Heat kernel estimates for an operator with unbounded diffusion coefficients in \(\mathbb R\) and \(\mathbb R^{2}\)
The author considers the operator \(L = (1 + |x|^2)^{\alpha/2} \Delta\) for \(\alpha > 0\) in the whole space \(\mathbb{R}^N\) for \(N=1\) and \(N=2\). It is known that in these cases \(L\) generates a semigroup \((T(t))_{t \geq 0}\) in \(C_b(\mathbb{R}^N)\) which can represented by an integral kernel \[ T(t)f(x) = \int_{\mathbb{R}^N} p(x,y,t) f(y) \, dy, \qquad f \in C_b(\mathbb{R}^N). \] The author establishes pointwise upper bounds for \(p\), which were already obtained in [\textit{G. Metafune} and the author, Discrete Contin. Dyn. Syst. 32, No. 6, 2285--2299 (2012; Zbl 1252.47038)] for the case \(N \geq 3\). To exemplify the obtained results, we cite the obtained bounds for \(p\) in the case \(N=2\). In this case one has \[ p(x,y,t) \leq \frac{C}{t} (1 + |y|^2)^{-\alpha/2} \qquad \text{for } 0 < t \leq 1, x,y \in \mathbb{R}^2. \]
scientific article; zbMATH DE number 6147891