Ritva Hurri-Syrjänen

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.

Aspects of local-to-global results

We establish local-to-global results for a function space which is larger than the well-known bounded mean oscillation space, and was also introduced by John and Nirenberg

A quasiconformal analogue of brennan's conjecture

In this paper we generalize Brennans conjecture to the setting where we consider quasi conformal mappings ⊘ from a simply connected domain D to the unit disk. Examples are given to demonstrate that the range of exponents p for which is sharp.

Pointwise estimates to the modified Riesz potential

In a smooth domain a function can be estimated pointwise by the classical Riesz potential of its gradient. Combining this estimate with the boundedness of the classical Riesz potential yields the optimal Sobolev–Poincaré inequality. We show that this method gives a Sobolev–Poincaré inequality also for irregular domains whenever we use the modified Riesz potential which arise naturally from the geometry of the domain. The exponent of the Sobolev–Poincaré inequality depends on the domain. The Sobolev–Poincaré inequality given by this approach is not sharp for irregular domains, although the embedding for the modified Riesz potential is optimal. In order to obtain the results we prove a new pointwise estimate for the Hardy–Littlewood maximal operator.