Abstract
Let Z be a subordinate Brownian motion in \(\mathbb {R}^{d}\), d ≥ 2, via a subordinator with Laplace exponent ϕ. We kill the process Z upon exiting a bounded open set \(D\subset \mathbb {R}^{d}\) to obtain the killed process ZD, and then we subordinate the process ZD by a subordinator with Laplace exponent ψ. The resulting process is denoted by YD. Both ϕ and ψ are assumed to satisfy certain weak scaling conditions at infinity. We study the potential theory of YD, in particular the boundary theory. First, in case that D is a κ-fat bounded open set, we show that the Harnack inequality holds. If, in addition, D satisfies the local exterior volume condition, then we prove the Carleson estimate. In case D is a smooth open set and the lower weak scaling index of ψ is strictly larger than 1/2, we establish the boundary Harnack principle with explicit decay rate near the boundary of D. On the other hand, when ψ(λ) = λγ with γ ∈ (0, 1/2], we show that the boundary Harnack principle near the boundary of D fails for any bounded C1,1 open set D. Our results give the first example where the Carleson estimate holds true, but the boundary Harnack principle does not. One of the main ingredients in the proofs is the sharp two-sided estimates of the Green function of YD. Under an additional condition on ψ, we establish sharp two-sided estimates of the jumping kernel of YD which exhibit some unexpected boundary behavior. We also prove a boundary Harnack principle for non-negative functions harmonic in a smooth open set E strictly contained in D, showing that the behavior of YD in the interior of D is determined by the composition ψ ∘ ϕ.
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We thank the referee for carefully reading the manuscript and providing some useful suggestions.
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Panki Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIP) (No. 2016R1E1A1A01941893).
Renming Song was supported in part by a grant from the Simons Foundation (#429343, Renming Song).
Zoran Vondraček was supported in part by the Croatian Science Foundation under the project 3526.
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Kim, P., Song, R. & Vondraček, Z. On the Boundary Theory of Subordinate Killed Lévy Processes. Potential Anal 53, 131–181 (2020). https://doi.org/10.1007/s11118-019-09762-2
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DOI: https://doi.org/10.1007/s11118-019-09762-2