On hypoellipticity of generators of L´ evy processes

DOI: 10.1007/s11512-009-0099-z

c 2009 by Institut Mittag-Leffler. All rights reserved

On hypoellipticity of generators of L´ evy processes

Helmut Abels and Ryad Husseini

Abstract. We give a sufficient condition on a L´evy measure μwhich ensures that the generatorLof the corresponding pure jump L´evy process is (locally) hypoelliptic, i.e., sing suppu⊆ sing suppLufor all admissibleu. In particular, we assume thatμ|_R^d_\{0}∈C^∞(R^d\{0}). We also show that this condition is necessary provided that suppμis compact.

1. Introduction

Hypoellipticity of elliptic partial differential operators or, more generally, pseu- dodifferential operators is one of the classical topics in the theory of partial differ- ential equations. Briefly, an operatorLis calledhypoellipticif the singular support ofuis contained in the singular support ofLufor alluin the domain ofL. In par- ticular, hypoellipticity comprises the C^∞-regularity of functions on their domains ofL-harmonicity where we callu:R^d→RL-harmonic on Ω ifLu=0 on Ω.

This analytic notion has, ifLgenerates a strong Markov process (Xt)t≥0in an appropriate way, a probabilistic counterpart. More precisely, a bounded measurable functionuis said to be harmonic on Ω with respect to (X_t)_t≥0 ifu(X_t∧τΩ) is, for allx∈R^d, a local P^x-martingale. Here τ_Ω denotes the first exit time of Ω and P^x is the probability measure under which the process starts in x, i.e.,X0=xa.s. If (Xt)t≥0is a Brownian motion, this yields the mean-value property of classical har- monic functions. In fact, harmonicity with respect to a reasonable Markov process can always be defined by a generalized mean-value property, see for example [6].

Functions harmonic in this sense play an important role in the potential theory of Markov processes. This motivates an increasing interest for example in questions of regularity of these operators by probabilists. Since, by the theorem of Courr`ege [7],

The second author’s research was financed by DFG (German Science Foundation) through SFB 611.

generators of strong Markov processes are pseudodifferential operators with contin- uous negative definite symbols as described for example in [14] or [8], generally they do not fit in the framework of classical symbol classes, for example the H¨ormander classS_1,0^m.

Regularity of functions which are harmonic with respect to jump processes has been an object of intense studies in the last years. Let us mention here for example [2], [4], [9], [11], [15], [16], [18] and [19]. Most of these papers deal with a-priori continuity estimates in the broader context of processes with space-dependent jump measures. They rely on a delicate interplay between lower and upper bounds on the jump measures, i.e., they deal with fixed order operators where the small jumps are in principle comparable to those of a rotationally symmetricα-stable L´evy process.

The variable order case is far more difficult as it is for example emphasized by a counterexample in [1].

In the present paper we concentrate on stochastic processes with stationary and independent increments, so-called L´evy processes. Their generators are translation- invariant and mapC₀^∞(R^d) continuously toC^∞(R^d). Hence they act as convolution with a distribution. We show that in this case essentially smoothness away from the origin and a lower bound on the L´evy measure are enough to yield smoothness of harmonic functions.

Let us give a precise formulation of our results: Letν be a L´evy measure, i.e., a nonnegative Borel measure on R^d such thatν({0})=0 and

R^dmin(1,|h|²)ν(dh)<

∞. Moreover, we assume that ν is absolutely continuous with respect to the Lebesgue measure with a densitynsatisfying the following assumptions:

(A1) n∈C^∞(R^d\{0}) andn|_R^d_\B1(0)∈H^∞(R^d\B1(0)).

(A2) There existr0, c>0 andα∈(0,2) so that for allω∈S^d−1={x∈R^d:|x|=1} and 0<r≤r0:

(1)

|h|≤r

|h·ω|²ν(dh)≥cr^2−α.

