Vanishing exponential integrability for functions whose gradients belong to L(log(e+L)) |
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Authors: | David R Adams Ritva Hurri-Syrjänen |
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Institution: | a Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, KY 40506-0027, USA b Department of Mathematics, University of Helsinki, P.O. Box 4, FIN-00014, Finland |
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Abstract: | If the gradient of u(x) is nth power locally integrable on Euclidean n-space, then the integral average over a ball B of the exponential of a constant multiple of |u(x)−uB|n/(n−1), uB=average of u over B, tends to 1 as the radius of B shrinks to zero—for quasi almost all center points. This refines a result of N. Trudinger (1967). We prove here a similar result for the class of gradients in Ln(log(e+L))α, 0?α?n−1. The results depend on a capacitary strong-type inequality for these spaces. |
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Keywords: | 46E35 31B15 |
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