Rectangular Summation of Multiple Fourier Series and Multi-parametric Capacity

https://doi.org/10.1007/s11118-020-09861-5

Rectangular Summation of Multiple Fourier Series and Multi-parametric Capacity

Karl-Mikael Perfekt¹

Received: 26 August 2019 / Accepted: 1 July 2020 /

©The Author(s) 2020

Abstract

We consider the class of multiple Fourier series associated with functions in the Dirich- let 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 con- struct 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.

Keywords Dirichlet space·Polydisc·Multiple Fourier series·Capacity·Multi-parameter Mathematics Subject Classification (2010) 31B15·32A40

1 Introduction

This article will consider unrestricted rectangular summation and other multi-parameter summation methods of the multiple Fourier series

f (θ )∼

α∈Zⁿ

aαe^{i(α1θ1+···αnθn)}. (1) To clarify this objective, note that there are several natural ways to form the partial sums of a multiple Fourier series. For example, one can attempt to sum the series viasquare partial sums,

M→∞lim

|αj|≤M

a_αe^{i(α1θ1+···αnθn)},

spherical partial sums,

R→∞lim

α₁²+···+α²n≤R

a_αe^{i(α1θ1+···αnθn)},

Karl-Mikael Perfekt [email protected]

1 Department of Mathematics and Statistics, University of Reading, Reading RG6 6AX, UK Published online: 17 July 2020

orunrestricted rectangular partial sums,

NⁿN→∞lim

|αj|≤Nj

a_αe^{i(α1θ1+···αnθn)}, (2) whereN → ∞means that min_1≤j≤nN_j → ∞, with no assumption made on the rela- tionship betweenNj andNk, 1 ≤ j, k ≤ n. These three modes of convergence behave quite differently, and typically require different techniques to treat. The first two summa- tion methods only depend on one parameter (MorR), while the the third is an example of a multi-parameter summation method. We refer to [4] and [23, Ch. XVII] for an introduction to multi-parameter summation methods for Fourier series.

Carleson [10] famously proved that the Fourier series of a functionf ∈L²(T)converges for almost everyθ ∈ [0,2π ). This can be exploited to show that the Fourier series of a functionf ∈L²(Tⁿ),n≥2, converges with respect to square partial sums for almost every θ ∈ [0,2π )ⁿ[2,12,21,22]. On the other hand, C. Fefferman [13] constructed a continuous functionf ∈C(T²)whose Fourier series diverges with respect to unrestricted rectangular sums for everyθ ∈ [0,2π )². Under spherical summation, the convergence question is still open for Fourier series off ∈L²(Tⁿ),n≥2, but we refer to [16] for some related negative results.

Let us now bring potential theory into the discussion. For a seriesf (θ )∼

k∈Za_ke^ikθ such that

k∈Z|k||a_k|² < ∞, Beurling [8] showed that f (θ ) is summable for every θ ∈T\E, whereEis a set of zero logarithmic capacity. This was given a one-parameter generalization to multiple Fourier series by Lippman and Shapiro [17]. They proved that if f ∈L¹(Tⁿ),n≥2, is as in Eq.1and satisfies that

α∈Zⁿ(α₁²+ · · · +α²_n)|aα|²<∞, then f (θ )is summable with respect to spherical partial sums, except for on a setE⊂Tⁿof zero ordinary capacity (logarithmic capacity forn=2 and Newtonian capacity forn≥3, under the identificationTⁿ(R/Z)ⁿ).

An interest in the multi-parameter summation method Eq.2thus leads us to seek a suit- able concept of capacity. A notion of multi-parametric logarithmic capacity has appeared recently in function-theoretic investigations of the Dirichlet spaceD(Dⁿ)of the polydisc [5–7,15]. In particular, in [3], it was proven that bi-parameter logarithmic capacity char- acterizes the Carleson measures ofD(D²). It is therefore natural to generalize Beurling’s result to this context.

