ESTIMATES FOR FELLER SEMIGROUPS GENERATED

BY PSEUDODIFFERENTIAL OPERATORS

Niels Jacob and René L. Schilling

It is well known that the generator A of a Feller semigroup {T_{t}}_{t≥0} on C_{∞}(ℝ^{n}) satisfies the positive maximum principle. Under mild regularity assumptions, Ph. Courrège proved in [3] that every operator satisfying the positive maximum principle is on C_{0}^{∞}(ℝ^{n}) of the form

(0.1)  Au(x) = −q(x, D)u(x) = −(2π)^{−n/2} ∫_{ℝ^{n}} e^{ixξ}q(x, ξ)û(ξ) dξ,

where q: ℝ^{n} × ℝ^{n} → ℂ is a continuous function such that for every fixed x ∈ ℝ^{n} the function q(x, ·): ℝ^{n} → ℂ is negative definite. An interesting problem is to find sufficient conditions on q(x, ξ) which ensure that −q(x, D) —defined on C_{0}^{∞}(ℝ^{n}) by (0.1)—extends to a generator of a Feller semigroup. Using the Lévy-Khinchine representation, A can be decomposed into the sum of a second-order differential operator and a non-local operator. We are interested in the case where no differential operator term occurs. In a series of papers the first-named author gave some sufficient conditions on q(x, ξ) under which −q(x, D) extends to a generator of a Feller semigroup {T_{t}}_{t≥0}, cf. [10], [11] and the references given there. Using a martingale problem approach, W. Hoh obtained similar results under somewhat weaker assumptions, see [6] and the references there. The essential idea in [10] was to decompose q(x, ξ) into a sum

(0.2)  q(x, ξ) = q(x_{0}, ξ) + (q(x, ξ) − q(x_{0}, ξ)) ≡ q_{1}(ξ) + q_{2}(x, ξ)

and to treat q_{2}(x, ξ) as a perturbation of the continuous negative definite function q_{1}(ξ). Observe that −q_{1}(D) itself generates a Feller semigroup (subsequently denoted by {S_{t}}_{t≥0}) which is even a convolution semigroup.

In this paper we will compare the semigroups {T_{t}}_{t≥0} and {S_{t}}_{t≥0}, that is, we want to see how the perturbation q_{2}(x, ξ) can give estimates for T_{t}u − S_{t}u for certain functions u. In the first section we recall some notation and definitions and in the second section we give precise conditions on q(x, ξ), q_{1}(ξ), and q_{2}(x, ξ), such that −q(x, D) extends to a generator of a Fellerian or sub-Markovian semigroup. These results are taken from [10], [11], [6], and [8]. In Section 3 we prove the following L^{2}-estimate for T_{t}u − S_{t}u,

(0.3)  ∥T_{t}u − S_{t}u∥_{0} ≤ \frac{t^{1−ρ}}{1 − ρ}d̃_{1} ∑_{|α|≤n+1} ∥ϕ_{α}∥_{L^{1}}∥u∥_{a^{2},1−ρ},

where d̃_{1} is a constant, the functions ϕ_{α}, |α| ≤ n + 1, express the smallness of q_{2}(x, ξ) with respect to q_{1}(ξ), and ∥ · ∥_{a^{2},1−ρ} is an appropriate anisotropic Sobolev norm. Section 4 contains a pointwise estimate,

(0.4)  |T_{t}u(x) − S_{t}u(x)| ≤ d′_{2} t ∥u∥_{n+3} + d′_{3} t^{2} ∥u∥_{n+5},

where d′_{2}, d′_{3} are constants depending again on the functions ϕ_{α}, cf. Corollary 4.3 below.

Since {T_{t}}_{t≥0} is generated by the pseudodifferential operator −q(x, D), it is reasonable to consider the operator e^{−tq(x,D)} with symbol e^{−tq(x,ξ)} as an approximation of T_{t}, see Maslov, Shishmareyev [16] or Kumano-go [15, Sect. 7.4], and as general references [9], [18]. In terms of a symbolic calculus for (classical) pseudodifferential operators this approach is the most natural. In our setting, however, no symbolic calculus is available, for, in general, ξ ↦ q(x, ξ) fails to be differentiable. Thus the methods of [9], [15], [16], [18] cannot be used here. It is still possible to study the pseudodifferential operator e^{−tq(x,D)}, especially its mapping properties, in dependence of the properties of q(x, ξ). This is done in Section 5, Theorem 5.6 and its corollaries. It turns out that e^{−tq(x,D)} maps L^{2}(ℝ^{n}) continuously into the anisotropic Sobolev space H^{a^{2},k}(ℝ^{n}), where k increases with the smoothness of x ↦ q_{2}(x, ξ). Since ⋂_{k∈ℕ} H^{a^{2},k}(ℝ^{n}) ⊂ C_{b}^{∞}(ℝ^{n}), the operator e^{−tq(x,D)} maps L^{2}(ℝ^{n}) into C_{b}^{∞}(ℝ^{n}) whenever x ↦ q_{2}(x, ξ) is of class C^{∞}. Moreover, we can show that

(0.5)  |T_{t}u(x) − e^{−tq(x,D)}u(x)| ≤ t γ′∥u∥_{n+1}

holds.

Acknowledgment. Part of this work was done while the first-named author was visiting the University of Warwick. We would like to thank Prof. K. D. Elworthy for his kind hospitality. The second-named author was supported by a European Union HCM post-doctoral fellowship.

1. Some notation

Let us recall some frequently used notation and definitions. We will need the following function spaces:

B_{b}(ℝ^{n}) bounded Borel measurable functions; C^{m}(ℝ^{n}) m-times continuously differentiable functions, 0 ≤ m ≤ ∞; for

m = 0 we write C(ℝ^{n}); C_{0}^{m}(ℝ^{n}) the elements of C^{m}(ℝ^{n}) with compact support; C_{b}^{m}(ℝ^{n}) the elements of C^{m}(ℝ^{n}) that are bounded with all their partial

derivatives up to the order m; C_{u}^{m}(ℝ^{n}) the elements of C_{b}^{m}(ℝ^{n}) which are uniformly continuous with

all their partial derivatives up to the order m; C_{∞}(ℝ^{n}) continuous functions vanishing at infinity; L_{p}(u) the usual Lebesgue space, 1 ≤ p ≤ ∞; H^{s}(ℝ^{n}) the usual L^{2}-Sobolev space of the order s ∈ ℝ. The norm in L_{p}(u) is denoted by ∥ · ∥_{L^{p}} but for p = 2 we use ∥ · ∥_{0} and

(·,·)_{0} for the corresponding inner product. In H^{s}(ℝ^{n}) we write ∥ · ∥_{s} for the norm and for the inner product (·,·)_{s}. The supremum-norm on B_{b}(ℝ^{n}) is denoted by ∥ · ∥_{∞}.

The Fourier transform of a function u is defined as

(1.1)  û(ξ) = (F_{x→ξ} u(x))(ξ) = (2π)^{−n/2} ∫_{ℝ^{n}} e^{−ixξ}u(x) dx,

and of a measure μ as

(1.2)  μ̂(ξ) = (2π)^{−n/2} ∫_{ℝ^{n}} e^{−ixξ} μ(dx).

The maximum of a, b ∈ ℝ is a ∨ b, the minimum is a ∧ b. A Feller semigroup is a family of linear contractions T_{t}: C_{∞}(ℝ^{n}) →

C_{∞}(ℝ^{n}), t ≥ 0, satisfying

T_{t}T_{s}u = T_{t+s}u and T_{0}u = u, (1.3)

lim_{t→0} ∥T_{t}u − u∥_{∞} = 0, (1.4)

and

(1.5)  0 ≤ u ≤ 1 implies 0 ≤ T_{t}u ≤ 1.