(A1) ensures that nis smooth onR^d\{0} with all its derivatives square-integrable away from 0. Note also, that (A2) only assumes a lower bound on the growth of ν near 0. For example, (A2) holds ifn(h)≥c|h|^−d−α near 0. The generator of the associated L´evy process Lis onC_b²(R^d) given by

(2) Lu(x) =

R^d

u(x+h)−u(x)−h·∇u(x) 1+|x|²

ν(dx).

It acts in the Fourier space as a multiplication operator with the continuous negative- definite function associated toνby the L´evy–Khinchin formula, cf. (5) below. More- over, it is of order 2 on certain weighted Sobolev spacesH^ψ,s(R^d), see Section2for precise definitions.

We say that L is locally hypoelliptic with respect to H=H^−∞(R^d) or H= H^s(R^d), s∈R, if for anyf∈H and a distribution u∈H^−∞(R^d) such thatLu=f in R^d, andU⊂R^d open we have

f|U∈C^∞(U) =⇒ u|U∈C^∞(U).

The translation invariance ofL implies that local hypoellipticity ofL onH^s0(R^d) for some s0∈R entails local hypoellipticity on H^−∞(R^d), cf. Lemma 2.4 below.

Therefore we will sometimes simply callL locally hypoelliptic in this case.

Our main results now reads as follows.

Theorem 1.1. Letν be an absolutely continuous L´evy measure with density n that satisfies(A1)–(A2). Then the generator of the pure-jump L´evy processLgiven by(2) is locally hypoelliptic on H^−∞(R^d).

Moreover, in the case thatν is a compactly supported L´evy measure satisfying (1) it is also necessary that ν is smooth on R^d\{0}. More precisely, we have the following result.

Theorem 1.2. Letν be a compactly supported L´evy measure that satisfies(A2).

Assume furthermore thatLis locally hypoelliptic onH^−∞(R^d). Thenν is absolutely continuous with a density which is smooth onR^d\{0}.

Note that a compactly supported L´evy measure with smooth density onR^d\{0} automatically satisfies (A1). Hence (A1) is also necessary in that case.

We want to finish this section by some examples.

Letα∈(0,2) andf:S^d−1→R⁺be a smooth function such that the support of f is not contained in any proper subspace of R^d. We set ν(dh)=|h|^−d−αf(h/|h|).

Thenν satisfies (A1) and (A2) and therefore the generator of the associated sym- metricα-stable L´evy process is hypoelliptic.

It is also interesting to remark the following: In [3] there is given a counterex- ample of a L´evy process which does not admit a scale-invariant Harnack inequality.

One can modify this construction in an obvious way such that our results apply.

Hence in this example the related L´evy process still has a hypoelliptic generator.

2. Prerequisites

We start by recalling some basic concepts. Details can be found for example in [5], [12], [13], [14] and [17].

We denote byS(R^d) the Schwartz space, byD(R^d) the space of distributions, i.e., the topological dual of C₀^∞(R^d), by E(R^d) the space of compactly supported distributions, and by S(R^d) the space of tempered distributions, i.e., the dual of S(R^d). Moreover, let H^s(R^d) be the usual L²-Sobolev space of order s∈R. Furthermore, we set H^∞(R^d)=

s∈RH^s(R^d) and H^−∞(R^d)=

s∈RH^s(R^d). We also write F for the Fourier transform and ˆu=F[u]. Note that by the Paley–

Wiener theorem [10, Theorem 7.3.1]E(R^d)⊂H^−∞(R^d).

A functionφ: R^d→C is called negative definite if the matrix (φ(ξ_i)+φ(ξ_j)− φ(ξi−ξj))^k_i,j=1 is positive definite for each choice of k∈Nand ξ1, ..., ξk∈R^d. Then φsatisfies for example φ(ξ)=φ(−ξ), Reφ(ξ)≥φ(0) and the Peetre-type inequality

(3) (1+|φ(ξ)|)^s

(1+|φ(η)|)^s≤2^|s|(1+|φ(ξ−η)|)^|s|. Note also the estimate

(4) |φ(ξ)−φ(η)| ≤4(1+|φ(ξ−η)|)

1+ Reφ(ξ) .

