Rectangular summation of multiple Fourier series and multi-parametric capacity

We consider the class of multiple Fourier series associated with functions in the Dirichlet space of the polydisc. We prove that every such series is summable with respect to unrestricted rectangular partial sums, everywhere except for a set of zero multi-parametric logarithmic capacity. Conversely, given a compact set in the torus of zero capacity, we construct a Fourier series in the class which diverges on this set, in the sense of Pringsheim. We also prove that the multi-parametric logarithmic capacity characterizes the exceptional sets for the radial variation and radial limits of Dirichlet space functions. As a by-product of the methods of proof, the results also hold in the vector-valued setting.


Introduction
This article will consider unrestricted rectangular summation and other multiparameter summation methods of the multiple Fourier series (1) f ∼ α∈Z n a α e i(α1θ1+···αnθn) .
C. Fefferman [10] constructed a continuous function f ∈ C(T 2 ) whose Fourier series diverges with respect to unrestricted rectangular sums for every θ ∈ [0, 2π) 2 . See also [13] for examples of continuous functions with almost everywhere divergent Fourier series under other partial summation schemes. On the other hand, if f ∈ L 2 (T n ), then the iterated Abel means of f do converge unrestrictedly for almost every θ ∈ [0, 2π) n . We refer to [3] and [18,Ch. XVII] for an introduction to multi-parameter summation methods for Fourier series.
For a series f ∼ k∈Z a k e ikθ such that k∈Z |k||a k | 2 < ∞, Beurling [7] showed that f (θ) is summable for every θ ∈ T \ E, where E is a set of zero logarithmic capacity. This was given a one-parameter generalization to multiple Fourier series by Lippman and Shapiro [14]. They proved that if f ∈ L 1 (T n ), n ≥ 2, is as in (1) and satisfies that α∈Z n (α 2 1 + · · · α 2 n )|a α | 2 < ∞, then f (θ) is summable with respect to spherical partial sums, except for on a set E ⊂ T n of ordinary capacity zero.
Interest in multi-parameter summation methods thus leads us to consider multiparameter notions of capacity. We will focus exclusively on multi-parametric logarithmic capacity in this paper. This capacity has appeared recently in functiontheoretic investigations of the Dirichlet space D(D n ) of the polydisc [4,5,6,12]. In [2], it was proven that bi-parameter logarithmic capacity characterizes the Carleson measures of D(D 2 ). Let us fix some notation for the rest of the paper. For a positive integer n, consider the multiple Fourier series f ∼ α∈N n a α e i(α,θ) , where N = {0, 1, 2, . . .}, θ ∈ [0, 2π) n , and the coefficients belong to some Hilbert space H, a α ∈ H. We say that f belongs to the Dirichlet space of the n-disc, f ∈ D(D n , H), if α∈N n (α 1 + 1) · · · (α n + 1) a α 2 H < ∞.
If H = C, we simply write D(D n ). Occasionally, it will be very useful for us to view for example the Dirichlet space of the bidisc as a Dirichlet space-valued one-variable Dirichlet space, ). This is the reason that we consider the vector-valued setting.
Through iterated Poisson extension, any f ∈ D(D n , H) defines an H-valued holomorphic function in z = (r 1 e iθ1 , . . . , r n e iθn ) ∈ D n , We will freely identify [0, 2π) n with the n-torus T n .
For a positive measurable function f on T n , let where dψ denotes the normalized Lebesgue measure on T n . For a set E ⊂ T n in the n-torus, we then define the following outer capacity: When n = 1 and E is a capacitable set, in particular when E is a Borel set, C(E) is equivalent to the usual (gently modified) logarithmic capacity of E. For n ≥ 2, C(E) is a multi-parameter analogue of logarithmic capacity. C(·) is a "true" capacity, roughly fitting into the general framework described by Choquet. However, a number of familiar properties from the one-parameter setting do not hold. Most notably, the associated n-logarithmic potentials defined in Section 2 generally fail to satisfy any kind of boundedness principle [2].
Following [3], we say that the series f (θ) converges in the sense of Pringsheim if Note that (3) describes unrestricted rectangular summation, since no assumption is made on the relationship between N i and N j , 1 ≤ i, j ≤ n. Finally, we say that a property holds quasi-everywhere if it holds everywhere on T n but for a set of capacity 0. Our first main result is the following. Our second main theorem shows that Theorem 1 is sharp.
