We prove an existence and uniqueness theorem for the Dirichlet problem for the equation $\text{div}\left(a\left(x\right)\nabla u\right)=\text{div}f$ in an open cube $\mathrm{\Omega }\subset {\mathbb{R}}^N$, when $f$ belongs to some $L^p\left(\mathrm{\Omega }\right)$, with $p$ close to 2. Here we assume that the coefficient $a$ belongs to the space BMO($\mathrm{\Omega }$) of functions of bounded mean oscillation and verifies the condition $a\left(x\right)\ge {\lambda }_0>0$ for a.e. $x\in \mathrm{\Omega }$.

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.

We extend a result of Coifman and Dahlberg [, Proc. Sympos. Pure Math., Vol. 35, pp. 231–234; Amer. Math. Soc., Providence, 1979] on the characterization of $H^p$ spaces by singular integrals of ${ℝ}^n$ with a nonisotropic metric. Then we apply it to produce singular integral versions of generalized BMO spaces. More precisely, if $T_{\lambda }$ is the family of dilations in ${ℝ}^n$ induced by a matrix with a nonnegative eigenvalue, then there exist $2n$ singular integral operators homogeneous with respect to the dilations...