This follows from the third inequality of Lemma 3.6.21 in [14] which implies

|φ(ξ)−φ(η)| ≤ |φ(ξ)−φ(η)+φ(η−ξ)|+|φ(η−ξ)|

≤2 Reφ(η−ξ) Reφ(ξ)+|φ(η−ξ)|

1+ Reφ(ξ) .

Ifφis locally bounded we have in addition|φ(ξ)|≤c(1+|ξ|²). The set of continuous negative definite functions CN(R^d) is a convex cone closed in the topology of uni- form convergence on compact sets. Eachφ∈CN(R^d) has the unique L´evy–Khinchin representation

(5) φ(ξ) =b+Aξ·ξ+iξ·γ+

R^d

1−e^ih·ξ+ ih·ξ 1+|h|²

ν(dh).

Here, b≥0, A is a positive definite matrix, γ∈R^d and ν is a L´evy measure, i.e., a nonnegative Borel measure onR^d withν({0})=0 and

R^d(1∧|h|²)ν(dh)<∞. By the theorem of Sch¨onberg [5, Theorem 7.8], the elementsφ∈CN(R^d) satis- fyingφ(0)=0 are in one-to-one correspondence with L´evy processes (Xt)t≥0. More precisely, the Fourier transform of the distribution ofXt inR^d is given by e^−tφ(ξ), on the other hand the generator of (X_t)_t≥0 is the pseudodifferential operator with symbol−φ(ξ),

−φ(Dx)u(x) =−Fξ→x[φ(ξ)ˆu(ξ)] =−

R^d

e^ix·ξφ(ξ)ˆu(ξ)d¯ξ,

whered¯ξ=(2π)^−ddξ. IfA=b=γ=0, thenLu=−φ(Dx)u, whereLuis as in (2).

An important example for continuous negative definite functions are the func- tionsξ→|ξ|^α, whereα∈(0,2]. The corresponding L´evy processes are the rotation- ally invariant α-stable L´evy processes for α∈(0,2) and in particular a Brownian motion forα=2.

In our framework it is useful to introduce weighted (or anisotropic) Sobolev spaces tailored on the operators we consider here. We fix a continuous negative definite reference functionψ:R^d→Cwhich satisfies lim_|ξ|→∞|ψ(ξ)|=∞and set

H^ψ,s(R^d) ={u∈L²(R^d) :uψ,s:=(1+|ψ(·)|)^s/2uˆL²(R^d)<∞}. Define also for an open set Ω⊂R^d,

H_loc^ψ,s(Ω) ={u: Ω→R:χu∈H^ψ,s(R^d) for all χ∈C₀^∞(Ω)}

andH_loc^s (Ω)=H_loc^|ξ|2,s(Ω). We haveH^s(R^d)=H^|ξ|2,s(R^d) andH^s(R^d)⊂H^ψ,s(R^d) due to |ψ(s)|≤C(1+|ξ|²). Note also, thatH^ψ,s(R^d)^∗∼=H^ψ,−s(R^d): uis in H^ψ,s(R^d) if and only if (u, v)L²≤cvψ,−s for allv∈H^ψ,−s(R^d).

Letφ∈CN(R^d) satisfy the following conditions:

(S1) There exists ˇ1>0 such that|φ(ξ)|≤ˇ1(1+|ψ(ξ)|).

(S2) There exist ˇ2, r₀>0 such that|φ(ξ)|≥ˇ2|ψ(ξ)|if|ξ|≥r₀. Then, by (S1), φ(Dx) mapsH^ψ,s+2(R^d) continuously toH^ψ,s(R^d).

Theorem 2.1. Let φ satisfy (S1) and (S2). Let t∈R and f∈H^ψ,t(R^d). If u∈H^−∞(R^d)is a solution of φ(Dx)u=f in S(R^d), thenu∈H^ψ,t+2(R^d).