Before stating the main results, let us fix some notation. For a positive integern, consider the multiple Fourier series

f (θ )∼

α∈Nⁿ

a_αe^{i(α,θ )},

whereN = {0,1,2, . . .},θ ∈ [0,2π )ⁿ, and the coefficients belong to some Hilbert space H,a_α∈H. We say thatfbelongs to the Dirichlet space of then-disc,f ∈D(Dⁿ,H), if

α∈Nⁿ

(α₁+1)· · ·(α_n+1)a_α²_H<∞.

IfH=C, we simply writeD(Dⁿ). 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,

D(D²)=D(D,D(D)).

This is the reason that we consider the vector-valued setting.

Through iterated Poisson extension, anyf ∈D(Dⁿ,H)defines anH-valued holomor- phic function inz=(r1e^iθ1, . . . , rne^iθn)∈Dⁿ,

f (z)=f_r(θ )=

α∈Nⁿ

a_αr^αe^{i(α,θ )}, r∈ [0,1)ⁿ, θ ∈ [0,2π )ⁿ. We will freely identify[0,2π )ⁿwith then-torusTⁿ.

For a positive measurable functionf onTⁿ, let Bf (θ )=

Tⁿ

1

|e^iθ1−e^iψ1|¹² · · · 1

|e^iθn−e^iψn|¹²f (ψ)dψ,

wheredψdenotes the normalized Lebesgue measure onTⁿ. For a setE⊂Tⁿin then-torus, we then define the following outer capacity:

C(E)=inf

f²_L2(Tⁿ):f ≥0, Bf (θ )≥1 for allθ∈E

. (3)

Whenn=1 andEis a Borel set (or more generally a capacitable set, see Section2),C(E) is equivalent to the usual (gently modified) logarithmic capacity of E. Forn ≥ 2,C(E) is a multi-parameter analogue of logarithmic capacity. The capacityC(·) fits the general theory of [1, Ch. 2.3–2.5], allowing us to access certain basic tools of potential theory such as equilibrium measures. However, we warn the reader that a number of familiar properties from the one-parameter setting do not hold. Notably, the associatedn-logarithmic potentials defined in Section2generally fail to satisfy any kind of boundedness principle [3].

We shall actually prove convergence in a stronger sense than that given by Eq.2. We say that the seriesf (θ )converges in the sense of Pringsheimif it converges with respect to unrestricted rectangular partial sums,

f (θ )= lim

NⁿN→∞

N1

α₁=0

· · ·

Nn

αn=0

a_αe^{i(α,θ )}, (4)

and it holds that

sup

N∈Nⁿ

N₁

α₁=0

· · ·

N_n

α_n=0

a_αe^{i(α,θ )} H

<∞. (5)

Finally, we say that a property holds quasi-everywhere if it holds everywhere onTⁿbut for a set of capacity 0. Our first main result is the following.

Theorem 1 Iff ∈ D(Dⁿ,H), then for quasi-everyθ ∈ [0,2π )ⁿ,f (θ )converges in the sense of Pringsheim.

Our second main theorem shows that Theorem 1 is sharp.

Theorem 2 IfE ⊂Tⁿis compact andC(E)=0, then there exists a functionf ∈D(Dⁿ) such thatf (θ )diverges in the sense of Pringsheim forθ ∈E.

To prove Theorems 1 and 2, we will first prove that multi-parametric logarithmic capacity characterizes the exceptional sets for the radial variationV_nf (θ )off ∈D(Dⁿ,H),

Vnf (θ )=

[0,1]ⁿ∂rfr(θ )Hdr, where∂_r =∂_r1· · ·∂_rnanddr=dr₁· · ·dr_n.

Theorem 3 Iff ∈D(Dⁿ,H), thenV_nf (θ )is finite for quasi-everyθ.

Remark Whenn=2 andH=C, this theorem is an immediate corollary of the work in [3]. In that paper, the Carleson measures forD(D²), which also turn out to be embedding measures for the radial variation, were given a potential-theoretic characterization. How- ever, the characterization of Carleson measures is a much more complicated problem than the characterization of exceptional sets for the radial variation—see [14,18].

Applying Theorem 3, we obtain the following corollary on unrestricted iterated Abel summation, that is, on the radial limits of a functionf ∈D(Dⁿ,H).

