Publishing house / Journals and Serials / Studia Mathematica / All issues

Abstract

Let L be the full laplacian on the Heisenberg group $ℍ^{n}$ of arbitrary dimension n. Then for $f ∈ L^{2}(ℍ^{n})$ such that $(I-L)^{s/2}f ∈ L^{2}(ℍ^{n})$, s > 3/4, for a $ϕ ∈ C_{c}(ℍ^{n})$ we have $ʃ_{ℍ^{n}} |ϕ(x)| sup_{0 < t≤1} |e^{(√-1)tL}f(x)|^{2} dx ≤ C_{ϕ} ∥f∥_{W^{s}}^{2}$. On the other hand, the above maximal estimate fails for s < 1/4. If Δ is the sublaplacian on the Heisenberg group $ℍ^{n}$, then for every s < 1 there exists a sequence $f_{n} ∈ L^{2}(ℍ^{n})$ and $C_{n} > 0$ such that $(I-L)^{s/2} f_{n} ∈ L^{2}(ℍ^{n})$ and for a $ϕ ∈ C_{c}(ℍ^{n})$ we have $ʃ_{ℍ^{n}} |ϕ(x)| sup_{0 < t≤1} |e^{(√-1)tΔ} f_{n}(x)|^{2} dx ≥ C_{n} ∥f_{n}∥_{W^{s}}^{2}, lim_{n→∞}C_{n} = +∞$.