Proof. Without loss of generality we may assumef∈L²(R^d). Thenφˆu∈L²(R^d), ˆ

u∈L²_loc(R^d), and lim_|ξ|→∞|φ(ξ)|=∞ imply (1+|φ|)ˆu∈L²(R^d). Thus (S2) implies the statement of the theorem.

Moreover, the commutator [φ(Dx), χ] of φ(Dx) and the multiplication with χ∈C₀^∞(R^d) acts with order 1 inH^ψ,−∞(R^d), i.e.:

Lemma 2.2. Let φsatisfy(S1) and (S2),t∈Randχ∈C₀^∞(R^d). Then for all u∈H^ψ,t+1(R^d) we have

[φ(Dx), χ]ut,ψ≤Cut+1,ψ, whereC is independent of u.

Proof. Letu, v∈S(R^d). Then on one hand we have F([φ(D), χ]u)(ξ) =

R^dχ(ξ −η)(φ(ξ)−φ(η))ˆu(η)dη.

By the theorem of Plancherel, (S1), (4) and (3) we estimate:

|(φ(D_x), χ]u, v)_L2(Ω)| ≤

R^2d|χ(ξ −η)| |φ(ξ)−φ(η)| |u(η)ˆ | |ˆv(ξ)|dη dξ

≤C

R^2d|χ(ξ−η)|(1+|ξ−η|²)(1+|ψ(ξ)|)^1/2|u(η)ˆ | |v(ξ)ˆ |dη dξ

=C

R^2d|χ(ξ−η)|(1+|ξ−η|²)

1+|ψ(ξ)| 1+|ψ(η)|

(t+1)/2

×(1+|ψ(η)|)^(t+1)/2|u(η)ˆ |(1+|ψ(ξ)|)^−t/2|ˆv(ξ)|dη dξ

≤C

R^2d|χ(ξ−η)|(1+|ξ−η|²)^(|t|+3)/2(1+|ψ(η)|)^(t+1)/2|ˆu(η)|

×(1+|ψ(ξ)|)^−t/2|v(ξ)ˆ |dη dξ

≤C(1+|ξ|²)^(|t|+3)/2χ(ξ) L₁(R^d)uψ,t+1vψ,−t

≤Cuψ,t+1vψ,−t.

The assertion now follows by continuity and the characterization of H^ψ,s(R^d) as dual ofH^ψ,−s(R^d).

A direct application of this commutator estimate yields local regularity of the following type.

Theorem 2.3. Let φsatisfy (S1) and (S2). If χ∈C₀^∞(R^d), t∈R, f∈L²(R^d), with χf∈H^ψ,t(R^d),andu∈H^ψ,t+1(R^d)solves φ(D)u=f, thenχu∈H^ψ,t+2(R^d).

Unfortunately, we cannot expect to iterate this result without additional as- sumptions as is illustrated by Theorem1.2.

We finish this section by showing that the notion of local hypoellipticity with respect to H^s(R^d) is independent ofs∈R.

Lemma 2.4. Let s0∈R. If L is locally hypoelliptic with respect to H^s0(R^d), thenL is locally hypoelliptic with respect to H^−∞(R^d).

Proof. Let L be locally hypoelliptic with respect to H^s0(R^d). In order to prove that L is locally hypoelliptic with respect toH^s(R^d), s∈R, let f∈H^s(R^d), u∈S(R^d) with Lu=f and U⊂R^d open be such that f|U∈C^∞(U). Then f= D_x^s0−sf∈H^s(R^d), where ξ=(1+|ξ|²)^1/2 and D_x^s denotes the pseudodiffer- ential operator with symbol ξ^s. Moreover, Lu=f with u=Dx^s0−su since L

commutes withDx^s0−s. Sinceξ^s0−s∈S_1,0^s0−s(R^d×R^d), Dx^s0−s is pseudolocal, i.e., sing suppf=sing suppDx^s0−sf⊆sing suppf, cf. e.g. [10, Theorem 18.1.16].

