Harmonic Analysis and Ultracontractivity

Abstract

Let ${({T_t})_{t > 0}}$ be a symmetric contraction semigroup on the spaces ${L^p}(M)\;(1 \leq p \leq \infty )$, 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 ${\left \| {{T_t}f} \right \|_\infty } \leq C\phi {(t)^{ - 1}}{\left \| f \right \|_1}$ for all $f$ in ${L^1}(M)$ and all $t$ in ${{\mathbf {R}}^ + }$, if and only if the infinitesimal generator $\mathcal {G}$ has Sobolev embedding properties, namely, ${\left \| {\psi {{(\mathcal {G})}^{ - \alpha }}f} \right \|_q} \leq C{\left \| f \right \|_p}$ for all $f$ in ${L^p}(M)$, whenever $1 < p < q < \infty$ 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 $m(\mathcal {G})$ to map ${L^p}(M)$ to ${L^q}(M)$, and for the example where there exists $\mu$ in ${{\mathbf {R}}^ + }$ such that $\phi (t) = {t^\mu }$ for all $t$ in ${{\mathbf {R}}^ + }$ , we give conditions which ensure that the maximal function ${\sup _{t > 0}}|{t^\alpha }{T_t}f( \bullet )|$ is bounded.