Cover lemma from Wiener

The Überdeckungslemma Wiener ( English Wiener covering lemma ) is a mathematical theorem which in the transition field between the areas of the topology , the measure theory and the harmonic analysis is based. This lemma is attributed to the American mathematician Norbert Wiener and deals with a question about open coverages of compact subsets in Euclidean space and in spaces of homogeneous type . It is related to a similar coverage lemma that goes back to the Italian mathematician Giuseppe Vitali . Both lemmas are significant for the derivation of sentences on the question of the point-wise convergence of Fourier series .

formulation

The lemma can be stated as follows:

Let be the n-dimensional Euclidean space or - more generally - a space of the homogeneous type for which the constant appearing in the quasi- triangle inequality should be. ${\displaystyle X}$ ${\displaystyle \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$ ${\displaystyle K}$

In a compact subset are given and also a family of open balls which overlap. ${\displaystyle X}$ ${\displaystyle Y\neq \emptyset }$ ${\displaystyle {\mathcal {B}}=(B(x_{j},r_{j}))_{j\in J}}$ ${\displaystyle X}$ ${\displaystyle Y}$

Then:

There is a subfamily consisting of a finite number of pairwise disjoint -balls in such a way that for the -fold enlarged -balls form an overlap of . ${\displaystyle {\mathcal {B}}}$ ${\displaystyle X}$ ${\displaystyle (B(x_{t},r_{t}))_{t\in T},\;T\subset J,\;|T|<\infty ,}$ ${\displaystyle M=2K^{2}+K}$ ${\displaystyle M}$ ${\displaystyle X}$ ${\displaystyle (B(x_{t},Mr_{t}))_{t\in T}}$ ${\displaystyle Y}$

In the event, you can choose here and with it. ${\displaystyle X=\mathbb {R} ^{n}}$ ${\displaystyle K=1}$ ${\displaystyle M=3}$

Explanations and Notes

• A space homogeneous type ( English space of homogeneous type ) is a mathematical space structure to a non-empty basic set such that a semimetrischer space and a measure space , wherein the following additional conditions apply: ${\displaystyle X=(X,d,\mu )}$ ${\displaystyle X}$ ${\displaystyle (X,d)}$ ${\displaystyle (X,\mu )}$
  • The semi metric which the topological structure of generated depends on a constant , allowing for always the quasi-triangle ( english quasi-triangle inequality ) is satisfied. ${\displaystyle d\colon {X\times X}\to [0,\infty ]}$ ${\displaystyle (X,d)}$ ${\displaystyle K\geq 1}$ ${\displaystyle x,y,z\in X}$ ${\displaystyle d(x,y)\leq K(d(x,z)+d(z,y))}$
  • The measure space structure of is based on a σ-algebra over the basic set , which contains the Borelian σ-algebra of and all spheres . ${\displaystyle (X,\mu )}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle B(x,r)=\{p\in X:d(x,p)<r\};\;(x\in X,\;r>0)}$
  • ${\displaystyle \mu \colon {\mathcal {A}}\to [0,\infty ]}$is a measure of , ${\displaystyle {\mathcal {A}}}$
    • which on the one hand satisfies the inequalities for each sphere , ${\displaystyle X}$ ${\displaystyle B(x,r)}$ ${\displaystyle 0<\mu {\bigl (}B(x,r){\bigr )}<\infty }$
    • which, on the other hand, has a constant so that every -ball has the doubling property , ${\displaystyle L>0}$ ${\displaystyle X}$ ${\displaystyle B(x,r)}$ ${\displaystyle \mu {\bigl (}B(x,2r){\bigr )}<L\mu {\bigl (}B(x,r){\bigr )}}$
    • and which finally always satisfies the condition for the points . ${\displaystyle x\in X}$ ${\displaystyle \mu (\{x\})=0}$
• In the case , the usual Euclidean metric and the Lebesgue measure are generally assumed to be given. ${\displaystyle X=\mathbb {R} ^{n}}$ ${\displaystyle d}$ ${\displaystyle \mu }$ ${\displaystyle \lambda ^{n}}$
• The basic conception of the homogeneous type rooms is based on ideas that Kennan T. Smith and Lars Hörmander developed and which, in their current form, were mainly worked out by Ronald Raphael Coifman and Guido Weiss . Steven G. Krantz gave a more generalized view of the concept in his monograph Explorations in Harmonic Analysis .
• The homogeneous type rooms are not to be confused with the homogeneous rooms .

