In this paper we introduce spaces of \(BLO \) -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 \(\alpha >-\frac{1}{2}\) . For every \(a>0\) , the space \(BLO _a((0,\infty ),\gamma _\alpha )\) 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 \(BLO _a((0,\infty ),\gamma _\alpha )\) is a subspace of the space \(BMO _a((0,\infty ),\gamma _\alpha )\) 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 \(BMO _a((0,\infty ),\gamma _\alpha )\) into \(BLO _a((0,\infty ),\gamma _\alpha )\) . 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 \(BLO _a((0,\infty ),\gamma _\alpha )\) . Also, we obtain a similar result for the maximal operator of local truncations for spectral Laplace transform type multipliers.
