ABSTRACT
In this chapter we prove some uniform estimates for the derivatives of the function T (t)f , when f ∈ Cb(RN ) and {T (t)} is the semigroup associated with the uniformly elliptic operator
A = N∑
qij(x)Dij +
bj(x)Dj ,
with unbounded coefficients in RN . The problem of estimating the derivatives of T (t)f has been studied in
literature with both analytic ([13, 15, 108]) and probabilistic methods ([29, 30, 143]). Here, we present the results of [18]. More precisely, we prove uniform es-
timates for the first-, second-and third-order derivatives of T (t)f . First, we show that, for any ω > 0 and any k, l ∈ N, with 0 ≤ k ≤ l ≤ 3, there exists a positive constant Ck,l = Ck,l(ω) such that
||T (t)f ||Cl b (RN ) ≤ Ck,lt
b (RN ), f ∈ C
N ), t > 0. (6.0.1)
Although we limit ourselves to the case when l ≤ 3, the techniques that we present work as well for l > 3 under suitable additional assumptions on the coefficients. To prove (6.0.1) we use the Bernstein method (see [14]) and approximate
T (t)f by solutions of Cauchy problems in bounded domains. We assume weak dissipativity-type and growth conditions on the coefficients of A. We notice that some dissipativity condition is necessary, because in general the estimate (6.0.1) fails; see Example 6.1.11. By interpolation, we can then extend the estimate (6.0.1) to the case when
k, l ∈ R+, 0 ≤ k ≤ l ≤ 3. This allows us to prove optimal Schauder estimates for the solution of the nonhomogeneous Cauchy problem{
Dtu(t, x) = Au(t, x) + g(t, x), t > 0, x ∈ RN ,
u(0, x) = f(x), x ∈ RN ,
as well as for the elliptic equation λu−Au = f , (λ > 0).
