Totally null sets and capacity in Dirichlet type spaces

In the context of Dirichlet type spaces on the unit ball of $\mathbb{C}^d$, also known as Hardy-Sobolev or Besov-Sobolev spaces, we compare two notions of smallness for compact subsets of the unit sphere. We show that the functional analytic notion of being totally null agrees with the potential theoretic notion of having capacity zero. In particular, this applies to the classical Dirichlet space on the unit disc and logarithmic capacity. In combination with a peak interpolation result of Davidson and the second named author, we obtain strengthenings of boundary interpolation theorems of Peller and Khrushch\"{e}v and of Cohn and Verbitsky.

|f (re it )| 2 dt < ∞ be the classical Hardy space. It is well known that H 2 can be identified with the closed subspace of all functions in L 2 (∂D) whose negative Fourier coefficients vanish. Correspondingly, subsets of ∂D of linear Lebesgue measure zero frequently play the role of small or negligible sets in the theory of H 2 and related spaces. For instance, a classical theorem of Fatou shows that every function in H 2 has radial limits outside of a subset of ∂D of Lebesgue measure zero; see for instance [22,Chapter 3]. For the disc algebra the Rudin-Carleson theorem shows that every compact set E ⊂ ∂D of Lebesgue measure zero is an interpolation set for A(D), meaning that for each g ∈ C(E), there exists f ∈ A(D) with f E = g. In fact, one can achieve that |f (z)| < g ∞ for z ∈ D \ E (provided that g is not identically zero); this is called peak interpolation. In particular, there exists f ∈ A(D) with f E = 1 and |f (z)| < 1 for z ∈ D \ E, meaning that E is peak set for A(D). Conversely, every peak set and every interpolation set has Lebesgue measure zero. For background on this material, see [20,Chapter II].
In the theory of the classical Dirichlet space where A denotes the planar Lebesgue measure, a frequently used notion of smallness of subsets of ∂D is that of having logarithmic capacity zero; see [18,Chapter II] for an introduction. This notion is particularly important in the potential theoretic approach to the Dirichlet space. We will review the definition in Section 2. A theorem of Beurling shows that every function in D has radial limits outside of a subset of ∂D of (outer) logarithmic capacity zero; see [18,Section 3.2]. In the context of boundary interpolation, Peller and Khrushchëv [23] showed that a compact set E ⊂ ∂D is an interpolation set for A(D)∩D if and only if E has logarithmic capacity zero. Many of these considerations have been extended to standard weighted Dirichlet spaces and their associated capacities, and more generally to Hardy-Sobolev spaces on the Euclidean unit ball B d of C d by Cohn [16] and by Cohn and Verbitsky [15]. The Hardy space H 2 , the Dirichlet space D and more generally Hardy-Sobolev spaces on the ball belong to a large class of reproducing kernel Hilbert spaces of holomorphic functions on the ball, called regular unitarily invariant spaces. We will recall the precise definition in Section 2.
In studying regular unitarily invariant spaces H and especially their multipliers, a functional analytic smallness condition of subsets of ∂B d has proved to be very useful in recent years. This smallness condition has its roots in the study of the ball algebra as explained in Rudin's book [ In the case of H 2 , the F. and M. Riesz theorem implies that a measure is Henkin if and only if it is absolutely continuous with respect to Lebesgue measure on ∂D. Hence the totally null sets are simply the sets of Lebesgue measure 0. Beyond the ball algebra, these notions were first studied by Clouâtre and Davidson for the Drury-Arveson space [13], and then for more general regular unitarily invariant spaces by Bickel, M c Carthy and the second named author [7]. Just as in the case of the ball algebra, Henkin measures and totally null sets appear naturally when studying the dual space of algebras of multipliers [13,17], ideals of multipliers [14,17], functional calculi [7,12], and peak interpolation problems for multipliers [13,17].