Theorem 2. If E ⊂ T n is compact and C(E) = 0, then there exists a function f ∈ D(D n ) such that f (θ) diverges in the sense of Pringsheim for θ ∈ E.
To prove Theorems 1 and 2, we will first prove that multi-parametric logarithmic capacity characterizes the exceptional sets for the radial variation V n f (θ) of f ∈ D(D n , H), where ∂ r = ∂ r1 · · · ∂ rn and dr = dr 1 · · · dr n .
Theorem 3. If f ∈ D(D n , H), then V n f (θ) is finite for quasi-every θ.
Remark. When n = 2 and H = C, this theorem is an immediate corollary of the work in [2]. In that paper, the Carleson measures for D(D 2 ), which also turn out to be embedding measures for the radial variation, were given a potential-theoretic characterization. However, the characterization of Carleson measures is a much more complicated problem than the characterization of exceptional sets for the radial variation -see [11,15].
Applying Theorem 3, we obtain the following corollary on unrestricted iterated Abel summation, that is, on the radial limits of a function f ∈ D(D n , H). exists, and furthermore that The value of f * (θ) coincides with the Pringsheim sum f (θ) quasi-everywhere.
Theorem 3 is also sharp.
To complete the analogy with Beurling's work [7], we shall also prove the following result on the strong differentiability of the integral of f . For θ ∈ [0, 2π) n and h ∈ (0, π) n , let for quasi-every θ.

2.1.
Multi-parametric capacity. First, let us slightly modify the kernel of B (without otherwise changing the notation). Letting we note that b(θ) ≥ 1 is convergent and continuous for θ > 0, and that , and for positive finite Borel measures µ on T n define this only changes the definition of C(·) up to constants, see (2). Note that the convolution of b with itself satisfies that The kernel H(θ) = h(θ 1 ) · · · h(θ n ) defines the n-logarithmic potential, The energy of a measure µ is thus given by Since B(θ) is lower semi-continuous on T n , our capacity C(·) fits in with the theory of [1, Ch. 2.3-2.5]. In particular, every Borel set E ⊂ T n is capacitable, For any capacitable set E, C(E) can be computed through the dual definition of capacity, which might give the reader a more familiar definition in the case of logarithmic capacity. More precisely, In particular, the set E has capacity 0, C(E) = 0, if and only if every non-zero positive finite measure µ with support in E has infinite energy, Furthermore, the following simple lemma, which we shall use without mention, is clear from (2) and (5).
Lemma 7. If E 1 , . . . , E n are Borel sets, then The final piece of information that we require is the existence of equilibrium measures. For any compact set K ⊂ T n , the extremal to the capacity problem is generated by a measure µ K such that: supp µ K ⊂ K, Hµ K (θ) ≤ 1 for θ ∈ supp µ K , Hµ(θ) ≥ 1 for quasi-every θ ∈ K and For a finite measure µ on T n , we denote by P µ the n-harmonic function where z = (r 1 e iθ1 , . . . , r n e iθn ) ∈ D n and P r (θ) denotes the usual Poisson kernel, We refer to [16,Ch. 2] for the fundamentals of n-harmonic functions and multiple Poisson integrals. We only need to know the following, which can be extracted from Theorems 2.1.3 and 2.3.1 in [16].
Lemma 8. If u ≥ 0 is n-harmonic and non-negative on D n , then there exists a function 0 ≤ g ∈ L 1 (T n ) and a singular measure σ ≥ 0 on T n such that Furthermore, for almost every θ ∈ [0, 2π) n , it holds that Remark. Since we will prove theorems about unrestricted summation and strong differentiability, we note that unlike the one-variable setting, the proof of the lemma does not specify for which points θ the limit exists. In general, localization fails for multiple Poisson integrals. In fact, let f 1 ∈ C ∞ (T) be such that f 1 (θ 1 ) = 0 for |θ 1 | ≤ ε, for some ε > 0, and such that there is a sequence Then the Fourier coefficients of f satisfy that In fact, the limit does not exist.

Convergence theorems
We begin by proving Theorem 3. Given f ∈ D(D n , H), note that The following proof is in the spirit of Salem and Zygmund's approach to exceptional sets for one-variable Dirichlet spaces [17].
We will also rely on the estimate Suppose now that the set E of (6) has positive capacity. Then there exists a non-zero finite measure µ, supported in E, such that The coefficients of F are square-summable, by (7), (8), and the fact that f ∈ D(D n , H). Thus F (θ) has meaning for almost every θ, and By our assumption on the support of the Fourier coefficients of f we have that and therefore by (9) that But then, by the assumption of finite energy, This is obviously a contradiction.