Corollary 4 Iff ∈D(Dⁿ,H), then for quasi-everyθit holds that f^∗(θ )= lim

r→(1,···,1)fr(θ ) exists, and furthermore that

sup

r fr(θ )H<∞.

The value off^∗(θ )coincides with the Pringsheim sumf (θ )quasi-everywhere.

Theorem 3 is also sharp.

Theorem 5 IfE⊂Tⁿis compact andC(E)=0, then there exists a functionf ∈D(Dⁿ) such that

zlim→ζRef (z)= ∞, ζ ∈E.

To complete the analogy with Beurling’s work [8], we shall also prove the following result on the strong differentiability of the integral off. Forθ ∈ [0,2π )ⁿandh∈(0, π )ⁿ, let

F_h(θ )= πⁿ h₁· · ·h_n

(θ1−h1,θ1+h1)· · ·

(θn−hn,θn+hn)

f (ψ)dψ. Theorem 6 Iff ∈D(Dⁿ,H), then

h→(0,...,0)lim Fh(θ )=f (θ ) for quasi-everyθ.

2 Preliminaries

2.1 Multi-parametric Capacity

First, let us slightly modify the kernel of B (without otherwise changing the notation).

Letting

b(θ )=3+^∞

k=1

coskθ k¹²

, θ ∈ [0,2π ),

we note thatb(θ )≥1 is convergent and continuous forθ >0, and that b(θ )≈

sinθ 2

⁻

1 2

.

See [23, Ch. V.1–V.2]. Hence, if we letB(θ )=b(θ1)· · ·b(θn), and for positive finite Borel measuresμonTⁿdefine

Bμ(θ )=

TⁿB(θ−ψ)dμ(ψ), θ ∈ [0,2π )ⁿ, this only changes the definition ofC(·)in Eq.3up to constants.

Note that the convolution ofbwith itself satisfies that h(θ ):=b∗b(θ )=9+1

2log 1

|1−e^iθ|. The kernelH (θ )=h(θ₁)· · ·h(θ_n)defines then-logarithmic potential,

H μ(θ )=

TⁿH (θ−ψ)dμ(ψ), θ ∈ [0,2π )ⁿ. The energy of a measureμis thus given by

Bμ²_L2(Tⁿ)=

TⁿH μ(θ )dμ(θ )=

Tⁿ

TⁿH (θ−ψ)dμ(ψ)dμ(θ ).

SinceB(θ )is lower semi-continuous onTⁿ, the theory of [1, Ch. 2.3–2.5] applies toC(·), as was mentioned in the introduction. In particular, every Borel setE⊂Tⁿis capacitable, that is,

C(E)=inf{C(G):G⊃Eopen} =sup{C(K):K⊂Ecompact}.

For any capacitable setE,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,

C(E)^1/2=sup μ(E):suppμ⊂E,BμL²(Tⁿ)≤1

. (6)

In particular, the setE has capacity 0,C(E) = 0, if and only if every non-zero positive finite measureμwith support inEhas infinite energy,

Tⁿ

TⁿH (θ−ψ)dμ(ψ)dμ(θ )= ∞.

Furthermore, the following simple lemma, which we shall use without mention, is clear from Eqs.3and6.

Lemma 7 IfE₁, . . . , E_nare Borel sets, then

C(E1× · · · ×En)=C(E1)· · ·C(En).

The final piece of information that we require is the existence of equilibrium measures.

For any compact set K ⊂ Tⁿ, the extremal to the capacity problem is generated by a measure μK such that: suppμK ⊂ K,H μK(θ ) ≤ 1 forθ ∈ suppμK,H μ(θ ) ≥ 1 for quasi-everyθ ∈Kand

μK(K)=

Tⁿ

TⁿH (θ−ψ)dμK(ψ)dμK(θ )=C(K).

