On Hardy type spaces in strictly pseudoconvex domains and the density, in these spaces, of certain classes of singular functions

J. Math. Anal. Appl. 484 (2020) 123697

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

On Hardy type spaces in strictly pseudoconvex domains and the density, in these spaces, of certain classes of singular functions Kyranna Kioulafa National and Kapodistrian University of Athens, Department of Mathematics, Panepistemiopolis, 157 84 Athens, Greece

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 26 June 2019 Available online 21 November 2019 Submitted by E.J. Straube

In this paper we study some Hardy type spaces in one or several complex variables and we prove that the set of the holomorphic functions which are totally unbounded in certain domains is dense and Gδ in these spaces. These totally unbounded functions are non-extendable, despite the fact that they have non-tangential limits at the boundary of the domain. Similarly we show that the set of the holomorphic functions in these spaces which are non-extendable is dense and Gδ in these spaces. We also consider local Hardy spaces and show that the set of the functions in these Hardy type spaces which do not belong – not even locally – to Hardy spaces of higher order is dense and Gδ . We first work in the case of the unit ball of C n where the calculations are easier and the results are somehow better, and then we extend them to the case of strictly pseudoconvex domains. © 2019 Elsevier Inc. All rights reserved.

Keywords: Hardy type spaces Strictly pseudoconvex domains Totally unbounded functions

1. Introduction and preliminaries Hardy type spaces for domains in C n and the boundary behaviour of the functions belonging to these spaces have been extensively studied by various authors. See for example [5,7,9,11–13]. More relevant to the results of this paper are those in [3], where analogous theorems were proved with Bergman type spaces in place of Hardy spaces. We also refer to [3] and the references given there for the techniques used in this present paper. 1.1. Hardy type spaces in the unit ball of C n Let B = {z ∈ C n : |z| < 1}. We recall that the Hardy space H p (B), 1 ≤ p < ∞, is defined to be the set of holomorphic functions f : B → C such that   f p = sup r<1

ζ∈∂B

E-mail address: [email protected]. https://doi.org/10.1016/j.jmaa.2019.123697 0022-247X/© 2019 Elsevier Inc. All rights reserved.

1/p |f (rζ)|p dσ(ζ) < +∞,

K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697

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where dσ is the Euclidean surface area measure on the sphere ∂B. The space H p (B) endowed with the norm  · p is a Banach space. We also recall that if a sequence fm ∈ H p (B) converges to f , in the above norm, then fm converges to f also uniformly on compact subsets of B. Indeed this follows from the inequality sup |f (z)| ≤ C(p, K)f p ,

z∈K

with K being a compact subset of B and C(p, K) is a constant depending on p and K. (See [9, Theorem 7.2.5].) Also, H ∞ (B) is the Banach space of bounded holomorphic functions f : B → C, with the norm f ∞ = sup |f (z)|. z∈B  For each q > 1, we also consider the space H p (B), which becomes a complete metric space with 1≤p
the metric d(f, g) =

∞  1 f − gpj , 2j 1 + f − gpj j=1

where 1 < p1 < p2 < · · · < pj < · · · < q and pj → q (j → ∞). Although this metric depends on the  sequence pj , the topology induced by this metric in the space H p (B) is independent of the choice of the 1≤p
for every p < q. Indeed, if the sequence fk converges to f , i.e., d(fk , f ) → 0, then clearly fk − f pj → 0 for every j = 1, 2, 3, . . . . But if p < q, we may choose a j0 so that p < pj0 < q. Then fk − f pj0 → 0 and therefore fk − f p → 0. Conversely, we will show that if fk − f p → 0, for every p < q, then ∞  1 ε d(fk , f ) → 0. Let ε > 0. We choose N = N (ε) ∈ N so that < . Since fk − f pj → 0 for j 2 2 j=N +1 ε for k ≥ k0 (ε) and 1 ≤ j ≤ N . Then it is 1 ≤ j ≤ N , we may choose k0 (ε) ∈ N so that fk − f pj < 2N easy to check that d(fk , f ) < ε for k ≥ k0 (ε). This shows that d(fk , f ) → 0.  Similarly, a sequence fk in H p (B) is Cauchy with respect to the metric d, i.e., d(fk , fl ) → 0 1≤p
(k, l → ∞) if and  only if fk − fl p → 0 for every p < q. Therefore, the completeness of the metric space  H p (B), d follows from the fact that each H p (B) is complete. 1≤p
1.2. Totally unbounded holomorphic functions A holomorphic function f : B → C is called totally unbounded if for every ζ ∈ ∂B and every ε > 0, the restriction f |B(ζ,ε)∩B of f to the set B(ζ, ε) ∩ B = {z ∈ B : |z − ζ| < ε} is unbounded, i.e., sup

|f (z)| = ∞.

z∈B(ζ,ε)∩B

Such a function is singular at every boundary point of the sphere ∂B. (See [3, Section 2].)  In the first part of the paper we will show that the set of the functions in the space H p (B) which 1≤p
are totally unbounded is dense and Gδ (Theorem 2.1). (We recall that a subset of a metric space is called Gδ if it is countable intersection of open subsets of the space.) 1.3. Local Hardy spaces Following a suggestion of Nestoridis, we also consider local Hardy spaces H p (B, G), for open subsets G of the sphere ∂B (the precise definition is given in Section 2) as another way of measuring how singular a