1.2. Main results. In this article, we will compare the functional analytic notion of being totally null with the potential theoretic notion of having capacity zero. As was pointed out in [17], for the Dirichlet space D, the energy characterization of logarithmic capacity easily implies that every compact subset of ∂D that is Mult(D)-totally null necessarily has logarithmic capacity zero. We will show that for Hardy-Sobolev spaces on the ball, including the Dirichlet space on the disc, the two notions of smallness in fact agree.
To state the result, let us recall the definition of Hardy-Sobolev spaces (a.k.a. Besov-Sobolev spaces) on the ball. Let σ denote the normalized surface measure on ∂B d and let be the Hardy space on the unit ball. Let s ∈ R. If f ∈ O(B d ) has a homogeneous decomposition f = ∞ n=0 f n , we let and define Thus, if R s f = ∞ n=1 n s f n denotes the fractional radial derivative, then There are also natural L p -versions of these spaces, but we will exclusively work in the Hilbert space setting. If s < 0, then H s is a weighted Bergman space on the ball, and clearly H 0 = H 2 (B d ).  [6] and in the theory of complete Pick spaces [2]. For more background on these spaces, we refer the reader to [25].
For each of the spaces H s for d−1 2 < s ≤ d 2 , there is a natural notion of (non-isotropic Bessel) capacity C s,2 (·), introduced by Ahern and Cohn [3]. Equivalently, for the spaces D a , there is a notion of capacity that can be defined in terms of the reproducing kernel of D a . We will review these definitions and show their equivalence in Section 2.
Our main result concerning the Hardy-Sobolev spaces H s is the following. In particular, taking d = 1 and s = 1 2 , we see that in the context of the classical Dirichlet space D, a compact subset E ⊂ ∂D is Mult(D)-totally null if and only if it has logarithmic capacity zero.
A direct proof of Theorem 1.1 will be provided in Section 3. Moreover, we will prove an abstract result about totally null sets, which, in combination with work on exceptional sets by Ahern and Cohn [3] and by Cohn and Verbitsky [15], will yield a second proof of Theorem 1.1. This result applies to some spaces that are not covered by Theorem 1.1, such as the Drury-Arveson space.
It is possible to interpret the capacity zero condition as a condition involving the reproducing kernel Hilbert space H (cf. Proposition 3.2 below), whereas the totally null condition is a condition on the multiplier algebra Mult(H). Complete Pick spaces form a class of spaces in which it is frequently possible to go back and forth between H and Mult(H); see the book [2] and Section 2 of the present article for more background. For now, let us simply mention that the spaces D a for 0 ≤ a ≤ 1 are complete Pick spaces. (For a = 0, one needs to pass to a suitable equivalent norm.) If H is a reproducing kernel Hilbert space on B d , let us say that a compact subset E ⊂ ∂B d is an unboundedness set for H if there exists f ∈ H so that lim rր1 |f (rζ)| = ∞ for all ζ ∈ E. The following result covers the spaces in A refinement of this result will be proved in Section 4. The results of Ahern and Cohn [3] and of Cohn and Verbitsky [15] on exceptional sets show that in the case of the spaces H s for d−1 2 < s ≤ d 2 , a compact subset E ⊂ ∂B d is an unboundedness set for H s if and only if C s,2 (E) = 0. Indeed, the "only if" part follows from [3, Theorem B], the "if" part is contained in the construction on p. 443 of [3]; see also [15, p. 94]. Thus, we obtain another proof of Theorem 1.1.

1.3.
Applications. We close the introduction by mentioning some applications of Theorem 1.1. The first application concerns peak interpolation. Extending the work of Peller and Khrushchëv [23] on boundary interpolation in the Dirichlet space, Cohn and Verbitsky [15,Theorem 3] showed that Combining Theorem 1.1 with a peak interpolation result for totally null sets of Davidson and the second named author [17], we can strengthen the result of Cohn and Verbitsky in two ways. Firstly, we replace H s ∩ A(B d ) with the smaller space A(H s ), which is defined to be the multiplier norm closure of the polynomials in Mult(H s ). Thus, with contractive inclusions. Secondly, we obtain a strict pointwise inequality off of E.
