Abstract
We show that a function in the variable exponent Sobolev spaces coincides with a Hölder continuous Sobolev function outside a small exceptional set. This gives us a method to approximate a Sobolev function with Hölder continuous functions in the Sobolev norm. Our argument is based on a Whitney-type extension and maximal function estimates. The size of the exceptional set is estimated in terms of Lebesgue measure and a capacity. In these estimates, we use the fractional maximal function as a test function for the capacity.
Highlights
Our main objective is to study the pointwise behaviour and Lusin-type approximation of functions which belong to a variable exponent Sobolev space
Variable exponent Sobolev spaces have been used in the modeling of electrorheological fluids, see, for example, [3,4,5,6,7] and references therein
We apply the fact that the fractional maximal function is smoother than the original function and it can be used as a test function for the capacity
Summary
Our main objective is to study the pointwise behaviour and Lusin-type approximation of functions which belong to a variable exponent Sobolev space. The rough philosophy behind the variable exponent Sobolev space W1,p(·)(Rn) is that the standard Lebesgue norm is replaced with the quantity u(x) p(x)dx, Rn (1.1). Variable exponent Sobolev spaces have been used in the modeling of electrorheological fluids, see, for example, [3,4,5,6,7] and references therein. Instead of approximating by smooth functions, we are interested in Lusin-type approximation of variable exponent Sobolev functions. Our result implies that every variable exponent Sobolev function can be approximated in the Lusin sense by Holder continuous Sobolev functions in the variable exponent Sobolev space norm. We apply the fact that the fractional maximal function is smoother than the original function and it can be used as a test function for the capacity
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