K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697

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holomorphic function is near a boundary point. One extreme of this concept is the property of being totally unbounded. (A similar definition – although slightly different – is given in [7].) In this context we show that  the set of the functions in the space H p (B) which do not belong to any local H q - space is dense and 1≤p
Gδ (Theorem 2.4). In Sections 4 and 5, we will extend these results from the ball to the case of strictly pseudoconvex domains. In this more general case we have to modify the definition of local Hardy spaces which we give in the case of the ball. Thus if Ω is a strictly pseudoconvex domain in C n , we consider the space H p (Ω, U ), where U is an open subset of C n so that U ∩ (∂Ω) = ∅. (For the precise definition, see Section 3.) 2. The case of the unit ball of C n In this section we will first prove the following theorem. Theorem 2.1. Let q ∈ (1, +∞]. Then the set of the functions in the space



H p (B) which are totally

1≤p
unbounded in B is dense and Gδ in this space. The proof of this theorem will be based on the following lemmas. Lemma 2.2. For each point ζ ∈ S = ∂B, we consider the functions fζ (z) =

1 = 1 − z, ζ

1−

1 n  j=1

, hζ (z) = log fζ (z) and ϕq,ζ (z) = exp ζ j zj

 n hζ (z) q

defined for z ∈ B.  Then (i) fζ ∈ H p (B) and fζ ∈ / H n (B). 1≤p
Proof. By [9, Proposition 1.4.10], if p < n, the integral 

dσ(w) , |1 − z, w |p

w∈S

as a function of z, remains bounded for z ∈ B, and, therefore since rζ ∈ B for r < 1,  sup 0
Thus fζ ∈



 |fζ (rz)| dσ(z) = sup p

0
dσ(z) = sup |1 − rz, ζ |p 0


dσ(z) < ∞. |1 − rζ, z |p

z∈S

H p (B). Next we show that

1≤p
 sup 0
 |fζ (rz)|n dσ(z) = sup

Indeed, by [9, Proposition 1.4.10], the integral

0
dσ(z) = ∞. |1 − rz, ζ |n

K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697

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dσ(z) 1 behaves as log for w ∈ B, n |1 − w, z | 1 − |w|2

z∈S

and therefore  sup 0
dσ(z) = sup |1 − rz, ζ |n 0


dσ(z) 1 = sup log = ∞. |1 − rζ, z |n 1 − r2 0
z∈S

This proves (i). Next, observing that Re(1 − z, ζ ) > 0, for z ∈ B, we see that Refζ (z) > 0 and therefore we may define hζ (z) = log fζ (z) using the principal branch of the logarithm with −π < arg ≤ π. Then |Im[log fζ (z)]| < π/2, i.e., hζ (z) = log |fζ (z)| +iθ(z) with |θ(z)| < π/2. Since |1 − rz, ζ | = |ζ − rz, ζ | ≤ |ζ − rz|, it follows that if the point rz ∈ B and is sufficiently close to ζ then p  p/2 2 1 1 2 |hζ (rz)| = log log + θ (rz) + iθ(rz) = |1 − rz, ζ | |1 − rz, ζ |  p 1 1 p/2 ≤2 ≤ 2p/2 (k!)p/k . log |1 − rz, ζ | |1 − rz, ζ |p/k p

The last inequality follows from the fact that (log x)p ≤ (k!)p/k xp/k (for x > 1, p ≥ 1 and k ∈ N) which we applied with x = 1/|1 − rz, ζ |. Thus, fixing p < ∞ and choosing k > p/n (so that n > p/k), we  see that (i) implies hζ ∈ H p (B) whence we obtain hζ ∈ H p (B). (We also used the fact that, since 1≤p<∞

|1 − rz, ζ | > 0 for rz ∈ B away from the point ζ, the quantity | log |1 − rz, ζ | is bounded.) Since obviously lim hζ (z) = ∞, (ii) follows. Finally, observing that |ϕq,ζ | = |fζ |n/q , we easily obtain (iii).  z∈B,z → ζ

The following lemma is proved in [10]. Lemma 2.3. Let V be a topological vector space over C, X a non-empty set, and let C X denote the vector space of all complex-valued functions on X. Suppose T : V → C X is a sublinear operator with the property that, for every x ∈ X, the functional Tx : V → C, defined by Tx (f ) := T (f )(x), for f ∈ V, is continuous. Let E = {f ∈ V : T (f ) is unbounded on X}. Then either E = ∅ or E is dense and Gδ set in the space V. Proof of Theorem 2.1. Let us consider a ball b, with sufficiently small radius, whose center lies on ∂B, and  let us set X = b ∩ B and V = H p (B). We define the linear operator 1≤p
T : V → C X with T (f )(z) = f (z) for z ∈ X and f ∈ V. For each fixed z ∈ X, the functional Tz : V → C defined by Tz (f ) = f (z), f ∈ V, is continuous. It is easy to see that the set E = {f ∈ V : T (f ) is unbounded on X} in this case is equal to

E(b) =

f∈

 1≤p
 H p (B) : sup |f (z)| = ∞ . z∈b∩B

Also, by Lemma 2.2 (ii), E(b) = ∅, since hζ ∈ E(b) for ζ ∈ b ∩ ∂B. Therefore, by Lemma 2.3, E(b) is dense  and Gδ set in the space H p (B). 1≤p
In order to complete the proof, we consider a countable dense subset {w1 , w2 , w3 , . . .} of ∂B, a decreasing sequence εs , s = 1, 2, 3, . . ., of positive numbers with εs → 0, and the balls b(wj , εs ), centered at wj and

K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697

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with radii εs . By the first part of the proof, each of the sets E(b(wj , εs )) is dense and Gδ set in

H p (B).

1≤p
It follows from Baire’s theorem that the set Y=

∞  ∞ 

E(b(wj , εs )) is dense and Gδ in the space

j=1 s=1



H p (B).