Proof. According to [17,Theorem 1.4], the conclusion holds when H s is replaced with any regular unitarily invariant space H and E is Mult(H)totally null. Combined with Theorem 1.1, the result follows.
In fact, in the setting of Theorem 1.3, there exists an isometric linear operator L : C(E) → A(H s ) of peak interpolation; see [17,Theorem 8.3]. In a similar fashion, one can now apply other results of [17] in the context of the spaces H s , replacing the totally null condition with the capacity zero condition. In particular, this yields a joint Pick and peak interpolation result (cf. [17,Theorem 1.5]) and a result about boundary interpolation in the context of interpolation sequences (cf. [17, Theorem 6.6]).
Our second application concerns cyclic functions. Recall that a function f ∈ H s is said to be cyclic if the space of polynomial multiples of f in dense in H s . It is a theorem of Brown and Cohn [8] that if E ⊂ ∂D has logarithmic capacity zero, then there exists a function f ∈ D ∩ A(D) that is cyclic for D so that f E = 0; see also [19] for an extension to other Dirichlet type spaces on the disc. The following result extends the theorem of Brown and Cohn to the spaces H s on the ball, and moreover achieves that f ∈ A(H s ), so in particular, f is a multiplier.
where a 0 = 1, a n > 0 for all n ∈ N and lim n→∞ an a n+1 = 1. We think of the last condition as a regularity condition, as it is natural to assume that the power series defining K has radius of convergence 1, since H is a space of functions on the ball of radius 1. Under this assumption, the limit, if it exists, necessarily equals 1. We recover H 2 and D by choosing d = 1 and a n = 1, respectively a n = 1 n+1 , for all n ∈ N. Expanding the reproducing kernels of D a into a power series, one easily checks that D a is a regular unitarily invariant space for all a ≥ 0. While the class of regular unitarily invariant spaces is not stable under passing to an equivalent norm, one can also check that the spaces H s are regular unitarily invariant spaces for all s ∈ R. Indeed, each space H s has a reproducing kernel as in (1), where a n = z n 1 −2 Hs . More background on these spaces can be found in [7,17,21].
Let H be a regular unitarily invariant space. We let Mult(H) denote the multiplier algebra of H. Identifying a multiplier ϕ with the corresponding multiplication operator on H, we can regard Mult(H) as a WOT closed subalgebra of B(H), the algebra of all bounded linear operators on H. By trace duality, Mult(H) becomes a dual space in this way, and hence is equipped with a weak- * topology. The density of the linear span of kernel functions in H implies that on bounded subsets of Mult(H), the weak- * topology agrees with the topology of pointwise convergence on B d . In a few places, we will use the following basic and well known fact, which we state as a lemma for easier reference. For a proof, see for instance [17,Lemma 2.2].
Let M (∂B d ) be the space of complex regular Borel measures on ∂B d . Finally, we require the notion of a complete Pick space. Complete Pick spaces are reproducing kernel Hilbert spaces that are defined in terms of an interpolation condition for multipliers; see the book [2] for more background. In the context of regular unitarily invariant spaces, there is a concrete characterization in terms of the reproducing kernel. If the reproducing kernel of H is K(z, w) = ∞ n=0 a n z, w n , then H is a complete Pick space if and only if the sequence (b n ) ∞ n=1 defined by the power series identity ∞ n=1 b n t n = 1 − 1 ∞ n=0 a n t n satisfies b n ≥ 0 for all n ∈ N (this is a straightforward generalization of [ The non-isotropic Riesz potential of µ is We extend the definition to non-negative measurable functions f ∈ L 1 (∂B d , dσ) by letting I s (f ) = I s (f dσ). (b) The non-isotropic Bessel capacity of E is defined by is called the energy of µ. By [1, Theorem 2.5.1], we have the following "dual" expression for the capacity C s,2 (·), In particular, C s,2 (E) > 0 if and only if E supports a probability measure of finite energy. A different approach, which can be justified by regarding H s as a reproducing kernel Hilbert space, is the following; cf. [18,Chapter 2]. Recall that if a = d − 2s, then H s = D a with equivalent norms. Moreover, we have k 2s = |K a |.   It appears to be well known to experts that the capacities cap(·, D a ) and C s,2 (·) are equivalent if a = d − 2s, the point being that the corresponding energies are comparable. A proof in the case d = 1, s = 1 2 can be found in [11,Lemma 2.2]. In the case s = d 2 , the crucial estimate is stated in [10, Remark 2.1] without proof. A proof of the estimate in one direction in this case is contained in [3, p. 442-442]. For the sake of completeness, we provide an argument that applies to all cases under consideration. We adapt the proof in [11, Lemma 2.2] to the non-isotropic geometry of ∂B d .
