## 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:

 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

### Keywords:

Sobolev spaces; maximal functions; capacity

Zbl 0859.46022
Full Text:

### References:

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