Proof of Corollary 4. We give the proof for n = 2. The proof is the same for n ≥ 3, but the notation is more difficult. Given f ∈ D(D 2 , H), define f 1 , f 2 ∈ D(D, H) by Let F = E∪(E 1 ×T)∪(T∪E 2 ). Then C(F ) = 0, by three applications of Theorem 3. Suppose now that θ / ∈ F , and for r, r ′ ∈ [0, 1) 2 , write by analyticity Since V 2 f (θ), V 1 f 1 (θ 1 ), and V 1 f 2 (θ 2 ) are all finite, it follows that Hence f * (θ) = lim r→(1,1) f r (θ) exists, for every θ outside the capacity zero set F . Letting r ′ = 0 in the estimate also shows that f r (θ) H is uniformly bounded in r.
We postpone the proof that f * (θ) coincides with the sum f (θ) quasi-everywhere to the proof of Theorem 1.
For n = 1 and H = C, a series f ∈ D(D) is summable at θ ∈ [0, 2π) if and only if it is Abel summable at θ. This is sometimes known as Féjer's Tauberian theorem. Thus, in this case Theorem 3 immediately implies Theorem 1. To prove Theorem 1 for n ≥ 2, we begin by stating a vector-valued version of Féjer's theorem. Then there is an absolute constant C > 0 such that H) . Moreover, for every fixed f we have that Proof. Let r = 1 − 1/N , and note that 1 − r k ≤ k/N , to see that By first choosing M large, and then N , we see that 1 In the proof of Theorem 1 we will consider tensors of the operators S N,θ and P N,θ , interpreted in the obvious way. For instance, if N ∈ N n , θ ∈ [0, 2π) n , and f ∈ D(D n , H), then a α e i(α,θ) , and (P N1,θ1 ⊗ · · · ⊗ P Nn,θn )f = f (1−1/N1,...,1−1/Nn) (θ) Similarly, we consider mixed tensor products, such as Proof of Theorem 1. We already know that the theorem is true for n = 1, by Theorem 3 and Lemma 9. Thus we first consider the case n = 2. By Corollary 4, there is a Borel set E ⊂ T 2 such that C(T 2 \ E) = 0, and for every θ = (θ 1 , θ 2 ) ∈ E we have that (P N1,θ1 ⊗ P N2,θ2 )f is uniformly bounded in N 1 , N 2 and convergent to f * (θ) as N 1 , N 2 → ∞. To prove the theorem, it is thus sufficient to provide a set F ⊂ E such that C(E \ F ) = 0 and such that for every θ ∈ F it holds that (10) lim Constructing such a set F of course also proves that f * (θ) = f (θ) quasi-everywhere, as claimed in Corollary 4. We write Now, by the n = 1 case of the theorem, applied to f ∈ D(D, D(D, H)), there is a set G 2 ⊂ T such that C(T \ G 2 ) = 0, and such that for every θ 2 ∈ G 2 we have the existence of N2,θ2 f ∈ D(D, H).
By a very similar argument (after reordering the variables θ 1 and θ 2 ), there is a set G 1 ⊂ T such that C(T \ G 1 ) = 0, and such that for every θ 1 ∈ G 1 and θ 2 ∈ T, the term (P N1,θ1 ⊗ (S N2,θ2 − P N2,θ2 ))f is uniformly bounded in N 1 , N 2 and tends to zero as N 1 , N 2 → ∞. Thus the proof for n = 2 is finished by letting Note that in the course of the proof we have also established that (P N1,θ1 ⊗ S N2,θ2 )f is uniformly bounded in N 1 , N 2 and converges to f * (θ) as N 1 , N 2 → ∞, for θ ∈ F . For n = 3, Corollary 4 gives us a set E ⊂ T 3 such that C(T 3 \ E) = 0 and on which (P N1,θ1 ⊗ P N2,θ2 ⊗ P N3,θ3 )f converges and is uniformly bounded. We then write Now we apply the n = 2 case of the theorem, together with the remark at the end of its proof, three separate times to f ∈ D(D 2 , D(D, H)). Arguing with Lemma 9 as before, this produces three sets H 1 , H 2 , H 3 ⊂ T 3 such that C(T 3 \H j ) = 0, and such that, for θ ∈ H j , the j:th term is uniformly bounded in N 1 , N 2 , N 3 and converges to zero as N 1 , N 2 , N 3 → ∞. Thus (S N1,θ1 ⊗ S N2,θ2 ⊗ S N3,θ3 )f is uniformly bounded and converges as N 1 , N 2 , N 3 → ∞, for θ ∈ E ∩ H 1 ∩ H 2 ∩ H 3 . Furthermore, the same is true of (P N1,θ1 ⊗ S N2,θ2 ⊗ S N3,θ3 )f and (P N1,θ1 ⊗ P N2,θ2 ⊗ S N3,θ3 )f .
