Let $S^{\text{'}}$ be the class of tempered distributions. For $f\in S^{\text{'}}$ we denote by $J^{-\alpha }f$ the Bessel potential of $f$ of order $\alpha$. We prove that if $J^{-\alpha }f\in \mathrm{B}MO$, then for any $\lambda \in (0,1)$, $J^{-\alpha }{\left(f\right)}_{\lambda }\in \mathrm{B}MO$, where ${\left(f\right)}_{\lambda }={\lambda }^{-n}f\left(\phi ({\lambda }^{-1}·)\right)$, $\phi \in S$. Also, we give necessary and sufficient conditions in order that the Bessel potential of a tempered distribution of order $\alpha >0$ belongs to the $\mathrm{V}MO$ space.

Let $A_1,\cdots ,A_m$ be $n\times n$ real matrices such that for each $1\le i\le m,$ $A_i$ is invertible and $A_i-A_j$ is invertible for $i\ne j$. In this paper we study integral operators of the form $$Tf\left(x\right)=\int k_1(x-A_{1}y)k_2(x-A_{2}y)\cdots k_m(x-A_{m}y)f\left(y\right)\mathrm{d}y,$$ $k_i\left(y\right)={\sum }_{j\in \mathbb{Z}}{2}^{jn/q_i}{\varphi }_{i,j}\left(2^{j}y\right)$, $1\le q_i<\infty ,$ $1/q_1+1/q_2+\cdots +1/q_m=1-r,$ $0\le r<1,$ and ${\varphi }_{i,j}$ satisfying suitable regularity conditions. We obtain the boundedness of $T:H^p\left({ℝ}^n\right)\to L^q\left({ℝ}^n\right)$ for $0<p<1/r$ and $1/q=1/p-r.$ We also show that we can not expect the $H^p$-$H^q$ boundedness of this kind of operators.

If $\Omega \subset {ℝ}^n$ is a bounded domain with Lipschitz boundary $\partial \Omega$ and $\Gamma$ is an open subset of $\partial \Omega$, we prove that the following inequality $${\left({\int }_{\Omega }{\left|A\left(x\right)\nabla u\left(x\right)\right|}^{p}dx\right)}^{1/p}+{\left({\int }_{\Gamma }{\left|u\left(x\right)\right|}^{p}d{\mathscr{H}}^{n-1}\left(x\right)\right)}^{1/p}\ge c{\parallel u\parallel }_{W^{1,p}\left(\Omega \right)}$$ holds for all $u\in W^{1,p}(\Omega ;{ℝ}^m)$ and $1<p<\infty$, where $${\left(A\left(x\right)\nabla u\left(x\right)\right)}_k=\sum _{i=1}^m\sum _{j=1}^n{a}_k^{ij}\left(x\right)\frac{\partial u_i}{\partial x_j}\left(x\right)(k=1,2,...,r;r\ge m)$$ defines an elliptic differential operator of first order with continuous coefficients on $\overline{\Omega }$. As a special case we obtain $${\int }_{\Omega }{\left|\nabla u\left(x\right)F\left(x\right)+{\left(\nabla u\left(x\right)F\left(x\right)\right)}^T\right|}^{p}dx\ge c{\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^{p}dx,(*)$$ for all $u\in W^{1,p}(\Omega ;{ℝ}^n)$ vanishing on $\Gamma$, where $F:\overline{\Omega }\to M^{n\times n}\left(ℝ\right)$ is a continuous mapping with $detF\left(x\right)\ge \mu >0$. Next we show that $(*)$ is not valid if $n\ge 3$, $F\in L^{\infty }\left(\Omega \right)$ and $detF\left(x\right)=1$, but does hold if $p=2$, $\Gamma =\partial \Omega$ and $F\left(x\right)$ is symmetric and positive definite in $\Omega$.