Uniform Complex Time Heat Kernel Estimates Without Gaussian Bounds

In this paper, first we consider the uniform complex time heat kernel estimates of $e^{-z(-\Delta)^{\frac{\alpha}{2}}}$ for $\alpha>0, z\in \mathbb{C}^+$. When $\frac{\alpha}{2}$ is not an integer, generally the heat kernel doest not have the Gaussian upper bounds for real time. Thus the Phragm\'en-Lindel\"of methods fail to give the uniform complex time estimates. Instead, our first result gives the asymptotic estimates for $P(z, x)$ as $z$ tending to the imaginary axis. Then we prove the uniform complex time heat kernel estimates. Finally we also show the uniform estimates of analytic semigroup generated by $H=(-\Delta)^{\frac{\alpha}{2}}+V$ where $V$ belongs to higher order Kato class.

where c n is a constant determined by the dimension.Recently, the fractional Laplace operator has been extensively studied due to its wide applications in nonlinear optics, plasma physics and other areas.See for example [6,7,11,12,15,17,18,20] and references therein.
In this paper, first we focus on the uniform estimates of the heat kernel P (z, x) for z ∈ C + .Now we recall some known facts about the heat kernel.
Moreover, when α are even numbers, the upper bounds can be improved into the sub-Gaussian type upper bounds in the following sense, for some positive constants C 1 , C 2 > 0. See, for example [2,10,17].
One important way to deduce the uniform complex time heat kernel estimates from the real time heat kernel estimates is by the Phragmén-Lindelöf theorems.Davies in [8] introduced this method to obtain the uniform complex time heat kernel estimates from the Gaussian upper bounds for the real time.Further, Carron et al in [4] proved the uniform complex time estimates for heat kernel satisfying the sub-Gaussian upper bounds for real time.In particular, by [4,Proposition 4.1] and (1.3), there exist positive constants C 1 , C 2 > 0 such that (1.4) where θ = arg z and α are even numbers.For more results concerning the Phragmén-Lindelöf methods and their applications, we refer the readers to [5,8,9,13,24] and references therein.
However, when α > 0 is not an even number, the sub-Gaussian estimates do not hold in general and hence the Phragmén-Lindelöf methods in [5,8] fail to give the uniform complex time heat kernel estimates.On the other hand, by simple calculations, there exists C > 0 such that To the best of our knowledge, we can not find the uniform complex time estimates in the literature for general α > 0 except the trivial estimates (1.5), even though the estimates for P (z, x) are well known when z are real numbers or pure imaginary numbers.
To get the desired results without the Gaussian upper bounds, we investigate the asymptotic behavior of P (z, x) as |x| → 0 and |x| → ∞ uniformly for z satisfying 0 < ω ≤ | arg z| < π 2 .Then our first results are as follows.Theorem 1.1.Let α > 0 and P (z, x) be defined by (1.1).
(1) When 0 < α < 1, then for any 0 < ε ≪ 1, there exists constant C > 0 and µ ε,V depending on V, ε, such that (2) When α > 1, then for any 0 < ε ≪ 1, there exists constant The paper is organized as follows: In section 2, we will show the asymptotic behavior of P (z, x) as |x| → 0 and |x| → ∞ uniformly for z satisfying 0 < ω ≤ | arg z| < π 2 .The proof relies heavily on properties of Bessel functions and we mainly apply the integration by parts as well as stationary phase methods.The calculation however is complicate.Section 3 is devoted to Theorem 1.1, Theorem 1.3.We will apply the heat kernel estimates in Theorem 1.1 and some global characterizations of K α (R n ) to show Theorem 1.3.In the appendix ,we gather some basic properties of Bessel functions.
Note that the constants δ, c, C, C k , C ′ k for k ∈ N may change from line to line.

