ABSTRACT
This chapter is devoted to prove some pointwise estimates for the first-, second-and third-order derivatives of T (t)f , when f ∈ Ckb (R
N ) (k = 0, 1, 2, 3). Under the same assumptions on the coefficients as in Chapter 6, we prove that, for any k = 1, 2, 3 and any p ∈ (1,+∞), there exists a constant Mk,p > 0 such that (
|DiT (t)f)(x)|2 ) p
≤Mk,p
( T (t)
( k∑
|Dif |2)
) p 2 ) (x), (7.0.1)
for any t > 0, any x ∈ RN and any f ∈ Ckb (R N ). Under somewhat heavier
assumptions, we show that( k∑
|DiT (t)f)(x)|2 ) p
≤ Mˆk,pe σk,pt
( T (t)
( k∑ i=1
|Dif |2)
) p 2 ) (x), (7.0.2)
for any t > 0 and any x ∈ RN , where Mˆk,p and σk,p are, respectively, a positive and a negative constant. The estimates (7.0.1) and (7.0.2) are then used to prove the sharper estimate
|(DkT (t)f)(x)|p ≤
( σ˜k,min{p,2}
1− e−eσk,min{p,2}t ϕk,min{p,2}(t)
× ( T (t)((|f |2 + . . .+ |Dk−1f |2)
2 ) ) (x), (7.0.3)
holding for any (t, x) ∈ (0,+∞) × RN , any f ∈ Ck−1b (R N ) and any p > 1,
where σ˜k,p is a real constant and ϕk,r ∈ C([0,+∞)) is a suitable function which behaves as t1−r/2 near 0 and it is such that the term in the first brackets in the right-hand side of (7.0.3) stays bounded at infinity, or it decreases to 0 exponentially. Taking the semigroup property into account, from (7.0.3) we get
|(DkT (t)f)(x)|p ≤ Ck,p eωk,pt
tpk/2 (T (t)(|f |p))(x), t > 0, x ∈ RN , (7.0.4)
T (t)f
for any f ∈ Cb(RN ), any p > 1 and some constants Ck,p > 0, blowing up as p tends to 1, and ωk,p ∈ R. In the particular case when qij(x) = δij , i.e., when A = ∆ +
∑ bi(x)Di,
we prove the estimate (7.0.1) also for p = 1. Such pointwise estimates are typical for transition semigroups of Markov processes, and they have been already studied for the first-order derivatives (k = 1); see [11, 13, 15]. Here, we present the results of [18]. On the contrary, the estimate (7.0.4) cannot be extended, in general, to
the case when p = 1. Counterexamples are easily obtained in the simple case A = ∆ (see Example 7.3.3). In the case when ω1,p ≤ 0, the estimate (7.0.4) with k = 1 allows us to
obtain a Liouville type theorem, namely, in such a situation we can show that if Au = 0, then u is constant. If ω1,p > 0, in general, such a result fails. Counterexamples are given in [128] also in the one-dimensional case. Sometimes in what follows, when there is no damage of confusion, we write
u instead of T (·)f .