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Difference between revisions of "Riesz convexity theorem"

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$$  
 
$$  
\sum _ { i= } 1 ^ { m }  \sum _ { j= } 1 ^ { n }  a _ {ij} x _ {i} y _ {j}  $$
+
\sum_{i=1} ^ { m }  \sum_{j=1} ^ { n }  a _ {ij} x _ {i} y _ {j}  $$
  
 
on the set
 
on the set
  
 
$$  
 
$$  
\sum _ { i= } 1 ^ { m }  | x _ {i} | ^ {1/ \alpha }  \leq  1,\ \  
+
\sum_{i=1} ^ { m }  | x _ {i} | ^ {1/ \alpha }  \leq  1,\ \  
\sum _ { j= } 1 ^ { m }  | y _ {j} | ^ {1/ \beta }  \leq  1
+
\sum_{j=1} ^ { m }  | y _ {j} | ^ {1/ \beta }  \leq  1
 
$$
 
$$
  
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the continuity of the operator  $  T:  L _ {p _ {t}  } \rightarrow L _ {q _ {t}  } $,  
 
the continuity of the operator  $  T:  L _ {p _ {t}  } \rightarrow L _ {q _ {t}  } $,  
 
$  t \in ( 0, 1) $,  
 
$  t \in ( 0, 1) $,  
under weaker assumptions than those of the Riesz–Thorin theorem. See also [[Interpolation of operators|Interpolation of operators]].
+
under weaker assumptions than those of the Riesz–Thorin theorem. See also [[Interpolation of operators]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Riesz,  "Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires"  ''Acta Math.'' , '''49'''  (1926)  pp. 465–497</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,  J.E. Littlewood,  G. Pólya,  "Inequalities" , Cambridge Univ. Press  (1934)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.O. Thorin,  "An extension of a convexity theorem due to M. Riesz"  ''K. Fysiogr. Saallskap. i Lund Forh.'' , '''8''' :  14  (1936)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.M. Stein,  G. Weiss,  "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press  (1971)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Marcinkiewicz,  "Sur l'interpolation d'opérateurs"  ''C.R. Acad. Sci. Paris'' , '''208'''  (1939)  pp. 1272–1273</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.K. Krein,  "Interpolation of linear operators" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  H. Triebel,  "Interpolation theory" , Springer  (1978)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Riesz,  "Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires"  ''Acta Math.'' , '''49'''  (1926)  pp. 465–497</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,  J.E. Littlewood,  G. Pólya,  "Inequalities" , Cambridge Univ. Press  (1934)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.O. Thorin,  "An extension of a convexity theorem due to M. Riesz"  ''K. Fysiogr. Saallskap. i Lund Forh.'' , '''8''' :  14  (1936)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.M. Stein,  G. Weiss,  "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press  (1971)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Marcinkiewicz,  "Sur l'interpolation d'opérateurs"  ''C.R. Acad. Sci. Paris'' , '''208'''  (1939)  pp. 1272–1273</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.K. Krein,  "Interpolation of linear operators" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  H. Triebel,  "Interpolation theory" , Springer  (1978)</TD></TR></table>

Latest revision as of 13:00, 6 January 2024


The logarithm, $ \mathop{\rm ln} M( \alpha , \beta ) $, of the least upper bound of the modulus $ M( \alpha , \beta ) $ of the bilinear form

$$ \sum_{i=1} ^ { m } \sum_{j=1} ^ { n } a _ {ij} x _ {i} y _ {j} $$

on the set

$$ \sum_{i=1} ^ { m } | x _ {i} | ^ {1/ \alpha } \leq 1,\ \ \sum_{j=1} ^ { m } | y _ {j} | ^ {1/ \beta } \leq 1 $$

(if $ \alpha = 0 $ or $ \beta = 0 $, then, respectively, $ | x _ {i} | \leq 1 $, $ i = 1 \dots m $ or $ | y _ {j} | \leq 1 $, $ j = 1 \dots n $) is a convex function (of a real variable) of the parameters $ \alpha $ and $ \beta $ in the domain $ \alpha \geq 0 $, $ \beta \geq 0 $ if the form is real $ ( a _ {ij} , x _ {i} , y _ {j} \in \mathbf R _ {+} ) $, and it is a convex function (of a real variable) in the domain $ 0 \leq \alpha , \beta \leq 1 $, $ \alpha + \beta \geq 1 $ if the form is complex $ ( a _ {ij} , x _ {i} , y _ {j} \in \mathbf C ) $. This theorem was proved by M. Riesz [1].

A generalization of this theorem to linear operators is (see [3]): Let $ L _ {p} $, $ 1 \leq p \leq \infty $, be the set of all complex-valued functions on some measure space that are summable to the $ p $- th power for $ 1 \leq p < \infty $ and that are essentially bounded for $ p = \infty $. Let, further, $ T: L _ {p _ {i} } \rightarrow L _ {q _ {i} } $, $ 1 \leq p _ {i} , q _ {j} \leq \infty $, $ i = 0, 1 $, be a continuous linear operator. Then $ T $ is a continuous operator from $ L _ {p _ {t} } $ to $ L _ {q _ {t} } $, where

$$ \frac{1}{p _ {t} } = 1- \frac{t}{p _ {0} } + \frac{t}{p _ {1} } ,\ \ \frac{1}{q _ {t} } = 1- \frac{t}{q _ {0} } + \frac{t}{q _ {1} } ,\ \ t \in [ 0, 1], $$

and where the norm $ k _ {t} $ of $ T $( as an operator from $ L _ {p _ {t} } $ to $ L _ {q _ {t} } $) satisfies the inequality $ k _ {t} \leq k _ {0} ^ {1-} t k _ {1} ^ {t} $( i.e. it is a logarithmically convex function). This theorem is called the Riesz–Thorin interpolation theorem, and sometimes also the Riesz convexity theorem [4].

The Riesz convexity theorem is at the origin of a whole trend of analysis in which one studies interpolation properties of linear operators. Among the first generalizations of the Riesz convexity theorem is the Marcinkiewicz interpolation theorem [5], which ensures for $ 1 \leq p _ {i} \leq q _ {i} \leq \infty $, $ i = 0, 1 $, the continuity of the operator $ T: L _ {p _ {t} } \rightarrow L _ {q _ {t} } $, $ t \in ( 0, 1) $, under weaker assumptions than those of the Riesz–Thorin theorem. See also Interpolation of operators.

References

[1] M. Riesz, "Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires" Acta Math. , 49 (1926) pp. 465–497
[2] G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)
[3] G.O. Thorin, "An extension of a convexity theorem due to M. Riesz" K. Fysiogr. Saallskap. i Lund Forh. , 8 : 14 (1936)
[4] E.M. Stein, G. Weiss, "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1971)
[5] J. Marcinkiewicz, "Sur l'interpolation d'opérateurs" C.R. Acad. Sci. Paris , 208 (1939) pp. 1272–1273
[6] S.K. Krein, "Interpolation of linear operators" , Amer. Math. Soc. (1982) (Translated from Russian)
[7] H. Triebel, "Interpolation theory" , Springer (1978)
How to Cite This Entry:
Riesz convexity theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_convexity_theorem&oldid=54883
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article