BLO spaces associated with Laguerre polynomials expansions

TL;DR

This paper introduces BLO-type spaces related to Laguerre polynomial expansions, studies their properties, and proves boundedness of various operators like maximal functions and Riesz transforms within these spaces.

Contribution

It defines BLO spaces associated with Laguerre expansions and establishes boundedness of maximal, variation, and oscillation operators on these spaces.

Findings

BLO spaces are subspaces of BMO spaces in the Laguerre setting.

The local maximal function maps BMO to BLO.

Operators like Riesz transforms are bounded from L-infinity to BLO.

Abstract

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 $d\gamma_\alpha(x)=\frac{2}{\Gamma(\alpha+1)}e^{-x^2}x^{2\alpha+1}dx$ with $\alpha>-\frac12$. For every $a>0$, the space $\textup{BLO}_a((0,\infty),\gamma_\alpha)$ consists of all those measurable functions defined on $(0,\infty)$ having bounded lower oscillation with respect to $\gamma_\alpha$ over an admissible family $\mathcal{B}_a$ of intervals in $(0,\infty)$. The space $\textup{BLO}_a((0,\infty),\gamma_\alpha)$ is a subspace of the space $\textup{BMO}_a((0,\infty),\gamma_\alpha)$ of bounded mean oscillation functions with respect to $\gamma_\alpha$ and $\mathcal{B}_a$. The natural $a$-local centered maximal function defined by $\gamma_\alpha$ is bounded from $\textup{BMO}_a((0,\infty),\gamma_\alpha)$ into $\textup{BLO}_a((0,\infty),\gamma_\alpha)$. We prove that the maximal operator, the $\rho$-variation and the oscillation operators associated with local truncations of the Riesz transforms in the Laguerre setting are bounded from $L^\infty((0,\infty),\gamma_\alpha)$ into $\textup{BLO}_a((0,\infty),\gamma_\alpha)$. Also, we obtain a similar result for the maximal operator of local truncations for spectral Laplace transform type multipliers.