Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

Maximal functions and transference for groups of operators

Gordon Blowera1

a1 Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK

Abstract

Let Δ be the Laplace operator on xs211Dd and 1 < δ < 2. Using transference methods we show that, for max {q, q/(q – 1)} < 4d/(2d + 1 – δ), the maximal function S0013091500020691_inline1 for the Schrödinger group is in Lq, for f xs2208 Lq with Δδ/2 f xs2208Lq. We obtain a similar result for the Airy group exp it Δ3/2. An abstract version of these results is obtained for bounded C0-groups eitL on subspaces of Lp spaces. Certain results extend to maximal functions defined for functions with values in U M D Banach spaces.

(Received September 03 1997)

Keywords

  • transference;
  • operator groups;
  • Schrödinger equation