JournalsrmiVol. 36, No. 2pp. 455–484

Sharp LpL^p estimates for Schrödinger groups on spaces of homogeneous type

  • The Anh Bui

    Macquarie University, Sydney, Australia
  • Piero D'Ancona

    Università di Roma La Sapienza, Italy
  • Fabio Nicola

    Politecnico di Torino, Italy
Sharp $L^p$ estimates for Schrödinger groups on spaces of homogeneous type cover

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Abstract

We prove an LpL^{p} estimate

eitLφ(L)fp(1+t)sfp,tR,s=n121p\| e^{-itL} \varphi(L)f \|_{p} \lesssim (1+|t|)^s \|f\|_p, \quad t\in \mathbb{R}, \quad s=n\Big|\frac{1}{2}-\frac{1}{p}\Big|

for the Schrödinger group generated by a semibounded, self-adjoint operator LL on a metric measure space X\mathcal{X} of homogeneous type (where nn is the doubling dimension of X\mathcal{X}). The assumptions on LL are a mild Lp0Lp0L^{p_{0}}\to L^{p_{0}'} smoothing estimate and a mild L2L2L^{2}\to L^{2} off-diagonal estimate for the corresponding heat kernel etLe^{-tL}. The estimate is uniform for φ\varphi varying in bounded sets of S(R)\mathscr{S}(\mathbb{R}),or more generally of a suitable weighted Sobolev space.

We also prove, under slightly stronger assumptions on LL, that the estimate extends to

eitLφ(θL)fp(1+θ1t)sfp,θ>0,tR,\|e^{-itL} \varphi(\theta L) f\|_{p} \lesssim (1+\theta^{-1}|t|)^s \|f\|_p, \quad \theta > 0, \quad t\in \mathbb{R},

with uniformity also for θ\theta varying in bounded subsets of (0,+)(0,+\infty). For nonnegative operators uniformity holds for all θ>0\theta > 0.

Cite this article

The Anh Bui, Piero D'Ancona, Fabio Nicola, Sharp LpL^p estimates for Schrödinger groups on spaces of homogeneous type. Rev. Mat. Iberoam. 36 (2019), no. 2, pp. 455–484

DOI 10.4171/RMI/1136