Vanishing exponential integrability for functions whose gradients belong to Ln(log(e+L))α

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)-u(B)\n/((n-1)), u(B)=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 L-n(log(e+L))(alpha), 0less than or equal toalphaless than or equal ton-1. The results depend on a capacitary strong-type inequality for these spaces. (C) 2002 Elsevier Science (USA). All rights reserved.