413:
186:
101:
The one-dimensional case of
Rademacher's theorem is a standard result in introductory texts on measure-theoretic analysis. In this context, it is natural to prove the more general statement that any single-variable function of
408:{\displaystyle \int _{\mathbf {R} ^{n}}{\frac {u(x+h\nu )-u(x)}{h}}\zeta (z)\,d{\mathcal {L}}^{n}(x)=-\int _{\mathbf {R} ^{n}}{\frac {\zeta (x)-\zeta (x-h\nu )}{h}}u(x)\,d{\mathcal {L}}^{n}(x).}
1186:. Proceedings of the Centre for Mathematical Analysis, Australian National University. Vol. 3. Canberra: Australian National University, Centre for Mathematical Analysis.
93:
zero. Differentiability here refers to infinitesimal approximability by a linear map, which in particular asserts the existence of the coordinate-wise partial derivatives.
610:, the proof of which can be modeled on that of the Rademacher theorem. In some special cases, the Rademacher theorem is even used as part of the proof.
602:
remain more subtle, since their usual definitions require more differentiability than is achieved by the
Rademacher theorem. In the presence of
163:
fails to exist (which must be proved to be measurable), the latter condition is met due to the one-dimensional case of
Rademacher's theorem.
451:
106:
is differentiable almost everywhere. (This one-dimensional generalization of
Rademacher's theorem fails to extend to higher dimensions.)
716:
477:-directional derivative almost everywhere (although perhaps on a smaller set). Hence, for any countable collection of unit vectors
1092:"Ăśber partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und ĂĽber die Transformation der Doppelintegrale"
1241:
1061:
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1359:
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There is a version of
Rademacher's theorem that holds for Lipschitz functions from a Euclidean space into an arbitrary
1292:
1191:
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649:
575:
holding in its standard form. With appropriate modification, this also extends to the more general
Sobolev spaces
1272:
423:
696:
530:
to be dense in the unit sphere, it is possible to use the
Lipschitz condition to prove the existence of
954:
653:
583:
1354:
903:. Die Grundlehren der mathematischen Wissenschaften. Vol. 153. Berlin–Heidelberg–New York:
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1327:
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20:
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1021:
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62:
1310:
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1209:
1166:
1117:
1079:
1029:
979:
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888:
821:
819:
545:. Another proof, also via a reduction to the one-dimensional case, uses the technology of
538:, together with its representation as the dot product of the gradient with the direction.
8:
672:
656:. Rademacher's theorem is a special case, due to the fact that any Lipschitz function on
427:
853:. Textbooks in Mathematics (Revised edition of 1992 original ed.). Boca Raton, FL:
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953:. Pure and Applied Mathematics (Second edition of 1984 original ed.). New York:
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Weakly differentiable functions. Sobolev spaces and functions of bounded variation
461:-tuple of partial derivatives) is guaranteed to exist almost everywhere; for each
149:
has (one-dimensional) Lebesgue measure zero. Considering in particular the set in
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590:, as it allows the analysis of first-order differential geometry, specifically
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110:
1284:
1233:
1053:
1044:. Die Grundlehren der mathematischen Wissenschaften. Vol. 130. New York:
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434:-directional derivatives. Then, based upon the known linear dependence of the
1348:
987:
632:
595:
591:
565:
1133:
668:
166:
The second step of Morrey's proof establishes the linear dependence of the
109:
One of the standard proofs of the general
Rademacher theorem was found by
571:
is preserved under a bi-Lipschitz transformation of the domain, with the
466:
135:
exists almost everywhere. This is a consequence of a special case of the
123:. The first step of the proof is to show that, for any fixed unit vector
1224:. Grundlehren der mathematischen Wissenschaften. Vol. 338. Berlin:
1174:
1109:
572:
145:
has
Lebesgue measure zero if its restriction to every line parallel to
854:
599:
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36:
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1340:(Rademacher's theorem with a proof is on page 18 and further.)
1091:
776:
764:
740:
509:-directional derivative exist everywhere on the complement of
752:
648:. CalderĂłn's theorem is a relatively direct corollary of the
1335:
Lectures at the 14th Jyväskylä Summer School in August 2004
582:
Rademacher's theorem is also significant in the study of
457:
At this point in the proof, the gradient (defined as the
557:
Rademacher's theorem can be used to prove that, for any
951:
Real analysis. Modern techniques and their applications
534:
directional derivative everywhere on the complement of
1140:(Third edition of 1966 original ed.). New York:
189:
606:, second-order differentiability is achieved by the
638:is differentiable almost everywhere, provided that
541:Morrey's proof can also be put into the context of
513:, and are linked by the dot product. By selecting
407:
1346:
1042:Multiple integrals in the calculus of variations
498:of measure zero such that the gradient and each
851:Measure theory and fine properties of functions
430:in the above expression by the corresponding
180:. This is based upon the following identity:
845:
825:
794:
722:
452:fundamental lemma of calculus of variations
1086:
117:denote a Lipschitz-continuous function on
375:
266:
1325:
986:
829:
1216:
945:
895:
810:
758:
730:
706:
702:
1347:
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746:
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992:Lectures on analysis on metric spaces
726:
710:
621:proved the more general fact that if
1180:Lectures on geometric measure theory
13:
613:
426:can be applied to replace the two
418:Using the Lipschitz assumption on
382:
273:
96:
14:
1371:
1319:
675:instead of the usual derivative.
