Functions of vanishing mean oscillation associated with operators and applications (Q1001941)

The authors introduce new function spaces \(\mathbf{VMO}_L (\mathbb{R}^n)\) of vanishing mean oscillation associated with infinitesimal generators of analytic semigroups on \(L^2 (\mathbb{R}^n)\). Such spaces generalize the classical \(\mathbf{VMO}\) space. Let \(L^*\) denote the adjoint of the operator \(L\). The main result of the paper is that under some conditions imposed on the operator \(L\), the dual of the space \(\mathbf{VMO}_{L^*} (\mathbb{R}^n)\) coincides with the Hardy space \(H^1_L(\mathbb{R}^n)\). Moreover, a number of spaces \(\mathbf{VMO}_L (\mathbb{R}^n)\) generated by specific differential operators \(L\) are considered and relations to the classic \(\mathbf{VMO}(\mathbb{R}^n)\) and to other relevant spaces are presented.