Let \begin{document}$ {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $\end{document} denote a BMO space on \mathbb{R}^{n} associated to Neumann operator \mathcal{L}: = -\Delta_{N} . In this article we will show that function f\in is the trace of solution {\mathbb L}u u_{t}+ \mathcal{L} u 0, u(x, 0) f(x), where satisfies Carleson-type condition \begin{eqnarray*} \sup\limits_{x_B, r_B} r_B^{-n}\int_0^{r_...

The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients a are assumed to be measurable in one direction and have small BMO semi-norms in the other directions. For equations in a bounded domain, additionally we assume...

We consider the Dirichlet problem { Lu = 0 in D u = g on ∂D for two second order elliptic operators Lku = ∑n i,j=1 a i,j k (x) ∂iju(x), k = 0, 1, in a bounded Lipschitz domain D ⊂ IR. The coefficients a k belong to the space of bounded mean oscillation BMO with a suitable small BMO modulus. We assume that L0 is regular in L(∂D, dσ) for some p, 1 < p < ∞, that is, ‖Nu‖Lp ≤ C ‖g‖Lp for all contin...

Let μ be a Radon measure on R, which may be non doubling. The only condition that μ must satisfy is the size condition μ(B(x, r)) ≤ C r, for some fixed 0 < n ≤ d. Recently, the author introduced spaces of type BMO(μ) and H(μ) with properties similar to ones of the classical spaces BMO and H defined for doubling measures. These new spaces proved to be useful to study the L(μ) boundedness of Cald...