1≤p


We claim that the set Y is exactly the set of the functions f ∈

H p (B) which are totally unbounded in

1≤p
B. Indeed, if f ∈ Y and U is an open set with U ∩ ∂B = ∅, we may choose a point wj0 ∈ U ∩ ∂B and an εs0 so that b(wj0 , εs0 ) ⊂ U . Since sup{|f (z)| : z ∈ b(wj0 , εs0 ) ∩B} = ∞, it follows that sup{|f (z)| : z ∈ U ∩B} = ∞.  Conversely, if f ∈ H p (B) and is totally unbounded then it is obvious that f ∈ Y. This completes the 1≤p
proof. 

Next we define Hardy type spaces associated to open subsets of the sphere S = ∂B. These are local  versions of the usual Hardy spaces and the main result is that, in general, the functions in H p (B) do 1≤p
not belong to Hardy spaces of higher order, not even locally. Local Hardy spaces in the unit ball of C n Let G ⊂ S be a non-empty open set (open in S) and 1 ≤ p < ∞. A holomorphic function f : B → C is said to belong to the space H p (B, G) if  sup r<1 z∈G

|f (rz)|p dσ(z) < ∞.

Now we can state the following theorem. Theorem 2.4. Let q ∈ (1, +∞). Then the set

Aq =

g∈





H (B) : g ∈ / H (B, S ∩ b(ζ, ε)) for any ζ ∈ S and any ε > 0 p

q

1≤p
is dense and Gδ in the space



H p (B).

1≤p
For the proof we will need the following lemma. Lemma 2.5. If ζ ∈ G then for the functions fζ and ϕq,ζ , defined in Lemma 2.2, we have: (i) fζ ∈ / H n (B, G), (ii) ϕq,ζ ∈ / H q (B, G) for 1 < q < ∞. Proof. Writing 





|fζ (rz)|n dσ(z) = z∈S

|fζ (rz)|n dσ(z) + z∈G

|fζ (rz)|n dσ(z) for r < 1,

z∈S−G

and taking into consideration the fact that  sup r<1 z∈S

|fζ (rz)|n dσ(z) = ∞,

K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697

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we see that it suffices to show that

 sup r<1 z∈S−G

|fζ (rz)|n dσ(z) < ∞.

For this, let us notice that |1 − rz, ζ | ≥ 1 − Re(rz, ζ ) = 1 − r(z · ζ). (z · ζ = Rez, ζ is the inner product in R2n = C n .) Thus if z · ζ < 0 then |1 − rz, ζ | ≥ 1, and therefore  |fζ (rz)|n dσ(z) ≤ σ(S). z∈(S−G)∩{z·ζ<0}

On the other hand if z · ζ ≥ 0 then |1 − rz, ζ | ≥ 1 − z · ζ. But 1 − z · ζ > 0 for z ∈ (S − G) ∩ {z · ζ ≥ 0}, since ζ ∈ G and z ∈ (S − G) ∩ {z · ζ ≥ 0} imply that z cannot be equal to λζ for any λ > 0, and therefore z · ζ < |z| |ζ| = 1. By the compactness of the set (S − G) ∩ {z · ζ ≥ 0}, α =: inf{1 − z · ζ : z ∈ (S − G) ∩ {z · ζ ≥ 0}} > 0, whence  |fζ (rz)|n dσ(z) ≤

σ(S) for every r < 1. αn

z∈S−G

This proves (i). Now (ii) follows from (i), if we notice that |ϕq,ζ | = |fζ |n/q .



Proof of Theorem 2.4. Let us fix a point w ∈ S and δ > 0. With X = {r : 0 < r < 1} and V =



H p (B),

1≤p
we consider the sublinear operator T : V → C X defined as follows: 



T (f )(r) =

1/q |f (rz)|q dσ(z) for f ∈ V and r ∈ X.

z∈S∩B(w,δ)

Then, for each fixed r ∈ X, the functional Tr : V → C, Tr (f ) = T (f )(r), f ∈ V, is continuous. Indeed, if fm ∈ V and fm → f (in V) then fm converges to f uniformly on compact subsets of B, as we pointed out in Subsection 1.1. Since r < 1, it follows that 

 |fm (rz)|q dσ(z) →

z∈S∩B(w,δ)

|f (rz)|q dσ(z), m → ∞, z∈S∩B(w,δ)

i.e., Tr (fm ) → Tr (f ). On the other hand, by Lemma 2.5 (ii), the set E(w, δ) := {f ∈ V : sup{T (f )(r) : r ∈ X} = ∞} = ∅. Therefore, by Lemma 2.3, the set E(w, δ) is dense and Gδ in the space

 1≤p
H p (B).

K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697

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In order to complete the proof, we consider a countable dense subset {w1 , w2 , w3 , . . .} of ∂B and a decreasing sequence δs , s = 1, 2, 3, . . ., of positive numbers with δs → 0. By the first part of the proof and Baire’s theorem, the set Y=

∞  ∞ 



E(wj , δs ) is dense and Gδ in the space

j=1 s=1

H p (B).

1≤p
We claim that Y = Aq . Indeed, if f ∈ Y, ζ ∈ ∂B and ε > 0, we may choose wj0 ∈ B(ζ, ε) and δs0 so that B(wj0 , δs0 ) ⊂ B(ζ, ε), and since  sup 0
it follows that sup



0
|f (rz)|q dσ(z) = ∞,

|f (rz)|q dσ(z) = ∞, i.e., f ∈ / H q (B, S ∩ B(ζ, ε)). Thus Y ⊂ Aq , and since it

is obvious that Aq ⊂ Y, the proof is complete.  3. Hardy type spaces First let us recall the definition of Hardy spaces in the case of bounded open sets with smooth boundary. Let Ω ⊂ C n be a bounded open set with C 2 boundary and let ρ be a defining function for this set, i.e., ρ : C n → R is a C 2 function so that Ω = {ρ < 0}, ∂Ω = {ρ = 0}, C n − Ω = {ρ > 0} and ∇ρ = 0 at the points of ∂Ω. For p ≥ 1, the Hardy space H p (Ω) is defined as follows:

