A probabilistic proof of apriori l^p estimates for a class of divergence form elliptic operators

2020/02/10 by Tymoteusz Chojecki, Chojecki, Tymoteusz, Tomasz Komorowski +1
Economics, Econometrics and Finance · Mathematics · #Advanced Harmonic Analysis Research #Numerical methods in inverse problems #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2002.03611

openalex publication_date 2020/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose that \cal L is a divergence form differential operator of the form \cal Lf:=(1/2) e^U∇_x⋅[e^-U(I+H)∇_x f], where U is scalar valued, I identity matrix and H an anti-symmetric matrix valued function. The coefficients are not assumed to be bounded, but are C² regular. We show that if Z=∫_ℝ^de^-U(x) dx0 such that ‖ f‖_W^2,p(μ)≤ C(‖\cal Lf‖_L^q(μ)+‖f‖_L^q(μ)) for f∈ C₀^∞(ℝ^d). Here W^2,p(μ) is the Sobolev space of functions that are L^p(μ) integrable with two derivatives. Our proof is probabilistic and relies on an application of the Malliavin calculus.

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