Besov functions and vanishing exponential integrability
The main result of the paper reads as follows. There is a number \(\beta >0\) such that for all elements \(u \in B^s_{pq} (\mathbb{R}^n)\) of the indicated Besov spaces with \(1< p,q < \infty\) and \(sp=n\), \[ \lim_{r \to 0} r^{-n} \int\limits_{| x-x^0| \leq r} \left[ \exp \left( \beta | u(x) - u(x^0)| ^{1/q'} \right) - 1 \right] \, dx =0 \] for all \(x^0 \in \mathbb R^n \backslash E\), where \(E\) is an exceptional set expressed in terms of capacity. Proofs are based on capacity arguments. There are generalisations where \(u(x^0)\) is replaced by Taylor polynomials of order \(m\); then \((s-m)p=n\).
scientific article; zbMATH DE number 2028863