The variable exponent Sobolev capacity and quasi-fine properties of Sobolev functions in the case

In this article we extend the known results concerning the subadditivity of capacity and the Lebesgue points of functions of the variable exponent Sobolev spaces to cover also the case p − = 1 . We show that the variable exponent Sobolev capacity is subadditive for variable exponents satisfying 1 ⩽ p < ∞ . Furthermore, we show that if the exponent is log-Hölder continuous, then the functions of the variable exponent Sobolev spaces have Lebesgue points quasieverywhere and they have quasicontinuous representatives also in the case p − = 1 . To gain these results we develop methods that are not reliant on reflexivity or maximal function arguments.