Pointwise conditions for membership of functions in weighted Sobolev classes
It is known that the Sobolev space \(W^{1,p}(U)\) could be defined as a class of functions \(f\in L^p(U)\) so that for any element of this class there is a function \(g\in L^p(U)\) such that the following pointwise estimate \[ |f(x)-f(y)| \leq |x-y|(g(x)+g(y)) \] holds for almost all \(x,y\in U\). This characterization was employed to define Sobolev spaces on metric measure spaces [\textit{P. Hajłasz}, Potential Anal. 5, No. 4, 403--415 (1996; Zbl 0859.46022)]. Now, the author of the present paper considers the Sobolev space \(D^{1,p}_H(\mu)\) of functions defined on a Hilbert space \(H\), and with some measure \(\mu\). The question is whether the pointwise estimate defines such a Sobolev space. In particular, can we obtain the existence of the gradient from the estimate? The author gives an affirmative answer to such questions. In the case of the Gaussian Sobolev space \(W^{1,p}(\gamma)\), the main result reads as following. {Theorem 3.1.} Let \(\gamma\) be a centered Radon Gaussian measure on a locally convex space \(X\), and \(H\) be the Cameron-Martin space. Suppose that a \(\gamma\)-measurable function \(f\) satisfies the following condition: there is a sequence \((h_n)\), everywhere dense in \(H\), such that \[ |f(x+h_n) - f(x)| \leq |h_n|(g(x+h_n) - g(x))\text{ a.e.} \] where \(g\in L^p(\gamma)\). Then \(f\in W^{1,p}(\gamma)\) and \(|\nabla_H f(x)|_H \leq 2g(x)\) a.e.
scientific article; zbMATH DE number 7599891
scientific article