AI 中文总结
研究有界李普希茨区域中分数阶拉普拉斯算子的达尔伯格定理与\(L^{2}\)可解性，通过建立定量理论、得到反向赫尔德估计等方法，得出外部狄利克雷问题可解性等结果，并应用于相关问题推导最优正则性估计。
AI 中文摘要
给定\(s\in(0,1)\)以及有界李普希茨区域\(\Omega\subset\mathbb{R}^{n}\)，我们为\(\Omega\)的\(s\)调和测度\(\omega_s^x\)建立了一个定量的达尔伯格理论。在非局部情形下，自然参考测度是\(\Omega^{c}\)中的一个积分权重\(\sigma_{s}\)，它在边界附近的行为类似于\((1 - s)\text{dist}(\cdot,\partial\Omega)^{-s}\)。我们的主要结果是关于边界中心球上密度\(d\omega_{s}^{x}/d\sigma_{s}\)的尺度不变反向赫尔德估计。由此，我们得到外部狄利克雷问题的\(L^2(\Omega^c,\sigma_s)\)可解性，以及非局部非切向极大函数的估计和自然分布类中的唯一性。加权盖林论证将反向赫尔德指数提高到\(2\)以上，从而在严格低于\(2\)的一系列指数范围内得到\(L^{q}\)可解性。我们的结果适用于与分数阶拉普拉斯算子可比的一般对称稳定算子。此外，证明与极限\(s\to1^-\)兼容，从而在非局部到局部极限中得到拉普拉斯算子的相应结果。主要的新步骤是将格林函数的分数阶泊松扎耶夫恒等式转化为李普希茨边界距离水平集上的一致平方估计。作为应用，我们推导了齐次加权狄利克雷问题和具有零外部数据的非齐次泊松问题的最优索伯列夫正则性估计。
英文摘要
Given $s\in (0,1)$ and a bounded Lipschitz domain $Ω\subset \mathbb{R}^{n}$, we establish a quantitative Dahlberg theory for the $s$-harmonic measure of $Ω$, $ω_s^x$. In the nonlocal setting, the natural reference measure is an integral weight $σ_{s}$ in $Ω^{c}$ that behaves like $(1-s)\text{dist}(\cdot, \partial Ω)^{-s}$ close to the boundary. Our main result is a scale-invariant reverse-Hölder estimate for the density $dω_{s}^{x}/dσ_{s}$ on boundary-centered balls. As a consequence, we obtain $L^2(Ω^c,σ_s)$-solvability of the exterior Dirichlet problem, with estimates for a nonlocal non-tangential maximal function and uniqueness in the natural distributional class. A weighted Gehring argument improves the reverse-Hölder exponent beyond $2$ and consequently yields $L^{q}$-solvability for a range of exponents extending strictly below $2$. Our results apply to general symmetric stable operators comparable to the fractional Laplacian. Moreover, the proofs are compatible with the limit $s\to 1^-$ and thus yield the corresponding results for the Laplacian in the nonlocal-to-local limit. The main new step is to convert a fractional Pohozaev identity for the Green function into uniform square estimates on distance level sets of a Lipschitz boundary. As applications, we derive optimal Sobolev regularity estimates for the homogeneous weighted Dirichlet problem and for the inhomogeneous Poisson problem with zero exterior data.
Comments39 pages, 3 figures
