dx.
 +
$$
  
<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  $  f  \mathop{\rm ln}  ^ {+}  f \in L _ {1} ( a, b) $,
 +
then
  
3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046370/h04637016.png" />, then
+
$$
 +
\int\limits _ { a } ^ { b }
 +
\theta ( x) dx  \leq  4
 +
\int\limits _ { a } ^ { b }
 +
f ( x)  \mathop{\rm ln}  ^ {+}  f ( x)  dx + A,
 +
$$
  
<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  $  A $
 +
depends only on  $  b - a $.  
 +
Here
  
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
+
$$
 +
\mathop{\rm ln}  ^ {+}  u  = \
 +
\left \{
 +
\begin{array}{ll}
 +
0 & \textrm{ if }  u < 1,  \\
 +
\mathop{\rm ln}  u  & \textrm{ if }  u \geq  1. \\
 +
\end{array}
  
<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>
+
\right .$$
  
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
+
Let $  f $
 +
be a $  2 \pi $-
 +
periodic function that is summable on $  [- \pi , \pi ] $,  
 +
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>
+
$$
 +
M ( x)  = \
 +
M _ {f} ( x)  = \
 +
\sup _ {0 < | t | \leq  \pi } \
 +
{
 +
\frac{1}{t}
 +
}
 +
\int\limits _ { x } ^ { {x }  + t }
 +
| f ( u) |  du.
 +
$$
  
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" />.
+
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====
 
====References====
Line 52: Line 152:
  
 
====Comments====
 
====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" />.
+
The function $  M _ {f} $
 +
is called the Hardy–Littlewood maximal function for $  f $.
  
 
====References====
 
====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>
 
<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>

Latest revision as of 19:43, 5 June 2020


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)
How to Cite This Entry:
Hardy-Littlewood theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hardy-Littlewood_theorem&oldid=13786
This article was adapted from an original article by E.D. Solomentsev, A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article