Throughout, we write A B to mean that there exists a constant C ∈ (0, ∞) so that A ≤ CB, and A ≈ B to mean that A B and B A. Proof. We will show that The statement then follows by integrating both sides with respect to µ and using Fubini's theorem.
Let d(z, w) = |1 − z, w | we split the domain of integration ∂B d as follows We denote by I, I ′ , II, II ′ the corresponding integrals. By the symmetry of the problem it suffices to estimate I and II.
For I, we note that if ζ ∈ Q δ (z), then δ ≤ d(ζ, w) ≤ 3δ by the triangle inequality for d. Hence, integrating with the help of the the distribution function, we find that Next, using the fact that d(z, ζ) ≤ √ 2 for all z, ζ ∈ ∂B d , we see that Combining the estimates for I and II and recalling the definition of δ we see that To establish the lower bound, it suffices to consider z, w ∈ ∂B d for which d(z, w) is small. In the case s < d 2 , the lower bound follows from the treatment of the integral I above. Let s = d 2 . Notice that in the region U z,w = {ζ ∈ ∂B d : d(z, w) ≤ d(w, ζ)}, the triangle inequality yields d(ζ, z) ≤ 2d(ζ, w). Hence integrating again with the distribution function and writing δ = d(z, w), we estimate where c 0 , c 1 > 0 are constants depending only on the dimension d. This shows the lower bound for small δ, which concludes the proof.
From this lemma the equivalence of the capacities C s,2 (·) and cap(·, D a ) for a = d − 2s follows easily.
Here, all implied constants only depend on d and s.
Proof. For a measure µ ∈ M + (∂B d ), we compute Thus, Lemma 2.5 yields that Since the energies involved are comparable, so are the capacities by (2).

Direct proof of Theorem 1.1
To prove Theorem 1.1, we will make use of holomorphic potentials. Since several of our proofs involve reproducing kernel arguments, it is slightly more convenient to work with the spaces D a rather than with H s . Definition 3.1. Let 0 ≤ a < 1 and let µ ∈ M + (∂B d ). The holomorphic potential of µ is the function denote the associated integration functional.
The following functional analytic interpretation of the holomorphic potential and of capacity will show that every totally null set has capacity zero. In the case of the Dirichlet space on the disc, it is closely related to the energy formula for logarithmic capacity in terms of the Fourier coefficients of a measure; see for instance [18,Theorem 2.4.4].
