Gundy-Varopoulos martingale transforms and their projection operators on manifolds and vector bundles (Q2200769)

The authors prove the \(L^p\) boundness of generalized first order Riesz transforms obtained as conditional expectations of martingale transforms of Gundy-Varopoupos type for diffusions on manifolds and vector bundles. The main result (Theorem~1.1) is as follows: For a smooth manifold \(\mathbb{M}\) with a smooth measure \(\mu\) and locally Lipschitz vector fields \(X_1,\ldots,X_d\) on \(\mathbb{M}\) consider the operator \[ L:=-\sum_{i=1}^d X_i^*X_i+V, \] where \(X_i^*\), \(i=1,\ldots,d\) are formal adjoints to \(X_i\) w.r.t. \(\mu\), \(V\,:\,\mathbb{M}\to\mathbb{R}\) is a non-positive smooth potential. Assume that \(L\) is essentially self-adjoint on the Schwartz space \(S(\mathbb{M})\). Let \(P_y:=e^{-y\sqrt{-L}}\), \(y\geq0\), be the Poisson semigroup. For \(i=1,\ldots,d\) define \[ T_if:=\int_0^{+\infty}yP_y\left( \sqrt{-L}X_i-X_i^*\sqrt{-L} \right)P_yf\,dy,\qquad f\in S(\mathbb{M}). \] Let \(p\in(1,\infty)\). Let \(p^*:=\text{max}\left\{ p,\frac{p}{p-1} \right\}\). Then for every \(f\in S(\mathbb{M})\) holds \[ \|T_i f\|_p\leq\frac{3(p^*-1)}{2}\|f\|_p. \] If the potential satisfies \(V\equiv0\), then \[ \|T_i f\|_p\leq\frac{1}{2}\cot\left(\frac{\pi}{2p^*} \right)\|f\|_p. \] As examples, the authors present the particular form of this general result in the cases when \(\mathbb{M}\) is a compact Lie group, the Heisenberg group, the group \(\mathbb{SU}(2)\). Further, the authors generalize Theorem~1.1 to the case of vector bundles and consider, as special cases, Riesz transform on forms and on spinors.