H (Ω) = f : Ω → C, f holomorphic in Ω so that p



|f (z)|

f p,ρ := sup ε>0

1/p

 p

dσερ (z)

 <∞ ,

{ρ=−ε}

where dσερ is the Euclidean surface area measure of the hypersurface {z ∈ C n : ρ(z) = −ε} (with ε > 0 and sufficiently small). Then H p (Ω) is independent of the defining function ρ. In fact if λ is another defining function for Ω, the norms f p,p and f p,λ are equivalent. This follows from the proof of [11, Lemma 3]. Let us also observe that for compact subsets K of Ω, sup |f (z)| ≤ A(K, ρ)f p,p , f ∈ H p (Ω),

(1)

z∈K

for some constant A(K, ρ). To prove this inequality we may use the representation  f (z) = f (ζ)Pε (ζ, z)dσερ (ζ), z ∈ K, ∂Ωε

where Pε (ζ, z) is the Poisson kernel of Ωε := {ρ < −ε} and ε > 0 and sufficiently small. (Once chosen, ε is fixed.) Since Pε (ζ, z)  [dist(z, ∂Ωε )]−2n , Hölder’s inequality gives   |f (z)| ≤ ζ∈∂Ωε



1/p   |f (ζ)| dσε (ζ) p

1/p˜ |Pε (ζ, z)| dσε (ζ)

ζ∈∂Ωε

f p,ρ f p,ρ  , [dist(z, ∂Ωε )]2n [dist(z, ∂Ω)]2n

p˜

K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697

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(with the point z restricted to the compact set K) and the inequality (1) follows, if p > 1. (For details concerning the Poisson kernel, see [5,11].) The case p = 1 is simpler. Furthermore, H p (Ω) becomes a Banach space with the norm f p,ρ . This follows from (1) as in the case of the unit ball. (See [13, Corollary 4.19].) From the same inequality also follows the fact that convergence in H p (Ω) implies uniform convergence on compact subsets of Ω.  As in the case of the unit ball, we define a metric in the space H p (Ω), for a fixed q > 1, as follows. 1≤p
We consider a sequence 1 < p1 < p2 < · · · < pj < · · · < q and pj → q, and we define the metric d(f, g) =

∞  1 f − gpj ,ρ . j 1 + f − g 2 pj ,ρ j=1



H p (Ω) does not depend on the choice of the 1≤p
1≤p
if and only if fk − f p,ρ → 0, for every p < q. 3.1. Local Hardy spaces

With Ω and ρ being as above, we consider an open set U ⊂ C n with U ∩ ∂Ω = ∅ and we define the space  to be the set of holomorphic functions f : Ω → C so that sup |f (z)|p dσερ (z) < ∞. The

Hρp (Ω, U )

ε>0 {ρ=−ε}∩U

space Hρp (Ω, U ) may depend on ρ. However, we have the following lemma. Lemma 3.1. Let ρ and λ be two defining functions for Ω. If U and V are two open subsets of C n with U ∩ ∂Ω = ∅ and V ∩ ∂Ω = ∅, and if V ⊂⊂ U then Hρp (Ω, U ) ⊂ Hλp (Ω, V ). Proof. The following proof is essentially the proof of [11, Lemma 3] with some minor modifications. There exist positive constants κ, κ1 and κ2 (independent of ε) so that if z ∈ {λ = −ε} (i.e. λ(z) = −ε) then B(z, κε) ⊂ Λε := {w ∈ C n : −κ1 ε < ρ(w) < −κ2 ε}. (The positive parameter ε is assumed to be sufficiently small so that the various assertions in this proof hold true.) By the submean value property, if f ∈ Hρp (Ω, U ), κ3 |f (z)| ≤ 2n ε

 χε (z, w)|f (w)|p dw for z ∈ {λ = −ε},

p

w∈C n

where χε (z, w) = 1 for w ∈ B(z, κε) and χε (z, w) = 0 for w ∈ C n −B(z, κε). In what follows, κj , j = 3, 4, 5, 6, are appropriate positive constants independent of ε. Then  |f (z)|p dσελ (z) ≤ {λ=−ε}∩V

κ3 ε2n

 w∈C n





 χε (z, w)dσελ (z) |f (w)|p dw,

{λ=−ε}∩V

where we used Fubini’s theorem and the measurability of the function χε(z, w) for (z, w) ∈ {λ = −ε} × C n with respect to the product measure dσελ (z) × dw. Since V ⊂⊂ U , making ε smaller – if necessary – we may assume that

K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697

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B(z, κε) ⊂ Λε ∩ U for z ∈ {λ = −ε} ∩ V. Then  χε (z, w)dσελ (z) = 0 if w ∈ C n − (Λε ∩ U ) and {λ=−ε}∩V



χε (z, w)dσελ (z) ≤ κ4 ε2n−1 for w ∈ Λε ∩ U. {λ=−ε}∩V

It follows that  |f (z)|p dσελ (z) ≤ {λ=−ε}∩V

κ5 ε

κ6 ≤ ε

 |f (w)|p dw w∈Λε ∩U

κ1 ε κ2 ε



 |f (w)|p dσηρ (w) dη.

