Conical square functionals on Riemannian manifolds

TL;DR

This paper establishes boundedness of conical square functionals related to Schr{"o}dinger operators on Riemannian manifolds under various geometric and analytic assumptions, extending classical harmonic analysis tools to more general settings.

Contribution

It introduces new boundedness results for conical square functionals associated with Schr{"o}dinger operators on Riemannian manifolds, including generalized and Poisson semigroup variants.

Findings

Boundedness on L^p for p ≥ 2 under volume doubling condition.

Boundedness on L^p for p in (1,2) with additional diagonal estimates.

Extension to generalized conical vertical square functionals with holomorphic functions.

Abstract

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.