Advanced Search

A Carleson condition on the difference function for the coefficients of two elliptic-caloric operators is shown to give absolute continuity of one measure with respect to the other on the lateral boundary. The elliptic operators can have time dependent coefficients and only one of them is assumed to have a measure which is doubling. This theorem is an extension of a result of B. Dahlberg [4] on absolute continuity for elliptic measures to the case of the heat equation. The method of proof is an...

If Ω is a Lip(1,1/2) domain, μ a doubling measure on ${\partial }_p\Omega ,\partial /\partial t-L_i$, i = 0,1, are two parabolic-type operators with coefficients bounded and measurable, 2 ≤ q < ∞, then the associated measures ${\omega }_0$, ${\omega }_1$ have the property that ${\omega }_0\in B^q\left(\mu \right)$ implies ${\omega }_1$ is absolutely continuous with respect to ${\omega }_0$ whenever a certain Carleson-type condition holds on the difference function of the coefficients of $L_1$ and $L_0$. Also ${\omega }_0\in B^q\left(\mu \right)$ implies ${\omega }_1\in B^q\left(\mu \right)$ whenever both measures are center-doubling measures. This is B. Dahlberg’s result for elliptic measures extended...

For two strictly elliptic operators L₀ and L₁ on the unit ball in ℝⁿ, whose coefficients have a difference function that satisfies a Carleson-type condition, it is shown that a pointwise comparison concerning Lusin area integrals is valid. This result is used to prove that if L₁u₁ = 0 in B₁(0) and $Su₁\in L^{\infty }\left(S^{n-1}\right)$ then ${u₁|}_{S^{n-1}}=f$ lies in the exponential square class whenever L₀ is an operator so that L₀u₀ = 0 and $Su₀\in L^{\infty }$ implies ${u₀|}_{S^{n-1}}$ is in the exponential square class; here S is the Lusin area integral. The exponential square theorem,...

Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ ℝⁿ. Let u be a solution to $Lu=div\overrightarrow{f}$ in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the $L^q(\Omega ,d\mu )$ norm of |∇u| is dominated by the $L^p(\Omega ,dv)$ norms of $div\overrightarrow{f}$ and $|\overrightarrow{f}|$. If we replace |∇u| by a local Hölder norm of u, the conditions on μ and ν can be significantly weaker.

We prove sufficiency of conditions on pairs of measures μ and ν, defined respectively on the interior and the boundary of a bounded Lipschitz domain Ω in d-dimensional Euclidean space, which ensure that, if u is the solution of the Dirichlet problem. Δu = 0 in Ω, ${u|}_{\partial \Omega }=f$, with f belonging to a reasonable test class, then $({\int }_{\Omega }{\left|\nabla u\right|}^q{d\mu )}^{1/q}\le ({\int }_{\partial \Omega }{\left|f\right|}^p{d\nu )}^{1/p}$, where 1 < p ≤ q < ∞ and q ≥ 2. Our sufficiency conditions resemble those found by Wheeden and Wilson for the Dirichlet problem on $ℝ{₊}^{d+1}$. As in that case we attack the problem by...