Capacities, removable sets and $L^p$-uniqueness on Wiener spaces

We prove the equivalence of two different types of capacities in abstract Wiener spaces. This yields a criterion for the $L^p$-uniqueness of the Ornstein-Uhlenbeck operator and its integer powers defined on suitable algebras of functions vanishing in a neighborhood of a given closed set $\Sigma$ of zero Gaussian measure. To prove the equivalence we show the $W^{r,p}(B,\mu)$-boundedness of certain smooth nonlinear truncation operators acting on potentials of nonnegative functions. We also give connections to Gaussian Hausdorff measures. Roughly speaking, if $L^p$-uniqueness holds then the 'removed' set $\Sigma$ must have sufficiently large codimension, in the case of the Ornstein-Uhlenbeck operator for instance at least $2p$.


Introduction
The present article deals with capacities associated with Ornstein-Uhlenbeck operators on abstract Wiener spaces (B, µ, H), [8,11,22,30,31,32,33,42,46], and applications to L p -uniqueness problems for Ornstein-Uhlenbeck operators and their integer powers, endowed with algebras of functions vanishing in a neighborhood of a small closed set.
( 1,2 ) Research supported by the DFG IRTG 2235: 'Searching for the regular in the irregular: Analysis of singular and random systems'.
Our original motivation comes from L p -uniqueness problems for operators L endowed with a suitable algebra A of functions, the special case p = 2 is the problem of essential self-adjointness. For the 'globally defined' operator L on the entire space L p -uniqueness is well understood, see for instance [18] and the references cited there. If the globally defined operator is L p -unique one can ask whether the removal of a small set (or, in other words, the introduction of a small boundary) destroys this uniqueness or not. A loss of uniqueness means that extensions to generators of C 0 -semigroups, [38], with different boundary conditions exist. The answer to this question depends on the size of the removed set. The most classical example may be the essential self-adjointness problem for the Laplacian ∆ on R n , endowed with the algebra C ∞ c (R n \ {0}) of smooth compactly supported functions on R n with the origin {0} removed. It is well known that this operator is essential self-adjoint in L 2 (R n ) if and only if n ≥ 4, [47, p.114] and [39,Theorem X.11,p.161]. Generalizations of this example to manifolds have been provided in [12] and [34], more general examples on Euclidean spaces can be found in [5] and [25], further generalizations to manifolds and metric measure spaces will be discussed in [26]. For the Laplacian on R n one main observation is that, if a compact set Σ of zero measure is removed from R n , the essential self-adjointness of (∆, C ∞ c (R n \ Σ)) in L 2 (R n ) implies that dim H Σ ≤ n − 4, where dim H denotes the Hausdorff dimension. See [5,Theorems 10.3 and 10.5] or [25,Theorem 2]. This necessary 'codimension four' condition can be rephrased by saying that we must have H n−d (Σ) = 0 for all d < 4, where H n−d denotes the Hausdorff measure of dimension n − d.
Having in mind coefficient regularity or boundary value problems for operators in infinite dimensional spaces, see e.g. [10,13,14,23,24], one may wonder whether a similar 'codimension four' condition can be observed in infinite dimensional situations. For the case of Ornstein-Uhlenbeck operators on abstract Wiener spaces an affirmative answer to this question follows from the present results in the special case p = 2.
