Let $E$ be a locally compact metric space and $\mu$ a non-negative Radon measure on $E$ (we also assume that the support is $E$).


I am concerned with holomorphic semigroups on $L^1(E,\mu)$. In particular, I assume the situation where the semigroup is determined by a symmetric Markov process on $E$. So, the semigroup is an extension of a holomorphic (contraction) semigroup on $L^2(E,\mu)$.


I know that holomorphic (contraction) semigroups on $L^2(E,\mu)$ are extended to holomorphic semigroups on $L^p(E,\mu)$ with $1<p<\infty$. However, under what conditions would the semigroups be extended to holomorphic semigroups on $L^1(E,\mu)$.　


I would appreciate if you could tell me the well-known conditions (even if there are strong restrictions).


I don't have a clear basis, but I think it is correct in the situation where $0$-order resolvents of symmetric Markov processes are bounded linear operators on $L^\infty(E,\mu)$.



Holomorphic semigroups on $L^1$ spaces
**General reference:**
A very useful overview about extrapolation properties of semigroups on the $L^p$-scale is given in Chapter 7, and in particular Section 7.2, of the survey "[Wolfgang Arendt: Semigroups and Evolution Equations: Functional
Calculus, Regularity and Kernel Estimates](https://doi.org/10.1016/S1874-5717(04)80003-3)" (this survey is Chapter 1 of the "Handbook of Differential Equations: Evolutionary Equations (2002)").


**Specific results:**


* In the third paragraph on page 64 of the survey, it is mentioned that there exists an unbounded domain $\Omega$ in $\mathbb{R}^d$ with rough boundary such that the semigroup generated by the Neumann Laplace operator is not analytic on $L^1(\Omega)$. This example is attributed to Kunstmann there (the reference given is "[Peer C. Kunstmann: $L\_p$-spectral properties of the Neumann Laplacian on horns, comets and stars (Math. Z., 2002)](https://doi.org/10.1007/s002090100313)"). I don't have access to this article right now, but it might be worthwhile to have a look at it since the aforementioned counterexample probably gives a good indication of what is *not* true.
* For open sets $\Omega \subseteq \mathbb{R}^d$ an important concept for semigroups on $L^2(\Omega)$ are *Gaussian estimates*. They are discussed in Section 7.4 of the survey article by Arendt. For semigroups that satisfy Gaussian estimates, holomorphy extrapolates from $L^2(\Omega)$ to $L^1(\Omega)$; see Subsection 7.4.3 of the survey.


Unfortunately, I don't know any criteria on $L^2(E)$ for more general spaces $E$ (but, obviously, this does not imply that such criteria don't exist).


