双曲调和映射的横向Hölder准则与Dini–Zygmund端点正则性
Transversal Hölder Criteria and Dini--Zygmund Endpoint Regularity for Hyperbolic Harmonic Mappings
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中文总结 AI 辅助
该研究针对双曲调和映射，证明了0<α<1时竖直线上α-Hölder连续性与全局等价，α=1时结论不成立，需Dini–Zygmund条件恢复端点结论，通过傅里叶-贝塞尔乘子等方法完成证明。
中文摘要 AI 辅助
设n≥3，u为上半空间上有界映射，满足实双曲拉普拉斯算子的调和条件。对于0<α<1，u在竖直线上的一致α-Hölder连续性与全局α-Hölder连续性等价；当u为实值时，|u|沿竖直线趋近边界模即可满足该条件。α=1时上述结论不成立：存在间隙迹生成的双曲调和延拓，其竖直线上为Lipschitz但全局非Lipschitz；在Dini–Zygmund（等价于B_{∞,1}^1）可和性条件下可恢复端点结论。证明结合双曲泊松核的傅里叶-贝塞尔乘子、逆逼近与临界Besov估计。
英文摘要
Let $n\ge3$ and let $u$ be a bounded mapping on the upper half-space that is harmonic for the real hyperbolic Laplacian. For $0<α<1$, uniform $α$-Hölder continuity of $u$ on the vertical lines is shown to be quantitatively equivalent to global $α$-Hölder continuity. For real-valued $u$, the vertical approach of $|u|$ to its boundary modulus already suffices. Both statements fail when $α=1$: a lacunary trace produces a hyperbolic harmonic extension that is vertically Lipschitz but not globally Lipschitz. Endpoint conclusions are recovered under a Dini--Zygmund, equivalently $B_{\infty,1}^{1}$, summability condition. The proofs combine the Fourier--Bessel multiplier of the hyperbolic Poisson kernel with inverse approximation and critical Besov estimates.