The basic tools to describe the critical size of a removed set Σ ⊂ B are capacities associated with the Sobolev spaces W r,p (B, µ) for the Hderivative respectively the Ornstein-Uhlenbeck semigroup, [8,11,22,30,31,32,33,42,46]. Such capacities can be introduced following usual concepts of potential theory, [11,20,33,41,42,43,44,46], see Definition 3.1 below, and they are known to be connected to Gaussian Hausdorff measures, [21]. Uniqueness problems connect easier to another, slightly different definition of capacities, where the functions taken into account in the definition are recruited from the initial algebra A and, roughly speaking, are required to be equal to one on the set in question, see Definitions 3.2 and 3.3. This type of definition connects them to an algebraic ideal property which is helpful to investigate extensions of operators initially defined on ideals of A. For Euclidean Sobolev spaces these two types of capacities are known to be equivalent, see for instance [2,Section 2.7]. The proofs of these equivalences go back to Mazja, Khavin, Adams, Hedberg, Polking and others, [1,2,3,35,36,37], and rely on bounds in Sobolev norms for certain nonlinear composition operators acting on the cone of nonnegative Sobolev functions, see e.g. [1,Theorem 3], or the cone of potentials of nonnegative functions, see e.g. [1,Theorem 2] or [2,Theorem 3.3.3]. Apart from the first order case r = 1 this is nontrivial, because in finite dimensions Sobolev spaces are not stable under such compositions, see for instance [2,Theorem 3.3.2]. Apart from the case p = 2, where one can also use an integration by parts argument, [1, Theorem 3], the desired bounds are shown using suitable Gagliardo-Nirenberg inequalities, [3,35], or suitable multiplicative estimates of Riesz or Bessel potential operators involving Hardy-Littelwood maximal functions and the L p -boundedness of the latter, [ [21] on Gaussian Hausdorff measures, we then observe that the L p -uniqueness of this Ornstein-Uhlenbeck operator 'after the removal of Σ' implies that the Gaussian Hausdorff measure ̺ d (Σ) of codimension d of Σ must be zero for all d < 2p, see Corollary 6.2. In particular, if the operator is essentially self-adjoint on L 2 (B, µ), then ̺ d (Σ) must be zero for all d < 4, what is an analog of the necessary 'codimension four' condition knwon from the Euclidean case.
In the next section we recall standard items from the analysis on abstract Wiener spaces. In Section 3 we define Sobolev capacities and prove their equivalence, based on the norm bound on nonlinear compositions, which is proved in Section 4. Section 5 contains the mentioned L p -uniqueness results. The connection to Gaussian Hausdorff measures is briefly discussed in Section 6, followed by some remarks on related Kakutani theorems for multiparameter processes in Section 7.

Acknowledgements
The authors would like to thank Masanori Hino, Jun Masamune, Michael Röckner and Gerald Trutnau for inspiring and helpful discussions.

Preliminaries
Following the presentation in [42], we provide some basic definitions and facts.
Let (B, µ, H) be an abstract Wiener space. That is, B is a real separable Banach space, H is a real separable Hilbert space which is embedded densely and continuously on B, and µ is a Gaussian measure on B with see for instance [42,Definition 1.2]. Here we identify H * with H as usual, so that B * ⊂ H ⊂ B. Since every ϕ ∈ B * is N(0, ϕ 2 H )distributed, it is an element of L 2 (B; µ) and the map ϕ → ϕ, · is an isometry from B * , equipped with the scalar product ·, · H , into L 2 (B, µ). It extends uniquely to an isometry for all h 1 , . . . , h k ∈ H. If so, Φ x is unique and denoted by D k f (x). A function f : B → R is called a (smooth) cylindrical function if there exist an integer n ≥ 1, linear functionals l 1 , ..., l n ∈ B * and a function The space of all such cylindrical functions on B we denote by F C ∞ b . Clearly F C ∞ b is an algebra under pointwise multiplication and stable under the composition with functions T ∈ C ∞ b (R). A cylindrical function f ∈ F C ∞ b as in (2) is infinitely many times H-differentiable at any x ∈ B, and for any k ≥ 1 we have where ∂ j denotes the j-th partial differentiation in the Euclidean sense. The space F C ∞ b is dense in L p (B, µ) for any 1 ≤ p < +∞, see e.g. [7, Lemma 2.1].
We write H 0 := R, H 1 := H and generalizing this, denote by H k the space of k-linear maps A : H k → R such that where (e i ) ∞ i=1 is an orthonormal basis in H. The value of this norm does not depend on the choice of this basis. See [9, p.3]. Clearly every such k-linear map A can also be seen as a linear map A : H ⊗k → R, where H ⊗k denotes the k-fold tensor product of H, with this interpretation we have A(e j 1 ⊗ ... ⊗ e j k ) = A(e j 1 , . . . , e j k ) and by (3) the operator A is a Hilbert-Schmidt operator. For later use we record the following fact.
h k are members of an orthonormal system in H, not necessarily distinct} .
Proof. By Parseval's identity and Cauchy-Schwarz in H ⊗k we have A H k = sup |Ay| : y ∈ H ⊗k and y H ⊗k = 1 .
