在A. Zygmund微分猜想
On A. Zygmund differentiation conjecture
摘要：
考虑$v$是$R^n$上的Lipschitz单位向量场，而$K$是它的Lipschitz常数。我们证明了映射$S_s:S_s(X) = X + sv(X)$对于$0\leq |s|<1/K$是可逆的，并定义了非奇异点变换。利用这些性质，我们首先证明了$1\le p<\infty.$的L^p模微分，然后证明了存在测度为1/K的泛值集$s\in [-1/2K,1/2K]$，使得Lipschitz单位向量场$v\circ S_s^{-1}$对$L^p(\R^n)$中的所有函数满足Zygmund猜想，对每个p，$1\leq p< \infty.$
英文摘要：
Consider $v$ a Lipschitz unit vector field on $R^n$ and $K$ its Lipschitz constant. We show that the maps $S_s:S_s(X) = X + sv(X)$ are invertible for $0\leq |s|<1/K$ and define nonsingular point transformations. We use these properties to prove first the differentiation in L^p norm for $1\le p<\infty.$ Then we show the existence of a universal set of values $s\in [-1/2K,1/2K]$ of measure 1/K for which the Lipschitz unit vector fields $v\circ S_s^{-1}$ satisfy Zygmund's conjecture for all functions in $L^p(\R^n)$ and for each p, $1\leq p< \infty.$
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