Fine properties of functions from Hajłasz–Sobolev classes M α p , p > 0, I. Lebesgue points

Journal of Contemporary Mathematical Analysis(2016)

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Let X be a metric measure space satisfying the doubling condition of order γ > 0. For a function f ∈ L loc p ( X ), p > 0 and a ball B ⊂ X by I B ( p ) f we denote the best approximation by constants in the space L p ( B ). In this paper, for functions f from Hajłasz–Sobolev classes M α p ( X ), p > 0, α > 0, we investigate the size of the set E of points for which the limit lim r →+0 I B ( x , r ) ( p ) f = f *( x ). exists. We prove that the complement of the set E has zero outer measure for some general class of outer measures (in particular, it has zero capacity). A sharp estimate of the Hausdorff dimension of this complement is given. Besides, it is shown that for x ∈ E lim_r → + 0∫_B( x,r)| f - f*( x )| ^qdμ = 0,1 / . - q = 1 / . - p - α/ . - r. Similar results are also proved for the sets where the "means" I B ( x , r ) ( p ) f converge with a specified rate.
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Key words
Metric measure space,doubling condition,Sobolev space,Lebesgue point,capacity,outer measure,Hausdorff measure and dimension
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