  Let $H=-\Delta+V$ be a Schr\"odinger operator on $\mathbb{R}^n$. We show that
gradient estimates for the heat kernel of $H$ with upper Gaussian bounds imply
polynomial decay for the kernels of certain smooth dyadic spectral operators.
The latter decay property has been known to play an important role in the
Littlewood-Paley theory for $L^p$ and Sobolev spaces. We are able to establish
the result by modifying Hebisch and the author's recent proofs. We give a
counterexample in one dimension to show that there exists $V$ in the Schwartz
class such that the long time gradient heat kernel estimate fails.
