与Laguerre多项式展开相关的BLO空间
BLO spaces associated with Laguerre polynomials expansions
摘要：
本文引入与Laguerre多项式展开有关的$\textup{BLO}$型空间。本文考虑了由$dγ_α(x)=\frac{2}{Γ(α+1)}e^{-x^2}x^{2α+1}dx$和$α>-\frac12$定义的$(0,\infty)$上的概率测度.对于每个$a>0$，空间$\textup{BLO}_a((0,\infty),γ_α)$由定义在$(0,\infty)$上的所有那些可测函数组成，这些可测函数在$(0,\infty)$中的容许区间族$\mathcal{B}_a$上相对于$γ_α$具有有界下振荡。空间$\textup{BLO}_a((0,\infty),γ_α)$是关于$γ_α$和$\mathcal{B}_a$的有界平均振荡函数的空间$\textup{BMO}_a((0,\infty),γ_α)$的子空间。由$γ_α$定义的自然$a$ -局部中心极大函数从$\textup{BMO}_a((0,\infty),γ_α)$到$\textup{BLO}_a((0,\infty),γ_α)$是有界的。证明了在Laguerre背景下，Riesz变换的极大算子、$ρ$ -变分和与局部截断相关的振荡算子从$L^\infty((0,\infty),γ_α)$到$\textup{BLO}_a((0,\infty),γ_α)$是有界的.对于谱Laplace变换型乘子的局部截断极大算子，也得到了类似的结果。
英文摘要：
In this paper we introduce spaces of $\textup{BLO}$-type related to Laguerre polynomial expansions. We consider the probability measure on $(0,\infty)$ defined by $dγ_α(x)=\frac{2}{Γ(α+1)}e^{-x^2}x^{2α+1}dx$ with $α>-\frac12$. For every $a>0$, the space $\textup{BLO}_a((0,\infty),γ_α)$ consists of all those measurable functions defined on $(0,\infty)$ having bounded lower oscillation with respect to $γ_α$ over an admissible family $\mathcal{B}_a$ of intervals in $(0,\infty)$. The space $\textup{BLO}_a((0,\infty),γ_α)$ is a subspace of the space $\textup{BMO}_a((0,\infty),γ_α)$ of bounded mean oscillation functions with respect to $γ_α$ and $\mathcal{B}_a$. The natural $a$-local centered maximal function defined by $γ_α$ is bounded from $\textup{BMO}_a((0,\infty),γ_α)$ into $\textup{BLO}_a((0,\infty),γ_α)$. We prove that the maximal operator, the $ρ$-variation and the oscillation operators associated with local truncations of the Riesz transforms in the Laguerre setting are bounded from $L^\infty((0,\infty),γ_α)$ into $\textup{BLO}_a((0,\infty),γ_α)$. Also, we obtain a similar result for the maximal operator of local truncations for spectral Laplace transform type multipliers.
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