FA ] 2 2 N ov 2 01 9 Hörmander functional calculus on UMD lattice valued L p spaces under generalised Gaussian estimates

We consider self-adjoint semigroups Tt = exp(−tA) acting on L (Ω) and satisfying (generalised) Gaussian estimates, where Ω is a metric measure space of homogeneous type of dimension d. The aim of the article is to show that A ⊗ IdY admits a Hörmander type H β 2 functional calculus on L (Ω; Y ) where Y is a UMD lattice, thus extending the wellknown Hörmander calculus of A on L(Ω). We show that if Tt is lattice positive (or merely admits an H∞ calculus on L(Ω; 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 HardyLittlewood maximal operator is bounded on L(Ω; Y ). Moreover, our spectral multipliers satisfy square function estimates in L(Ω; Y ). In a variant, we show that if e satisfies a dispersive L(Ω) → L∞(Ω) 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.