Choose an element y = y 1 ⊗ ... ⊗ y k ∈ H ⊗k such that y H ⊗k = 1 and A H k ≤ 2|Ay|. Without loss of generality we may assume that y j H = 1, 1 ≤ j ≤ k. Choosing an orthonormal basis (b i ) n i=1 in the subspace span {y 1 , ..., y k } of H we observe n ≤ k and y j = n i=1 b i λ ij with some |λ ij | ≤ 1. Since this implies we obtain the desired result.
We recall the definition of Sobolev spaces on B. For any 1 ≤ p < +∞ and k ≥ 0 let L p (B, µ, H k ) denote the L p -space of functions from B into H k . For any 1 ≤ p < +∞ and integer r ≥ 0 set The Sobolev class W r,p (B, µ) is defined as the completion of F C ∞ b in this norm, see [8,Section 5.2] or [9, Section 8.1]. In particular, W 0,p (B, µ) = L p (B, µ). For f ∈ W r,p (B, µ) the derivatives D k f , k ≤ r, are well defined as elements of L p (B, µ), see [8,Section 5.2]. By definition the spaces W r,p (B, µ) are Banach spaces, Hilbert if p = 2. The space W ∞ of Watanabe test functions is defined as In contrast to Sobolev spaces over finite dimensional spaces, [2, Theorem 3.3.2], also the Sobolev classes W r,p (B, µ), r ≥ 2, are known to be stable under compositions u → T (u) = T • u with functions T ∈ C ∞ b (R), as follows from the evaluation of an integration by parts identity together with the chain rule, applied to cylindrical functions. See [8, Remark 5.2.1 (i)] or [9,Proposition 8.7.5]. In particular, the space W ∞ is stable under compositions with functions from C ∞ b (R).
Also, it is an algebra with respect to pointwise multiplication, [33,Corollary 5.8]. Given a bounded (or nonnegative) Borel function f : B → R and t > 0 set The function P t f is again bounded (resp. nonnegative) Borel on B and the operators P t form a semigroup, i.e. that for any s, t > 0 we have P t+s = P t P s . The semigroup (P t ) t>0 is called the Ornstein-Uhlenbeck semigroup on B. For any 1 ≤ p ≤ +∞ it extends to a contraction semi- We will always write P t and L instead of P (p) t and L (p) , the meaning will be clear from the context. Given r > 0 and a bounded (or nonnegative) where Γ denotes the Euler Gamma function. The function V r f is again bounded (resp. nonnegative) Borel, and for any 1 ≤ p < ∞ the operators V r form a strongly continuous contraction semigroup (V r ) r>0 on L p (B, µ), see [8,Corollary 5.3.3] or [42,Proposition 4.7], symmetric for p = 2. In any of these spaces the operators V r are the powers (I −L) −r/2 of order r/2 of the respective 1-resolvent operators (I − L) −1 . Meyer's equivalence, [9,Theorem 8.5.2], [42,Theorem 4.4], states that for any integer r ≥ 1 and any 1 < p < +∞ and any u ∈ W r,p (B, µ) we have with constants c 1 > 0 and c 2 > 0 depending only on r and p. By the continuity of the V r and the density of cylindrical functions we observe The operator V r acts as an isometry from W s,p (B, µ) onto W s+r,p (B, µ), [11, Chapter II, Theorem 7.3.1]. For later use we record the following well known fact.
, and by applying Gram-Schmidt we may assume {l 1 , ..., l n } is an orthonormal system in H. The Ornstein-Uhlenbeck . Given x ∈ B and writing ξ = ( x, l 1 H , ..., x, l n H ), we have , and using (6) and Although different in nature both F C ∞ b and W ∞ can serve as natural replacements in infinite dimensions for algebras of smooth differentiable functions in Euclidean spaces or on manifolds.

Capacities and their equivalence
We define two types of capacities related to W r,p (B, µ)-spaces and verify their equivalence.
The following definition is standard, see for instance [20,41].
We give two further definitions of (r, p)-capacities. The first one is based on cylindrical functions and resembles [ The capacities cap have useful 'algebraic' properties which we will use in Section 5.