Hence f|U∈C^∞(U) and therefore u|U∈C^∞(U) due to the local hypoellipticity with respect toH^s0(R^d). Finally, sinceD_x^s−s0 is pseudolocal too,

sing suppu= sing suppDx^s−s0u⊆sing suppu.

Thusu|U∈C^∞(U). This shows thatLis locally hypoelliptic with respect toH^s(R^d) for anys∈R. HenceLis locally hypoelliptic with respect to H^−∞(R^d).

3. Proof of Theorem 1.1

Let ψ be the continuous negative definite function associated by (5) to the pure-jump L´evy process with L´evy measureν and letL be its generator. The real part ofψ is

(6) Reψ(ξ) =

R^d

(1−cos(h·ξ))ν(dh).

Lemma 3.1. Assume(A2). Then there exists somec>0such that 1+|ψ(ξ)|≥

c|ξ|^α.

Proof. Using (A2) and the inequality 1−cosx≥x²/4 for |x|≤¹₂, we estimate for all|ξ|≥(2r1)⁻¹,

|ψ(ξ)| ≥Reψ(ξ)≥

|h|≤(2|ξ|)⁻¹

(1−cos(h·ξ))ν(dh)

≥|ξ|² 4

|h|≤(2|ξ|)⁻¹

h·|ξ|⁻¹ξ²ν(dh)≥c|ξ|^α.

Therefore for all s>0 the anisotropic Sobolev space H^ψ,s(R^d) is continuously embedded intoH^αs/2(R^d).

Observe that by (A2) the asymptotic behavior of |ψ(ξ)| as |ξ|→∞ remains unchanged if one cuts off the large jumps of the L´evy process in the following sense:

Fix for r>0 a function ρr∈C₀^∞(B2r(0)) with 0≤ρr≤1 and ρr≡1 on Br(0). Then ψcan be decomposed asψ=ψr,long+ψr,short, where

ψr,long(ξ) =

R^d

(1−e^ih·ξ)(1−ρr(h))ν(dh), ψr,short(ξ) =

R^d

1−e^ih·ξ+ ih·ξ 1+|h|²

ρr(h)ν(dh)+iξ·

R^d

h(1−ρr(h)) 1+|h|² ν(dh).

Becauseψr,long is bounded, Lemma3.1implies thatψr,shortsatisfies (S1) and (S2).

Note also that the operator associated toψr,short is 2r-local in the sense that suppψr,short(Dx)u⊂B2r(suppu) ={x∈R^d: dist(x,suppu)<2r}.

For the following we also assume that ν(dh)=n(h)dh and the density n satis- fies (A1). The key step in our argument is the following regularity result.

Lemma 3.2. Let Ω⊂R^d be open. If f∈L²(R^d) with f|Ω∈H_loc^ψ,t(Ω) and u∈ H^ψ,1(R^d)with u|Ω∈H_loc^ψ,t+1(Ω) solve ψ(D_x)u=f inS(R^d),then u|Ω∈H_loc^ψ,t+2(Ω).

Proof. Let χ₁∈C₀^∞(Ω). We fix r>0 such that 4r<dist(R^d\Ω,suppχ₁) and choose a cut-off function χ2∈C₀^∞(Ω) with χ2≡1 on B4r(suppχ1). If ψr,short and ψr,long are as above, thenχ2usolves

ψr,short(Dx)(χ2u) =f−ψr,long(Dx)u−ψr,short(Dx)((1−χ2)u) = ˜f in S(R^d), where ˜f is in L²(R^d). By (A1), ψr,long(Dx)u is the sum of a convolution of u with (1−ρr)n∈H^∞(R^d)—which is smooth—and a constant multiple of u. Since supp(1−χ₂)u⊆R^d\B_4r(suppχ₁) the support ofψ_r,short(D_x)((1−χ₂)u) is contained in R^d\suppχ1. Hence χ1f˜∈H^ψ,t(R^d) and Theorem 2.3 yields χ1χ2u=χ1u∈ H^ψ,t+2(R^d).

