Conical square functionals on Riemannian manifolds

Let $L = \Delta + V$ be Schr{\"o}dinger operator with a non-negative potential $V$ on a complete Riemannian manifold $M$. We prove that the conical square functional associated with $L$ is bounded on $L^p$ under different assumptions. This functional is defined by $$\mathcal{G}_L (f) (x) = \left( \int_0^\infty \int_{B(x,t^{1/2})} |\nabla e^{-tL} f(y)|^2 + V |e^{-tL} f(y)|^2 \frac{\mathrm{d}t \mathrm{d}y}{Vol(y,t^{1/2})} \right)^{1/2}.$$For $p \in [2,+\infty)$ we show that it is sufficient to assume that the manifold has the volume doubling property whereas for $p \in (1,2)$ we need extra assumptions of $L^p-L^2$ of diagonal estimates for $\{ \sqrt{t} \nabla e^{-tL}, t\geq 0 \}$ and $\{ \sqrt{t} \sqrt{V} e^{-tL} , t \geq 0\}$.Given a bounded holomorphic function $F$ on some angular sector, we introduce the generalized conical vertical square functional$$\mathcal{G}_L^F (f) (x) = \left( \int_0^\infty \int_{B(x,t^{1/2})} |\nabla F(tL) f(y)|^2 + V |F(tL) f(y)|^2 \frac{\mathrm{d}t \mathrm{d}y}{Vol(y,t^{1/2})} \right)^{1/2}$$ and prove its boundedness on $L^p$ if $F$ has sufficient decay at zero and infinity. We also consider conical square functions associated with the Poisson semigroup, lower bounds, and make a link with the Riesz transform.