Cores of second order differential linear operators with unbounded coefficients on \(\mathbb R^N\)
Consider the linear second order differential operator \[ Au(x) = \sum_{i,j=1}^N a_{ij}(x)D_{ij}u(x) + \sum_{i=1}^N b_i(x) D_iu(x), \quad x \in {\mathbb R}^N, \tag{1} \] with real-valued continuous coefficients and satisfying the ellipticity condition \[ \sum_{i,j=1}^N a_{ij}(x)\xi_i \xi_j \geq \nu(x)|\xi|^2,\quad \xi, x \in {\mathbb R}^N, \tag{2} \] with \(\inf_K \nu > 0\) for every \(K\) compact. Denote by \(C_b({\mathbb R}^N)\) the space of real valued continuous and bounded functions and by \(C_c^\infty({\mathbb R}^N)\) the space of infinitely differentiable functions with compact support. By previous results of Metafune--Pallara--Wacker, it is known that, under appropriate assumptions, \(\lambda - A\) is bijective from \(D_{max}(A)\) to \(C_b({\mathbb R}^N)\) for every \(\lambda > 0\), and the inverse operator \(R(\lambda,A)= (\lambda - A)^{-1}\) satisfies \(\|R(\lambda,A)\|_\infty \leq 1/\lambda\). Here, \[ D_{max}(A) := \{u \in C_b(\mathbb R^N)\cap W^{2,p}_{loc}({\mathbb R}^N)| \text{ for all } 1 \leq p \leq \infty : Au \in C_b(\mathbb R^N)\}. \] The authors consider the problem whether the space \(C_c^\infty({\mathbb R}^N)\) is a bicore for \(A: D_{max}(A) \to C_b({\mathbb R}^N)\). This means the following: for every \(f \in D_{max}(A)\), there exists a sequence \((f_n)\) in \(C_c^\infty({\mathbb R}^N)\), such that \((f_n)\) and \((Af_n)\) are \(\|.\|_\infty\)-bounded and and converge respectively to \(f\) and \(Af\) uniformly in compact subsets. First, it is shown by a counterexample that this is not always the case. Next, sufficient conditions are found, implying the desired result. These results are employed to prove a Lie--Trotter product formula with a sum of operators of the form (1), improving previous results by Tessitore--Zabczyk.
scientific article; zbMATH DE number 2173807