Similarly, a family of linear contractions T_{t}: L^{2}(ℝ^{n}) → L^{2}(ℝ^{n}), t ≥ 0, is a sub-Markovian semigroup if

T_{t}T_{s}u = T_{t+s}u and T_{0}u = u, (1.6)

lim_{t→0} ∥T_{t}u − u∥_{0} = 0, (1.7)

and

(1.8)  0 ≤ u ≤ 1 a.e. implies 0 ≤ T_{t}u ≤ 1 a.e.

The infinitesimal generator A of a Fellerian or sub-Markovian semigroup is given by the strong limit

(1.9)  Au = lim_{t→0} \frac{T_{t}u − u}{t}

on

(1.10)  D(A) = {u: the limit (1.9) exists in the norm sense}.

Generators of Feller semigroups are closely related to continuous negative definite functions. A function ψ: ℝ^{n} → ℂ is negative definite if ψ(0) ≥ 0 and ξ → e^{−tψ(ξ)} is positive definite for all t > 0. Many properties of negative definite functions are given in [1]. We recall some results for realvalued continuous negative definite functions ψ: ℝ^{n} → ℝ. Because of the continuity and subadditivity of \sqrt{ψ} we have

(1.11)  0 ≤ ψ(ξ) ≤ c_{ψ}(1 + |ξ|^{2}).

Since \sqrt{ψ} is subadditive, the following Peetre-type inequality holds

(1.12)  \frac{1}{2 (1 + ψ(ξ − η))} ≤ \frac{1 + ψ(ξ)}{1 + ψ(η)} ≤ 2 (1 + ψ(ξ − η)).

We write ({X_{t}}_{t≥0}, ℙ^{x})_{x∈ℝ^{n}} for the Markov process associated with the Feller semigroup {T_{t}}_{t≥0}, and 𝔼^{x}(·) for the expectation w.r.t. ℙ^{x}.

Finally, we assume some knowledge of the multi-index notation. Further standard notation that is not explained here can be found in [12].

2. On a class of pseudodifferential operators

In this section we introduce the pseudodifferential operators we will work with later on and we also recall some results from [10] and [6].

Let a^{2} : ℝ^{n} → ℝ be a continuous negative definite function such that

(2.1)  a^{2}(ξ) ≥ κ_{0} |ξ|^{r_{0}} for large |ξ|

with some 0 < r_{0} ≤ 2 and

(2.2)  1 + a^{2}(ξ) ≤ κ_{1}(1 + |ξ|^{2}).

Note that (2.1) is a non-trivial assumption whereas (2.2) is a restatement of (1.11) with κ_{1} ≃ c_{ψ}, which follows from the continuity and subadditivity of a = \sqrt{a}^{2}.

Furthermore, let q: ℝ^{n} × ℝ^{n} → ℝ be a continuous function such that for a fixed x ∈ ℝ^{n} the function ξ ↦ q(x, ξ) is negative definite. We assume that q(x, ξ) has a decomposition

(2.3)  q(x, ξ) = q(x_{0}, ξ) + (q(x, ξ) − q(x_{0}, ξ)) ≡ q_{1}(ξ) + q_{2}(x, ξ),

where q_{1} : ℝ^{n} → ℝ is a continuous negative definite function. Note that therefore q(x, ξ) ≥ 0 and q_{1}(ξ) ≥ 0 hold for all x, ξ ∈ ℝ^{n}. The crucial idea is to consider q_{2}(x, ξ) as a perturbation of q_{1}(ξ), and to view objects (like semigroups or stochastic processes) related to q(x, ξ) as perturbations of the corresponding objects related to q_{1}(ξ). For such q(x, ξ) we define on C_{0}^{∞}(ℝ^{n}) the pseudodifferential operator q(x, D) by

(2.4)  q(x, D)u(x) = (2π)^{−n/2} ∫_{ℝ^{n}} e^{ixξ}q(x, ξ)û(ξ) dξ.

For q(x, ·) is continuous and negative definite, it is polynomially bounded and q(x, D)u(x) is well-defined for every x ∈ ℝ^{n}. In a similar manner we define the operators q_{1}(D) and q_{2}(x, D). It is well known that −q_{1}(D) has extensions to a generator of a symmetric Fellerian semigroup and also to a generator of a symmetric sub-Markovian semigroup. Since these semigroups coincide on L^{2}(ℝ^{n}) ∩ C_{∞}(ℝ^{n}), we denote both by {S_{t}}_{t≥0}. We need some additional assumptions on q(x, ξ) in order to ensure that −q(x, D) extends to a generator of a Fellerian, resp., sub-Markovian semigroup. In this paper we state all conditions w.r.t. the fixed reference function a^{2}(ξ); different set-ups are considered in [14], [17], [19].

Basically there are two types of conditions: boundedness conditions in terms of a^{2} and ellipticity conditions in terms of a^{2}. We list some of these below and state the appropriate combinations in subsequent theorems.

Ellipticity conditions.

(2.5)  1 + q_{1}(ξ) ≥ γ_{1} (1 + a^{2}(ξ)) for all ξ ∈ ℝ^{n};
(2.6)  1 + q(x, ξ) ≥ γ_{2} (1 + a^{2}(ξ)) for all x, ξ ∈ ℝ^{n};
(2.7)  q(x, ξ) ≥ γ̃(x)a^{2}(ξ) for all x ∈ ℝ^{n}, |ξ| ≥ ρ > 0.

Here, γ_{1}, γ_{2} > 0 are constants and γ̃(x) is a continuous, strictly positive function.

Note that it is possible (and in our situation even equivalent) to assume for all x ∈ ℝ and large |ξ| that q_{1}(ξ) ≥ γ′_{1} a^{2}(ξ) or q(x, ξ) ≥ γ′_{2} a^{2}(ξ) rather than (2.5) or (2.6).

Boundedness conditions.

(2.8)  1 + q_{1}(ξ) ≤ γ_{0} (1 + a^{2}(ξ)) for all ξ ∈ ℝ^{n};

(2.9)  \{ \begin{matrix}q(x, ξ) ≤ γ′(ξ) ≤ c (1 + |ξ|^{2}) for all x, ξ ∈ ℝ^{n} \\ with a continuous function γ′: ℝ^{n} → ℝ s.t. γ′(0) = 0; \end{matrix}

(2.10.m) \{ \begin{matrix}q_{2}(·, ξ) ∈ C^{m}(ℝ^{n}) and |∂_{x}^{α}q_{2}(x, ξ)| ≤ ϕ_{α}(x) (1 + a^{2}(ξ)) \\ for all x, ξ ∈ ℝ^{n} and all α ∈ ℕ_{0}^{n}, |α| ≤ m, \\ with functions ϕ_{α} ∈ L^{1}(ℝ^{n}) ∩ L^{∞}(ℝ^{n}); \end{matrix}

(2.11.m) \{ \begin{matrix}q(·, ξ) ∈ C^{m}(ℝ^{n}) and |∂_{x}^{α}q(x, ξ)| ≤ γ_{3} (1 + a^{2}(ξ)) \\ for all x, ξ ∈ ℝ^{n} and α ∈ ℕ_{0}^{α}, |α| ≤ m. \end{matrix}

Note that (2.8) is equivalent to the more naturally looking condition q_{1}(ξ) ≤ γ′_{0} (1 + a^{2}(ξ)) but (2.8) suits our purposes better. Sometimes we also need

(2.12)  q(x, 0) = 0 for all x ∈ ℝ^{n}.

Theorem 2.1 ([10]). Suppose that (2.5), (2.8), and (2.10.m) hold for some sufficiently large m. If ∑_{|α|≤m} ∥ϕ_{α}∥_{L^{1}} is small compared to γ_{1}, then −q(x, D) extends to a generator of a Feller semigroup {T_{t}}_{t≥0}.

If −q(x, D) is symmetric, then it also extends to a generator of a sub-Markovian semigroup {T_{t}}_{t≥0}, and we have the estimate

(2.13)  ∥T_{t}u∥_{L^{∞}} ≤ c′ \frac{e^{ct}}{t^{n/2r_{0}}} ∥u∥_{L^{1}}

for all u ∈ C_{∞}(ℝ^{n}) ∩ L^{1}(ℝ^{n}).

