arXiv 1905.032322019-05math.CA

Lorentz空间上极大算子的有界性

Boundedness properties of maximal operators on Lorentz spaces

Dariusz Kosz

备注：: 20 pages, 2 figures

摘要：

研究了中心Hardy-Littlewood极大算子$\mathcal{M}$在Lorentz空间上的映射性质.给定$p \in (1,\infty)$和度量测度空间$\mathfrak{X}$，我们让$Ω^p_{\rm HL}(\mathfrak{X}) \subset [0,1]^2$是所有对$(\frac{1}{q},\frac{1}{r})$的集合，使得$\mathcal{M}$从$L^{p,q}(\mathfrak{X})$到$L^{p,r}(\mathfrak{X})$有界。对于每个固定的$p$，表征$Ω^p_{\rm HL}(\mathfrak{X})$的所有可能形状。也就是说，我们证明了$Ω^p_{\rm HL}(\mathfrak{X})$的边界要么是空的，要么是$$\{ δ\} \times [0, \lim_{u \rightarrow δ} F(u)] \ \cup \ \{(u, F(u)) : u \in (δ, 1] \},$$的形式，其中$δ\in [0,1]$和$F \colon [δ, 1] \rightarrow [0,1]$是凹的，非减的，并满足$F(u) \leq u$。相反，对于每个这样的$F$，我们发现$\mathfrak{X}$使得$\mathcal{M}$从$L^{p,q}(\mathfrak{X})$到$L^{p,r}(\mathfrak{X})$有界当且仅当点$(\frac{1}{q}, \frac{1}{r})$位于$F$的图上或下，即$\frac{1}{q} \geq δ$和$\frac{1}{r} \leq F\big(\frac{1}{q}\big)$。

英文摘要：

We study mapping properties of the centered Hardy--Littlewood maximal operator $\mathcal{M}$ acting on Lorentz spaces. Given $p \in (1,\infty)$ and a metric measure space $\mathfrak{X}$ we let $Ω^p_{\rm HL}(\mathfrak{X}) \subset [0,1]^2$ be the set of all pairs $(\frac{1}{q},\frac{1}{r})$ such that $\mathcal{M}$ is bounded from $L^{p,q}(\mathfrak{X})$ to $L^{p,r}(\mathfrak{X})$. For each fixed $p$ all possible shapes of $Ω^p_{\rm HL}(\mathfrak{X})$ are characterized. Namely, we show that the boundary of $Ω^p_{\rm HL}(\mathfrak{X})$ either is empty or takes the form $$\{ δ\} \times [0, \lim_{u \rightarrow δ} F(u)] \ \cup \ \{(u, F(u)) : u \in (δ, 1] \},$$ where $δ\in [0,1]$ and $F \colon [δ, 1] \rightarrow [0,1]$ is concave, non-decreasing, and satisfying $F(u) \leq u$. Conversely, for each such $F$ we find $\mathfrak{X}$ such that $\mathcal{M}$ is bounded from $L^{p,q}(\mathfrak{X})$ to $L^{p,r}(\mathfrak{X})$ if and only if the point $(\frac{1}{q}, \frac{1}{r})$ lies on or under the graph of $F$, that is, $\frac{1}{q} \geq δ$ and $\frac{1}{r} \leq F\big(\frac{1}{q}\big)$.

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摘要：

英文摘要：

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