2.2 n-Harmonic Functions

A continuous function onDⁿisn-harmonic if it is harmonic in each variablez_jseparately, z = (z₁, . . . , z_n) ∈ Dⁿ. For a finite measureμonTⁿ, we denote byP μthen-harmonic function

P μ(z)=P μ(r, θ )=

TⁿP_r1(θ₁−ψ₁)· · ·P_rn(θ_n−ψ_n)dμ(ψ), wherez=(r1e^iθ1, . . . , rne^iθn)∈DⁿandPr(θ )denotes the usual Poisson kernel,

Pr(θ )= 1−r² 1−2rcosθ+r².

We refer to [19, Ch. 2] for the fundamentals ofn-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 [19].

Lemma 8 Ifu ≥ 0 isn-harmonic and non-negative onDⁿ, then there exists a function 0≤g∈L¹(Tⁿ)and a singular measureσ≥0onTⁿsuch that

u(z)=P ν(z), dν=gdθ+dσ, z∈Dⁿ. Furthermore, for almost everyθ ∈ [0,2π )ⁿ, it holds that

t→1lim⁻u(te^iθ1, . . . te^iθn)=g(θ ).

Remark Since we will prove theorems about unrestricted summation and strong differentia- bility, we note that unlike the one-variable setting, the proof of the lemma does not specify for which pointsθ the limit exists. In general, localization fails for multiple Poisson inte- grals. In fact, letf¹ ∈C^∞(T)be such thatf¹(θ1)=0 for|θ1| ≤ε, for someε >0, and such that there is a sequencetj →1 for whichP[f¹dθ1](tj,0) > 0. Letf² ∈D(D)be any function such that limt→1ReP[f²dθ2](t,0)= ∞. Let

f (θ )=f¹(θ₁)f²(θ₂)∼

α∈Z²

a_αe^{i(α,θ )}. Then the Fourier coefficients off satisfy that

α∈Z²

(|α₁| +1)(|α₂| +1)|a_α|²<∞, andf (θ )vanishes in an open neighborhood of 0, but still

lim

(r1,r2)→(1,1)P[f dθ](r,0)=0.

In fact, the limit does not exist.

3 Convergence Theorems

We begin by proving Theorem 3. Givenf ∈D(Dⁿ,H), note that E= {θ :Vnf (θ )= ∞} =

i≥1

j≥1

θ:

[0,1−1/j]ⁿ∂rfr(θ )Hdr > i

(7)

is aG_δ-set, hence capacitable. The following proof is in the spirit of Salem and Zygmund’s approach to exceptional sets for one-variable Dirichlet spaces [20].

Proof of Theorem 3 We may assume that the Fourier coefficients of f are supported in (Z≥1)ⁿ,f ∼

α∈(Z≥1)ⁿaαe^{i(α,θ )}. Fork≥0, let ck=

k−1/2 k

= 1

√π k^1/2

1+O(k⁻¹)

, (8)

so that

˜ b(θ ):=

∞ k=0

c_kcoskθ=Re 1

(1−e^iθ)^1/2, 0< θ <2π,

see [23, Ch. V.2]. Note thatb(θ )˜ is another uniformly positive function with the same singu- lar behavior asb(θ ). Leth˜= ˜b∗ ˜b. Thenh˜≥c >0 for somec, and by Eq.8we see thath(θ )˜ has the same logarithmic singularity ash(θ ), when sin^θ₂ →0. LetB(θ )˜ = ˜b(θ₁)· · · ˜b(θ_n), H (θ )˜ = ˜h(θ₁)· · · ˜h(θ_n), and forr∈ [0,1)ⁿ,

B˜r(θ )=P[ ˜B(ψ)dψ](r, θ )=:

α∈Zⁿ

Cαr₁^|α1|· · ·r_n^|αn|e^{i(α,θ )}. Note that

C_α= c_α1· · ·c_αn

2ⁿ , α∈(Z≥1)ⁿ. (9)

We will also rely on the estimate

[0,1]ⁿ|∂_rB˜_r(θ )|dr

[0,1]ⁿ

1

|1−r₁e^iθ1|^3/2 · · · 1

|1−r_ne^iθn|^3/2dr

sinθ₁ 2

⁻

1 2· · ·

sinθ_n 2

⁻

1

2 B(θ ).˜ (10)

Suppose now that the setEof Eq.7has positive capacity. Then there exists a non-zero finite measureμ, supported inE, such that

˜Bμ²_L2(Tⁿ)=

Tⁿ

TⁿH (θ˜ −ψ)dμ(ψ)dμ(θ ) <∞, whereBμ(θ )˜ =

TⁿB(θ˜ −ψ)dμ(ψ). LetF be theH-valued series F (θ )∼

α∈(Z≥1)ⁿ

C⁻_α¹aαe^{i(α,θ )}.

