Lebesgue points for Sobolev functions on metric spaces. (English) Zbl 1037.46031

Let \(M^{1,p}(X)\) be the Sobolev space of functions \(u\) defined on a metric measure space \((X,d)\) with a doubling measure \(\mu\). The space was introduced by P. Hajlasz [Potential Anal. 5, 403–415 (1999; Zbl 0859.46022)] and it is based on the Lipschitz type representation \(|(u(x)- u(y)|\leq d(x,y)(g(x)+ g(y))\), where \(g\) can be regarded as a derivative of \(u\). It is shown that the quasicontinuous representative of a function \(u\) has Lebesgue points except on a set of \(p\)-capacity zero. The proof makes use of a discrete Hardy-Littlewood maximal function and capacitary weak type estimates; the Besicovitch covering theorem and representations formulas for classical Sobolev functions are not available in metric spaces.


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems


Zbl 0859.46022
Full Text: DOI EuDML


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