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

• Autorzy: Caroline Sweezy
• Języki publikacji: EN

Abstrakty

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.

Kategorie tematyczne

• 42B25: Maximal functions, Littlewood-Paley theory
• 35J25: Boundary value problems for second-order elliptic equations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2004
• Tom: 164
• Numer: 1
• Strony: 1-28

Daty

wydano

2004

Twórcy

autor

Caroline Sweezy

• Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, U.S.A.

Identyfikatory

DOI

10.4064/sm164-1-1