Uniform Asymptotic Behavior of P (z, x)
In this section, we denote by e iθ = z for simplicity.Recall that, when z = iℑz is pure imaginary number, the estimates for P (iℑz, x) are well known.As we shall see, the behaviors of P (iℑz, x) are quite different from that of the real time heat kernel.To state the results, we make some reduction.By scaling property, we obtain where θ = arg z.Moreover, since P (e −iθ , −y) = P (e iθ , y), it is sufficient to consider P (e iθ , y) for 0 First of all, we have where ϕ(t) is a smooth cutoff function which equals 1 for 0 ≤ t ≤ 1 2 and 0 for t ≥ 1.For P 1 (i, y), there exists constant C > 0 such that Note that the above estimates are essentially known in various literature.For completeness, we still give a proof below.See for example the proof of (2.1).
When α > 1, P 2 (i, y) is smooth in R n and we have 1 is a constant determined by α, n.Then the asymptotic behaviors of P (i, y) will totally be determined by P 1 (i, y), P 2 (i, y).
As we have seen, the asymptotic behaviors of P (e iθ , y) are quite different between θ = 0 and θ = π 2 as |y| → 0 and |y| → ∞.Moreover, in the case that α 2 is integer, the upper bounds in (1.4) tends to infinity as θ → π 2 .They fail to give the upper bounds for P (i, y).Thus it is natural to ask the question that how P (e iθ , y) changes as θ → π 2 .Now we give an example which is heuristic for our problems.Consider where z ∈ C + and m ≥ 2 is an integer.Integration by parts gives It follows that I(z) has different behaviors between z = is and z = s as s → +∞.However, the main contribution to I(z, m) as |z| → +∞ is z −m − e −z z −1 uniformly for ℜz ≥ 0.
As we have shown in the example, to determine the asymptotic behavior of P (e iθ , y) uniformly for 0 < ω ≤ |θ| < π 2 , we need to find a balance between the two cases: θ = 0, θ = π where ϕ(t) is a smooth cutoff function which equals 1 for 0 ≤ t ≤ 1 2 and 0 for t ≥ 1.Then our first result is as follows.

Proof of Proposition 2.1 (1).
Proof of (2.1).Set It is direct to check L(y, D)e iy•ξ = e iy•ξ and L * is the conjugate operator to L. Thus integration by parts gives where ϕ is smooth cutoff and δ > 0 will be determined later.
For I, we obtain In the last inequality, we have used the facts that ϕ ′ (|ξ|) is supported in Now we turn to the estimates of II.Integration by parts for N times with some Combing the estimates of I and II gives As a result, (2.1) follows since P 1 (z, y) is bounded for y ∈ R n .2.2.Proof of of Proposition 2.1 (2).Note that it is sufficient to consider 0 < ω ≤ θ < π 2 , since P (z, y) = P (z, −y).Proof of (2.2).Since ϕ(|ξ|) is supported in |ξ| ≤ 1, we have It is clear that Thus it is sufficient to consider where A = |y| α α−1 and we have changed the variable r = |y| 1 α−1 s in the second equality.Note that A → +∞ as |y| → 0 when 0 < α < 1. 7 Then we have where 2), we need further to estimate I. (2.9) First we have, According to (4.1), we have where and Moreover, integration by parts gives Combing these estimates gives (2.10) where H 1 (A) satisfies for some positive constant C > 0.
On the other hand, integration by parts gives After [ n α ] + 1 steps of integrating by parts, we obtain Furthermore there exists C > 0 determined by α, n such that where H 2 (A) satisfies Thus (2.10) and (2.11) imply where as in (2.11).
To estimates J 3 , we separate the integral into two parts It is clear that Integration by parts for N = [ n+1 2 ] + 1 times gives where β k ≥ 0 are integers satisfying
For J 2 , we will apply the oscillatory integral theories.For this purpose, J 2 can be written as
To start with, consider where s 0 = (α sin θ) 1 1−α and δ will be determined later.Set N 1 = [α + n+1 2 ] + 1 and we have where When We have used the facts As a result, the following estimate holds for |H 4 |(A), Together with the following estimates and In turn, we obtain that (2.28) where C 1 , C 2 , C 3 > 0 are only determined by n, α, ω.Since ψ(sA Then the proof are almost the same as in the case 0 < α < 1 and we omit the details.It follows that (2.31) (2.32) The estimates for are easer than the above proof due to the facts there is no critical points.The proof are minor correction to the above arguments and we omit the detail.Combing (2.28), (2.29), (2.30), (2.31), (2.32) implies (2.4).
Now we are ready to consider the fractional Schrödinger operator with Kato potentials.We adopt the methods in [3,17] to prove Theorem 1.3.Set According to (1.6), we have In the second step, we have used the facts Then there exist constants D 1 , D 2 > 0 depending only on n, α such that Next we only prove (1.8) in details cause minor correction of the proof will show (1.9).Following [17], we need some characterizations of Kato potentials.
Proof.The proof can be found in [17].
Denote by H = e iθ (−∆) The proof of the last inequality can be found in [17].Then we have by induction, Let ω = eD 1 D 3 D 4 and by the definition of KV (ζ) we get the desired result.
To proceed, let where f ∈ L 1 .Then we have the following lemma.
Then the following holds for every |z| > 0 Proof.Note first that e iθ (−∆) Then H generates an analytic semigroup and hence can be represented as for certain proper path Γ Moreover there exist large enough ω > 0 and ε > 0 such that the following holds for As a result, for µ ∈ ω + Σ π−|θ| we have Thus we have completed the proof.

Appendix
In this section, we gather some facts about the Bessel functions as well as the auxiliary functions which are frequently used.
Denote by J ν (z) the bessel function for ℜν > − In our proof, the following properties of the auxiliary functions have been used.