One can give a similar definition based on the space W ∞ . To do so, we recall some potential theoretic notions. If a property holds outside a set E ⊂ B with Cap r,p (E) = 0 then we say it holds (r, p)-quasi everywhere (q.e.). We follow [33, Chapter IV, Section 1.2] and call a set E ⊂ B slim if Cap r,p (E) = 0 for all 1 < p < +∞ and all integer r > 0, and if a property holds outside a slim set, we say it holds quasi surely (q.s.). A function u : B → R is said to be (r, p)-quasi continuous if for any ε > 0 we can find an open set U ε ⊂ B such that Cap r,p (U) < ε and the restriction u| U c ε of u to U c ε is continuous. Every function u ∈ W r,p (B, µ) admits a (r, p)-quasi-continuous version u, unique in the sense that two different quasi continuous versions can differ only on a set of zero (r, p)-capacity. Since continuous functions are dense in W r,p (B, µ) this follows by standard arguments, see for instance [11, Chapter I, Section 8.2]. Now one can follow [33, Chapter IV, Section 2.4] to see that for any u ∈ W ∞ there exists a function u : B → R such that u = u µ-a.e. and for all r and p the function u is (r, p)-quasi continuous. It is referred to as the quasi-sure redefinition of u and it is unique in the sense that the difference of two quasi-sure redefinitions of u is zero (r, p)-quasi everywhere for all r and p, [33].
where u denotes the quasi-sure redefinition of u with respect to the capacities from Definition 3.1, and for an arbitrary set A ⊂ B, This definition may seem a bit odd because it refers to Definition 3.1. However, for some applications capacities based on the algebra W ∞ may be more suitable that those based on cylcindrical functions.
The following equivalence can be observed.
Theorem 3.4. Let 1 < p < +∞ and let r > 0 be an integer. Then there are positive constants c 3 and c 4 depending only on p and r such that for any set A ⊂ B we have One ingredient of our proof of Theorem 3.4 is a bound in W r,p (B, µ)norm for compositions with suitable smooth truncation functions. For the spaces W 1,p (B, µ) such a bound is clear from the chain rule for D respectively from general Dirichlet form theory, see [11]. Norm estimates in W r,p (B, µ) for compositions T • u of elements u ∈ W r,p (B, µ) with suitable smooth functions T : R → R can be obtained via the chain rule. For instance, in the special case r = 2 the chain and product rules and the definition of the generator L imply By (7) it would now suffice to show a suitable bound for LT (u) in L p , and the summand more difficult to handle is the one involving the first derivatives Du. In the finite dimensional Euclidean case an L pestimate for it follows immediately from a simple integration by parts argument, [1, Theorem 3], or by a use of a suitable Gagliardo-Nirenberg inequality, [3,35]. Integration by parts for Gaussian measures comes with an additional 'boundary' term involving the direction h ∈ H of differentiation that spoiles the original trick, and the classical proof of the Gagliardo-Nirenberg inequality involves dimension dependent constants. A simple alternative approach, suitable for any integer r > 0, is to prove truncation results for potentials in a similar way as in [2, Theorem 3.3.3], so that a quick evaluation of the first order term above follows from estimates in terms of the maximal function, [ Lemma 3.5. Assume 1 < p < +∞ and let r > 0 be an integer. Let T ∈ C ∞ (R + ) and suppose that T satisfies Then for every nonnegative f ∈ L p (B, µ) the function T • V r f is an element of W r,p (B, µ), and there is a constant c T > 0 depending only on p, r and L such that for every nonnegative f ∈ L p (B, µ) we have Another useful tool in our proof of Theorem 3.4 is the following 'intermediate' description of Cap r,p . By F C ∞ b,+ we denote the cone of nonnegative elements of F C ∞ b . Lemma 3.6. Let 1 ≤ p < +∞ and let r > 0 be an integer. For any open set U ⊂ B we have Consequently, if f ∈ L p is such that V r f ≥ 1 + δ µ-a.e. on U, then also V r (f + ) ≥ 1 + δ µ-a.e. on U, and clearly f + L p ≤ f L p . Given ε > 0 choose a nonnegative function f ∈ L p (B, µ) such that u := V r f ≥ 1 + δ µ-a.e. on U with some δ > 0 and Approximating f by bounded nonnegative functions in L p (B, µ), taking their cylindrical approximations, which are nonnegative as well, and smoothing by convolution in finite dimensional spaces, we can approximate f in L p (B, µ) by a sequence of nonnegative functions . Clearly the functions u n := V r f n satisfy lim n u n = u in W r,p (B, µ). By (13) and the convergence in W r,p (B, µ) we can now choose a subsequence (u for all i = 1, 2, ... For any k = 1, 2, ... let now Consequently, setting u(x) := lim n→∞ u n (x) for all x ∈ ∞ k=1 A c k and u(x) = 0 for all other x, we obtain a µ-version u of u.