Remark 2.2. (i) The largeness condition on m and the smallness condition on ∑_{|α|≤m} ∥ϕ_{α}∥_{L^{1}} are explicitly given in [10]. Here we can do without the exact and rather technical conditions. One should, however, note that these conditions depend only on the dimension n and the constant r_{0} of (2.1).

(ii) If ∥ϕ_{0}∥_{L^{∞}} < γ_{1}, it is easy to see that the assumptions of Theorem 2.1 imply (2.6) with a suitable constant γ_{2}.

For λ ∈ ℝ we put q_{λ}(x, D) ≡ q(x, D) + λ id.

Corollary 2.3 ([11]). Under the assumptions of Theorem 2.1 there exists a λ_{1} ≥ 0 such that for all λ ≥ λ_{1} the operator −q_{λ}(x, D) extends to a generator of a Feller semigroup and also to a generator of a sub-Markovian semigroup.

Note that in Corollary 2.3 no symmetry condition is required.

Theorem 2.4 ([6]). Suppose that (2.7), (2.12), and (2.11.m) hold for some m > 2 (\frac{n}{r_{0}} ∨2)+n+2. If q(x, D) maps C_{0}^{∞}(ℝ^{n}) into C_{∞}(ℝ^{n}), then −q(x, D) has an extension that generates a Feller semigroup.

Note that, e.g., the condition (2.10.m) with m > n − 1 + n/r_{0} implies q(x, D)(C_{0}^{∞}(ℝ^{n})) ⊂ C_{∞}(ℝ^{n}).

If q(x, D) is symmetric, Theorem 2.4 also guarantees the existence of a sub-Markovian semigroup.

Theorem 2.5 ([8]). Suppose that (2.5), (2.8), and (2.10.m) hold for a sufficiently large m. If ∑_{|α|≤m} ∥ϕ_{α}∥_{L^{1}} is small w.r.t. γ_{1}, and if, in addition, (2.9) and (2.13) hold, then for any λ > 0 the pseudodifferential operator −q_{λ}(x, D) extends to a generator of a strong Feller semigroup.

In view of the above theorems we may always assume that the pseudodifferential operator −q(x, D)|C_{0}^{∞}(ℝ^{n}) with symbol −q(x, ξ) as in (2.3) extends to a generator of a Fellerian and a sub-Markovian semigroup. We will denote both semigroups by {T_{t}}_{t≥0}. For the rest of the paper we will make the following General Assumption.

General Assumption. Whenever we consider the operator q(x, D) = q_{1}(D) + q_{2}(x, D), the symbols will fulfil (2.5) and (2.8) with a fixed continuous negative definite function a^{2} satisfying (2.1) and (2.2).

For our estimates we need a class of anisotropic Sobolev spaces associated with the function a^{2}. We put for s ≥ 0

(2.14)  ∥u∥_{a^{2},s}^{2} = ∫_{ℝ^{n}}(1 + a^{2}(ξ))^{2s}|û(ξ)|^{2} dξ,

and

(2.15)  H^{a^{2},s}(ℝ^{n}) = {u ∈ L^{2}(ℝ^{n}): ∥u∥_{a^{2},s} < ∞}.

The space H^{a^{2},s}(ℝ^{n}) is a Hilbert space with the inner product (·,·)_{a^{2},s}, and we clearly have H^{a^{2},0}(ℝ^{n}) = L^{2}(ℝ^{n}). Moreover, it is possible to define the spaces H^{a^{2},−s}(ℝ^{n}), s ≥ 0, in the usual way using the P. Lax’ negative norm (with some obvious modifications), cf. [20], and to identify H^{a^{2},−s}(ℝ^{n}) with the dual of H^{a^{2},s}. In particular, (2.14) with s ≤ 0 holds for u ∈ L^{2}(ℝ^{n}).

Under our General Assumption q_{1}(D) is a closed operator on L^{2}(ℝ^{n}) with domain H^{a^{2},1}(ℝ^{n}) and we have

(2.16)  γ_{1}^{s}∥u∥_{a^{2},s} ≤ ∥(1 + q_{1}(D))^{s}u∥_{0} ≤ γ_{2}^{s}∥u∥_{a^{2},s}

for all u ∈ H^{a^{2},s}(ℝ^{n}).

Proposition 2.6 ([10]). Under the assumptions of Theorem 2.1 q(x, D) is a closed operator on L^{2}(ℝ^{n}) with domain H^{a^{2},1}(ℝ^{n}). In particular, if −q_{λ}(x, D) defined on C_{0}^{∞}(ℝ^{n}) extends to a generator of a sub-Markovian semigroup, then the domain of this extension is H^{a^{2},1}(ℝ^{n}).

For the operator q_{2}(x, D) we have the following mapping property.

Lemma 2.7. Suppose that (2.10.m) holds with some m ≥ n + 1. Then q_{2}(x, D) maps H^{a^{2},s}(ℝ^{n}) continuously into H^{a^{2},s−1}(ℝ^{n}) for all s ∈ ℝ s.t. |s − 1| < (m − n)/2. Moreover, we have the following estimate

(2.17)  ∥q_{2}(x, D)u∥_{a^{2},s−1} ≤ d_{1}(n, s, κ_{1}) ∑_{|α|≤m} ∥ϕ_{α}∥_{L^{1}} ∥u∥_{a^{2},s}.

In the proof of Lemma 2.7 we will use the Peetre-type inequality (1.12), and the inequality

(2.18)  |q̂_{2}(ξ, η)| ≤ γ̃_{k,n} ∑_{|α|≤k} ∥ϕ_{α}∥_{L^{1}} (1 + |ξ|^{2})^{−k/2} (1 + a^{2}(η)),

which follows from (2.10.m) for any k ≤ m, see [10]. Here q̂_{2}(ξ, η) stands for the Fourier transform w.r.t. the first variable, i.e.,

q̂_{2}(ξ, η) = (F_{x↦ξ} q_{2}(x, η))(ξ, η).

Proof of Lemma 2.7 (cf. also [10, Lemma 3.1]). For u, v ∈ C_{0}^{∞}(ℝ^{n}) we find

|(q_{2}(x, D)u, v)_{0}| = | ∫_{ℝ^{n}}∫_{ℝ^{n}} q̂_{2}(ξ − η, η)û(η)\overline{v̂(ξ)} dη dξ|

≤ γ̃_{m,n} ∑_{|α|≤m} ∥ϕ_{α}∥_{L^{1}}∫_{ℝ^{n}}∫_{ℝ^{n}}(1 + |ξ − η|^{2})^{−m/2} (1 + a^{2}(η))|û(η)| |v̂(ξ)| dη dξ

≤ 2^{|s−1|} γ̃_{m,n} ∑_{|α|≤m} ∥ϕ_{α}∥_{L^{1}}∫_{ℝ^{n}}∫_{ℝ^{n}}(1 + |ξ − η|^{2})^{−m/2}(1 + a^{2}(ξ − η))^{|s−1|}

× (1 + a^{2}(η))^{s}|û(η)| (1 + a^{2}(ξ))^{1−s}|v̂(ξ)| dη dξ

≤ 2^{|s−1|} κ_{1}^{|s−1|} γ̃_{m,n} ∑_{|α|≤m} ∥ϕ_{α}∥_{L^{1}}∫_{ℝ^{n}}∫_{ℝ^{n}}(1 + |ξ − η|^{2})^{|s−1|−m/2}

× (1 + a^{2}(η))^{s}|û(η)| (1 + a^{2}(ξ))^{1−s}|v̂(ξ)| dη dξ

≤ 2^{|s−1|} κ_{1}^{|s−1|} γ̃_{m,n} ∑_{|α|≤m} ∥ϕ_{α}∥_{L^{1}}∫_{ℝ^{n}}(1 + |τ|^{2})^{|s−1|−m/2} dτ ∥u∥_{a^{2},s} ∥v∥_{a^{2},1−s},

that is,

|(q_{2}(x, D)u, v)_{0}| ≤ d_{1}(n, s, κ_{1}) ∑_{|α|≤m} ∥ϕ_{α}∥_{L^{1}} ∥u∥_{a^{2},s} ∥v∥_{a^{2},−(s−1)},

which, in turn, gives (2.17). □

3. An L^{2}-comparison of semigroups

In this section we assume that −q(x, D) generates a sub-Markovian and a Fellerian semigroup, both denoted by {T_{t}}_{t≥0}, and by {S_{t}}_{t≥0} we denote the corresponding semigroups generated by −q_{1}(D). The semigroup {S_{t}}_{t≥0} is a convolution semigroup, i.e.,

(3.1)  S_{t}u(x) = ∫_{ℝ^{n}} u(x + y) μ_{t}(dy),

where μ_{t} is determined by

(3.2)  μ̂_{t}(ξ) = (2π)^{−n/2} e^{−tq_{1}(ξ)}.

