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.
Categories: math.AP math.FA