AI 中文总结
本文研究了三进制鞅等周轮廓函数，证明了其为Takagi型Bellman函数，并给出了n-adic情况下鞅的下界及端点估计。
AI 中文摘要
设\(S_1\)是与\([0,1)\)上的正则\(n\)进鞅过滤相关的一阶变差。研究鞅等周轮廓\[V_n(x):=\inf_{\substack{A\subset[0,1)\ 可测\\ |A|=x}} \|S_1(\mathbbm 1_A)\|_1\]。对于三元过滤，精确确定了此轮廓，即\[V_3(x)=T_3(x):=\sum_{j=0}^{\infty}3^{-j}\psi_3(\{3^j x\})\]，其中\(\psi_3(t)=\min\left\{\frac{1 + 2\left|t-\frac{1}{2}\right|}{3},\frac{2 - 4\left|t-\frac{1}{2}\right|}{3}\right\}\)，\(0\leq t\leq1\)。所以尖锐三元轮廓是 Takagi 型 Bellman 函数，但不是通常的三元 Takagi - van der Waerden 函数\(\omega_3\)。对于一般\(n\geq2\)，证明了每个可测\(A\subset[0,1)\)满足\(\|S_1(\mathbbm 1_A)\|_1\geq\omega_n(|A|^*) \asymp_n |A|^*\log\frac{1}{|A|^*}\)，且此对数阶是尖锐的。最后，对于\(0\lt\alpha\lt1\)，证明了端点估计\(\|S_1(\mathbbm 1_A)\|_\alpha\geq |A|^*\)，且它也是尖锐的。
英文摘要
Let $S_1$ be the one-variation associated with the regular $n$-adic martingale filtration on $[0,1)$. We study the martingale isoperimetric profile \[ V_n(x):= \inf_{\substack{A\subset[0,1)\ {\rm measurable}\\ |A|=x}} \|S_1(\mathbbm 1_A)\|_1 . \] For the ternary filtration we determine this profile exactly. Namely, \[ V_3(x)=T_3(x):= \sum_{j=0}^{\infty}3^{-j}ψ_3(\{3^j x\}), \] where \[ ψ_3(t)= \min\left\{ \frac{1+2\left|t-\frac12\right|}{3}, \frac{2-4\left|t-\frac12\right|}{3} \right\}, \qquad 0\le t\le1 . \] Thus the sharp ternary profile is a Takagi-type Bellman function. It is, however, not the usual ternary Takagi--van der Waerden function $ω_3$; for example, \[ T_3(1/3)=4/9, \qquad ω_3(1/3)=1/3 . \] For general $n\ge2$, we prove that every measurable $A\subset[0,1)$ satisfies \[ \|S_1(\mathbbm 1_A)\|_1 \ge ω_n(|A|^*) \asymp_n |A|^*\log\frac1{|A|^*}, \qquad |A|^*:=\min\{|A|,1-|A|\}. \] Moreover, this logarithmic order is sharp up to a constant depending only on $n$. Finally, for every $0<α<1$, we prove the endpoint estimate \[ \|S_1(\mathbbm 1_A)\|_α\ge |A|^*, \] and show that it is sharp up to a constant depending only on $α$ and $n$.