{ρ=−η}∩U

(The existence of the constant κ6 follows from the coarea formula (see [2, Theorem 3.13].) Thus  sup ε>0 {λ=−ε}∩V

 |f (z)|p dσελ (z) ≤ κ6 (κ1 − κ2 ) sup

η>0 {ρ=−η}∩U

|f (z)|p dσηρ (z),

and this implies that f ∈ Hλp (Ω, V ).  4. The case of strictly pseudoconvex domains In this section we will show that some functions which are defined in terms of Henkin’s support function belong to certain Hardy spaces. First we describe Henkin’s support function Φ(z, ζ) which is constructed in [4]. Following Henkin and Leiterer [4], let us consider an open set Θ ⊂⊂ C n and a C 2 strictly plurisubharmonic function ρ in a neighbourhood of Θ. If we set 1 β = min 3



∂ 2 ρ(ζ) ξj ξ k : ζ ∈ Θ, ξ ∈ C n with |ξ| = 1 ∂ζ ∂ζ j k 1≤j,k≤n



then β > 0 and there exist C 1 functions ajk in a neighbourhood of Θ such that

 ∂ 2 ρ(ζ) β max ajk (ζ) − : ζ ∈ Θ < 2. ∂ζj ∂ζk n (See [4, Lemma 2.4.2].) Let η > 0 be sufficiently small so that

2  ∂ ρ(ζ) ∂ 2 ρ(z) β max − : ζ, z ∈ Θ with |ζ − z| ≤ η < 2 for j, k = 1, 2, . . . , 2n, ∂xj ∂xk ∂xj ∂xk 2n where xj = xj (ξ) are the real coordinates of ξ ∈ C n such that ξj = xj (ξ) + ixj+n (ξ). For z, ζ ∈ Θ we consider the modified Levi polynomial

K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697

10

 n ∂ρ(ζ) Q(z, ζ) = − 2 (zj − ζj ) + ∂ζj j=1



 ajk (zj − ζj )(zk − ζk ) .

1≤j,k≤n

Then we have the estimate ReQ(z, ζ) ≥ ρ(ζ) − ρ(z) + β|ζ − z|2 for z, ζ ∈ Θ with |ζ − z| ≤ η. (See [4, Lemma 1.4.13].) The following theorem is proved in [4, Theorem 2.4.3]. Theorem 4.1. Let Ω ⊂⊂ C n be a strictly pseudoconvex open set with C 2 boundary, let Θ be an open neighbourhood of ∂Ω, and let ρ be a C 2 strictly plurisubharmonic function in a neighbourhood of Θ such that ∇ρ = 0 at the points of ∂Ω and Ω ∩ Θ = {z ∈ Θ : ρ(z) < 0}. Let us choose η, β, and Q(z, ζ), as above, and let us make the positive number η smaller so that {z ∈ C n : |ζ − z| ≤ 2η} ⊆ Θ for every ζ ∈ ∂Ω. Then there exists a function Φ(z, ζ) defined for ζ in some open neighbourhood U∂Ω ⊆ Θ of ∂Ω and z ∈ UΩ = Ω ∪ U∂Ω , which is C 1 in (z, ζ) ∈ UΩ × U∂Ω , holomorphic in z ∈ UΩ , and such that Φ(z, ζ) = 0 for (z, ζ) ∈ UΩ × U∂Ω with |ζ − z| ≥ η, and Φ(z, ζ) = Q(z, ζ)C(z, ζ) for (z, ζ) ∈ UΩ × U∂Ω with |ζ − z| ≤ η, for some C 1 -function C(z, ζ) defined for (z, ζ) ∈ UΩ × U∂Ω and = 0 when |ζ − z| ≤ η. In this setting we will prove the following lemma. We use a set of coordinates – the Levi coordinates – which are appropriate when we are dealing with integrals involving the function Φ(z, ζ). (See [4,8,12].) As a matter of fact we will use a slight modification of the Levi coordinates. Lemma 4.2. For each fixed point ζ ∈ ∂Ω,  sup ε>0 {ρ(z)=−ε}

dσε (z) < ∞ when 1 < p < n, |Φ(z, ζ)|p

(2)

and  sup ε>0 {ρ(z)=−ε}

Therefore,

 1 ∈ H p (Ω) and Φ(·, ζ) 1≤p
1 Φ(·,ζ)

dσε (z) = ∞. |Φ(z, ζ)|2n−1

(3)

∈ / H 2n−1 (Ω).

Proof. We consider a coordinate system t = (t1 , t2 , t3 , . . . , t2n ) = (t1 (z), t2 (z), t3 (z), . . . , t2n (z)), of real C 1 -functions, for points z ∈ C n = R2n , which are sufficiently close to the point ζ, as follows: We set t1 (z) = −ρ(z) and t2 (z) = ImQ(z, ζ).

K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697 n  Then dz Q(z, ζ) z=ζ = −2

j=1



∂ρ(ζ) ∂ζj dzj

11

= −2∂ρ(ζ) and, therefore, z=ζ

dz t2 (z) z=ζ = dz [ImQ(z, ζ)] z=ζ = i[∂ρ(ζ) − ∂ρ(ζ)]. On the other hand, dz t1 (z) z=ζ = dz [−ρ(z)] z=ζ = −[∂ρ(ζ) + ∂ρ(ζ)]. It follows that     dz t1 (z) z=ζ ∧ dz t2 (z) z=ζ = 2i∂ρ(ζ) ∧ ∂ρ(ζ) = 0. Now the existence of C 1 -functions t3 (z), . . . , t2n (z) such that the mapping z → (t1 (z), t2 (z), t3 (z), . . . , t2n (z)) is a C 1 -diffeomorphism, from an open neighbourhood of the point ζ to an open neighbourhood of 0 ∈ C n = R2n (with t(ζ) = 0), follows from the inverse function theorem. Also let us point out that, for z sufficiently close to ζ, z ∈ Ω if and only if t1 = −ρ(z) > 0. For points z ∈ Ω which are sufficiently close to ζ, |Φ(z, ζ)| ≈ |Q(z, ζ)| ≈ |ReQ(z, ζ)| + |ImQ(z, ζ)| ≥ −ρ(z) + β|ζ − z|2 + |ImQ(z, ζ)| and |ζ − z|2 ≈ t21 + t22 + t23 + · · · + t22n . (When we write A ≈ B, we mean that κB ≤ A ≤ μB, for some positive constants κ and μ which are independent of z.) Therefore (for z ∈ Ω and sufficiently close to ζ) |Φ(z, ζ)|  t1 + t21 + t22 + t2 + 3 + · · · + t22n + |t2 |. Therefore (2) follows from  sup ε>0 {t1 =ε,t22 +···+t22n <1}

