Riesz transforms for a non-symmetric Ornstein--Uhlenbeck semigroup
Let \(({\mathcal H}_t)_{t\geq 0}\) be the Ornstein--Uhlenbeck semigroup on \(\mathbb{R}^d\) with covariance matrix \(Q\) and drift \(B\), where \(B\) is any real matrix whose eigenvalues have a negative real part, and let \(\gamma_\infty\) be the invariant measure. The corresponding Riesz transform \({\mathcal R}_\alpha\) of order \(\alpha\), where \(\alpha\) is a \(d\)-dimensional multi-index, is defined by \[ {\mathcal R}_\alpha f:= (1/\Gamma(|\alpha|/2))\,\partial^\alpha\int^\infty_0 t^{|\alpha|/2- 1}{\mathcal H}_t f\,dt \] for \(f\in L^p_0(\gamma_\infty)\), where \(L^p_0(\gamma_\infty)\) is the subspace of all functions with \(\gamma_\infty\)-mean \(0\) in \(L^p(\gamma_\infty)\). For \(1<p<\infty\), the Riesz transforms of any order are bounded as mappings from \(L^p_0(\gamma_\infty)\) to \(L^p(\gamma_\infty)\). If \(Q= I\) and \(B= -\lambda(I+ R)\), where \(\lambda> 0\) and \(R\) is a skew adjoint matrix generating a one-parameter group of periodic rotations, then the first-order Riesz transforms are of weak type 1 with respect to \(\gamma_\infty\), i.e., there exists a constant \(C\) such that \[ \gamma_\infty\{x\in \mathbb{R}^d:|{\mathcal R}_\alpha f(x)|> s\}\leq C\| f\|_1/s \] for each a with \(|\alpha|= 1\), every \(f\in L^2_0(\gamma_\infty)\), and all \(s>0\).
scientific article; zbMATH DE number 5530524
scientific article