Proof of Theorem 1.1. Because of Lemma 2.4 it is sufficient to prove that L is locally hypoelliptic with respect to L²(R^d). To this end let Ω⊂R^d and let u∈ H^−∞(R^d) be a solution ofψ(D_x)u=f inS(R^d) withf∈L²(R^d) andf|Ω∈C^∞(Ω).

Sincef|Ω∈H_loc^ψ,∞(Ω), iterating Lemma3.2impliesu∈H_loc^ψ,∞(Ω). By Lemma3.1we have u∈H_loc^∞(Ω) and therefore, by Sobolev embedding,u∈C^∞(Ω).

4. Proof of Theorem1.2

The proof of Theorem1.2is based on the results of Chapter 16 in [10], which will be summarized below. Let us first fix some notation: IfE is a set, then chE denotes its convex hull. The supporting function of a convex, compact subsetE⊂R^d is given by

H_E(x) = sup

y∈E

x·y,

whereH_∅≡−∞by definition. This gives a one-to-one correspondence between con- vex compact subsets and the setHof convex, subadditive, positively homogeneous functions. For each u∈E(R^d) we denote by H(u)⊂H the same set as defined in [10, Definition 16.3.2]. We omit the precise definition at this point since it is a bit

involved and not needed for our purposes. In the following we will only use some of the properties ofH(u), which we now summarize.

Theorem 4.1. Let u∈E(R^d) and let H be the supporting function of ch sing suppu. Then

H(x) = sup

h∈H(u)

h(x).

The latter theorem coincides with [10, Theorem 16.3.4].

Theorem 4.2. Let u∈E(R^d) and h∈H(u). Then there is some w∈E(R^d)∩ C⁰(R^d)satisfying sing suppw={0} such thath is the supporting function of

ch sing suppu∗w.

The statement of the theorem is just the first statement of [10, Theorem 16.3.13]

with the only difference that the statement is formulated withw∈E(R^d) only. That indeed there is somew∈E(R^d)∩C⁰(R^d) with the stated properties is shown in the proof of [10, Theorem 16.3.13].

Finally, we note that u is called invertible if −∞∈H/ (u), cf. [10, Defini- tion 16.3.12]. The following condition forunot to be invertible will be used several times.

Theorem 4.3. Letμ∈E(R^d). Then the following statements are equivalent:

(1) −∞∈H(μ);

(2) For everyx∈R^d there is somew∈C⁰(R^d)\C¹(R^d)such that sing suppw=

{x} andμ∗w∈C^∞(R^d);

(3) There is somew∈E(R^d)such that μ∗w∈C^∞(R^d)butw /∈C^∞(R^d).

The latter theorem coincides with [10, Theorem 16.3.9].

Theorem 4.4. Letμ∈E(R^d). Then the following statements are equivalent:

(1) u∈D(R^d)andμ∗u∈C^∞(R^d)impliesu∈C^∞(R^d);

(2) μis hypoelliptic in the sense of [10], i.e., μ is invertible and

|Imζ|

log|ζ| → ∞, as|ζ|→ ∞, on{ζ∈C^d: ˆμ(ζ) = 0};

(3) There is some E∈E(R^d) such that E∗μ−δ∈C^∞(R^d) and sing suppE=

−sing suppμ.

Proof. The theorem follows directly from the equivalent conditions (i), (ii), and (v) of [10, Theorem 16.6.5], where we note that hypoellipticity is defined in Definition 16.6.4 of [10].

Note the following: If ν is a L´evy measure, then the associated operator L:C₀^∞(R^d)→C^∞(R^d) is linear, translation invariant and continuous and can there- fore be written as a convolution with a distribution μ∈D(R^d), cf. [10, Theo- rem 4.2.1]. Moreover, foru∈C₀^∞(R^d) andx /∈suppuit follows that

Lu(x) =

R^d

u(x+h)−u(x)−h·∇u(x) 1+|x|²

ν(dh) =

R^d

u(x+h)ν(dh) = ˜ν∗u, where ˜ν denotes the reflection of ν, i.e.,ν, ϕ˜ :=−ν,ϕand ϕ(x)=ϕ( −x) for all ϕ∈C₀^∞(R^d). Thusμand ˜ν agree onR^d\{0}. As a consequence suppμ=−suppν is compact and sing suppμ\{0}=−sing suppν\{0}. The following proposition relates local hypoellipticity forLas we have defined it above and to hypoellipticity ofμin the sense of H¨ormander, cf. Theorem4.4.