Moreover, we assume that our General Assumption and (2.10.m) hold for some m > n + 1.

Note that, in order to construct a semigroup by starting with −q(x, D), it might be necessary to enlarge m or to impose a smallness condition on ∑_{|α|≤m} ∥ϕ_{α}∥_{L^{1}}.

Now, however, our point of view is that −q(x, D) does already generate the semigroup {T_{t}}_{t≥0}, and we want to study the effect of q_{2}(x, D) viewed as a perturbation of q_{1}(D). Our aim is to compare {T_{t}}_{t≥0} with {S_{t}}_{t≥0} for small t > 0. We will need the elementary estimate

(3.3)  e^{−x} ≤ \frac{1}{1 + x} for all x ≥ 0.

Theorem 3.1. Let {T_{t}}_{t≥0}, {S_{t}}_{t≥0} and q(x, ξ) be as above, in particular assume (2.10.m) for some m > n + 1. Then we have for 0 < t ≤ 1 and 0 ≤ ρ < 1

(3.4) ∥T_{t}u − S_{t}u∥_{0} ≤ \frac{t^{1−ρ}}{1 − ρ}d_{1}(n, 1, κ_{1}) \frac{γ_{0}^{1−ρ}}{γ_{1}} ∑_{|α|≤n+1} ∥ϕ_{α}∥_{L^{1}} ∥u∥_{a^{2},1−ρ}.

If ρ = 0, we have for all t ≥ 0

(3.5)  ∥T_{t}u − S_{t}u∥_{0} ≤ t d_{1}(n, 1, κ_{1}) \frac{γ_{0}}{γ_{1}} ∑_{|α|≤n+1} ∥ϕ_{α}∥_{L^{1}} ∥u_{a^{2},1}∥.

Proof. For u ∈ H^{a^{2},s}(ℝ^{n}) we see that

(3.6) ∥T_{t}u − S_{t}u∥_{0} = ∥ ∫_{0}^{t} T_{t−s}q_{2}(x, D)S_{s}u ds∥_{0}

≤ ∫_{0}^{t} ∥T_{t−s}∥_{L^{2}−L^{2}}∥q_{2}(x, D)S_{s}u∥_{0} ds

≤ d_{1}(n, 1, κ_{1}) ∑_{|α|≤n+1} ∥ϕ_{α}∥_{L^{1}} ∫_{0}^{t} ∥S_{s}u∥_{a^{2},1} ds.

To estimate ∥S_{s}u∥_{a^{2},1} we assume first that 0 < t ≤ 1 and 0 ≤ ρ < 1. Then

ESTIMATES FOR FELLER SEMIGROUPS 37

we obtain for s ≤ t

(3.7)  ∥S_{s}u∥_{a^{2},1}^{2} ≤ \frac{1}{γ_{1}^{2}} ∥(1 + q_{1}(D))S_{s}u∥_{0}^{2}

≤ \frac{1}{γ_{1}^{2}} ∫_{ℝ^{n}}(1 + q_{1}(ξ))^{2}(e^{−sq_{1}(ξ)})^{2}|û(ξ)|^{2} dξ

≤ \frac{1}{γ_{1}^{2}} ∫_{ℝ^{n}}(1 + q_{1}(ξ))^{2} \frac{1}{(1 + sq_{1}(ξ))^{2}} |û(ξ)|^{2} dξ

≤ \frac{1}{s^{2ρ}} \frac{1}{γ_{1}^{2}} ∫_{ℝ^{n}}(1 + q_{1}(ξ))^{2(1−ρ)}|û(ξ)|^{2} dξ

≤ \frac{1}{s^{2ρ}} \frac{γ_{0}^{2(1−ρ)}}{γ_{1}^{2}} ∫_{ℝ^{n}}(1 + a^{2}(ξ))^{2(1−ρ)}|û(ξ)|^{2} dξ

= \frac{1}{s^{2ρ}} \frac{γ_{0}^{2(1−ρ)}}{γ_{1}^{2}} ∥u∥_{a^{2},1−ρ}^{2}

Combining (3.6) and (3.7) we conclude that

∥T_{t}u − S_{t}u∥_{0} ≤ d_{1}(n, 1, κ_{1}) \frac{γ_{0}^{1−ρ}}{γ_{1}} ∑_{|α|≤n+1} ∥ϕ_{α}∥_{L^{1}} ∫_{0}^{t} \frac{ds}{s^{ρ}} ∥u∥_{a^{2},1−ρ}

= \frac{t^{1−ρ}}{1 − ρ} d_{1}(n, 1, κ_{1}) \frac{γ_{0}^{1−ρ}}{γ_{1}} ∑_{|α|≤n+1} ∥ϕ_{α}∥_{L^{1}} ∥u∥_{a^{2},1−ρ},

and the first part of the assertion is established. By a similar calculation we get

∥S_{s}∥_{a^{2},1}^{2} ≤ \frac{1}{γ_{1}^{2}} ∫_{ℝ^{n}}(1 + q_{1}(ξ))^{2}(e^{−sq_{1}(ξ)})^{2}|û(ξ)|^{2} dξ

≤ \frac{1}{γ_{1}^{2}} ∫_{ℝ^{n}}(1 + q_{1}(ξ))^{2}|û(ξ)|^{2} dξ

≤ \frac{γ_{0}^{2}}{γ_{1}^{2}} ∫_{ℝ^{n}}(1 + a^{2}(ξ))^{2}|û(ξ)|^{2} dξ

= \frac{γ_{0}^{2}}{γ_{1}^{2}} ∥u∥_{a^{2},1}^{2},

which implies (3.5). □

By (2.1) we have H^{r_{0}}(ℝ^{n}) ⊂ H^{a^{2},1}(ℝ^{n}) and H^{r_{0}(1−ρ)}(ℝ^{n}) ⊂ H^{a^{2},1−ρ}(ℝ^{n}). For n = 1 and r_{0}(1−ρ) < 1/2, that is, ρ > (2r_{0}−1)/(2r_{0}), it follows that the indicator function of any bounded interval belongs to H^{a^{2},1−ρ}(ℝ^{1}). In this case Theorem 3.1 gives a direct L^{2}-comparison of the transition functions of the stochastic processes corresponding to {T_{t}}_{t≥0} and {S_{t}}_{t≥0}, respectively.

4. Some pointwise estimates

In [13] it was proved that for any Feller semigroup {T_{t}}_{t≥0} that extends to a map from C_{b}(ℝ^{n}) into C_{b}(ℝ^{n}) the operators T_{t}|C_{0}^{∞}(ℝ^{n}) itself are pseudodifferential operators with the symbols (2π)^{n/2}λ_{t}(x, ξ) where

(4.1)  λ_{t}(x, ξ) = 𝔼^{x}(e^{iξ(X_{t}−x)}).

