Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 143-150, 2017


Generalized Lebesgue points for Sobolev functions

Nijjwal Karak

Received July 27, 2015.  First published February 24, 2017.

Abstract:  In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X,d,\mu)$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B(x,r)$ converge to $f(x)$ when $r$ converges to $0$. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function. We show that a function $f\in M^{s,p}(X)$, $0<s\leq1$, $0<p<1$, where $X$ is a doubling metric measure space, has generalized Lebesgue points outside a set of $\mathcal{H}^h$-Hausdorff measure zero for a suitable gauge function $h$.
Keywords:  Sobolev space; metric measure space; median; generalized Lebesgue point
Classification MSC:  46E35, 28A78
DOI:  10.21136/CMJ.2017.0405-15


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Affiliations:   Nijjwal Karak, Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FI-40014, Jyväskylä, Finland, e-mail: nijjwal@gmail.com

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