Proposition 4.5. Let L be a L´evy operator that is locally hypoelliptic and satisfies (A2). Then for any u∈D(R^d) and f∈C^∞(R^d) such that Lu=f we have u∈C^∞(R^d).

Proof. Let M >0 be such that suppμ⊆BM(0). In order to show that u∈ C^∞(R^d) it is sufficient to show that u|BR(0)∈C^∞(BR(0)) for anyR>0. Therefore letR>0 be arbitrary and let η∈C₀^∞(R^d) be such that η≡1 on BR+M(0). Then

Lu(x) =μ∗u(x) =μ∗(ηu)(x) for allx∈B_R(0),

whereηu∈E(R^d). Now there is somes∈Rsuch thatηu∈H^s(R^d). ThusL(ηu)=f∈ H^s−2(R^d) since|ψ(ξ)|≤C(1+|ξ|)²for every continuous negative definite functionψ.

Moreover,f|BR(0)=Lu|BR(0)=f|BR(0)∈C^∞(BR(0)), which implies thatηu|BR(0)∈ C^∞(BR(0)) because of the local hypoellipticity of L. Since R>0 was arbitrary we obtainu∈C^∞(R^d).

Proof of Theorem 1.2. First of all, because of Proposition4.5, the first state- ment of Theorem4.4is true. Therefore there is a parametrixE∈E(R^d) such that

k=E∗μ−δ∈C^∞(R^d) and sing suppE=−sing suppμ.

Here even k∈C₀^∞(R^d) since E and μ have compact support. As μ is in turn a parametrix for E, E is also hypoelliptic due to Theorem 4.4 again. In particular this implies thatE is invertible, i.e., −∞∈H/ (E).

Next we show thatH(E)={0}. To this end leth∈H(E). By Theorem4.2there is some w∈E(R^d)∩C0(R^d) with sing suppw={0} such that h is the supporting function of ch sing suppE∗w. In particular,w∈L²(R^d). Next let

v:=F⁻¹[η(ξ)ψ(ξ)⁻¹fˆ(ξ)],

whereη∈C^∞(R^d) such thatη(ξ)=1 for|ξ|≥ρ+1 andη(ξ)=0 for|ξ|≤ρ, whereρis as in (S2). Thenv∈H^ψ,2(R^d) and

Lv=f+k, wherek∈C^∞(R^d).

Now, ifu=E∗w, thenμ∗(u−v)=k∗w−k∈C₀^∞(R^d). Thereforeu−v∈H^∞(R^d) and thereforeu∈H^ψ,2(R^d). Because convolution withμis by assumption locally hypoel- liptic, we have

sing suppE∗w⊆sing suppμ∗E∗w= sing supp(w+k∗w) = sing suppw={0}. As noted above −∞∈H/ (E), and therefore h cannot be the supporting function of ∅. We conclude that ch sing suppE∗w={0} which implies h≡0. This shows thatH(E)={0}.

Thus the supporting function of ch sing suppEisH(x)=sup_h∈H(E)h(x)=0 and finally

{0}= ch sing suppE= sing suppE=−sing suppμ.

This completes the proof.

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Helmut Abels

Max Planck Institute for Mathematics in the Sciences

Inselstrasse 22–26 DE-04103 Leipzig Germany

abels@mis.mpg.de

Ryad Husseini

Institute for Applied Mathematics University of Bonn

Poppelsdorfer Allee 82 DE-53115 Bonn Germany

ryad@uni-bonn.de

Received August 18, 2008 published online June 3, 2009