Density of smooth functions in variable exponent Sobolev spaces. (November 2015) - Kostopoulos, Thanasis; Yannakakis, Nikos
Abstract: We show that if p − ≥ 2, then a sufficient condition for the density of smooth functions with compact support, in the variable exponent Sobolev space W 1, p ( ⋅ ) ( R n ), is that the Riesz potentials of compactly supported functions of L p ( ⋅ ) ( R n ), are also elements of L p ( ⋅ ) ( R n ) . Using this result we then prove that the above density holds if (i) p − ≥ n or if (ii) 2 ≤ p − < n and p + < n p − n − p − . Moreover our result allows us to give an alternative proof, for the case p − ≥ 2, that the local boundedness of the maximal operator and hence local log-Hölder continuity imply the density of smooth functions with compact support, in the variable exponent Sobolev space W 1, p ( ⋅ ) ( R n ) .