Let H be the Hermite operator $-\Delta +|x|^2$ on $\mathbb {R}^n$. We prove a weighted $L^2$ estimate of the maximal commutator operator $\sup _{R>0}|[b, S_R^\lambda (H)](f)|$, where $ [b, S_R^\lambda (H)](f) = bS_R^\lambda (H) f - S_R^\lambda (H)(bf) $ is the commutator of a BMO function b and the Bochner–Riesz means $S_R^\lambda (H)$ for the Hermite operator H. As an application, we obtain the almost everywhere convergence of $[b, S_R^\lambda (H)](f)$ for large $\lambda $ and $f\in L^p(\mathbb {R}^n)$.

We prove that the minimal operator and the maximal operator of the Hermite operator arethe same on Lp(ℝn), 4 / 3<p< 4. Thedomain and the spectrum of the minimal operator (=maximal operator) of the Hermiteoperator on Lp(ℝn),4/3 <p<4, are computed. In addition, we can give anestimate for the Lp-norm of thesolution to the initial value problem for the heat equation governed by the minimal(maximal) operator for 4/3<p<4.

Let , ρ > 2, be the ρ-variation of the heat semigroup associated to the harmonic oscillator H = ½(−Δ + |x|2). We show that if f ∈ L∞ (ℝ), the (f)(x) < ∞, a.e. x ∈ ℝ. However, we find a function G ∈ L∞ (ℝ), such that (G)(x) ∉ L∞ (ℝ). We also analyse the local behaviour in L∞ of the operator . We find that its growth is smaller than that of a standard singular integral operator. As a by-product of our work we obtain an L∞ (ℝ) function F, such that the square function

a.e. x ∈ ℝ, where is the classical Poisson kernal in ℝ.