ON AN EXTENSION OF JUMP-TYPE SYMMETRIC DIRICHLET FORMS

We show that any element from the ( L 2 -)maximal domain of a jump-type symmetric Dirichlet form can be approximated by test functions under some conditions. This gives us a direct proof of the fact that the test functions is dense in Bessel potential spaces.


Introduction
In this note, we are concerned with the following symmetric quadratic form (E, D(E)) defined on L 2 (R d ): x =y (u(x) − u(y))(v(x) − v(y)) n(x, y) dx dy, where n(x, y) is a positive measurable function on x = y.
In order that the form (E, D(E)) makes sense, we assume that the set {(x, y) ∈ R d × R d : n(x, y) = ∞} is a Lebesgue null set. In fact, under this condition, we have already shown that the form (E, D(E)) is a Dirichlet form on L 2 (R d ) in the wide sense (see [11] and [6]). Moreover if we set C 0,1 0 (R d ) the totality of all uniformly Lipschitz continuous functions defined on R d with compact support, then D(E) ⊃ C 0,1 0 (R d ) if and only if the following conditions are satisfied(see [12], [6] and also [ where j(x, y) = n(x, y) + n(y, x). Then under (A) and (B), the quadratic form (E, F) becomes a regular symmetric Dirichlet form on L 2 (R d ), where F is the closure of C 0,1 0 (R d ) with respect to the norm E(•, •) + ||•|| 2 L 2 . Note that, from the integral representation of the form E, we can adopt the test functions, C ∞ 0 (R d ), as a core instead of C 0,1 0 (R d ) under the conditions (A) and (B). We now give some examples(see e.g., [11,12]): Then (A) and (B) hold if and only if 0 < α < 2 and c > 0. This is nothing but the Dirichlet form corresponding to a symmetric α-stable process on R d .
(2) (symmetric stable-like process) For a measurable function α(x) defined on R d , set Then (A) and (B) hold if and only if the following three conditions are satisfied: In general, we do not know whether the set F coincides with D(E). Determining the domains of the Dirichlet form corresponds, in some sense, to solve the boundary problem of the associated Markov processes. This analytic structure was investigated first by Silverstein in [7] and [8], and then by Chen [1] and Kuwae [5].

Identification of the domains
In order to classify the domains of the forms, we will consider the following conditions: there exists a positive constant C > 0 such that and that is, any element in D(E) can be approximated from elements of C ∞ 0 (R d ) with respect to E 1 .
For u ∈ D(E), set the convolution of u and ρ 1/n : satisfies the following inequality: Then we see that Now we estimate (I).
In the first inequality, we used the Jensen inequality for the measure ρ 1/n (z)dz, while the second is from the Fubini theorem, the third is by translation and the fourth is obtained by the assumption (A"). ( Summarizing the calculus done above, we see That is, E(w n v n , w n v n ) are uniformly bounded. Moreover we have seen that ||w n v n || L 2 are also uniform bounded and w n v n converges to u in L 2 (R d ). Thus the Cesàro means of a subsequence of {w n v n } are E 1 -Cauchy and convergent to u a.e. Hence u ∈ F. Thus

Then this satisfies the conditions (A") and (B'). A Markov process corresponding to the Dirichlet form (E, D(E)) is called "stable-like process" by Chen-Kumagai[2].
For a subclass B of all measurable functions on R d , we denote by B b the bounded functions in B. In the following, we always assume that (A) and (B) hold. Then a symmetric Dirichlet form (η, D(η)) on L 2 (R d ) is said to be an extension of the Dirichlet form (E, F) if D(η) ⊃ F and η(u, u) = E(u, u) whenever u ∈ F. Denote by A(E, F) the totality of the extensions of (E, F). By this definition, (E, D(E)) is an element of A(E, F). An element (η, D(η)) of A(E, F) is called a Silverstein extension if F b is an algebraic ideal in D(η) b . For the probabilistic counterpart or an application of Silverstein extensions, see, for example, [8], [10] and [4]. Proof: It is enough to show that u · f ∈ F b whenever u ∈ D(E) b and f ∈ C ∞ 0 (R d ). Let ρ and ρ ε be the same functions in the proof of the preceding theorem. Take the convolution of functions uf and ρ 1/n : w n = ρ 1/n * (uf ). Then w n ∈ C ∞ 0 (R d ), w n converges to uf in the L 2 -space and the inequality ||w n || L ∞ ≤ ||uf || L ∞ holds. Denote by K the support of the function f . As in the proof of the preceding theorem, we estimate E(w n , w n ) as follows: Since j(x, y) = j(y, x), we see where K n = {x + y ∈ R d : x ∈ K, y ∈ B(0, 1/n)}. Now we estimate (I).
Combining the estimates (II) and (I), we have So E(w n , w n ) are uniformly bounded. We have already known that w n ∈ C ∞ 0 (R d ) converges to uf in L 2 . Then by making use of the Banach-Saks theorem, the Cesàro means of a subsequence of {w n } are E 1 -Cauchy and converges to uf a.e. Hence uf ∈ F. This shows that F b is an ideal of D(E) b , whence (E, D(E)) is a Silverstein extension of (E, F).

