Author:
Baumgarth Robert,Devyver Baptiste,Güneysu Batu
Abstract
AbstractWith $$\vec {\Delta }_j\ge 0$$
Δ
→
j
≥
0
is the uniquely determined self-adjoint realization of the Laplace operator acting on j-forms on a geodesically complete Riemannian manifold M and $$\nabla $$
∇
the Levi-Civita covariant derivative, we prove among other things
a Gaussian heat kernel bound for $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$
∇
e
-
t
Δ
→
j
, if the curvature tensor of M and its covariant derivative are bounded,
an exponentially weighted $$L^p$$
L
p
-bound for the heat kernel of $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$
∇
e
-
t
Δ
→
j
, if the curvature tensor of M and its covariant derivative are bounded,
that $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$
∇
e
-
t
Δ
→
j
is bounded in $$L^p$$
L
p
for all $$1\le p<\infty $$
1
≤
p
<
∞
, if the curvature tensor of M and its covariant derivative are bounded,
a second order Davies-Gaffney estimate (in terms of $$\nabla $$
∇
and $$\vec {\Delta }_j$$
Δ
→
j
) for $$\mathrm {e}^{ -t\vec {\Delta }_j }$$
e
-
t
Δ
→
j
for small times, if the j-th degree Bochner-Lichnerowicz potential $$V_j=\vec {\Delta }_j-\nabla ^{\dagger }\nabla $$
V
j
=
Δ
→
j
-
∇
†
∇
of M is bounded from below (where $$V_1=\mathrm {Ric}$$
V
1
=
Ric
), which is shown to fail for large time, if $$V_j$$
V
j
is bounded.
Based on these results, we formulate a conjecture on the boundedness of the covariant local Riesz-transform $$\nabla (\vec {\Delta }_j+\kappa )^{-1/2}$$
∇
(
Δ
→
j
+
κ
)
-
1
/
2
in $$L^p$$
L
p
for all $$1\le p<\infty $$
1
≤
p
<
∞
(which we prove for $$1\le p\le 2$$
1
≤
p
≤
2
), and explain its implications to geometric analysis, such as the $$L^p$$
L
p
-Calderón-Zygmund inequality. Our main technical tool is a Bismut derivative formula for $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$
∇
e
-
t
Δ
→
j
.
Funder
Technische Universität Chemnitz
Publisher
Springer Science and Business Media LLC