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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.

MSC:

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

Citations:

Zbl 0859.46022
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References:

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