发表机构
Faculty of Applied Sciences, Ho Chi Minh City University of Industry and Trade（胡志明市工业贸易大学应用科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
在支持加倍测度和弱(1,2)-庞加莱不等式的无界完备度量测度空间上，开发局部到全局方法比较正拉东测度势与倒数尺度乘数，核心是混合振荡积分，经系列证明和验证得到多种结果，还明确了自然结构假设及应用。
AI 中文摘要
我们在支持加倍测度和弱(1,2)-庞加莱不等式的无界完备度量测度空间上，开发了一种从局部到全局的方法，用于比较正拉东测度势与倒数尺度乘数。核心局部对象是真正的混合振荡积分。当π奇异时，普通的dωdω庞加莱不等式无法控制它。我们首先证明了每个正余维增益下的固定外域内容 - 容量估计。π的球增长条件进而产生容量支配和马兹亚型迹不等式，从而在归一化球上得到所需的混合庞加莱估计。有界重叠归一化覆盖使局部结果全局化，并给出双边费弗曼 - 冯不等式以及相应能量完备化的等价性。该抽象理论在具有广义薛定谔测度势的欧几里得\(A_2\)权重、反向赫尔德函数势、卡诺群以及低维奇异测度上得到验证。我们还表明自然结构假设是二次PI性质：\(A_2\)不是必需的，而仅\(p>2\)的\(A_p\)假设是不够的。欧几里得应用产生形式域等价、光滑形式核、自伴实现、预解式能量估计和局部临界乘数界。最后，该方法提供了先前发表的\(A_2\)广义薛定谔论证中缺失的混合测度步骤，并给出了自然增强测度的固定膨胀有限尺度\(A_2\)扩展，用于后来\(A_1\)理论中使用的广义庞加莱机制。
英文摘要
Let $(X,d,ω)$ be an unbounded complete doubling metric measure space supporting a weak $(1,2)$-Poincaré inequality, and let $π$ be a positive Radon measure. For a ball $B=B(x,R)$, suppose that for some $δ>0$ and $C_{\mathrm{gr}}>0$, \[ π(B(y,r))\le C_{\mathrm{gr}} \left(\frac rR\right)^δ\frac{ω(B(y,r))}{r^2}, \qquad y\in B,\quad 0<r\le R. \] Let $\operatorname{cap}_{2,ω}(K;Ω)$ denote the relative variational $2$-capacity of $K$ in an open set $Ω$. We prove that, for every fixed $Λ_0>2$, there is a constant $C>0$ such that \[ π(K)\le C\operatorname{cap}_{2,ω}(K;Λ_0B), \qquad K\subset B\ \text{compact}, \] with the same outer ball for all compact $K$. No doubling or lower-Ahlfors regularity is imposed on $π$. This fixed-domain capacitary domination yields representative-independent $L^2(dπ)$ traces and, under a lower normalization $π(B)\ge c_0ω(B)/R^2$ with $c_0>0$, the mixed $dω\,dπ$ Poincaré estimate. Summation over bounded-overlap normalized covers gives two-sided global Fefferman--Phong inequalities and identifies the corresponding homogeneous energy spaces. In $\mathbb R^d$ with a Muckenhoupt $A_2$ weight $w$, scale decay and controlled enlargement of $π$ generate the required critical-radius cover for every sufficiently large fixed threshold $A$; no additional lower reverse-doubling hypothesis on $w$ is needed. The resulting energy comparison gives equality of the measure-potential and critical-multiplier form domains, a smooth form core, a nonnegative self-adjoint realization, resolvent estimates, and local critical-multiplier bounds. The framework also covers reverse-Hölder function potentials, Carnot groups, and lower-dimensional singular measures, and provides a representative-aware capacity derivation of an earlier generalized Schrödinger energy estimate.