Vitali's coverage lemma

The Überdeckungslemma Vitali ( english Vitali covering lemma ) can be formulated as follows:

Is a non-empty family of real intervals , all the interval belong and doing a Lebesgue measurable set cover, it can be from a finite or infinite sequence of pairwise disjoint intervals select which with respect to the Lebesgue measure the inequality ${\displaystyle (I_{j})_{j\in J}}$ ${\displaystyle \mathbb {T} =[0,2\pi )}$ ${\displaystyle A\subseteq \mathbb {T} }$ ${\displaystyle (I_{j_{m}})_{m=1,2,3,\dotsc }}$

${\displaystyle \lambda ^{1}{\Bigl (}\bigcup _{m=1,2,3,\dotsc }{I_{j_{m}}}{\Bigr )}>{\frac {1}{4}}\lambda ^{1}(A)}$

Fulfills.

literature

• Ronald R. Coifman, Guido L. Weiss: Analyze harmonique non-commutative sur certains espaces homogènes: étude de certaines intégrales singulières . Étude de certaines intégrales singulières (=  Lecture Notes in Mathematics . Volume 242 ). Springer Verlag , Berlin, New York 1971 ( MR0499948 ).
• Ronald R. Coifman, Guido Weiss: Extensions of Hardy spaces and their use in analysis . In: Bulletin of the American Mathematical Society . tape 83 , 1977, pp. 569-645 ( MR0447954 ).
• Donggao Deng, Yongsheng Han: Harmonic Analysis on Spaces of Homogeneous Type . With a preface by Yves Meyer. Springer Verlag, Berlin, Heidelberg 2009, ISBN 978-3-540-88744-7 ( MR2039503 ).
• Henry Helson : Harmonic Analysis . Addison-Wesley, Reading, Mass (et al.) 1983, ISBN 0-201-12752-0 ( MR0729682 ).
• Lars Hörmander: L ^p estimates for (pluri-) subharmonic functions . In: Mathematica Scandinavica . tape 20 , 1967, p. 65-78 ( MR0234002 ).
• Roberto A. Macías , Carlos Segovia : Lipschitz functions on spaces of homogeneous type . In: Advances in Mathematics . tape 33 , 1979, pp. 257-270 ( MR0546295 ).
• Yitzhak Katznelson : An Introduction to Harmonic Analysis (=  Cambridge Mathematical Library ). 3. Edition. Cambridge University Press , Cambridge 2004, ISBN 0-521-54359-2 ( MR1710388 ).
• Steven G. Krantz: A Panorama of Harmonic Analysis (=  Carus Mathematical Monographs . Volume 27 ). The Mathematical Association of America , Washington, DC 1999, ISBN 0-88385-031-1 ( MR1710388 ).
• Steven G. Krantz: Explorations in Harmonic Analysis (=  Applied and Numerical Harmonic Analysis . Volume 27 ). Birkhäuser Verlag , Boston, Basel, Berlin 2009, ISBN 978-0-8176-4668-4 ( MR2508404 ).
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).
• KT Smith: A generalization of an inequality of Hardy and Littlewood . In: Canadian Journal of Mathematics . tape 8 , 1956, pp. 157-170 ( MR0086889 ).
• Norbert Wiener: The Fourier Integral and Certain of its Applications . Dover Publications, Inc. , New York 1959 ( MR0100201 - An unaltered republication of the 1933 edition [University Press, Cambridge]).

Individual evidence

1. ^ Steven G. Krantz: A Panorama of Harmonic Analysis. 1999, pp. 71, 235 ff, 357
2. ↑ Donggao Deng, Yongsheng Han: Harmonic Analysis on Spaces of Homogeneous Type. 1999, p. 13
3. ^ Yitzhak Katznelson: An Introduction to Harmonic Analysis. 2004, p. 96 ff
4. ^ ^{A b} Henry Helson: Harmonic Analysis. 1983, p. 130
5. ↑ Krantz, op.cit., Pp. 71, 246
6. ↑ Deng / Han, op.cit., P. 13
7. ↑ Because of this inequality, the English-language specialist literature also speaks of a quasi-metric in the local context . The concept of quasi-metrics is, however, sometimes interpreted differently in the German-language specialist literature - such as in Horst Schubert's Topologie (4th edition, p. 114) - namely in such a way that both the distance and the distance are permitted for two different points , that otherwise the quasi- metric has all the usual properties of a metric and in particular fulfills the triangle inequality. ${\displaystyle 0}$ ${\displaystyle \infty }$
8. ↑ The balls are in the case of unnecessary open subsets of . ${\displaystyle X}$ ${\displaystyle K>1}$ ${\displaystyle (X,d)}$
9. ↑ In the English literature refers to the doubling property ( English doubling property ) as a doubling condition ( English doubling condition ).
10. ^ Steven G. Krantz: Explorations in Harmonic Analysis. 2009, p. 192 ff
11. ↑ Katznelson, op.cit., P. 97