BLO spaces associated with Laguerre polynomials expansions

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.