JournalsrmiVol. 32, No. 3pp. 1019–1038

Sharp LpL^p estimates for Schrödinger groups

  • Piero D'Ancona

    Università di Roma La Sapienza, Italy
  • Fabio Nicola

    Politecnico di Torino, Italy
Sharp $L^p$ estimates for Schrödinger groups cover
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Abstract

Consider a non-negative self-adjoint operator HH in L2(Rd)L^2(\mathbb R^d). We suppose that its heat operator etHe^{-tH} satisfies an off-diagonal algebraic decay estimate, for some exponents p0[0,2)p_0 \in [0,2). Then we prove sharp LpLpL^p \to L^p frequency truncated estimates for the Schrödinger group eitHe^{itH} for p[p0,p0]p \in [p_0,p'_0].

In particular, our results apply to every operator of the form H=(i+A)2+VH=(i\nabla+A)^2+V, with a magnetic potential ALloc2(Rd,Rd)A \in L^2_{\mathrm {loc}}(\mathbb R^d,\mathbb R^d) and an electric potential VV whose positive and negative parts are in the local Kato class and in the Kato class, respectively.

Cite this article

Piero D'Ancona, Fabio Nicola, Sharp LpL^p estimates for Schrödinger groups. Rev. Mat. Iberoam. 32 (2016), no. 3, pp. 1019–1038

DOI 10.4171/RMI/907