The coefficients ofF are square-summable, by Eqs.8,9, and the fact thatf ∈D(Dⁿ,H).

ThusF (θ )has meaning for almost everyθ, and

TⁿF (θ )²_Hdθ <∞.

By our assumption on the support of the Fourier coefficients off we have that

∂rfr(θ )=

TⁿF (ψ)∂rB˜r(θ−ψ)dψ, and therefore by Eq.10that

Vnf (θ )

TⁿF (ψ)HB(θ˜ −ψ)dψ.

But then, by the assumption of finite energy,

TⁿV_nf (θ )dμ(θ ) 2

TⁿF (ψ)HBμ(ψ)dψ˜ 2

≤ ˜Bμ²_L2(Tⁿ)

TⁿF (ψ)²_Hdψ <∞. This is obviously a contradiction.

Proof of Corollary 4 We give the proof forn=2. The proof is the same forn≥3, but the notation is more difficult. Givenf ∈D(D²,H), definef¹, f² ∈D(D,H)by

f¹(z)=f (z,0), f²(w)=f (0, w), z, w∈D. Let

E=

θ ∈ [0,2π )²:V2f (θ )= ∞ , and

E1 =

θ1∈ [0,2π ):V1f¹(θ1)= ∞

, E2 =

θ2∈ [0,2π ):V1f²(θ2)= ∞ . LetF =E∪(E1×T)∪(T∪E2). ThenC(F )=0, by three applications of Theorem 3.

Suppose now thatθ /∈F, and forr, r∈ [0,1)², write by analyticity f_r(θ )−f_r(θ )=

_r1

0

_r2

0

∂_ρf_ρ(θ )dρ− _r

1 0

_r

2 0

∂_ρf_ρ(θ )dρ

+ _r1

r₁

∂_ρ1f_ρ¹

1(θ₁)dρ₁+ _r2

r₂

∂_ρ2f_ρ²

2(θ₂)dρ₂. Thus

fr(θ )−f_r(θ )H≤ 1 min(r1,r₁)

1

0 ∂ρfρ(θ )Hdρ+ 1 0

1

min(r2,r₂)∂ρfρ(θ )Hdρ + 1

min(r₁,r₁)∂_ρ1f_ρ¹

1(θ1)Hdρ1+ 1

min(r₂,r₂)∂_ρ2f_ρ²

2(θ2)Hdρ2. SinceV2f (θ ),V1f¹(θ1), andV1f²(θ2)are all finite, it follows that

f_r(θ )−f_r(θ )H→0, r, r→(1,1).

Hencef^∗(θ )=lim_r→(1,1)fr(θ )exists, for everyθ outside the capacity zero setF. Letting r=0 in the estimate also shows thatfr(θ )His uniformly bounded inr.

We postpone the proof thatf^∗(θ )coincides with the sumf (θ )quasi-everywhere to the proof of Theorem 1.

Forn=1 andH=C, a seriesf ∈D(D)is summable atθ ∈ [0,2π )if and only if it is Abel summable atθ. This is sometimes known as Fej´er’s Tauberian theorem. Thus, in this case Theorem 3 immediately implies Theorem 1. To prove Theorem 1 forn≥2, we begin by stating a vector-valued version of Fej´er’s theorem.

Lemma 9 ForN∈Nandθ∈ [0,2π ), defineS_N,θ^H , P_N,θ^H :D(D,H)→Hby S_N,θ^H f =

N k=0

ake^ikθ, P_N,θ^H f =f1−1/N(θ ), f ∈D(D,H).

