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1
Content available remote

Elliptic functions, area integrals and the exponential square class on B₁(0) ⊆ ℝⁿ, n > 2

100%
Sweezy C.
Studia Mathematica
|
2004
|
tom 164
|
nr 1
1-28
EN
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 $Su₁ ∈ L^{∞}(S^{n-1})$ then $u₁|_{S^{n-1}} = f$ lies in the exponential square class whenever L₀ is an operator so that L₀u₀ = 0 and $Su₀ ∈ L^{∞}$ implies $u₀|_{S^{n-1}}$ is in the exponential square class; here S is the Lusin area integral. The exponential square theorem, first proved by Thomas Wolff for harmonic functions in the upper half-space, is proved on B₁(0) for constant coefficient operator solutions, thus giving a family of operators for L₀. Methods of proof include martingales and stopping time arguments.
2
Content available remote

Absolute continuity for elliptic-caloric measures

100%
Sweezy C.
Studia Mathematica
|
1996
|
tom 120
|
nr 2
95-112
EN
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 adaptation of Fefferman, Kenig and Pipher's proof of Dahlberg's result [8].
3
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$B^q$ for parabolic measures

100%
Sweezy C.
Studia Mathematica
|
1998
|
tom 131
|
nr 2
115-135
EN
If Ω is a Lip(1,1/2) domain, μ a doubling measure on $∂_{p}Ω, ∂/∂t - L_{i}$, i = 0,1, are two parabolic-type operators with coefficients bounded and measurable, 2 ≤ q < ∞, then the associated measures $ω_{0}$, $ω_{1}$ have the property that $ω_{0} ∈ B^{q}(μ)$ implies $ω_{1}$ is absolutely continuous with respect to $ω_{0}$ whenever a certain Carleson-type condition holds on the difference function of the coefficients of $L_{1}$ and $L_{0}$. Also $ω_{0} ∈ B^{q}(μ) $ implies $ω_{1} ∈ B^{q}(μ)$ whenever both measures are center-doubling measures. This is B. Dahlberg's result for elliptic measures extended to parabolic-type measures on time-varying domains. The method of proof is that of Fefferman, Kenig and Pipher.
4
Content available remote

A Littlewood-Paley type inequality with applications to the elliptic Dirichlet problem

100%
Sweezy C.
Annales Polonici Mathematici
|
2007
|
tom 90
|
nr 2
105-130
EN
Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ ℝⁿ. Let u be a solution to $Lu = div \vec{f}$ in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the $L^{q}(Ω,dμ)$ norm of |∇u| is dominated by the $L^{p}(Ω,dv)$ norms of $div \vec{f}$ and $|\vec{f}|$. If we replace |∇u| by a local Hölder norm of u, the conditions on μ and ν can be significantly weaker.
5
Content available remote

Weighted inequalities for gradients on non-smooth domains

68%
Sweezy C., Wilson J.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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, $u|_{∂Ω} = f$, with f belonging to a reasonable test class, then $(∫_{Ω} |∇u|^{q} dμ) ^{1/q} ≤ (∫_{∂Ω} |f|^{p} dν)^{1/p}$, 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 means of Littlewood-Paley theory. However, the lack of translation invariance forces us to use a general result of Wilson, which must then be translated into the setting of homogeneous spaces. We also consider what can be proved when a strictly elliptic divergence form operator replaces the Laplacian.
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