The ρ-variation of the heat semigroup in the Hermitian setting: behaviour inL∞

Abstract

AbstractLet$\mathcal{V}_\rho(\re^{-tH})$, ρ > 2, be the ρ-variation of the heat semigroup associated to the harmonic oscillatorH= ½(−Δ + |x|2). We show that iff∈L∞(ℝ), the$\mathcal{V}_\rho(\re^{-tH})$(f)(x) < ∞, a.e.x∈ ℝ. However, we find a functionG∈L∞(ℝ), such that$\mathcal{V}_\rho(\re^{-tH})$(G)(x) ∉L∞(ℝ). We also analyse the local behaviour inL∞of the operator$\mathcal{V}_\rho(\re^{-tH})$. We find that its growth is smaller than that of a standard singular integral operator. As a by-product of our work we obtain anL∞(ℝ) functionF, such that the square functiona.e.x∈ ℝ, whereis the classical Poisson kernal in ℝ.