Then there is an absolute constantC >0such that

S_N,θ^H f−P_N,θ^H fH≤CfD(D,H). Moreover, for every fixedf we have that

S^H_N,θf−P_N,θ^H f →0, N→ ∞, uniformly inθ.

Proof Letr=1−1/N, and note that 1−r^k≤k/N, to see that S_N,θ^H f −P_N,θ^H fH≤ 1

N N k=1

kakH+^∞

k=N

akHr^k. ForM≤N, we estimate

1 N

N k=1

kakH≤ 1 N

M k=1

kakH+ 1 N

_N

k=M

kak²_H

^1/2 _N

k=M

k ^1/2

. By first choosingMlarge, and thenN, we see that_N¹ N

k=1ka_kH→0 asN→ ∞. For the second term we have that

∞ k=N

a_kHr^k≤ 1

√N _∞

k=N

ka_k²_H

1/2 _∞

k=N

r^2k 1/2

,

and thus this term also tends to 0 asN → ∞. This second estimate, together with the first estimate forM=0, also shows the uniform bound of the operator norm ofS_N,θ^H −P_N,θ^H .

In the proof of Theorem 1 we will consider tensors of the operators SN,θ andPN,θ, interpreted in the obvious way. For instance, ifN ∈Nⁿ,θ ∈ [0,2π )ⁿ, andf ∈D(Dⁿ,H), then

(SN₁,θ₁⊗ · · · ⊗SN_n,θ_n)f =

N1

α1=0

· · ·

Nn

αn=0

aαe^{i(α,θ )},

and

(P_N1,θ1⊗ · · · ⊗P_Nn,θn)f = f_{(1−1/N1,...,1−1/Nn)}(θ )

= ∞ α₁=0

· · · ∞ αn=0

a_α(1−1/N₁)^α1· · ·(1−1/N_n)^αne^{i(α,θ )}. Similarly, we consider mixed tensor products, such as

(S_N1,θ1⊗P_N2,θ2)f =

N1

α1=0

∞ α2=0

a_α1,α2(1−1/N₂)^α2e^{i(α,θ )}.

Proof of Theorem 1 We will deduce the result from Theorem 3, Lemma 9, and an inductive procedure which exploits the fact that

D(Dⁿ,H)=D(Dⁿ⁻¹,D(D,H)).

We already know that Theorem 1 is true forn=1, precisely by Theorem 3 and Lemma 9.

Thus we first consider the casen=2. By Corollary 4, there is a Borel setE⊂T²such that C(T² \E) = 0, and for everyθ = (θ₁, θ₂) ∈ E we have that(P_N1,θ1 ⊗P_N2,θ2)f

is uniformly bounded inN₁, N₂and convergent tof^∗(θ )asN₁, N₂ → ∞. To prove the theorem, it is thus sufficient to provide a setF ⊂Esuch thatC(E\F )=0 and such that for everyθ∈F it holds that

N₁,Nlim₂→∞(S_N1,θ1⊗S_N2,θ2−P_N1,θ1⊗P_N2,θ2)fH=0, (11) and

sup

N1,N2

(S_N1,θ1⊗S_N2,θ2−P_N1,θ1⊗P_N2,θ2)fH<∞. (12) Constructing such a setF of course also proves thatf^∗(θ ) =f (θ )quasi-everywhere, as claimed in Corollary 4.

We write

(S_N1,θ1⊗S_N2,θ2−P_N1,θ1⊗P_N2,θ2)f

=((SN₁,θ₁−PN₁,θ₁)⊗SN₂,θ₂)f +(PN₁,θ₁⊗(SN₂,θ₂−PN₂,θ₂))f.

Now, by then = 1 case of the theorem, applied tof ∈ D(D,D(D,H)), there is a set G₂⊂Tsuch thatC(T\G₂)=0, and such that for everyθ₂∈G₂we have the existence of

h_θ2 := lim

N2→∞S_N^D(D,H)

2,θ₂ f ∈D(D,H). (13)

Next, forθ2∈G2, note that

((S_N1,θ1−P_N1,θ1)⊗S_N2,θ2)f =(S_N^H

1,θ₁−P_N^H

1,θ₁)S_N^D(D,H)

2,θ2 f

=(S_N^H

1,θ₁−P_N^H

1,θ₁)(S_N^D(D,H)

2,θ2 f −h_θ2)+(S_N^H

1,θ₁−P_N^H

1,θ₁)h_θ2.

