THE HILLE–YOSIDA THEOREM FOR LOCAL CONVOLUTED SEMIGROUPS

The characterization theorem for the Banach-space-valued local Laplace transform established by Keyantuo, Müller and Vieten is used to obtain a real variable characterization of generators of local convoluted semigroups. The concept of local convoluted semigroups extends that of distribution as well as ultradistribution semigroups. Complete characterizations existed only for exponentially bounded semigroups integrated $\alpha$ times, whereas for the non-exponential case generation results had been obtained in terms of complex conditions only.AMS 2000 Mathematics subject classification: Primary 47D03; 47D06; 44A10

MAXIMAL FUNCTIONS AND SUBORDINATION FOR OPERATOR GROUPS

Let $E$ be a UMD Banach space, and $L$ a positive self-adjoint operator in $\mathrm{L}^2$ of Laplace type, for which the imaginary powers $L^{-\ri t}$ form a $C_0$-group of exponential growth $0\leq\alpha\lt \pi$ on $\mathrm{L}^p(E)$, where $1\lt p\lt\lt \infty$. Suppose $G(z)$ is holomorphic inside and on the boundary of the sector $\{z:z\neq0,\ |\arg z|\leq\phi\}$, and $z^\kappa G(z)\rightarrow0$ uniformly as $z\rightarrow\infty$ for some $\kappa\gt0$ and $\phi\gt\alpha$. Then $G(tL)$ $(t \gt0)$ defines a bounded family of linear operators on $\mathrm{L}^p(E)$; and the maximal operator $f\mapsto\sup_{t \gt 0}\|G(tL)f\|_E$ is bounded on the domain of $\log L$. The proof uses transference methods. These hypotheses hold for the maximal solution operators for the heat, wave and Schrödinger equations, and for Cesàro sums.AMS 2000 Mathematics subject classification: Primary 47D03; 42B25; 47D09