- LI-YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS 317
Applying the Sobolev inequality, we obtain for 0 < r' < r < oo
llvllL n;J;:^2 2p(Prt) :S C llvllL2P(Pr)'where Pr is a parabolic cylinder of radius r. Given ro > 0, iterating this
inequality for suitable choices of r' and r between ro and 2ro, we obtain the
L^2 mean value inequality (here we need to keep track of the constants in theiterations of the previous inequality; in regards to this, we use 2::::~ 1 k (3-k <
oo for (3 E (1,oo)) ·
llvllL=(Pro) :SC llvllv(Pzro) ·An improvement, based on combining this with IJvllL2 :S Jlvll~( llvllz~ and
an elementary iteration, yields the £^1 mean value inequality2. Li-Yau differential Harnack estimate for positive solutions of
heat-type equations with respect to evolving metrics
For a heat-type equation, a differential Harnack estimate of Li-Yau type
yields a space-time gradient estimate for a positive solution, which when
integrated compares the solution at different points in space and time. It is
one of the main ingredients in obtaining lower bounds for heat kernels (see
subsection 2.2 in Chapter 26).
In this section we shall prove differential Harnack estimates, with respect
to evolving metrics, by adjustments of Li and Yau's proof of Theorem 1.2
and Theorem 1.3 in [12l]. We state the estimates and some consequences
in subsections 2.1 and 2.2 and we begin the proofs in subsection 2.3. In the
case of the Ricci fl.ow the hypotheses one assumes are local; this is related to
the cutoff function in the maximum principle argument (i.e., localization)
and uses Perelman's changing distance estimate.
Before being immersed in the more technical exposition below, the reader
may wish to see the exposition of the Li-Yau inequality in the 'cleaner' case
of the heat equation with respect to a static metric on a closed manifold
with Re 2:: O; see Theorem 10.1 in [45].
2.1. Statement of the differential Harnack estimate and some
consequences.
Let g ( r), T E [O, T], be a smooth family of complete Riemannian metrics
on a manifold Mn. As in the previous section, define
(25.49)which is a time-dependent symmetric 2-tensor on M defined for TE [O, T],
and again let R ~ gijRij. Let
Q : M x [O, T] -+ JR.