The Lebesgue differentiation theorem revisited

E. Dubon; A. San Antolín

doi:10.1016/j.exmath.2019.02.001

A Fractional Differentiation Theorem for the Laplace Transform

Canadian Mathematical Bulletin ◽

10.4153/cmb-1975-107-2 ◽

1975 ◽

Vol 18 (4) ◽

pp. 605-606 ◽

Cited By ~ 1

Author(s):

J. Conlan ◽

E. L. Koh

Keyword(s):

Laplace Transform ◽

Systems Analysis ◽

Inverse Image ◽

Single Variable ◽

Fractional Differentiation ◽

The Laplace Transform ◽

Differentiation Theorem

In certain systems analysis ([1], [2], [3]), it is essential to invert the n-dimensional Laplace transform and specify the inverse image at a single variable t.

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Differentiation of Multiparameter Superadditive Processes

Canadian Journal of Mathematics ◽

10.4153/cjm-1985-023-6 ◽

1985 ◽

Vol 37 (3) ◽

pp. 385-404

Author(s):

Doğan Çömez

Keyword(s):

Continuous Semigroup ◽

Strongly Continuous Semigroup ◽

Semigroup Of Operators ◽

Markovian Semigroup ◽

Extra Assumption ◽

Differentiation Theorem

In this article our purpose is to prove a differentiation theorem for multiparameter processes which are strongly superadditive with respect to a strongly continuous semigroup of positive L1 contractions (see Section 1 for definitions).Recently, the differentiation theorem for superadditive processes with respect to a one-parameter semigroup of positive L1-contractions has been proved by D. Feyel [9]. Another proof is given by M. A. Akçoğlu [1]. R. Emilion and B. Hachem [7] also proved the same theorem, but with an extra assumption on the process (see also [1]). The proof of this theorem for superadditive processes with respect to a Markovian semigroup of operators on L1 is given by M. A. Akçoğlu and U. Krengel [4]. Thus [1] and [9] extend the result of [4] to the sub-Markovian setting. Here we will obtain the multiparameter sub-Markovian version of this theorem, namely Theorem 3.17 below

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The Lebesgue Differentiation Theorem via the Rising Sun Lemma

Real Analysis Exchange ◽

10.14321/realanalexch.29.2.0947 ◽

2004 ◽

Vol 29 (2) ◽

pp. 947 ◽

Cited By ~ 1

Author(s):

Claude-Alain Faure

Keyword(s):

Differentiation Theorem

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Besicovitch Covering Property on graded groups and applications to measure differentiation

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2016-0051 ◽

2019 ◽

Vol 2019 (750) ◽

pp. 241-297 ◽

Cited By ~ 2

Author(s):

Enrico Le Donne ◽

Séverine Rigot

Keyword(s):

Riemannian Manifold ◽

Borel Measure ◽

Blow Up ◽

General Study ◽

Covering Property ◽

Locally Finite ◽

Homogeneous Groups ◽

Finite Borel Measure ◽

Stratified Group ◽

Differentiation Theorem

Abstract We give a complete answer to which homogeneous groups admit homogeneous distances for which the Besicovitch Covering Property (BCP) holds. In particular, we prove that a stratified group admits homogeneous distances for which BCP holds if and only if the group has step 1 or 2. These results are obtained as consequences of a more general study of homogeneous quasi-distances on graded groups. Namely, we prove that a positively graded group admits continuous homogeneous quasi-distances satisfying BCP if and only if any two different layers of the associated positive grading of its Lie algebra commute. The validity of BCP has several consequences. Its connections with the theory of differentiation of measures is one of the main motivations of the present paper. As a consequence of our results, we get for instance that a stratified group can be equipped with some homogeneous distance so that the differentiation theorem holds for each locally finite Borel measure if and only if the group has step 1 or 2. The techniques developed in this paper allow also us to prove that sub-Riemannian distances on stratified groups of step 2 or higher never satisfy BCP. Using blow-up techniques this is shown to imply that on a sub-Riemannian manifold the differentiation theorem does not hold for some locally finite Borel measure.

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