Thus, by Lemma 9 and Eq.13it follows that, for any fixed(θ₁, θ₂) ∈ T×G₂, the term ((S_N1,θ1−P_N1,θ1)⊗S_N2,θ2)f is uniformly bounded inN₁, N₂and tends to 0 asN₁, N₂→

∞.

By a very similar argument (after reordering the variablesθ1andθ2), there is a setG1 ⊂ T such that C(T\G1) = 0, and such that for every θ1 ∈ G1 andθ2 ∈ T, the term (P_N1,θ1⊗(S_N2,θ2−P_N2,θ2))f is uniformly bounded inN₁, N₂and tends to zero asN₁, N₂→

∞. Thus the proof forn=2 is finished by letting

F =E∩(G1×T)∩(T×G2).

Note that in the course of the proof we have also established that(PN₁,θ₁ ⊗SN₂,θ₂)f is uniformly bounded inN1, N2and converges tof^∗(θ )asN1, N2→ ∞, forθ ∈F.

Forn=3, Corollary 4 gives us a setE ⊂T³such thatC(T³\E) =0 and on which (P_N1,θ1⊗P_N2,θ2⊗P_N3,θ3)f converges and is uniformly bounded. We then write

(SN1,θ1⊗SN2,θ2⊗SN3,θ3−PN1,θ1⊗PN2,θ2⊗PN3,θ3)f

=((S_N1,θ1−P_N1,θ1)⊗S_N2,θ2⊗S_N3,θ3)f +(P_N1,θ1⊗(S_N2,θ2−P_N2,θ2)⊗S_N3,θ3)f +(PN₁,θ₁⊗PN₂,θ₂⊗(SN₃,θ₃−PN₃,θ₃)f.

Now we apply then = 2 case of the theorem, together with the remark at the end of its proof, three separate times tof ∈D(D²,D(D,H)). Arguing with Lemma 9 as before, this produces three setsH₁, H₂, H₃⊂T³such thatC(T³\H_j)=0, and such that, forθ∈H_j, thej:th term is uniformly bounded inN1, N2, N3and converges to zero asN1, N2, N3 →

∞. Thus(SN₁,θ₁⊗SN₂,θ₂⊗SN₃,θ₃)f is uniformly bounded and converges asN1, N2, N3→

∞, forθ ∈E∩H1∩H2∩H3. Furthermore, the same is true of(PN₁,θ₁⊗SN₂,θ₂⊗SN₃,θ₃)f and(P_N1,θ1⊗P_N2,θ2⊗S_N3,θ3)f.

It is now clear how the construction extends by induction ton≥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 one-parameter 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 F_h(θ )=

α∈Nⁿ

a_αsin(α₁h)

α₁h · · ·sin(α_nh)

α_nh e^{i(α,θ )}. (14)

This is obviously true for polynomials, and for allf ∈ D(Dⁿ,H)by continuity. For this last statement, note that, with continuous dependence on f, the valuesf (θ ) are square- integrable onTⁿ, and the right-hand side of Eq.14is absolutely convergent.

The argument is now very similar to the proof of Theorem 1. First we consider the case n=1, letting

R_h,θ^Hf =^∞

k=0

a_ksin(kh)

kh e^ikθ, f ∈D(D,H),

forθ ∈ [0,2π )andh∈(0,1). Let 1≤N∈Nbe such that _N+1¹ ≤h < _N¹, and letM≤N. Then

R_h,θ^H f−S_N,θ^H fH N k=1

a_kH(kh)²+^∞

k=N

a_kH

kh ≤ M k=1

a_kH(kh)² +

_N

k=M

ka_k²_H ¹₂

h⁴ N k=M

k^{3 1}₂

+ _∞

k=N

ka_k²_H ¹₂

1 h²

∞ k=N

1 k³

¹₂ . By this estimate,R_h,θ^H −S_N,θ^H :D(D,H) →His uniformly bounded inN and converges pointwise to 0 asN→ ∞, as long as_N+1¹ ≤h < _N¹. Thus Theorem 1 implies Theorem 6 in the case thatn=1.