The expectation is to be understood with respect to the Feller process ({X_{t}}_{t≥0}, ℙ^{x})_{x∈ℝ^{n}} associated with {T_{t}}_{t≥0}. Thus we have on C_{0}^{∞}(ℝ^{n})

(4.2)  T_{t}u(x) = ∫_{ℝ^{n}} e^{ixξ}λ_{t}(x, ξ)û(ξ) dξ.

In [8] some sufficient criteria were given for a Feller semigroup generated by a pseudodifferential operator −q_{λ}(s, D) to be strongly Feller, i.e., to map B_{b}(ℝ^{n}) into C_{b}(ℝ^{n}), see Theorem 2.5. Clearly, for these (strong) Feller semigroups, (4.1) and (4.2) are valid. If −q(x, D) maps C_{u}^{2}(ℝ^{n}) into C_{u}(ℝ^{n}), we have

(4.3)  q(x, ξ) = (2π)^{n/2} \frac{d}{dt}λ_{t}(x, ξ)|_{t=0},

and for u ∈ C_{u}^{2}(ℝ^{n})

(4.4)  \frac{d}{dt}T_{t}u = T_{t}Au = AT_{t}u,

where A denotes the extension of −q(x, D)|C_{0}^{∞}(ℝ^{n}) onto C_{u}^{2}(ℝ^{n}).

By a result of Courrège [2] the operator −q_{1}(D) always fulfils this additional hypothesis. Using this result it is easy to find examples of operators −q(x, D) with variable coefficients satisfying this assumption, too.

Until now, however, we do not know a simple general criterion in order to decide whether −q(x, D) maps C_{u}^{2}(ℝ^{n}) into C_{u}(ℝ^{n}) or not, see also [4].

In this section we will assume that (4.1)–(4.4) hold.

Lemma 4.1. Let {T_{t}}_{t≥0} be a Fellerian semigroup generated by the pseudodifferential operator −q(x, D). We assume the General Assumption, (2.10.0), and (4.1)–(4.4). Then we have

(4.5)  |λ_{t}(x, ξ) − e^{−tq(x,ξ)}| ≤ 2t (γ_{0} + ∥ϕ_{0}∥_{L^{∞}}) (1 + a^{2}(ξ)).

Proof. Using Taylor’s formula in t, we obtain from (4.3) and (4.4)

|λ_{t}(x, ξ) − e^{−tq(x,ξ)}|

= t | ∫_{0}^{1} ( − e^{−ixξ} T_{ρt}(e^{i(·,ξ)}q(·, ξ))(x, ξ) + q(x, ξ)e^{−ρtq(x,ξ)}) dρ|

= t | ∫_{0}^{1} ( − e^{−ixξ} ∫_{ℝ^{n}} e^{iyξ}q(y, ξ) p_{ρt}(x, dy) + q(x, ξ)e^{−ρtq(x,ξ)}) dρ|

≤ t ∫_{0}^{1} ( ∫_{ℝ^{n}} q(y, ξ) p_{ρt}(x, dy) + q(x, ξ)e^{−ρtq(x,ξ)}) dρ

≤ t ( sup_{y∈ℝ^{n}} q(y, ξ) + q(x, ξ)),

where p_{t}(x, dy) denotes the transition kernel of the associated Feller process, i.e., we have for u ∈ C_{b}(ℝ^{n})

T_{t}u(x) = 𝔼^{x}(u(X_{t})) = ∫_{ℝ^{n}} u(y) p_{t}(x, dy).

From (2.8) and (2.10.0) we deduce that

sup_{y∈ℝ^{n}} q(y, ξ) + q(x, ξ) ≤ 2 (γ_{0} + ∥ϕ_{0}∥_{L^{∞}}) (1 + a^{2}(ξ)),

and we are done. □

In the next theorem we denote by D_{F} (q_{1}(D)) and D_{F} (q(x, D)) the domain of the extension of −q_{1}(D) and −q(x, D), which generate the Feller semigroups {S_{t}}_{t≥0} and {T_{t}}_{t≥0}, respectively.

Theorem 4.2. Suppose that the General Assumption, (2.10.n + 3), and (4.1)–(4.4) hold. Since C_{0}^{∞}(ℝ^{n}) ⊂ D_{F} (q_{1}(D)) ∩ D_{F} (q(x, D)), we have for all u ∈ C_{0}^{∞}(ℝ^{n})

(4.6) |T_{t}u(x) − S_{t}u(x)| ≤ d_{2} t ∥(1 + a^{2}) û∥_{L^{1}} + d_{3} t^{2} ∥(1 + a^{2})^{2} û∥_{L^{1}}

40 NIELS JACOB AND RENÉ L. SCHILLING

with the constants

(4.7)  d_{2} = (2π)^{−n} γ̃_{n+1,n} ∑_{|α|≤n+1} ∥ϕ_{α}∥_{L^{1}} ∫_{ℝ^{n}}(1 + |ξ|^{2})^{−(n+1)/2} dξ

and

(4.8)  d_{3} =2(2π)^{−n}κ_{1} (γ_{0} + ∥ϕ_{0}∥_{L^{∞}}) γ̃_{n+3,n}

× ∑_{|α|≤n+3} ∥ϕ_{α}∥_{L^{1}} ∫_{ℝ^{n}}(1 + |ξ|^{2})^{−(n+1)/2} dξ.

The constant γ̃_{k,n} is taken from (2.18), and κ_{1} is taken from (2.2). Proof. For u ∈ C_{0}^{∞}(ℝ^{n}) we get

(4.9)  T_{t}u(x) − S_{t}u(x) = ∫_{0}^{t} T_{t−s}(q_{1}(D) − q(x, D))S_{s}u(x) ds
= − ∫_{0}^{t} T_{t−s}q_{2}(x, D)S_{s}u(x) ds,

which yields

T_{t}u(x) − S_{t}u(x) =
= (2π)^{−n} ∫_{0}^{t}∫_{ℝ^{n}}∫_{ℝ^{n}} e^{ixξ}λ_{t−s}(x, ξ) q̂_{2}(ξ − η, η) e^{−sq_{1}(η)} û(η) dη dξ ds
= (2π)^{−n} ∫_{0}^{t}∫_{ℝ^{n}}∫_{ℝ^{n}} e^{ixξ}e^{−(t−s)q(x,ξ)} q̂_{2}(ξ − η, η) e^{−sq_{1}(η)} û(η) dη dξ ds
+ (2π)^{−n} ∫_{0}^{t}∫_{ℝ^{n}}∫_{ℝ^{n}} e^{ixξ}(λ_{t−s}(x, ξ) − e^{−(t−s)q(x,ξ)})
× q̂_{2}(ξ − η, η) e^{−sq_{1}(η)} û(η) dη dξ ds
≡ I_{1} + I_{2}.

As before, q̂_{2}(ξ, η) stands for the Fourier transform w.r.t. the first variable. We will now estimate I_{1}. For s ≤ t we have e^{−(t−s)q(x,ξ)} ≤ 1, leading to

|I_{1}| ≤ (2π)^{−n} ∫_{ℝ^{n}}∫_{ℝ^{n}} |q̂_{2}(ξ − η, η)| |û(η)| | ∫_{0}^{t} e^{−sq_{1}(η)} ds| dη dξ

≤ (2π)^{−n} t γ̃_{n+1,n} ∑_{|α|≤n+1} ∥ϕ_{α}∥_{L^{1}}

× ∫_{ℝ^{n}}∫_{ℝ^{n}}(1 + |ξ − η|^{2})^{−(n+1)/2}(1 + a^{2}(η))|û(η)| dη dξ ≤ d_{2} t ∥(1 + a^{2}) û∥_{L^{1}}.

