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 Levy process

It is also reasonable to ask if the conditions in the assumption (A.1) are satisfied by more general functions of the form φ(s) = 1 if s ∈ [0, 1], e −ms β s γ if s > 1, with m, β > 0, γ ∈ R. (2) 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 Section 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. [20,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 (the constant 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 f ( Under the assumptions (A.1) and (A.2) we may consider the operator Recall the following basic fact (see [25,Lemma 1]).

Lemma 1 If (A.1), (A.2) hold and the function
In the following we always assume that the condition (A.1) is satisfied. For every ε > 0 we denote Note that the operators A ε are bounded since In fact for every ε > 0 the family of operators {e tAε , t ≤ 0} is a semigroup on B b (R d ), i.e., e (t+s)Aε = e tAε e sAε for all t, s ≥ 0, ϕ ∈ B b (R d ). We note that e tAε is positive for all t ≥ 0, ε > 0 (see (5)). Our first result is the following theorem.
Theorem 1 If (A.1) -(A.4) are satisfied then there exist the constants C 1 and C 2 such that for every nonnegative ϕ ∈ B b (R d ) and ε ∈ (0, ε 0 ∧ 1) we have for every x ∈ R d .
The proof of Theorem 1 is given in Section 2. To study a limiting semigroup we will need some additional assumptions.
(A.6) A regarded as an operator on C ∞ (R d ) is closable and its closureĀ is a generator of a strongly continuous contraction semigroup of operators {P t , t ≥ 0} on C ∞ (R d ).
The following theorem is our main result.
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 [11,12,13,14,16]. Generally, smoothness of the coefficients q, l, c, ν in (4) is sufficient for the existence (see Theorem 5.24 in [10], Theorem 4.6.7 in [17] and Lemma 2 in [25]). Other conditions are given also in [22].
Z.-Q. Chen, P. Kim and T. Kumagai in [6,7,5] investigate the case of symmetric jump-type 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 [5]) analogous to (3). In the present paper we propose completely different approach which is based on general approximation scheme recently devised in [25]. 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 [7,5] 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 [25]. 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 condi-tion (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).
Other estimates of Lévy and Lévy-type transition densities are discussed in [18,19]. In [21,23] the derivatives of stable densities have been considered, while bounds of heat kernels of the fractional Laplacian perturbed by gradient operators were studied in [2]. An alternative approximation scheme is given in [4].

Approximation
In this section we apply an approximation scheme recently devised in [25]. We have This yields that A consequence of (5) is that we may consider the operator Γ ε and its powers instead of A ε . The fact that Γ ε is positive enables for more precise estimates.
where we let f 1,ε = f ε . By induction and Fubini-Tonelli theorem we get Also, it was proved in [25,Lemma 3] that for all ε > 0, x ∈ R d , and n ∈ N The next lemma is crucial for our further investigation. The significance of the inequalities below is that before the expressions on the right hand side we obtain precisely the constants equal to one.

Lemma 2
We have the following.
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. Denote θ(z) := |z − x| −α−d , |z − x| > 0.
We now obtain estimates of f n,ε (x, y). Our argument in the proof of the following lemma shows 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 + II.

By (A.1) (a) and (6), we have
By symmetry of f (see (A.3)), induction and Lemma 2 (1), we also have We get which ends the proof of part (1). We now complete the proof of the inequality in (2). We will prove that it holds with constants c 10 = c 1 max(c 9 , 2 α+d M) and c 11 = max(c 7 , c 8 + Mc 3 ). When |x − y| ≤ 2, then it directly follows from the part (1) and (A.1)(a). Assume now that |x − y| > 2. We have

By (A.1) (a) and (6), we get
By symmetry of f (see (A.3
Using the above lemmas we may estimate Γ n ε and in consequence also the exponent operator e tAε = e −tbε e tΓε .
Proof. By (7) and Lemma 3 for every ϕ such that x ∈ supp(ϕ) we get

Discussion of examples
We now prove the condition (A.1) (c) for functions φ of the form (2) for restricted set of parameters β and γ. First we recall some well known geometric fact, see e.g. [20,Lemma 5.3].