HARMONIC-ANALYSIS AND ULTRACONTRACTIVITY

Let (T(t))t>0 be a symmetric contraction semigroup on the spaces L(p)(M) (1 less-than-or-equal-to p less-than-or-equal-to infinity), and let the functions phi and psi be ''regularly related''. We show that (T(t))t>0 is phi-ultracontractive, i.e., that (T(t))t>0 satisfies the condition \\T(t)f\\ infinity less-than-or-equal-to C phi(t)-1\\f\\1 for all f in L1(M) and all t in R+, if and only if the infinitesimal generator G has Sobolev embedding properties, namely, \\psi(G)-alpha f\\q less-than-or-equal-to C\\f\\p for all f in L(p)(M), whenever 1 < p < q < infinity and alpha = 1/p - 1/q. We establish some new spectral multiplier theorems and maximal function estimates. In particular, we give sufficient conditions on m for m(G) to map L(p)(M) to L(q)(M), and for the example where there exists mu in R+ such that phi(t) = t(mu) for all t in R+ , we give conditions which ensure that the maximal function sup(t>0) \t(alpha)T(t)f(.)\ is bounded.