This paper studies a new maximal operator introduced by Hytönen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The L p -boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to σfinite measure spaces with filtrations and the L p -boundedness is shown not to depend on the underlying measure space or the filtration. Martingale techniques are applied to prove that a weak type inequality is sufficient for L p -boundedness and also to provide a characterization by concave functions.2010 Mathematics Subject Classification. Primary 46E40; Secondary 42B25, 46B09.

We consider self-adjoint semigroups T t = exp(−tA) acting on L 2 ( ) and satisfying (generalized) Gaussian estimates, where is a metric measure space of homogeneous type of dimension d. The aim of the article is to show that A ⊗ Id Y admits a Hörmander type H β 2 functional calculus on L p ( ; Y) where Y is a UMD lattice, thus extending the well-known Hörmander calculus of A on L p ( ). We show that if T t is lattice positive (or merely admits an H ∞ calculus on L p ( ; Y)) then this is indeed the case. Here the derivation exponent has to satisfy β > α•d+ 1 2 , where α ∈ (0, 1) depends on p, and on convexity and concavity exponents of Y. A part of the proof is the new result that the Hardy-Littlewood maximal operator is bounded on L p ( ; Y). Moreover, our spectral multipliers satisfy square function estimates in L p ( ; Y). In a variant, we show that if e itA satisfies a dispersive L 1 ( ) → L ∞ ( ) estimate, then β > d+1 2 above is admissible independent of convexity and concavity of Y. Finally, we illustrate these results in a variety of examples.

We study a class of spectral multipliers φ(L) for the Ornstein-Uhlenbeck operator L arising from the Gaussian measure on R n and find a sufficient condition for integrability of φ(L)f in terms of the admissible conical square function and a maximal function.2010 Mathematics Subject Classification. 42B25 (Primary); 42B15, 42B30 (Secondary).

We develop a theory of `non-uniformly local' tent spaces on metric measure spaces. As our main result, we give a remarkably simple proof of the atomic decomposition.Comment: 18 pages, revised version, to appear in Publicacions Matem\`atique