ESTIMATES FOR FELLER SEMIGROUPS 41

Next we turn to I_{2}. Using (4.5), we find that

|I_{2}| ≤ (2π)^{−n} ∫_{0}^{t}∫_{ℝ^{n}}∫_{ℝ^{n}} |λ_{t−s}(x, ξ) − e^{−(t−s)q(x,ξ)}|

× |q̂_{2}(ξ − η, η)| e^{−sq_{1}(η)} |û(η)| dη dξ ds

≤ 2(2π)^{−n} (γ_{0} + ∥ϕ_{0}∥_{L^{∞}}) ∫_{0}^{t}∫_{ℝ^{n}}∫_{ℝ^{n}}(t − s)(1 + a^{2}(ξ))|q̂_{2}(ξ − η, η)|

× e^{−sq_{1}(η)}|û(η)| dη dξ ds

≤ 2(2π)^{−n} γ̃_{n+3,n} (γ_{0} + ∥ϕ_{0}∥_{L^{∞}})

× ∑_{|α|≤n+3} ∥ϕ_{α}∥_{L^{1}} ∫_{ℝ^{n}}∫_{ℝ^{n}}(1 + |ξ − η|^{2})^{−(n+3)/2}

× (1 + a^{2}(ξ))(1 + a^{2}(η))|û(η)| ∫_{0}^{t}(t − s)e^{−sq_{1}(η)} ds dη dξ

≤ (2π)^{−n} γ̃_{n+3,n} (γ_{0} + ∥ϕ_{0}∥_{L^{∞}})

× ∑_{|α|≤n+3} ∥ϕ_{α}∥_{L^{1}}t^{2} ∫_{ℝ^{n}}∫_{ℝ^{n}}(1 + |ξ − η|^{2})^{−(n+3)/2}

× \frac{1 + a^{2}(ξ)}{1 + a^{2}(η)} (1 + a^{2}(η))^{2} |û(η)| dη dξ

≤ d_{3} t^{2} ∥(1 + a^{2})^{2} û∥_{L^{1}},

where we have used (1.12) and (2.2). □ Corollary 4.3. In the situation of Theorem 4.2 we have

(4.10)  |T_{t}u(x) − S_{t}u(x)| ≤ d′_{2} t ∥u∥_{n+3} + d′_{3} t^{2} ∥u∥_{n+5}.

Proof. Since 1 + a^{2}(η) ≤ κ_{1}(1 + |η|^{2}), we have

∫_{ℝ^{n}}(1 + a^{2}(η))|û(η)| dη
≤ κ_{1} ∫_{ℝ^{n}}(1 + |η|^{2})|û(η)| dη

≤ κ_{1} ∫_{ℝ^{n}}(1 + |η|^{2})^{−(n+1)/2}(1 + |η|^{2})^{(n+3)/2} |û(η)| dη ≤ κ_{1}( ∫_{ℝ^{n}}(1 + |η|^{2})^{−(n+1)/2} dη)^{1/2} ∥u∥_{n+3}, and analogously we get

∫_{ℝ^{n}}(1 + a^{2}(η))^{2} |û(η)| dη ≤ κ_{1}^{2}( ∫_{ℝ^{n}}(1 + |η|^{2})^{−(n+1)/2} dη)^{1/2} ∥u∥_{n+5},

which proves the corollary. □

5. Towards a symbolic calculus

In the theory of pseudodifferential operators it is useful to have a symbolic calculus, i.e., a calculus reducing operations on the level of operators to manipulations on the level of functions, namely the symbols. Differentiability properties w.r.t. the second variable of the symbol are essential when establishing such a calculus. In general, the symbols considered here are not differentiable in the second variable ξ. Therefore, a symbolic calculus does not seem to be available. In [7] W. Hoh, however, made a remarkable attempt to develop such a symbolic calculus for pseudodifferential operators with symbols q(x, ξ) which are for fixed x negative definite functions of ξ.

One of the most interesting problems in a symbolic calculus is the comparison of the semigroup {T_{t}}_{t≥0} generated by −q(x, D) with the one-parameter family of pseudodifferential operators with symbols (2π)^{n/2}e^{−tq(x,ξ)}, t ≥ 0,

(5.1)  e^{−tq(x,D)}u(x) = ∫_{ℝ^{n}} e^{ixξ} e^{−tq(x,ξ)} û(ξ) dξ.

Note that e^{−tq(x,D)} does not mean the exponential, i.e., the semigroup, of and related to the operator q(x, D) but something that could be called symbolic exponential. Of course, in the constant coefficient case, where q(x, ξ) is independent of x, both notions of exponentials coincide.

In order to estimate the difference between the proper and the symbolic exponentials we need some auxiliary results.

Lemma 5.1. For all x ≥ 0 and m ∈ ℕ we have

(5.2)  e^{−x} ≤ e(\frac{m}{e})^{m}(\frac{1}{1 + x})^{m} ≤ e(\frac{m}{e})^{m} \frac{1}{1 + x^{m}} .

(Note that by Stirling’s formula e(\frac{m}{e})^{m} ≤ m! .)

Proof. By elementary calculus we determine the maximum of f(x) = (1+x), e^{−x/m}. This proves the first inequality, whereas the second is obvious. □

Let us recall the chain rule for higher-order derivatives, cf. [5, 0.430].

Lemma 5.2. Let f, g ∈ C^{N}(ℝ). For any 1 ≤ k ≤ N we have

(5.3) ∂_{x}^{k}(f ∘ g)(x)

= ∑_{j=1}^{k} f^{(j)}(g(x)) ∑_{\begin{matrix}α_{1}+α_{2}+...+α_{ℓ}=j \\ α_{1}+2α_{2}+...+ℓα_{ℓ}=k \end{matrix}}\frac{k!}{α_{1}! … α_{ℓ}!} ∏_{i=1}^{ℓ} (\frac{g^{(i)}(x)}{i!})^{α_{i}}.

We want to estimate (x, ξ) ↦ e^{−tq(x,ξ)} and its derivatives w.r.t. the variable x.

Lemma 5.3. Suppose that the General Assumption, (2.6) and (2.10.m) hold for some m ∈ ℕ, and let N ≤ m. Then for all k ∈ ℤ, k ≥ −1, and β ∈ ℕ_{0}^{n} with |β| = 1 there exist functions Φ_{Nβ,j} ∈ L^{1}(ℝ^{n}) ∩ L^{∞}(ℝ^{n}), 1 ≤ j ≤ N, such that

(5.4) |∂_{x}^{Nβ}e^{−tq(x,ξ)}| ≤ \frac{1}{t^{k}} ( ∑_{j=1}^{N} e(\frac{k + j}{γ_{2} e} )^{k+j}Φ_{Nβ,j}(x))(\frac{1}{1 + a^{2}(ξ)})^{k}

holds for all 0 < t ≤ 1 and x, ξ ∈ ℝ^{n}.

Proof. By Lemma 5.2 we have

∂_{x}^{Nβ}e^{−tq(x,ξ)} (5.5)

= ∑_{j=1}^{N}(−t)^{j} e^{−tq(x,ξ)} ∑_{\begin{matrix}α_{1}+α_{2}+...+α_{ℓ}=j \\ α_{1}+2α_{2}+...+ℓα_{ℓ}=N \end{matrix}}\frac{N!}{α_{1}! … α_{ℓ}!} ∏_{i=1}^{ℓ} (\frac{∂_{x}^{iβ}q_{2}(x, ξ)}{i!})^{α_{i}},

where we have used the fact that ∂_{x}^{iβ}q(x, ξ) = ∂_{x}^{iβ}(q_{1}(ξ) + q_{2}(x, ξ)) = ∂_{x}^{iβ}q_{2}(x, ξ). From Lemma 5.1 we obtain

e^{−tq(x,ξ)} ≤ e(\frac{m}{e})^{m}(\frac{1}{1 + tq(x, ξ)})^{m}

for all m ∈ ℕ_{0} and for t ≤ 1 we have because of (2.6)

e^{−tq(x,ξ)} ≤ \frac{1}{t^{m}} e(\frac{m}{γ_{2} e})^{m}(\frac{1}{1 + a^{2}(ξ)})^{m}.

