Pointwise Differentiability and Absolute Continuity
نویسنده
چکیده
This paper is concerned with the relationships between L differentiability and Sobolev functions. It is shown that if / is a Sobolev function with weak derivatives up to order k in L , and 0 s / s k, then / has an L derivative of order / everywhere except for a set which is small in the sense of an appropriate capacity. It is also shown that if a function has an 2V derivative P everywhere except for a set small in capacity and if these derivatives are in L , then the function is a Sobolev function. A similar analysis is applied to determine general conditions under which the Gauss-Green theorem is valid.
منابع مشابه
Laplace Transform Using the Henstock-kurzweil Integral
We consider the Laplace transform as a Henstock-Kurzweil integral. We give conditions for the existence, continuity and differentiability of the Laplace transform. A Riemann-Lebesgue Lemma is given, and it is proved that the Laplace transform of a convolution is the pointwise product of Laplace transforms.
متن کاملQualitative Fuzzy Sets and Granular Computing
A real world fuzzy set should be able to tolerate ”small amounts” of perturbations. In type II fuzzy set represents each grade is represented by a fuzzy number. In other words, each grade may be perturbed independently. Such an ”independent perturbation” is inadequate, when a membership function is required to be continuous, differentiable, and etc. Pointwise perturbation cannot guarantee the c...
متن کاملProperties of eigenvalue function
For the eigenvalue function on symmetric matrices, we have gathered a number of it’s properties.We show that this map has the properties of continuity, strict continuity, directional differentiability, Frechet differentiability, continuous differentiability. Eigenvalue function will be extended to a larger set of matrices and then the listed properties will prove again.
متن کاملGirsanov theorem for anticipative shifts on Poisson space
We study the absolute continuity of the image measure of the canonical Poisson probability measure under nonlinear shifts. The Radon-Nykodim density function is expressed using a Carleman-Fredholm determinant and a divergence operator. Results are obtained for non-necessarily invertible transformations, under almost-sure differentiability hypothesis.
متن کاملMass Transportation on Sub-Riemannian Manifolds
We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann’s Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal m...
متن کامل