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We first prove that the realization $A_{\mathrm {min}}$ of $A:={\operatorname {\mathrm {div}}}(Q\nabla )-V$ in $L^2({\mathbb {R}}^d)$ with unbounded coefficients generates a symmetric sub-Markovian and ultracontractive semigroup on $L^2({\mathbb {R}}^d)$ which coincides on $L^2({\mathbb {R}}^d)\cap C_b({\mathbb {R}}^d)$ with the minimal semigroup generated by a realization of $A$ on $C_b({\mathbb {R}}^d)$. Moreover, using time-dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernel of $A$ and deduce some spectral properties of $A_{\min }$ in the case of polynomially and exponentially growing diffusion and potential coefficients.

Summary

Fractional calculus and the Riemann–Liouville integral are used to motivate the Hardy–Littlewood–Sobolev inequality. These are then generalised to the Riesz potential, and via heat kernels, toVaropoulos’s class of ultracontractive semigroups, which we show also satisfy this key inequality. As an application, we establish the Nash inequality for uniformly elliptic second-order diffusion operators and show that these are in the Varopoulos class.

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General kernel estimates of Schrödinger-type operators with unbounded diffusion terms

9 - Varopoulos Semigroups

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General kernel estimates of Schrödinger-type operators with unbounded diffusion terms

9 - Varopoulos Semigroups

Summary