Density of smooth functions in variable exponent Sobolev spaces

作者：Kostopoulos Thanasis; Yannakakis Nikos^*

来源：Nonlinear Analysis-Theory Methods & Applications, 2015, 127: 196-205.

DOI：10.1016/j.na.2015.07.007

• Variable exponent Lebesgue space
• Density of smooth functions
• Variable exponent Sobolev space
• Maximal operator
• Riesz potential

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+ < np-/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-Holder continuity imply the density of smooth functions with compact support, in the variable exponent Sobolev space W-1-,P() (Rn).

• 出版日期2015-11