Proposition 3.2. Let µ ∈ M + (∂B d ) and let 0 ≤ a < 1. The following assertions are equivalent: In this case, Da , where the implied constants only depend on a and d, and Proof. For ease of notation, we write f = f µ , ρ = ρ µ and k w (z) = K a (z, w). For 0 ≤ r < 1, define f r (z) = f (rz) and ρ r (f ) = ρ(f r ). Then each f r ∈ D a and each ρ r is a bounded functional on D a . First, we connect f r and ρ r , which will be useful in all parts of the proof. By the reproducing property of the kernel, we find that for all z ∈ B d . Since finite linear combinations of kernels are dense in D a , it follows that (3) ρ r (g) = g, f r for all g ∈ D a and hence ρ r (Da) * = f r Da . Next, we show the equivalence of (ii) and (iii). If f ∈ D a , then lim rր1 f r = f in D a and hence for all g ∈ A(B d ) ∩ D a , Equation (3) shows that so ρ is bounded on D a . In this case, ρ (Da) * = f Da , which establishes the final statement of the proposition as well. Conversely, if ρ is bounded on D a , then since ρ r (Da) * ≤ ρ (Da) * , it follows that sup 0≤r<1 f r Da ≤ ρ (Da) * , hence f ∈ D a . It remains to show the equivalence of (i) and (iii) and that E(µ, D a ) ≈ f 2 Da . With the help of Equation (3), we see that Taking real parts and using the fact that Re K a and |K a | are comparable, we find that Thus, if f ∈ D a , then Fatou's lemma shows that Conversely, if E(µ, D a ) < ∞, we use the basic inequality and the Lebesgue dominated convergence theorem to find that With this proposition in hand, we can prove the "only if" part of Theorem 1.1, which we restate in equivalent form (see Corollary 2.6 for the equivalence). The idea is the same as that in the proof of [17, Proposition 2.6]. Proof. Suppose that cap(E, D a ) > 0. Then E supports a probability measure µ of finite energy E(µ, D a ). By Proposition 3.2, we see that the integration functional ρ µ is bounded on D a . In particular, it is weak- * continuous on Mult(D a ). Hence E is not Mult(D a )-totally null.
To prove the converse, we require the following fundamental properties of the holomorphic potential of a capacitary extremal measure of a compact subset E ⊂ B d , i.e. a measure for which the supremum in (2) is achieved. If a > 0, these properties are contained in the proof of [3, Theorem 2.10], see also [11,Lemma 2.3] for a proof in the case d = 1 and a = 0. An argument that directly works with the capacity cap(·, D 0 ) in the case d = 1 and a = 0 can be found on pp. 40-41 of [18]. We briefly sketch the argument in general.  (3) show that for z ∈ ∂B d , we have so in combination with the basic inequality | 1 1−r z,w | ≤ 2| 1 1− z,w | for z, w ∈ ∂B d and 0 ≤ r < 1, we see that (c) holds.
To establish (b), notice that (c) implies that f µ ∈ H ∞ (B d ), so f µ has radial boundary limits f * µ almost everywhere with respect to σ, and f µ = P [f * µ ], the Poisson integral of f * µ . Fatou's lemma and the fact that Re K a and |K a | are comparable show that for σ-almost every z ∈ ∂B d , the estimate Now C s,2 (K) = 0 implies that σ(K) = 0 for compact sets K ⊂ ∂B d . (This is because σ K has finite energy, which for instance follows from Proposition 3.2 since D a is continuously contained in H 2 (B d ).) Therefore, Item (2) and Lemma 2.5 imply that Re f * µ (z) 1 for σ-almost every z ∈ E. In combination with Re f µ = P [Re f * µ ], this easily implies (b).
In [9], Cascante, Fàbrega and Ortega showed that if 0 < a < 1 and if the holomorphic potential f µ is bounded in B d , then it is a multiplier of D a . They also proved an L p -analogue of this statement. We will require an explicit estimate for the multiplier norm of f µ . It seems likely that the arguments in [9] could be used to obtain such an estimate. Instead, we will provide a different argument in the Hilbert space setting, based on the following result of Aleman, M c Carthy, Richter and the second named author for all t > 0. If Re V f ∞ = 0, then taking t → ∞ above yields f = 0. If Re V f ∞ = 0, then choosing t = Re V f that lim n→∞ cap(E n , D a ) = 0; see [18, Theorem 2.1.6]. Let µ n be a positive measure supported on E n as in Lemma 3.4 and let g (n) = f µn be the corresponding holomorphic potential. We claim that (1) lim inf rր1 Re g (n) (rζ) 1 for all ζ ∈ E and all n ∈ N; (2) the sequence (g (n) ) converges to 0 in the weak- * topology of Mult(D a ). Indeed, Part (1) is immediate from Lemma 3.4 (b). To see (2), we first observe that Lemma 3.4 (c) and Proposition 3.6 imply that the sequence (g (n) ) is bounded in multiplier norm. Using Lemma 3.4 (a), we see that g (n) 2 Da cap(E n , D a ), so (g (n) ) converges to zero in the norm of D a and in particular pointwise on B d , hence (2) holds.
