Advanced Search

Given a modulus of continuity $\omega$ and $q\in [1,\infty [$ then $H_q^{\omega }$ denotes the space of all functions $f$ with the period $1$ on $ℝ$ that are locally integrable in power $q$ and whose integral modulus of continuity of power $q$ (see(1)) is majorized by a multiple of $\omega$. The moduli of continuity $\omega$ are characterized for which $H_q^{\omega }$ contains “many” functions with infinite “essential” variation on an interval of length $1$.

With help of suitable anisotropic Minkowski’s contents and Hausdorff measures some results are obtained concerning removability of singularities for solutions of partial differential equations with anisotropic growth in the vicinity of the singular set.