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)−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.