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Hölder Quasicontinuity in Variable Exponent Sobolev Spaces

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.

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Correspondence to Petteri Harjulehto.

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Harjulehto, P., Kinnunen, J. & Tuhkanen, K. Hölder Quasicontinuity in Variable Exponent Sobolev Spaces. J Inequal Appl 2007, 032324 (2007). https://doi.org/10.1155/2007/32324

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Keywords

  • Continuous Function
  • Sobolev Space
  • Lebesgue Measure
  • Function Estimate
  • Maximal Function
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