Now choose l such that Cap ′ r,p (A l ) < ε 3 and then j large enough so that f n j − f Then u n j ≥ 1 µ-a.e. on some neighborhood V of U ∩ A c l . The topological support of µ is B, see for instance [8, Theorem 3.6.1, Definition 3.6.2 and the remark following it]. Since u n j is continuous by Proposition 2.2 we therefore have u n j ≥ 1 everywhere on V . Now, since Cap ′ r,p is clearly subadditive and monotone, Using Lemmas 3.5 and 3.6 we can now verify Theorem 3.4.

Proof. We show (8). It suffices to consider open sets
with c 2 as in (7), so that it suffices to show with a suitable constant c > 0 depending only on r and p.
Let T ∈ C ∞ (R) be a function such that 0 ≤ T ≤ 1, T (t) = 0 for 0 ≤ t ≤ 1/2 and T (t) = 1 for t ≥ 1, and let c T be as in Lemma 3.5. Given ǫ > 0, let f ∈ F C ∞ b,+ be such that u := V r f ≥ 1 on U and due to Lemma 3.6 such f can be found. Clearly T • u ∈ F C ∞ b and T • u = 1 on U. Therefore, using Lemma 3.5, we have and we arrive at (8) with (9) is an easy consequence.

Smooth truncations
To verify Lemma 3.5 we begin with the following generalization of Proposition 4.1. Assume p > 1 and f ∈ L p (B, µ). Then for any t > 0 and µ-a.e. x ∈ B the mapping h → P t f (x + h) from H to B is infinitely Fréchet differentiable, and given h 1 , ..., h k ∈ H we have where the functionsĥ i are as in (1) and Q : R n → R, n ≤ k, is a polynomial of degree k whose coefficients are constants or products of scalar products h i , h j H . If the h 1 , ..., h k are elements of an orthonormal system (g i ) k i=1 in H, not necessarily distinct, then each coefficient of Q depends only on the multiplicity according to which the respective element of (g i ) k i=1 occurs in {h 1 , ..., h k }.
Proof. The infinite differentiability was shown in [8, Proposition 5.4.8] as a consequence of the Cameron-Martin formula. By the same arguments we can see that A straightforward calculation shows that with a polynomial Q as stated.
The next inequality is a counterpart to [2, Proposition 3.1.8]. It provides a pointwise multiplicative estimate for derivatives of potentials in terms of powers of the potential and a suitable maximal function.
Lemma 4.2. Let 1 < q < +∞, let r > 0 be an integer and let k < r. Then for any nonnegative Borel function f on B and all x ∈ B we have Note that lemma 4.2 is interesting only for r ≥ 2.
Proof. Suppose h 1 , ..., h k ∈ H are members of an orthonormal system in H, not necessarily distinct. Then for any δ > 0 we have, by dominated convergence, ..,ĥ k (y)) µ(dy)dt with a polynomial Q of degree k as in Proposition 4.1. Now let β > 1 be a real number such that Hölder's inequality yields ×|Q(ĥ 1 (y), ...,ĥ k (y))| β µ(dy)dt Using the elementary inequality e −t t ≤ 1 − e −2t for t ≥ 0 and (15), so that another application of Hölder's inequality, now with q, shows that the first factor on the right hand side of (16) is bounded by .
According to Proposition 4.1 the coefficients of the polynomial Q are bounded for fixed k, and since its degree does not exceed k, it involves only finitely many distinct products of powers of the functionsĥ i . Together with the fact that eachĥ i is N(0, 1)-distributed, this implies that there is a constant c 1 (k, q, β) > 0, depending on k but not on the particular choice of the elements h 1 , ..., h k , such that B |Q(ĥ 1 (y), ...,ĥ k (y))| βq ′ µ(dy) Taking into account (15), we therefore obtain To estimate I 2 (δ) let In a similar fashion we can then obtain the estimate where c 2 (k, q, γ) > 0 is a constant depending on n but not on the particular choice of h 1 , ..., h k . We finally choose suitable δ > 0. The function can attain any value in (0, 1). Since Jensen's inequality implies (20) ( we have sup t>0 (P t (f q )(x)) 1/q ≥ V r f (x) and can choose δ > 0 such that note that the denominator cannot be zero unless f is zero µ-a.e. Combining with (17) and (19) we obtain for some constants c ′ 1 (k, q, β), c ′ 2 (k, q, γ). For any given r there exist only finitely many numbers k < r and for any such k numbers β and γ as in (15) and (18) can be fixed. Using Proposition 2.1 we can therefore find a constant c(k, q, r) depending only on k, q and r such that (14) holds.
