Upper estimates of transition densities for stable-dominated semigroups

We derive upper estimates of transition densities for Feller semigroups with jump intensities lighter than that of the rotation invariant stable Lévy process.


Introduction and preliminaries
Let α ∈ (0, 2) and d = 1, 2, . . . . For the rotation invariant α-stable Lévy process on R d with the Lévy measure In [29], estimates of semigroups of stable-dominated Feller operators are given. The corresponding Markov process is a Feller process and not necessarily a Lévy process. The name stable-dominated refers to the fact that the intensity of jumps for the investigated semigroup is dominated by (1). In the present paper, we extend the results obtained in [29] and give estimates from above for a wider class of semigroups with the intensity of jumps lighter than stable processes. We will now describe our results.
Let f : R d × R d → [0, ∞] be a Borel function. We consider the following assumptions on f .
It follows from (A.1) that there is also the constant Thus, (A.4) is a partial converse of (A.1) and we have Mc(α,d) . We note that the assumption (A.1)(c) is satisfied for every nonincreasing function for some positive constant c. Therefore, it is easy to verify that all the assumptions on φ are satisfied, e.g., for functions φ(s) = e (1−s β where γ > 0, φ(s) = 1/ log(e(s ∨ 1)), φ(s) = 1/ log log(e e (s ∨ 1)), and all their products and positive powers.
It is also reasonable to ask whether the conditions in the assumption (A.1) are satisfied by more general functions of the form In this case, both conditions (a) and (b) on φ hold for β ∈ (0, 1] with no further restrictions on parameters m and γ , while, as proven in Sect. 3, the condition (c) is satisfied when β ∈ (0, 1] and γ < d/2 + α − 1/2. Furthermore, this restriction on parameters is essential (see Remark 1 in Section 3). Note also that this range of β and γ in (2) covers, e.g., jump intensities dominated by those of isotropic relativistic stable processes (see, e.g., [23,Lemma 2.3]). For x ∈ R d and r > 0, we let B(x, r ) = {y ∈ R d : |y − x| < r }. B b (R d ) denotes the set of bounded Borel measurable functions, C k c (R d ) denotes the set of k times continuously differentiable functions with compact support, and C ∞ (R d ) is the set of continuous functions vanishing at infinity. We use c, C (with subscripts) to denote finite positive constants which depend only on φ, M, α, and the dimension d. Any additional dependence is explicitly indicated by writing, e.g., c = c(n). The value of c, C, when used without subscripts, may change from place to place. We write Under the assumptions (A.1) and (A.2), we may consider the operator

LEMMA 1. If (A.1) and (A.2) hold and the function x → f (x, y) is continuous on
In the following, we always assume that the condition (A.1) is satisfied. For every ε > 0, we denote We note that e tA ε is positive for all t ≥ 0, ε > 0 (see (5)). Our first result is the following theorem.
The proof of Theorem 1 is given in Sect. 2. To study a limiting semigroup, we will need some additional assumptions.
We note that A is conservative, i.e., for It is known that every generator G of a Feller semigroup with where is a Lévy measure (see [17,Chapter 4.5]).
The converse problem whether a given operator G generates a Feller semigroup is not completely resolved yet. For the interested reader, we remark that criteria are given, e.g., in [13][14][15][16]18]. Generally, smoothness of the coefficients q, l, c, ν in (4) is sufficient for the existence (see Theorem 5.24 in [12], Theorem 4.6.7 in [19] and Lemma 2 in [29]). Other conditions are given also in [26].
Chen et al. [6] and Chen and Kumagai [7,8] investigate the case of symmetric jumptype Markov processes on metric measure spaces by using Dirichlet forms. Under the assumption that the corresponding jump kernels are comparable with certain rotation invariant functions, they prove the existence and obtain estimates of the densities (see Theorem 1.2 in [6]) analogous to (3). In the present paper, we propose completely different approach which is based on general approximation scheme recently devised in [29]. In Theorem 2, we assume the estimate (A.1) from above but we use (A.4) as the only estimate for the size of f from below. We also emphasize that we obtain exactly φ(|x − y|) in (3) and from [8,6] follow estimates with φ(c|x − y|) for some constant c ∈ (0, 1). This seems to be essential especially in the case of exponentially localized Lévy measures. Our general framework, including a layout of lemmas, is similar to that in [29]. However, in the present case, the decay of the jump intensity may be significantly lighter than stable, and therefore, much more subtle argument is needed. Note that the new condition (A.1)(c), which is pivotal for our further investigations, is necessary for the two-sided sharp bounds similar to the right-hand side of (3).

Approximation
In this section, we apply an approximation scheme recently devised in [29] (we note that an alternative approximation scheme is given in [5]). We recall that We have This yields that A consequence of (5) is that we may consider the operator ε and its powers instead of A ε . The fact that ε is positive enables for more precise estimates. For n ∈ N, we define where we let f 1,ε = f ε . By induction and Fubini-Tonelli theorem, we get Also, it was proved in [29, Lemma 3] that for all ε > 0, x ∈ R d , and n ∈ N whenever ϕ ∈ B b (R d ). The next lemma is crucial for our further investigation. It is essential that we obtain precisely the constants equal to one before b ε (y) below.
Proof. First, we prove the statement (1). We have We only need to estimate the first integral on the right-hand side of the above equality.
Let now α ≥ 1. Again, using the Taylor expansion for θ , (9), (A.1) and (A.2), we obtain We thus see that by the two above estimates and by (A.4), we finally have for sufficiently small κ ∈ (0, 1). This ends the proof of (1). We now show the statement (2). Let |x − y| > 2. Similarly as before, we have Observe that it is enough to estimate the first integral on the right-hand side of the above-displayed equality. Denote η(z) Vol. 13 (2013) Stable-dominated semigroups 641 Using the Taylor expansion for η, (10), (A.1) and (A.2), we obtain which ends the proof.
We now obtain the estimates of f n,ε (x, y). Our argument in the proof of the following lemma shows the significance of assumptions on the dominating function φ.
Proof. We use induction. Clearly, for n = 1, both inequalities hold with constants c 9 = M , c 10 = M (and an arbitrary positive c 11 ), respectively. Consider first the inequality in (1). We will prove that it holds with constant c 9 = Mκ −α−d , where κ ∈ (0, 1) is the number from previous lemma. Let = I + I I.