Let now ν be a positive Mult(D a )-Henkin measure that is supported on E. We will finish the proof by showing that ν(E) = 0; see the discussion following Definition 2.2. Item (1) above and Fatou's lemma show that Since ν is Mult(D a )-Henkin, the associated integration functional ρ ν extends to a weak- * continuous functional on Mult(D a ), which we continue to denote by ρ ν . Since lim rր1 g (n) r = g (n) in the weak- * topology of Mult(D a ) by Lemma 2.1, we find that for all n ∈ N, Thus, ν(E) Re ρ ν (g n ) for all n ∈ N. Taking the limit n → ∞ and using Item (2), we see that ν(E) = 0, as desired.

Proof of Theorem 1.2
In this section, we prove a refined version of Theorem 1.2. Let H be a regular unitarily invariant space on B d . Recall that a compact set E ⊂ B d is said to be an unboundedness set for H if there exists f ∈ H with lim rր1 |f (rζ)| = ∞ for all ζ ∈ E. We also say that E is a weak unboundedness for H if there exists a separable auxiliary Hilbert space E and f ∈ H ⊗E so that lim rր1 f (rζ) = ∞ for all ζ ∈ E.  Thus, lim rր1 |f (rζ)| = ∞ for all ζ ∈ E, so E is an unboundedness set for H.
Let now µ be a positive Mult(H)-Henkin measure that is supported on E and let ρ µ denote the associated weak- * continuous integration functional on Mult(H). We have to show that µ(E) = 0; see the discussion following Definition 2.2. To this end, we write ψ r (z) = ψ(rz) and let n ∈ N. Applying the dominated convergence theorem and the fact that lim rր1 ψ n r = ψ n in the weak- * topology of Mult(H) (see Lemma 2.1), we find that µ(E) = lim rր1 E ψ n r dµ = lim rր1 ∂B d ψ n r dµ = ρ µ (ψ n ).
Since ψ is a contractive multiplier satisfying |ψ(z)| < 1 for all z ∈ B d , it follows that ψ n tends to zero in the weak- * topology of Mult(H). So taking the limit n → ∞ above, we conclude that µ(E) = 0, as desired.
Let us briefly compare the direct proof of the implication "capacity 0 implies totally null" given in Section 3 with the proof via Theorem 1.2. If E ⊂ ∂B d is a compact set with C s,2 (E) = 0, then the work of Ahern and Cohn [3] and of Cohn and Verbitsky [15] shows that E is unboundedness set for H s . To show this, they use holomorphic potentials and their fundamental properties (cf. Lemma 3.4) to construct a function f ∈ H s satisfying lim rր1 |f (rζ)| = ∞ for all ζ ∈ E. Proceeding via Theorem 1.2, one then applies the factorization result [5, Theorem 1.1] to f to obtain a multiplier ψ of H of norm at most 1 satisfying lim rր1 ψ(rζ) = 1 for all ζ ∈ E, from which the totally null property of E can be deduced.
The direct proof given in Section 3 uses holomorphic potentials as well, this time to construct a sequence of functions in H, which, roughly speaking, have large radial limits on E compared to their norm. It is then shown that the holomorphic potentials themselves form a bounded sequence of multipliers, from which the totally null property of E can once again be deduced.