If m = j + k, j ≥ 1 and k ≥ −1, then

|(−t)^{j}e^{−tq(x,ξ)}| ≤ \frac{1}{t^{k}} e(\frac{k + j}{γ_{2} e} )^{k+j}(\frac{1}{1 + a^{2}(ξ)})^{k+j}.

By (2.10.m) we have |∂_{x}^{iβ}q_{2}(x, ξ)| ≤ ϕ_{iβ}(x)(1 + a^{2}(ξ)) with ϕ_{iβ} ∈ L^{1}(ℝ^{n}) ∩ L^{∞}(ℝ^{n}), 1 ≤ i ≤ N, hence

| ∏_{i=1}^{ℓ} (\frac{∂_{x}^{iβ}q_{2}(x, ξ)}{i!})^{α_{i}} | ≤ ∏_{i=1}^{ℓ} (\frac{ϕ_{iβ}(x)}{i!})^{α_{i}}(1 + a^{2}(ξ))^{α_{i}}.

For t ≤ 1 and k ≥ −1 we have

|∂_{x}^{Nβ}e^{−tq(x,ξ)}| ≤ \frac{1}{t^{k}} ∑_{j=1}^{N} e(\frac{k + j}{γ_{2} e} )^{k+j}(\frac{1}{1 + a^{2}(ξ)})^{k+j}

× ∑_{\begin{matrix}α_{1}+α_{2}+...+α_{ℓ}=j \\ α_{1}+2α_{2}+...+ℓα_{ℓ}=N \end{matrix}}\frac{N!}{α_{1}! … α_{ℓ}!} ∏_{i=1}^{ℓ} (\frac{ϕ_{iβ}(x)}{i!})^{α_{i}}(1 + a^{2}(ξ))^{α_{i}}

= \frac{1}{t^{k}} ( ∑_{j=1}^{N} e(\frac{k + j}{γ_{2} e} )^{k+j}Φ_{Nβ,j}(x))(\frac{1}{1 + a^{2}(ξ)})^{k},

where

(5.6)  Φ_{Nβ,j}(x) = ∑_{\begin{matrix}α_{1}+α_{2}+...+α_{ℓ}=j \\ α_{1}+2α_{2}+...+ℓα_{ℓ}=N \end{matrix}}\frac{N!}{α_{!} … α_{ℓ}!} ∏_{i=1}^{ℓ} (\frac{ϕ_{iβ}(x)}{i!})^{α_{i}},

which by our assumptions belongs to L^{1}(ℝ^{n}) ∩ L^{∞}(ℝ^{n}). □

Corollary 5.4. In the situation of Theorem 5.3 we have

(5.7)  |∂_{x}^{Nβ}(e^{−tq(x,ξ)} − e^{−tq_{1}(ξ)})| ≤ \frac{1}{t^{k}} Ψ_{Nβ,k}(x)(\frac{1}{1 + a^{2}(ξ)})^{k}

for all 0 < t ≤ 1, k > −1, N = 0, 1, …, m, and β ∈ ℕ_{0}^{n}, |β| = 1, and a suitable function Ψ_{Nβ,k} ∈ L^{1}(ℝ^{n}) ∩ L^{∞}(ℝ^{n}).

Proof. Since e^{−tq_{1}(ξ)} is independent of x, we have only to consider the case N = 0, i.e., the case when there is no derivative at all. Observing that q(x, ξ) = q_{1}(ξ) + q_{2}(x, ξ), we get

|e^{−tq(x,ξ)} − e^{−tq_{1}(ξ)}| = e^{−t(q_{1}(ξ)∧q(x,ξ))} |e^{−t(q−q_{1}∧q)(x,ξ)} − e^{−t(q_{1}−q_{1}∧q)(x,ξ)}|
= e^{−t(q_{1}(ξ)∧q(x,ξ))} (1 − e^{−t |q_{2}(x,ξ)|})
≤ t |q_{2}(x, ξ)|(e^{−tq(x,ξ)} ∨ e^{−tq_{1}(ξ)}).

ESTIMATES FOR FELLER SEMIGROUPS 45

By (2.10.0), (2.5), (2.6), and Lemma 5.1 we have for t ≤ 1

|e^{−tq(x,ξ)} − e^{−tq_{1}(ξ)}| ≤ t ϕ_{0}(x)(1 + a^{2}(ξ))(e^{−tq(x,ξ)} ∨ e^{−tq_{1}(ξ)})

≤ t ϕ_{0}(x)(1 + a^{2}(ξ)) e(\frac{k + 1}{e})^{k+1}(\frac{1}{1 + tq(x, ξ)} ∨ \frac{1}{1 + tq_{1}(ξ)})^{k+1}

≤ \frac{1}{t^{k}} ϕ_{0}(x) e(\frac{k + 1}{e})^{k+1}(\frac{1}{γ_{2}} ∨ \frac{1}{γ_{1}} )^{k+1}(\frac{1}{1 + a^{2}(ξ)})^{k},

and the assertion follows with

(5.8)  Ψ_{Nβ,k}(x) = \{ \begin{matrix}e(\frac{k+1}{(γ_{1}∧γ_{2}) e})^{k+1} ϕ_{0}(x) & if N = 0 \\ ∑_{j=1}^{N} e(\frac{k+j}{γ_{2} e} )^{k+j}Φ_{Nβ,j}(x) & if N > 0, \end{matrix}

and Φ_{Nβ,j} as in (5.6). □

From Theorem 5.3 and Corollary 5.4 we get the following result. Corollary 5.5. Let q(x, ξ) be as in Theorem 5.3 and N ≤ m. Then we have for all 0 < t ≤ 1, x, ξ ∈ ℝ^{n}, and k ≥ −1

(5.9) |F_{x↦η}(e^{−tq(x,ξ)} −e^{−tq_{1}(ξ)})(η, ξ)| ≤ \frac{1}{t^{k}} C (\frac{1}{1 + a^{2}(ξ)})^{k}(\frac{1}{1 + |η|^{2}} )^{N/2}

with the constant C = C(N, n, k, Φ_{•}, ϕ_{•}) given by

(5.10)  C = (n + 1)^{\frac{N}{2} −1}(e(\frac{k + 1}{(γ_{1} ∧ γ_{2}) e})^{k+1}∥ϕ_{0}∥_{L^{1}}

+ ∑_{j=1}^{N} ∑_{|β|=1} e(\frac{k + j}{γ_{2} e} )^{k+j}∥Φ_{Nβ,j}∥_{L^{1}}).

Proof. For N ∈ ℕ we obtain from Hölder’s inequality

(5.11)  (1 + |η|^{2})^{\frac{N}{2}} ≤ (n + 1)^{\frac{N}{2} −1} (1 + |η_{1}|^{N} + … + |η_{n}|^{N}).

Therefore, we need only to estimate |η^{Nβ} ∫_{ℝ^{n}} e^{−ixη} (e^{−tq(x,η)} − e^{−tq_{1}(η)}) dx| for some β ∈ ℕ_{0}^{n}, |β| = 1, and N ∈ ℕ_{0}. We proceed as in [10, Lemma 2.1] to obtain

|η^{Nβ} ∫_{ℝ^{n}} e^{−ixη}(e^{−tq(x,η)} − e^{−tq_{1}(η)}) dx|

= | ∫_{ℝ^{n}} (∂_{x}^{Nβ}e^{−ixη})(e^{−tq(x,η)} − e^{−tq_{1}(η)}) dx|

= | ∫_{ℝ^{n}} e^{−ixη} ∂_{x}^{Nβ}(e^{−tq(x,η)} − e^{−tq_{1}(η)}) dx|

≤ \frac{1}{t^{k}} ∥Ψ_{Nβ,k}∥_{L^{1}}(\frac{1}{1 + a^{2}(ξ)})^{k}

for any N ∈ ℕ_{0} and |β| = 1. The assertion follows by summation over these indices. □

Using Corollary 5.5 we can study some mapping properties of the operators e^{−tq(x,D)} and e^{−tq(x,D)} − e^{−tq_{1}(D)}.