We prove Lemma 3.5, basically following the method of proof used for [2, Theorem 3.3.3].
Proof. If r = 1 then T has a bounded first derivative, and the desired bound is immediate from the definition of the norm · W 1,p , the chain rule for the gradient D and Meyer's equivalence, [42,Theorem 4.4]. In the following we therefore assume r ≥ 2.
We verify that for any k ≤ r the inequality holds with a constant c(k, L, p, r) > 0 depending only on k, L, p and r. If so, then summing up yields with a constant c T > 0 depending on L, p and r, as desired. To see (22) suppose k ≤ r and that h 1 , ..., h k are members of an orthonormal system (g i ) k i=1 , not necessarily distinct. To simplify notation, we use multiindices with respect to this orthonormal system: Given a multiindex α = (α 1 , ..., α k ) we write D α := ∂ α 1 g 1 · · · ∂ α k g k , where for β = 0, 1, 2, ..., a function u : B → R and an element g ∈ H we define ∂ β g u as the image of u under the application β differentiations in direction g, ∂ β g u(x) := ∂ g · · · ∂ g u(x) = D β u(x)(g, ..., g). Now let α be a multiindex such that D α = ∂ h 1 · · · ∂ h k . Then clearly |α| = k. Moreover, we have by the chain rule, where the interior sum is over all j-tuples (α 1 , ..., α j ) of multiindices α i such that |α i | ≥ 1 for all i and α 1 + α 2 + ... + α j = α.
The interior sum has k−1 j−1 summands. The C α 1 ,...,α j are real valued coefficients, and since there are only finitely many different C α 1 ,...,α j , there exists a constant C(k) > 0 which for all multiindices α with |α| = k dominates these constants, C α 1 ,...,α j ≤ C(k). In particular, C(k) does not depend on the particular choice of the elements h 1 , ..., h k . More explicit computations can for instance be obtained using [19].
The hypothesis (10) on T implies with a constant c(k) > 0 depending only on k and with L being as in (10).
where 1 < q < +∞ is arbitrary and c(k, q, r) > 0 is a constant depending only on k, q and r. For the case j = 1 we have Taking the supremum over all h 1 , ..., h k ∈ H as above we obtain with a constant c(k, L, q, r) > 0 by Proposition 2.1. Fixing 1 < q < p and using the boundedness of the semigroup maximal function, [42,Theorem 3.3], we see that there is a constant c(p, q) > 0 depending only on p and q such that On the other hand, by (7), we have Combining, we arrive at (22).

L p -uniqueness
We discuss related uniqueness problems for the Ornstein Uhlenbeck operator L and its integer powers.
Recall first that a densely defined operator (L, A) on L p (B, µ), 1 ≤ p < +∞ is said to be L p -unique if there is only one C 0 -semigroup on L p (B, µ) whose generator extends (L, A), see e.g. From (7) it follows that for any m = 1, 2, ... and 1 < p < +∞ we have D((−L) m ) = W 2m,p (B, µ). The density of F C ∞ b and W ∞ in the spaces W 2m,p (B, µ) and the completeness of the latter imply that ((−L) m , W 2m,p (B, µ)) is the closure in L p (B, µ) of ((−L) m , F C ∞ b ) and also of ((−L) m , W ∞ ).
