An. Şt. Univ. Ovidius Constanţa Vol. 112, 23, 119 126 A CLASS OF DIRICHLET FORMS GENERATED BY PSEUDO DIFFERENTIAL OPERATORS Emil Popescu Abstract We prove that under suitable assumptions it is possible to construct some Dirichlet forms starting with pseudo differential operators. 1. Introduction The notion of Dirichlet form had been introduced by A. Beurling and J. Deny with the aim to generalize Hilbert spaces methods from classical potential theory to more general situations. To any Dirichlet form one can associate a self-adoint operator, its generator. All properties of the form must be reflected in properties of this generator. If the form is local, this generator is a closed extension of a differential operator. In the case of translation invariant Dirichlet forms one has a complete description of all forms and their generators. One of the most important tool to get these results is the Fourier transform. In the case of a non-local form, it is only little known about nonlocal generators of Dirichlet forms. N. Jacob and co-workers have shown that under suitable assumptions it is possible to construct Dirichlet forms associated with pseudo differential operators. Starting from [2] and [4], we done a class of non-local Dirichlet forms generated by pseudo differential operators. 2. Sobolev spaces generated by a negative definite function We denote by C andc the set of all infinitely differentiable functions which have compact support, respective the set of all continuous functions which vanish at infinity on. For f L 1, we define the Fourier transform: f ξ =2π n 2 e ix ξ f xdx. Key Words: Dirichlet form, negative definite function, pseudo differential operator Mathematical Reviews subect classification: 31C25, 47D7, 6G99 119
12 Emil Popescu The well known formula of Parseval takes place: ϕψdx = ϕ ψdξ. If u L 2 then u L 2 and u = û, where is the norm in L 2. The general form of a pseudo-differential operator is p x, D ϕ x =2π n 2 e ix ξ px, ξ ϕξdξ, for ϕ C. px, ξ is called the symbol of this operator. A function a : C is said to be negative definite if for all m N and x 1,x 2,..., x m, x, 1 m and for all m-tuple c 1,c 2,..., c m C m we have i,=1 [ a x i + a x a x i x ] c i c. Let a be a continuous negative definite function.then exists C> such that aξ C 1+ ξ 2, holds for all ξ [1]. If a is a continuous negative definite function then the same holds for a 1/2. Definition. Let a be a continuous negative definite function on. We define { Ha 1/2 := u L 2 u 2 1/2,a := } 1 + aξ ûξ 2 dξ< 1 + aξ ûξ 2 dξ For aξ = ξ 2s we obtain the usual Sobolev spaces. For more details see [6], [7], [8]. Proposition 2.1. Let a be as previous. Then: 1 Ha 1/2 isahilbertspacewiththenorm u 1/2,a ; 2 for all u from Ha 1/2, u 2 1/2,a = u 2 + a1/2 D u 2,
A CLASS OF DIRICHLET FORMS GENERATED BY PSEUDO DIFFERENTIAL OPERATORS 121 where a 1/2 D u x := e ix ξ a 1/2 ξûξdξ; 3 C Rn is a dense subspace of H 1/2 a. Proposition 2.2. Let a, b be continuous negative definite functions on and suppose that exist ρ, C > such that aξ Cbξ, for all ξ with ξ ρ. Then Ha 1/2 is continuously embedded into H 1/2 b. In particular, H 1 Ha 1/2 L 2. Definition. ADirichlet form on L 2, is a closed symmetric non-negative bilinear form B with the domain D B such that u D B implies that u + 1 D B and B u + 1,u + 1 B u, u. The pair D B,B is called Dirichlet space. Theorem 2.3. Beurling-Deny Let B be a translation invariant symmetric Dirichlet form on. Then there exists a continuous real valued negative definite function a with a 1 L 1 loc Rn such that for all u, v D B C wehave B u, v = aξûξ vξdξ. Conversely, any continuous negative definite function with the property stated above defines a translation invariant symmetric Dirichlet form B with domain D B = u L2 aξ ûξ 2 dξ<. Assume that C Rn D B. We define for u C Rn a pseudodifferential operator by a D u x = e ix ξ aξûξdξ.
