AbstractFor any dimension n ≥ 2, we consider the maximal directional Hilbert transform $$\mathscr{H}_U$$HU on ℝn associated with a direction set $$U\subseteq\mathbb{S}^{n-1}$$U⊆Sn−1: $${H_U}f\left( x \right): = \frac{1}{\pi }\mathop {\sup }\limits_{v \in U} \left| {p.v.\int {f\left( {x - tv} \right)\frac{{dt}}{t}} } \right|.$$HUf(x):=1πsupv∈U|p.v.∫f(x−tv)dtt|. The main result in this article asserts that for any exponent p ∈ (1,∞), there exists a positive constant Cp,n such that for any finite direction set $$U\subseteq\mathbb{S}^{n-1}$$U⊆Sn−1, $${\left\| {{H_U}} \right\|_{p \to p}} \geq {C_{p,n}}\sqrt {\log \# U} $$∥HU∥p→p≥Cp,nlog#U where #U denotes the cardinality of U. As a consequence, the maximal directional Hilbert transform associated with an infinite set of directions cannot be bounded on Lp(ℝn) for any n ≥ 2 and any p ∈ (1,∞). This completes a result of Karagulyan [11], who proved a similar statement for n = 2 and p = 2.

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关于最大方向希尔伯特变换

摘要对于任何维度 n ≥ 2，我们考虑与方向集 $$U\subseteq\mathbb{S}^{n-1}$$ 相关的ℝn 上的最大方向希尔伯特变换 $$\mathscr{H}_U$$HU U⊆Sn−1: $${H_U}f\left( x \right): = \frac{1}{\pi }\mathop {\sup }\limits_{v \in U} \left| {pv\int {f\left( {x - tv} \right)\frac{{dt}}{t}} } \right|.$$HUf(x):=1πsupv∈U|pv∫f(x -tv)dtt|。本文的主要结果断言，对于任何指数 p ∈ (1,∞)，存在正常数 Cp,n 使得对于任何有限方向集 $$U\subseteq\mathbb{S}^{n-1} $$U⊆Sn−1, $${\left\| {{H_U}} \right\|_{p \to p}} \geq {C_{p,n}}\sqrt {\log \# U} $$∥HU∥p→p≥Cp,nlog#U其中#U 表示U 的基数。因此，对于任何n ≥ 2 和任何p ∈ (1,∞)，与无限方向集相关联的最大方向希尔伯特变换不能在Lp(ℝn) 上有界。