Since obviously (P t ) t>0 is a C 0 -semigroup, (L, F C ∞ b ) and (L, W ∞ ) are L p -unique in all L p (B, µ), 1 ≤ p < +∞. To discuss the its powers −(−L) m for m ≥ 2 we quote well known facts to provide a sufficient condition for them to generate C 0 -semigroups. Since (P t ) t>0 is a symmetric Markov semigroup on L 2 (B, µ), for any 1 < p < +∞ the operator L = L (p) generates a bounded holomorphic semigroups on L p (B, µ) with angle θ satisfying π 2 − θ ≤ π 2 | 2 p − 1|, see for instance [15,Theorem 1.4.2]. On the other hand [16,Theorem 4] tells that if L is the generator of a bounded holomorphic semigroup with angle θ satisfying π 2 − θ < π 2m , then also −(−L) m generates a bounded holomorphic semigroup. Combining, we can conclude that −(−L) m generates a bounded holomorphic semigroup on L p (B, µ) and therefore in particular a (bounded) C 0 -semigroup if [17, Theorem 8] shows that (up to a discussion of limit cases) this is a sharp condition for −(−L) m to generate a bounded C 0 -semigroup. For 1 < p < +∞ this also recovers the L p -uniqueness in the case m = 1. For p = 2 condition (23) is always satisfied. Alternatively we can conclude the generation of C 0 -semigroups on L 2 (B, µ) directly from the spectral theorem.
For later use we fix the following fact. Here we are interested in L p -uniqueness after the removal of a small closed set Σ ⊂ B of zero measure. This is similar to our discussion in [25] and, in a sense, similar to a removable singularities problem, see for instance [35] or [36] or [2,Section 2.7].
Let Σ ⊂ B be a closed set of zero Gaussian measure and N := B \ Σ. We define  p (B, µ)).
If in addition m satisfies (23) and it remains to show the converse inclusion.
and for each l the function v l equals one on an open neighborhood of Σ. Set w jl := (1−v l )u j to obtain functions w jl ∈ F C ∞ b (N). Now let j be fixed. For any 1 ≤ k ≤ 2m let h 1 , ..., h k be members of an orthonormal system (g i ) k i=1 , not necessarily distinct. As in the proof of Lemma 3.5 we use multiindex notation with respect to this orthonormal system. Let α be such that D α = ∂ h 1 · · · ∂ h k . Then, by the general Leibniz rule, where for two multiindices α and β we write β ≤ α if β i ≤ α i for all i = 1, ..., k. For any such β we clearly have and taking the supremum over all h 1 , ..., h k as above, To . Then its unique extension must be (−(−L) m , W 2m,p (B, µ)). Let u ∈ F C ∞ b be a function that equals one on a neighborhood of Σ. Since The proof for W ∞ is similar.

Comments on Gaussian Hausdorff measures
For finite dimensional Euclidean spaces the link between Sobolev type capacities and Hausdorff measures is well known and the critical size of a set Σ in order to have (r, p)-capacity zero or not is, roughly speaking, determined by its Hausdorff codimension, see e.g. [2,Chapter 5]. For Wiener spaces one can at least provide a partial result of this type.
Hausdorff measures on Wiener spaces of integer codimension had been introduced in [21, Section 1]. We briefly sketch their method but allow non-integer codimensions, this is an effortless generalization and immediate from their arguments.
Given an m-dimensional Euclidean space F and a real number 0 ≤ d ≤ m the spherical Hausdorff measure S d of dimension d can be defined as follows: For any ε > 0 set  [20,11. Théorème]. We write F for the kernel of p F . The spaces B and F × F are isomorphic under the map p F × (I − p F ). If A ⊂ B is analytical and for any x ∈ F the section with respect to the above product is denotes by A x ⊂ F , then for any a ∈ R the set {x ∈ F : Restricted to the Borel σ-algebra it is a Borel measure. The next result follows in the same way as [21, 9. Theorem] from [20,32. Théorème] and [37], see also [2, Theorem 5.1.13]. Combined with Theorem 5.2 this yields a necessary codimension condition which is similar as in the case of Laplacians on Euclidean spaces, [5,25]. In particular, if (L, F C ∞ b (N)) is essentially self-adjoint, then ̺ d (Σ) = 0 for all d < 4.
The same is true with W ∞ (N) in place of F C ∞ b (N).

Comments on stochastic processes
We finally like to briefly point out connections to known Kakutani type theorems for related multiparameter Ornstein-Uhlenbeck processes. The connection between Gaussian capacities, [20], and the hitting behavious of multiparameter processes, [27,28,29], has for instance been investigated in [6,43,44]. We briefly sketch the construction and main result of [44], later generalized in [6].
A more causal connection between uniqueness problems for operators and classical probability should involve certain branching diffusions rather than multiparameter processes, but even for finite dimensional Euclidean spaces the problem is not fully settled and remains a future project.