dt2 · · · dt2n <∞ (t1 + |t2 | + t21 + t22 + t23 + · · · + t22n )p

or equivalently from  sup ε>0 {t1 =ε,t22 +···+t22n <1}

dt2 · · · dt2n < ∞. (t1 + |t2 | + t23 + · · · + t22n )p

But  {t1 =ε,t22 +···+t22n <1}



{t22 +t23 +···+t22n <1}

dt2 · · · dt2n ≈ (t1 + |t2 | + t23 + · · · + t22n )p dt2 · · · dt2n . (ε + |t2 | + t23 + · · · + t22n )p

(4)

K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697

12

Also, by Fubini’s theorem,  {t22 +t23 +···+t22n <1}

dt2 · · · dt2n (ε + |t2 | + t23 + · · · + t22n )p  1

 ≈ {t23 +···+t22n <1}

 ≈ {t23 +···+t22n <1}

t2 =0

 dt2 dt3 · · · dt2n (ε + t2 + t23 + · · · + t22n )p

dt3 · · · dt2n . (ε + t23 + · · · + t22n )p−1

Integrating in polar coordinates we see that the last integral is equal to 1

1

r2n−3 dr ≤ (ε + r2 )p−1

r=0

r2n−2p−1 dr < ∞. r=0

This proves (4) and completes the proof of (2). In order to prove (3), let us observe that for points z ∈ Ω which are sufficiently close to ζ, |Φ(z, ζ)| ≈ |Q(z, ζ)|  |ζ − z| ≈ (t21 + t22 + t23 + · · · + t22n )1/2 , whence  {ρ(z)=−ε}

dσε (z) ≈ |Φ(z, ζ)|2n−1

 {t1 =ε,t22 +···+t22n <1}

  {t22 +t23 +···+t22n <1}

dt2 · · · dt2n (t21 + t22 + · · · + t22n )(2n−1)/2

dt2 · · · dt2n . (ε2 + t22 + · · · + t22n )(2n−1)/2

By introducing polar coordinates in the last integral, we see that this integral behaves as 1

r2n−2 dr ≈ (ε2 + r2 )(2n−1)/2

r=0

1

r2n−2 dr . (ε + r)2n−1

r=0

But as ε decreases, the function r2n−2 /(ε + r)2n−1 increases, and the monotone convergence theorem gives that 1 lim +

ε→0

r=0

r2n−2 dr = (ε + r)2n−1

1

dr = ∞, r

r=0

and proves (3).  Lemma 4.3. Let Ω ⊂⊂ C n be a strictly pseudoconvex open set with C 2 boundary and let ρ be a C 2 strictly plurisubharmonic defining function of Ω defined in a neighbourhood of Ω. If 1 < q < ∞ and U ⊂ C n with U ∩ ∂Ω = ∅, then there exists a function hq,U so that  hq,U ∈ H p (Ω) and hq,U ∈ / Hρ(2n−1)q/n (Ω, U ). 1≤p
K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697

13

Proof. Let us fix a point ζ ∈ U ∩ ∂Ω. Then, as it follows from Taylor’s theorem and the strict plurisubharmonicity of ρ (see [8]), the Levi polynomial of ρ

 n ∂ρ(ζ) F (z, ζ) = − 2 (zj − ζj ) + ∂ζj j=1

 1≤j,k≤n

 ∂ 2 ρ(ζ) (zj − ζj )(zk − ζk ) ∂ζj ∂ζk

satisfies the inequality ReF (z, ζ) ≥ ρ(ζ) − ρ(z) + γ|ζ − z|2 for z ∈ C n with |ζ − z| < δ, for some positive constants δ and γ. In particular, ReF (z, ζ) > 0 for z ∈ B(ζ, δ) ∩ Ω − {ζ}. It follows that the function log[1/F (z, ζ)] is defined and holomorphic for z ∈ B(ζ, δ) ∩ Ω, and that lim log[1/F (z, ζ)] = ∞. (Here log is the principal branch of the logarithm with | arg | ≤ π.) Also z∈Ω,z → ζ

we can prove, as in the proof of the Lemma 4.2, that if q < n, 

dσε (z) <∞ |F (z, ζ)|q

(5)

dσε (z) = ∞. |F (z, ζ)2n−1

(6)

sup ε>0 {ρ(z)=−ε}∩B(ζ,δ)

and  sup ε>0 {ρ(z)=−ε}∩B(ζ,δ)

Then, using (5), we obtain, as in Lemma 2.2,  sup ε>0 {ρ(z)=−ε}∩B(ζ,2δ/3)

 p

1 dσε (z) < ∞, for every p < ∞. log F (z, ζ)

(7)

Next we consider a C ∞ -function χ : C n → R, 0 ≤ χ(z) ≤ 1, with compact support contained in B(ζ, 2δ/3), and such that χ(z) = 1 when z ∈ B(ζ, δ/3). Now the function

χ(z) log

1 F (z, ζ)



is extended to a C ∞ -function in Ω, by defining it to be 0 in Ω − B(ζ, 2δ/3). Then the (0, 1)-form