122 Emil Popescu We observe a D u 2 = Hence if u C obtain: It follows that a 2 ξ ûξ 2 dξ C 1+ ξ 2 2 ûξ 2 dξ<. Rn thenad u L 2. From Plancherel s theorem we aξûξ vξdξ = a D u xv xdx. B u, v =a D u, v,u,v C Rn. Theorem 2.4. The Hilbert space Ha 1/2 with the bilinear form.,. 1/2,a is a Dirichlet space. Proof. Using the fact that u, v 1/2,a = 1 + aξûξ vξdξ and Theorem 2.3, the assertion holds immediatly. 3. A class of Dirichlet forms Let m N. For 1 m, n N, let a : R be a continuous negative definite function a ξ = 1 cos ξ x d µ x, ξ := ξ n, \{} where µ is a positive finite symmetric measure on \{}. We denote by The image of µ under the mapping n := n 1 + n 2 +... + n m. T :, x,...,,x,,..., is denoted by µ.letb L be independent of x. We denote by and we identify x =x 1,..., x 1,x +1,..., x m,x k := x nk, 1 k m, R := Rn1... 1 +1... m
A CLASS OF DIRICHLET FORMS GENERATED BY PSEUDO DIFFERENTIAL OPERATORS 123 with a subspace of ; b = b x. Let a : R be the continuous negative definite function a ξ = In this case Ha 1/2 := u L2 a ξ. =1 and Ha 1/2,.,. 1/2,a is a Dirichlet space. Ha 1/2 the function u + 1 Ha 1/2. For ϕ C Rn we define Lϕ x = 1 + a γ û γ 2 dξ< b x A ϕ x, =1 We retain that for u where A, 1 m, A ϕ x =2π n 2 e ix ξ a ξ ϕ ξdξ. We observe Lϕ x =2π n 2 e ix ξ b x a ξ ϕ ξdξ. =1 Since b is independent of x we can associate with L a symmetric bilinear form defined on C Rn by B ϕ, ψ =Lϕ, ψ = =1 where A 1/2 has the symbol a 1/2,1 m. b x 1/2 A ϕ x A 1/2 ψ xdx, Proposition 3.1. For all ϕ, ψ C wehave B ϕ, ψ C ϕ 1/2,a ψ 1/2,a.
124 Emil Popescu Therefore, B has a continuous extension to Ha 1/2. Proof. Since C Rn isdenseinha 1/2, we consider ϕ, ψ C Rn. We have B ϕ, ψ = =1 C b x A 1/2 A 1/2 =1 ϕ x,a 1/2 ϕ A 1/2 ψ. ψ x and A 1/2 ϕ C ϕ 1/2,a, A 1/2 ψ C ψ 1/2,a. We assume that there exists a constant c > such that b x c, =1,..., n. Proposition 3.2. For all ϕ Ha 1/2 wehave B ϕ, ϕ, Bϕ, ϕ c ϕ 2 1/2,a c ϕ 2. Proof. For ϕ Ha 1/2, we have B ϕ, ϕ = b x 1/2 A ϕ, A 1/2 ϕ m = =1 m C =1 A 1/2 ϕ x 2 dx. =1 b x A 1/2 ϕ x 2 dx Moreover, B ϕ, ϕ C a ξ ϕ ξ 2 dξ = c ϕ 2 1/2,a c ϕ 2. =1 Thus, B with the domain D B =Ha 1/2 is a closed symmetric bilinear form on L 2. Proposition 3.3. For all ϕ, ψ Ha 1/2 wehave B ϕ, ψ = [ϕ x + y ϕ x] [ψ x + y ψ x] J x, dydx,
A CLASS OF DIRICHLET FORMS GENERATED BY PSEUDO DIFFERENTIAL OPERATORS 125 where J x, dy = 1 2 n b x µ dy =1 and µ are measures on \{}. Proof. The idea of the proof is the formula B ϕ, ψ = n b x =1 R A ϕ x A ψ xdx d x and to apply the partial Fourier transform F ϕγ,x = e iξ x ϕx,x dx. G Theorem 3.4. B is a Dirichlet form with the domain D B =H 1/2 a on L 2. Proof. If ϕ H 1/2 a then, from Theorem 2.4, ϕ + 1 H 1/2 a. The fact that T 1 ϕ =infϕ +, 1 is a normal contraction, i.e. T 1 =, T 1 x T 1 y x y, implies that ϕ + 1 x + y ϕ + 1 x ϕ x + y ϕ x holds for all x, y. It follows that B ϕ + 1,ϕ + 1 B ϕ, ϕ. Remark 3.5. If n 1 = n 2 =... = n m = 1 then we obtain the results from [4]. REFERENCES [1] C. Berg and G. Forst, Potential theory on locally compact abelian groups. Springer, 1975. [2] K. Doppel and N. Jacob, On Dirichlet s boundary value problem for certain anisotropic differential and pseudo differential operators. Potential Theory Ed. J. Kral et al., Plenum Press 1988, 75-83.
126 Emil Popescu [3] M. Fukushima, Dirichlet forms and Markov processes. North Holland, 198. [4] W. Hoh and N. Jacob, Some Dirichlet forms generated by pseudo differential operators. Bull. Sc. Math. 16 1992, 383-398. [5] N. Jacob, Characteristic functions and symbols in the theory of Feller processes. Potential Analysis 8 1998, 61-68. [6] E. Popescu, PD-opérateurs k-elliptiques sur les groupes abéliens local compacts. Rev. Roumaine Math. Pures Appl. 39 1994, 487-499. [7] E. Popescu, A Note on Feller Semigroups. Potential Analysis 14 21, 27-29. [8] E. Popescu, On the symbols associated with a semigroup of operators. Rev. Roumaine Math. Pures Appl. 46 21, 77-84. Department of Mathematics Technical University of Civil Engineering B-dul Lacul Tei, Nr 124, Sector 2, RO-2396, Bucharest 38, Romania E-mail: emilvpopescu@yahoo.com