Generalized Lebesgue points for Sobolev functions

Karak, Nijjwal

Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 143-150.

Résumé

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 $f\in M^{s,p}(X)$, $0$, $0$, 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 $h$.

Détail

DOI : 10.21136/CMJ.2017.0405-15

Classification : 28A78, 46E35
Keywords: Sobolev space; metric measure space; median; generalized Lebesgue point