1 u(z) := ∂ χ(z) log F (z, ζ)



is defined and is C ∞ in an open neighbourhood Ω, it is zero for z ∈ B(ζ, δ/3) ∩ Ω, and, in particular, it has bounded coefficients in Ω. In fact u(z) extends to a C ∞ (0, 1)-form for z in an open neighbour 1  hood of Ω, since the function log is holomorphic in an open neighbourhood of the compact set F (z, ζ) [B(ζ, 2δ/3) − B(ζ, δ/3)] ∩ Ω. It follows that there exists a bounded C ∞ -function ψ : Ω → C which solves the equation ∂ψ = u in Ω. (See [9].) Then we may define the functions

K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697

14

 1 − ψ(z) fζ (z) := χ(z) log F (z, ζ)

and



n/q  n 1 n hq,ζ (z) := exp fζ (x) = exp χ(z) log − ψ(z) . q F (z, ζ) q Then the functions fζ (z) and hq,ζ (z) are holomorphic for z ∈ Ω. We claim that  sup ε>0 {ρ(z)=−ε}∩B(ζ,δ)

|hq,ζ (z)|p dσερ (z) < ∞ for p < q.

(8)

Notice that the behaviour of the above integral is not affected by the functions χ or ψ, since χ ≡ 1 near ζ and ψ is bounded in Ω. Thus (8) follows from (7). Also  sup |hq,ζ (z)|(2n−1)q/n dσερ (z) = ∞. ε>0 {ρ(z)=−ε}∩B(ζ,δ)

Indeed, this follows from (6), since χ ≡ 1 near ζ and exp(−ψ) is bounded away from zero in Ω. Thus, setting hq,U := hq,ζ we obtain the required function.  Remark 4.4. The function fζ (z) which was constructed in the proof of the previous lemma has the following properties: 

fζ ∈

H p (Ω) and

1≤p<∞

lim

z∈Ω,z → ζ

fζ (z) = ∞.

(The first part follows from (7).) Theorem 4.5. Let Ω ⊂⊂ C n be a strictly pseudoconvex open set with C 2 boundary and q ∈ R ∪ {∞}, q > 1. Then the following hold: 

(i) The set of the functions in the space

H p (Ω) which are totally unbounded in Ω is dense and Gδ in

1≤p
this space. (ii) The set of the functions in the space



H p (Ω) which are singular at every boundary point of Ω is

1≤p
dense and Gδ in this space.

Proof. Let us consider a “small” ball B whose center lies on ∂Ω, and let us set X = B ∩ Ω and V =  H p (Ω). We define the linear operator 1≤p
T : V → C X with T (f )(z) = f (z) for z ∈ X and f ∈ V. For each fixed z ∈ X, the functional Tz : V → C defined by Tz (f ) = f (z), f ∈ V, is continuous. It is easy to see that the set E = {f ∈ V : T (f ) is unbounded on X} in this case is equal to

E(B) =

f∈

 1≤p
 H (Ω) : sup |f (z)| = ∞ . p

z∈B∩Ω

K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697

15

Now we consider the function fζ which was constructed in the proof of Lemma 4.3. If the point ζ ∈ B ∩ ∂Ω then fζ ∈ E(B), and therefore E(B) = ∅. (See also Remark 4.4.) Therefore, by Lemma 2.3, E(B) is dense and Gδ set in the space V. In order to complete the proof, we consider a countable dense subset {w1 , w2 , w3 , . . .} of ∂Ω, a decreasing sequence εs , s = 1, 2, 3, . . ., of positive numbers with εs → 0, and the balls B(wj , εs ), centered at wj and with radii εs . By the first part of the proof, each of the sets E(B(wj , εs )) is dense and  Gδ set in H p (Ω). It follows from Baire’s theorem that the set 1≤p
Y=

∞ ∞  

E(B(wj , εs )) is dense and Gδ in the space

j=1 s=1



H p (Ω).

1≤p
 But Y is the set of the functions in the space H p (Ω) which are totally unbounded in Ω. Indeed, if 1≤p
1≤p
z∈B(wj ,εs )∩Ω

E(B(wj , εs )). Conversely, if B is a ball whose center lies on ∂Ω, then there exist j0 , s0 , so that

j=1 s=1

B(wj0 , εs0 ) ⊂ B. Thus, if f ∈

∞ ∞  

E(B(wj , εs )) then

j=1 s=1

sup

|f (z)| = ∞, whence

z∈B(wj0 ,εs0 )∩Ω

sup |f (z)| =

z∈B∩Ω

∞. This proves (i). Finally, the assertion (ii) follows from (i) and [6, Theorem 3.3], since a totally unbounded function in Ω is clearly singular (in Ω).  Theorem 4.6. Let Ω ⊂⊂ C n be a strictly pseudoconvex open set with C 2 boundary and let ρ be a C 2 strictly plurisubharmonic defining function of Ω defined in a neighbourhood of Ω. If q ∈ R, q > 1, then the set

  ρ p (2n−1)q/n Bq = g ∈ H (Ω) : g ∈ / Hρ (Ω, U ) for any open set U with U ∩ ∂Ω = ∅ 1≤p
is dense and Gδ in the space



H p (Ω).

1≤p
Proof. Let us fix a point w ∈ ∂Ω and a positive number δ. For fixed ε0 > 0 set X = {ε : 0 < ε < ε0 } and  consider the sublinear operator T : V → C X defined on V = H p (Ω) as follows: 1≤p




T (f )(ε) =

n  (2n−1)q |f (z)|(2n−1)q/n dσερ (z) for f ∈ V and ε ∈ X.