Remark 1
If the form (E, F) is moreover conservative, then, using a theorem from [5], we can show that the Silverstein extension is unique. Hence this implies that F = D(E). In [6], we showed that under some conditions (which includes the condition (B')), the form (E, F) is conservative. So, we have an alternative proof of Theorem 1 under (A") and (B').
In the following, we consider 'the homogeneous' Dirichlet space: where E is defined in §1 and L 0 (R d ) is the family of all measurable functions on R d . We assume (A) and (B) hold. Since E is defined as an integral form, we can easily see that D 0 (E) ∩ L ∞ (R d ) =: D ∞ (E) is dense in D 0 (E) with respect to quasi-norm E. We now want to consider when any function in D ∞ (E) (hence, in D 0 (E)) can be approximated from a sequence of the test functions with respect to E. Of couse, this relates the notion of 'the extended Dirichlet space' F e . In general, .
If the form (E, F) is transient, then F = F e ∩ L 2 (R d ) (see Theorem 1.5.2(iii) in [3]). It is not easy to see whether the 'homogeneous' domain D 0 (E) coincides with F e except the special cases. In order to consider this, we introduce a little bit stronger condition as follows: there exists a positive functionñ(x) defined on R d − {0} satisfying the condition in Example 1 (3) so that for some constants c i > 0 (i = 1, 2), Proposition 1 Suppose that (C) holds. Moreover, we assume the Dirichlet form (E, F) is recurrent. Then any element in D ∞ (R d ) (hence, in D 0 (E)) can be approximated from the test functions with respect to E. That is, D 0 (E) = F e .
Proof: First note that a similar argument developed in the proof of Theorem 2 gives us that ϕ · u ∈ D 0 (E) provided that u ∈ D 0 (E) and ϕ ∈ C ∞ 0 (R d ). Take the test function ρ defined in the proof of Theorem 1. And also consider the function ρ 1/n for each n. Then considering the convolution u n of u and ρ 1/n , we have the following estimate: In the first inequality, we used the Schwarz inequality, and the second follows from (C). Accordingly, we see that the sequence {u n } is E-bounded. Since ||u n − ϕu|| L 2 converges to 0, a subsequence of u n converges to ϕu almost everywhere. So we can find the Casaro mean {ũ n k } of some subsequence from {u n } n so that E(ũ n k − u,ũ n k − u) converges to 0 andũ n k → ϕ u a. e. This means that there exists a sequence from test functions which conveges to ϕ u with respect to E and with respect to almost everywhere convergence. On the other hand, the Dirichlet form (E, F) is recurrent, we can construct a sequence {ϕ k } ⊂ C ∞ 0 (R d ) satisfying 0 ≤ ϕ k → 1 a.e., ||ϕ k || L ∞ ≤ 1 and E(ϕ k , ϕ k ) → 0.
Note that ϕ k · u ∈ D(E) ∩ L 2 (R d ) for each k because ϕ k ∈ C ∞ 0 (R d ). Similarly, noting the following estimates and the property of ϕ k , we can see that the cesaro meansφ n k u of some subsequence of {ϕ k u} converges to u with respect to E and with respect to almost everywhere convergence: Now for each k, take f k ∈ C ∞ 0 (R d ) so that E(φ n k u − f k ,φ n k u − f k ) < 1/k, Then we see E(f k − u, f k − u) 1/2 ≤ E(f k −φ n k u, f k −φ n k u) 1/2 + E(φ n k u − u,φ n k u − u) 1/2 ≤ 1/k + E(φ n k u − u,φ n k u − u) 1/2 .
So, taking k → ∞, we see that f k converges to u with respect to the quasi-norm E. This concludes the proof.