Let $\omega :(0,\infty )\to (0,\infty )$ be a gauge function satisfying certain mid regularity conditions. A (signed) finite Borel measure $\mu \in {ℝ}^n$ is called $\omega$-Zygmund if there exists a positive constant $C$ such that $|\mu \left(Q_{+}\right)-\mu \left(Q_{-}\right)|\le C\omega \left(\ell \left(Q_{+}\right)\right)\left|Q_{+}\right|$ for any pair $Q_{+},Q_{-}\subset {ℝ}^n$ of adjacent cubes of the same size. Similarly, $\mu$ is called an $\omega$- symmetric measure if there exists a positive constant $C$ such that $|\mu \left(Q_{+}\right)/\mu \left(Q_{-}\right)-1|\le C\omega \left(\ell \left(Q_{+}\right)\right)$ for any pair $Q_{+},Q_{-}\subset {ℝ}^n$ of adjacent cubes of the same size, $\ell \left(Q_{+}\right)=\ell \left(Q_{-}\right)\<1$. We characterize Zygmund and symmetric measures in terms of their harmonic extensions. Also, we show that the quadratic condition...

By $F_n\left(X\right)$, $n\ge 1$, we denote the $n$-th symmetric product of a metric space $(X,d)$ as the space of the non-empty finite subsets of $X$ with at most $n$ elements endowed with the Hausdorff metric $d_H$. In this paper we shall describe that every isometry from the $n$-th symmetric product $F_n\left(X\right)$ into itself is induced by some isometry from $X$ into itself, where $X$ is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the $n$-th symmetric product of the Euclidean space up to bi-Lipschitz equivalence and...