Forn≥2 we proceed precisely as in the proof of Theorem 1. For instance, forn=2 we write

(S_N1,θ1⊗S_N2,θ2−R_h1,θ1⊗R_h2,θ2)f

=((SN₁,θ₁−Rh₁,θ₁)⊗SN₂,θ₂)f+(Rh₁,θ₁⊗(SN₂,θ₂−Rh₂,θ₂))f, whereN=(N₁, N₂)is related toh=(h₁, h₂)by the facts that_N¹

j+1 ≤h_j < _N¹

j,j=1,2.

The rest of the proof is essentially repetition.

4 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 [11, Theorem 3.4.1].

Proof of Theorem 5 SinceC(·)is outer andC(E) =0, we may choose a sequenceG₁ ⊃ G₂⊃G₃⊃ · · ·of open sets such thatE⊂G_j, for allj, and

∞ j=1

C(Gj)^1/2<∞.

SinceE is compact, we may additionally assume thatG_j+1 ⊂ G_j for everyj. Letting Fj = Gj, we thus have a decreasing sequence F1 ⊃ F2 ⊃ F3 ⊃ · · · of compact sets containingE, such that

∞ j=1

C(Fj)^1/2<∞. (15)

LetμF_j be the equilibrium measure ofFj, and definefj ∈D(Dⁿ)by the relationship fj(z)=

Tⁿ

C+log 1

1−z1e^−iψ1

· · ·

C+log 1

1−zne^−iψn

dμF_j(ψ),

forz∈Dⁿ. Let G(ψ)=

C+log 1 1−e^−iψ1

· · ·

C+log 1 1−e^−iψn

, ψ∈ [0,2π )ⁿ. It is key to the proof that if we chooseC >0 sufficiently large, then

ReG(ψ)≈H (ψ). (16)

In particular, Re

C+log 1

1−z₁e^−iψ1

· · ·

C+log 1

1−z_ne^−iψn

≥0,

forz∈Dⁿandψ∈ [0,2π )ⁿ, since the left-hand side is the Poisson integral of ReG(ψ−·).

Therefore we fixCas a constant such that Eq.16holds. The choice ofConly depends onn.

WithμF_j(α)=

Tⁿe^{−i(α,θ )}dμF_j(θ ), we then have that BμF_j²_L2(Tⁿ) =

Tⁿ

TⁿH (θ−ψ)dμF_j(ψ)dμF_j(θ )

≈ Re

Tⁿ

TⁿG(θ−ψ)dμ_Fj(ψ)dμ_Fj(θ )≈

α∈Nⁿ

|μ_Fj(α)|² (α1+1)· · ·(αn+1), where the last step follows by a computation with coefficients (including a straightforward approximation argument). A computation with Fourier coefficients also yields that

fj²_D(Dⁿ)≈

α∈Nⁿ

|μ_Fj(α)|²

(α1+1)· · ·(αn+1) ≈ BμFj²_L2(Tⁿ)=C(Fj).

In view of Eq.15we may therefore define the function f =

∞ j=1

f_j∈D(Dⁿ).

We will demonstrate that lim_z→ζRef (z)= ∞, for everyζ ∈E.

Since Ref_j is n-harmonic and non-negative, there is by Lemma 8 a measuredμ_j = gjdθ+dσj such that 0≤gj ∈L¹(Tⁿ),σj ≥0 is singular, and Refj(z) =P μj(z)for z ∈Dⁿ. Actuallyσj = 0, sincefj belongs to the Hardy spaceH²(Dⁿ)[19, Ch. 3.4], but we do not need to know this. By Corollary 4 the limit lim_t→1Ref_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

t→1limRefj(te^iθ1, . . . , te^iθn)≥Re

TⁿG(ψ−θ )dμFj(ψ)≈

TⁿH (θ−ψ)dμFj(ψ)≥1