It is now clear that the construction extends by induction to n ≥ 4.
To conclude this section, we consider Theorem 6. One potential approach is to use a capacitary weak type inequality for the strong maximal function, or for the iterate of one-variable maximal functions. See [1, Theorem 6.2.1] for the oneparameter case. Instead of pursuing this, we will give a different argument which directly connects Theorem 6 with Theorem 1.
Proof of Theorem 6. Note first that This is obviously true for polynomials, and for all f ∈ D(D n , H) by continuity. For this last statement, note that, with continuous dependence on f , the values f (θ) are square-integrable on T n , and the right-hand side of (13) is absolutely convergent. The argument is now very similar to the proof of Theorem 1. First we consider the case n = 1, letting for θ ∈ [0, 2π) and h ∈ (0, 1). Let 1 ≤ N ∈ N be such that 1 N +1 ≤ h < 1 N , and let M ≤ N . Then By this estimate, R H h,θ − S H N,θ : D(D, H) → H is uniformly bounded in N and converges pointwise to 0 as N → ∞, as long as 1 N +1 ≤ h < 1 N . Thus Theorem 1 implies Theorem 6 in the case that n = 1.
For n ≥ 2 we proceed precisely as in the proof of Theorem 1. For instance, for n = 2 we write The rest of the proof is essentially repetition.

Sharpness of results
To prove Theorem 5 in the multi-parameter setting, we adapt a one-variable construction of Carleson which is well described for example in [9,Theorem 3.4.1].
Proof of Theorem 5. Since C(·) is outer and C(E) = 0, we may choose a sequence G 1 ⊃ G 2 ⊃ G 3 ⊃ · · · of open sets such that E ⊂ G j , for all j, and ∞ j=1 C(G j ) 1/2 < ∞.
Since E is compact, we may additionally assume that G j+1 ⊂ G j for every j. Letting F j = G j , we thus have a decreasing sequence F 1 ⊃ F 2 ⊃ F 3 ⊃ · · · of compact sets containing E, such that Let µ Fj be the equilibrium measure of F j , and define f j ∈ D(D n ) by the relationship It is key to the proof that if we choose C > 0 sufficiently large, then (15) Re G(ψ) ≈ H(ψ).

In particular,
Re C + log 1 1 − z 1 e −iψ1 · · · C + log 1 1 − z n e −iψn ≥ 0, for z ∈ D n and ψ ∈ [0, 2π) n , since the left-hand side is the Poisson integral of Re G(ψ − ·). Therefore we fix C as a constant such that (15) holds. The choice of C only depends on n.
With µ Fj (α) = T n e −i(α,θ) dµ(θ), we then have that where the last step follows by a computation with coefficients (including a straightforward approximation argument). A computation with Fourier coefficients also yields that In view of (14) we may therefore define the function We will demonstrate that lim z→ζ Re f (z) = ∞, for every ζ ∈ E.
Since Re f j is n-harmonic and non-negative, there is by Lemma 8 a measure dµ j = g j dθ + dσ j such that 0 ≤ g j ∈ L 1 (T n ), σ j ≥ 0 is singular, and Re f j (z) = P µ(z) for z ∈ D n . By Corollary 4 the limit lim t→1 Re f j (te iθ1 , . . . , te iθn ) exists for quasi-every, and thus almost every, θ ∈ [0, 2π) n . Furthermore, by Fatou's lemma and the properties of an equilibrium measure, we have that for quasi-every θ ∈ F j . On the other hand, by Lemma 8, we have that lim t→1 Re f j (te iθ1 , . . . , te iθn ) = g j (θ) for almost every θ ∈ [0, 2π) n . We conclude that there is a constant c > 0, independent of j, such that g j (θ) ≥ c for almost every θ in the open set G j ⊃ E.