Theorem 5.6. Suppose that the General Assumption, (2.6), and (2.10.m) hold for some m > n − 2. Then e^{−tq(x,D)} − e^{−tq_{1}(D)} is for any fixed t a continuous operator from L^{2}(ℝ^{n}) into H^{a^{2},k}(ℝ^{n}), −1 ≤ k < m − n/2. Moreover, we have for 0 < t ≤ 1 the estimate

(5.12)  ∥(e^{−tq(x,D)} − e^{−tq_{1}(D)})u∥_{a^{2},k} ≤ C̃(N, n, k, Φ_{•}, ϕ_{•}) \frac{1}{t^{k}} ∥u∥_{0}.

(A similar estimate with a larger constant holds for any 0 < t ≤ const.)

Proof. It is clearly sufficient to prove (5.12). Choose u ∈ L^{2}(ℝ^{n}) and v ∈ C_{0}^{∞}(ℝ^{n}) and put

q̂_{t}(η, ξ) = F_{x↦η} (e^{−tq(x,ξ)} − e^{−tq_{1}(ξ)})(η, ξ).

Then Corollary 5.5 gives with C = C(m, n, k, Φ_{•}, ϕ_{•})

|([e^{−tq(x,D)} − e^{−tq_{1}(D)}]u, v)_{0}|
= (2π)^{−n}| ∫_{ℝ^{n}}∫_{ℝ^{n}} q̂_{t}(ξ − η, η)û(η)\overline{v̂(ξ)} dη dξ|
≤ (2π)^{−n}C \frac{1}{t^{k}} ∫_{ℝ^{n}}∫_{ℝ^{n}}(1 + |ξ − η|^{2})^{−m/2}

× (\frac{1}{1 + a^{2}(η)})^{k}|û(η)| |v̂(ξ)| dη dξ =

= (2π)^{−n}C \frac{1}{t^{k}} ∫_{ℝ^{n}}∫_{ℝ^{n}}(1 + |ξ − η|^{2})^{−m/2}

× (\frac{1 + a^{2}(ξ)}{1 + a^{2}(η)})^{k}|û(η)|(\frac{1}{1 + a^{2}(ξ)})^{k}|v̂(ξ)| dη dξ

≤ 2^{k} (2π)^{−n}C \frac{1}{t^{k}} ∫_{ℝ^{n}}∫_{ℝ^{n}}(1 + |ξ − η|^{2})^{(2k−m)/2} |û(η)|

× (\frac{1}{1 + a^{2}(ξ)})^{k}|v̂(ξ)| dη dξ

≤ 2^{k} (2π)^{−n}C \frac{1}{t^{k}} ( ∫_{ℝ^{n}}(1 + |τ|^{2})^{(2k−m)/2} dτ)∥u∥_{0} ∥v∥_{a^{2},−k}.

For −1 ≤ k < \frac{m−n}{2}, we obtain

\frac{|([e^{−tq(x,D)} − e^{−tq_{1}(D)}]u, v)_{0}|}{∥v∥_{a^{2},−k}}

≤ 2^{k} (2π)^{−n}C \frac{1}{t^{k}} ( ∫_{ℝ^{n}}(1 + |τ|^{2})^{(2k−m)/2} dτ)∥u∥_{0},

and the theorem follows. □

Since e^{−tq_{1}(D)} : L^{2}(ℝ^{n}) → H^{a^{2},k}(ℝ^{n}) is continuous, too, the next corollary easily follows from the triangular inequality.

Corollary 5.7. In the situation of Theorem 5.6 both e^{−tq(x,D)} − e^{−tq_{1}(D)} and e^{−tq(x,D)} are for any fixed t continuous operators from L^{2}(ℝ^{n}) into H^{a^{2},k}(ℝ^{n}), −1 ≤ k < \frac{m−n}{2}.

From our General Assumption, in particular from (2.1), we know that H^{a^{2},k}(ℝ^{n}) is continuously embedded into H^{kr_{0}}(ℝ^{n}). By Sobolev’s embedding theorem we conclude for k > (n − ℓ)/(2r_{0}), ℓ ∈ ℕ_{0}, that H^{a^{2},k} ⊂ C_{b}^{ℓ}(ℝ^{n}), thus ⋂_{k∈ℕ} H^{a^{2},k}(ℝ^{n}) ⊂ C_{b}^{∞}(ℝ^{n}). This already proves the following Corollary.

Corollary 5.8. Suppose that the assumptions of Theorem 5.6 hold with k > (n − ℓ)/(2 r_{0}), ℓ ∈ ℕ_{0}. Then the operators e^{−tq(x,D)} − e^{−tq_{1}(D)} and e^{−tq(x,D)} map L^{2}(ℝ^{n}) continuously into C_{b}^{ℓ}(ℝ^{n}) for any fixed t > 0. In case m = ∞, the operators map L^{2}(ℝ^{n}) continuously into C_{b}^{∞}(ℝ^{n}).

Note that Corollary 5.8 does not require any smallness condition on ∑_{|α|≤m} ∥ϕ_{α}∥_{L^{1}}.

We will now proceed with the comparison of {T_{t}}_{t≥0} and {e^{−tq(x,D)}}_{t≥0}.

Lemma 5.9. Let {T_{t}}_{t≥0} be a Fellerian semigroup generated by the pseudodifferential operator −q(x, D). We assume the General Assumption, (2.10.0), and (4.1)–(4.4). Then we have for all u ∈ C_{0}^{∞}(ℝ^{n}), t ≥ 0, and every x ∈ ℝ^{n} the estimate

(5.13)  |T_{t}u(x) − e^{−tq(x,D)}u(x)| ≤ 2 t (γ_{0} + ∥ϕ_{0}∥_{L^{∞}}) ∥(1 + a^{2}) û∥_{L^{1}}.

Proof. Observe that for every x ∈ ℝ^{n} and t ≥ 0 the operator e^{−tq(x,D)} is well-defined for those u with û ∈ L^{1}(ℝ^{n}) which is clearly the case if u ∈ C_{0}^{∞}(ℝ^{n}). This follows from the fact that ξ ↦ q(x, ξ) is negative definite, i.e., q(x, ξ) ≥ 0, and therefore e^{−tq(x,ξ)} ≤ 1. Now, with the notation of Section 4,

|T_{t}u(x) − e^{−tq(x,D)}u(x)| = | ∫_{ℝ^{n}} e^{ixξ}(λ_{t}(x, ξ) − e^{−tq(x,ξ)})û(ξ) dξ|
≤ ∫_{ℝ^{n}} |λ_{t}(x, ξ) − e^{−tq(x,ξ)}| |û(ξ)| dξ
≤ 2 t (γ_{0} + ∥ϕ_{0}_{L^{∞}}∥) ∥(1 + a^{2}) û∥_{L^{1}}.
□

As in Section 4 we have:

Corollary 5.10. Under the assumptions of Lemma 5.9 with m > n + 1 we have for all u ∈ C_{0}^{∞}(ℝ^{n})

(5.14)  |T_{t}u(x) − e^{−tq(x,D)}u(x)| ≤ t γ′ ∥u∥_{n+1},

where γ′ is given by

(5.15)  γ′ = 2 (γ_{0} + ∥ϕ_{0}∥_{L^{∞}}) κ_{1}( ∫_{ℝ^{n}}(1 + |τ|^{2})^{−(n+1)/2} dτ)^{1/2}.

The operator e^{−tq(x,D)} is therefore, at least for small t > 0, a reasonable approximation of T_{t}. If one wants to compare e^{−tq(x,D)}u and T_{t}u in L^{2}-norm, it is necessary to have some estimates for the partial derivatives of λ_{t}(x, ξ) w.r.t. the variable x. Such estimates are difficult to obtain from (4.1). Another approach using the resolvent of −q(x, D) rather than the semigroup seems to be possible. This will be considered in a forthcoming paper.

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Address of R. L. Schilling: Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom