Let \(\mathfrak g\) be a finite-dimensional Lie algebra. A family \(V=\{V(x); x\in\mathfrak g\}\) of closed operators on a Banach space \(B\) is a representation of \(\mathfrak g\) if the common domain \(B_{\infty}(V)\) of arbitrary finite products of elements from \(V\) is norm dense and if \(V\) respects the Lie brackets. Such a representation is denoted by \((B_{\infty},{\mathfrak g},V)\). If \(x_1,\ldots,x_n\) is a basis for \(\mathfrak g\) then the Laplacian of the representation \((B_{\infty},\mathfrak g,V)\) is defined by \(\Delta =- \sum^n_1 V(x_j)^2\). The authors consider various conditions under which the Laplacian is closable and generates a continuous one parameter semigroup \(S\). They derive growth conditions of the type \(\| V(x_{i_1})\cdots V(x_{i_k})S_ta\| \leq c_k\| a\| /t^{k/2}\) from (*) \(\| V(x_i)S_ta\| \leq c\| a\| /t^{1/2}\).
These estimates are used to show that under certain circumstances (*) implies that a representation of \(\mathfrak g\) can be integrated to a continuous representation of the simply connected Lie group \(G\) with Lie algebra \(\mathfrak g\). Conversely it is shown that the differential representation \(d\pi\) of \(\mathfrak g\) associated to a continuous representation \(\pi\) of \(G\) satisfies (*). The proof uses heat equation methods and the observation that the estimates essentially only have to be proved for the regular representation on \(L^1(G)\).
The paper contains an example section which illustrates some aspects of the main results for \(sl(2,\mathbb{R})\) and the Heisenberg algebra. Finally the \(L^1\)-estimates for the heat kernel derived earlier in the paper are extended to weighted \(L^1\)-spaces.