1328:"Lectures on Lipschitz Analysis"
650:Lebesgue differentiation theorem
598:. Higher-order concepts such as
306:
197:
552:
1222:Optimal transport. Old and new
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393:
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354:
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1:
1273:Graduate Texts in Mathematics
849:; Gariepy, Ronald F. (2015).
690:
424:dominated convergence theorem
89:differentiable form a set of
446:, the same can be proved of
7:
1275:. Vol. 120. New York:
1267:Ziemer, William P. (1989).
955:John Wiley & Sons, Inc.
678:
660:is an element of the space
631:then every function in the
438:-directional derivative of
170:-directional derivative of
159:-directional derivative of
131:-directional derivative of
31:, states the following: If
10:
1376:
1360:Theorems in measure theory
994:. Universitext. New York:
625:is an open bounded set in
1285:10.1007/978-1-4612-1015-3
1234:10.1007/978-3-540-71050-9
1138:Real and complex analysis
1054:10.1007/978-3-540-69952-1
1004:10.1007/978-1-4613-0131-8
913:10.1007/978-3-642-62010-2
801:, pp. 243, 249, 281.
654:Sobolev embedding theorem
77:; that is, the points in
901:Geometric measure theory
826:Evans & Gariepy 2015
795:Evans & Gariepy 2015
723:Evans & Gariepy 2015
584:geometric measure theory
494:, there is a single set
113:. In the following, let
1326:Heinonen, Juha (2004).
543:generalized derivatives
1038:Morrey, Charles B. Jr.
409:
139:: a measurable set in
1097:Mathematische Annalen
410:
21:mathematical analysis
1142:McGraw-Hill Book Co.
733:, Theorem 10.8(ii);
673:metric differentials
428:difference quotients
187:
63:Lipschitz continuous
25:Rademacher's theorem
16:Mathematical theorem
1110:10.1007/BF01498415
947:Folland, Gerald B.
847:Evans, Lawrence C.
705:, Theorem 2.9.19;
608:Alexandrov theorem
547:approximate limits
405:
69:is differentiable
1243:978-3-540-71049-3
1063:978-3-540-69915-6
922:978-3-540-60656-7
872:978-1-4822-4238-6
361:
252:
104:bounded variation
71:almost everywhere
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1088:Rademacher, Hans
1083:
1033:
983:
942:
897:Federer, Herbert
892:
833:
823:
814:
813:, Theorem 14.25.
808:
802:
792:
786:
785:, Theorem 3.1.7.
780:
774:
773:, Theorem 2.2.2.
768:
762:
756:
750:
749:, Theorem 3.1.6.
744:
738:
720:
714:
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685:Pansu derivative
663:
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619:Alberto CalderĂłn
588:rectifiable sets
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91:Lebesgue measure
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60:
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34:
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1277:Springer-Verlag
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1226:Springer-Verlag
1218:Villani, CĂ©dric
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1046:Springer-Verlag
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996:Springer-Verlag
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905:Springer-Verlag
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836:
828:, Section 4.2;
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817:
809:
805:
797:, p. 151;
793:
789:
781:
777:
769:
765:
757:
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729:, Section 2.1;
725:, Section 3.1;
721:
717:
709:, Section 3.5;
701:
697:
693:
681:
661:
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614:Generalizations
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132:
128:
124:
118:
114:
99:
97:Sketch of proof
82:
78:
74:
66:
48:
40:
32:
29:Hans Rademacher
17:
12:
11:
5:
1373:
1363:
1362:
1357:
1355:Lipschitz maps
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1342:
1321:
1320:External links
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1316:
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988:Heinonen, Juha
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893:
871:
863:10.1201/b18333
842:
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763:
761:, Section 3.1.
751:
739:
737:, Section 2.2.
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137:Fubini theorem
111:Charles Morrey
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27:, named after
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832:, Section 6.
811:Villani 2009
806:
790:
778:
766:
759:Federer 1969
754:
742:
731:Villani 2009
718:
713:, Chapter 7.
707:Folland 1999
703:Federer 1969
698:
671:in terms of
669:metric space
666:
644:
640:
627:
617:
581:
559:
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553:Applications
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42:
24:
18:
1175:Simon, Leon
799:Ziemer 1989
783:Morrey 1966
771:Ziemer 1989
747:Morrey 1966
735:Ziemer 1989
473:equals the
467:dot product
37:open subset
1349:Categories
1311:0692.46022
1260:1156.53003
1210:0546.49019
1167:0925.00005
1118:47.0243.01
1080:1213.49002
1030:0985.46008
980:0924.28001
939:0176.00801
889:1310.28001
727:Simon 1983
711:Rudin 1987
691:References
573:chain rule
155:where the
855:CRC Press
662:W(Ω)
636:W(Ω)
604:convexity
600:curvature
577:W(Ω)
569:W(Ω)
352:ν
346:−
337:ζ
334:−
322:ζ
301:∫
297:−
255:ζ
234:−
228:ν
192:∫
81:at which
1220:(2009).
1177:(1983).
1136:(1987).
1090:(1919).
1040:(1966).
990:(2001).
949:(1999).
899:(1969).
679:See also
450:via the
1303:1014685
1252:2459454
1202:0756417
1159:0924157
1126:1511935
1072:0202511
1022:1800917
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564:, the
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