{ρ=−ε}∩B(w,δ)

Then, for each fixed ε ∈ X, the functional Tε : V → C, Tε (f ) = T (f )(ε), f ∈ V, is continuous. Also, by Lemma 4.3, the set E(w, δ) := {f ∈ V : sup{T (f )(ε) : ε ∈ X} = ∞} = ∅. Therefore, by Lemma 2.3, the set E(w, δ) is dense and Gδ in the space V. In order to complete the proof, we consider a countable dense subset {w1 , w2 , w3 , . . .} of ∂Ω and a decreasing sequence δs , s = 1, 2, 3, . . ., of positive numbers with δs → 0. By the first part of the proof and Baire’s theorem, the set Y=

∞  ∞ 

E(wj , δs ) is dense and Gδ in the space

j=1 s=1

Now it is easy to see that Y = Bqρ , and this completes the proof. 

 1≤p
H p (Ω).

K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697

16

Combining Theorem 4.6 with Lemma 3.1, we will see that the set Bqρ is independent of ρ. Thus we have the following theorem. Theorem 4.7. Let Ω ⊂⊂ C n be a strictly pseudoconvex open set with C 2 boundary. If q ∈ R, q > 1, then the set Bq = {g ∈



(2n−1)q/n

H p (Ω) : g ∈ / Hλ

(Ω, U ) for any open set U with U ∩ ∂Ω = ∅

1≤p
and any defining function λ of Ω} is dense and Gδ in the space



H p (Ω).

1≤p
Proof. It is clear that Bq ⊂ Bqρ . Conversely, if g ∈ Bqρ , U is any open set with U ∩ ∂Ω = ∅ and λ is any defining function of Ω, let us consider an open set V with V ∩ ∂Ω = ∅ and V ⊂⊂ U . By Lemma 3.1, (2n−1)q/n (2n−1)q/n (2n−1)q/n Hλ (Ω, U ) ⊂ Hρ (Ω, V ). But g ∈ Bqρ implies that g ∈ / Hρ (Ω, V ), and therefore (2n−1)q/n

g∈ / Hλ

(Ω, U ). It follows that g ∈ Bq . Thus Bq = Bqρ .



It is easy to see that one can obtain results analogous to the ones ot Theorems 2.1, 2.4, 4.5, 4.7, with the  p spaces H p in place of the intersections H . Thus, we have the following theorem. p
Theorem 4.8. (i) For 1 ≤ p < ∞, the set of the functions in the space H p (B) which are totally unbounded in B is dense and Gδ in this space. (ii) For 1 ≤ p < q < ∞, the set {g ∈ H p (B) : g ∈ / H q (B, S ∩ B(ζ, ε)) for any ζ ∈ S and any ε > 0} is dense and Gδ in the space H p (B). (iii) If Ω ⊂⊂ C n is a strictly pseudoconvex open set with C 2 boundary and 1 ≤ p < ∞, the set of the functions in the space H p (Ω) which are totally unbounded in Ω is dense and Gδ in this space. (iv) If Ω ⊂⊂ C n is a strictly pseudoconvex open set with C 2 boundary, 1 ≤ p < ∞ and q > (2n − 1)p/n, then the set {g ∈ H p (Ω) : g ∈ / Hλq (Ω, U ) for any open set U with U ∩ ∂Ω = ∅ and any defining function λ of Ω} is dense and Gδ in the space H p (Ω). 5. Hardy spaces of harmonic functions The results of the previous sections can be extended to the case of harmonic functions in domains of Rn . To describe this extension, let us consider a bounded open set Ω ⊂ Rn with C 2 boundary. If ρ is a C 2 defining function of Ω then one can define the harmonic Hardy spaces hp (Ω), p ≥ 1 (see [1,11]),  p the intersections h (Ω), and the local Hardy spaces hpρ (Ω, U ), as before. (U ⊂ Rn is an open set with U ∩ ∂Ω = ∅.)

p
Lemma 5.1. Let n ≥ 3 and y ∈ ∂Ω. Then the function ϕy (x) = only if p <

n−1 / h(n−1)/(n−2) (Ω). . In particular ϕy ∈ n−2

1 (x = y) belongs to hp (Ω) if and |x − y|n−2

K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697

17

Proof. We may assume that y = 0 ∈ ∂Ω, in which case ϕy becomes the function ϕ0 (x) =

1 1 = 2 . 2 n−2 |x| (x1 + x2 + · · · + x2n )(n−2)/2

We must show that  sup ε>0 {ρ=−ε}

n−1 dσερ (x) < ∞ if and only if p < . n−2 |x|(n−2)p

(9)

Using a local diffeomorphism – near the point 0 of ∂Ω – we may assume that the hypersurface ∂Ω, near 0, is defined by the equation x1 = 0, and that x1 > 0 for x ∈ Ω (close to 0). Then (9) is equivalent to  sup ε>0 x22 +···+x2n <1

dx2 . . . dxn n−1 < ∞ if and only if p < . n−2 (ε2 + x22 + · · · + x2n )(n−2)p/2

(10)

Integrating in polar coordinates we see that the above integral behaves as 1 (ε2

rn−2 dr . + r2 )(n−2)p/2

r=0

By monotone convergence theorem, 1 sup ε>0 r=0

rn−2 dr = lim + 2 ε→0 (ε + r2 )(n−2)p/2

1

1

rn−2 dr ≤ 2 (ε + r2 )(n−2)p/2

r=0

rn−2 dr , r(n−2)p

r=0

and (10) follows.  (n−1)/(n−2)

Lemma 5.2. Let n ≥ 3 and y ∈ ∂Ω. Then ϕy ∈ / hρ

(Ω, U ) for y ∈ U .

Proof. It follows easily from the previous lemma.  With the above lemmas, we can prove the following theorems. Their proofs are similar to the proofs of Theorems 4.5 and 4.7. n−1 . Then the set of the functions in the space n−2 unbounded in Ω is dense and Gδ in this space. Theorem 5.3. Let 1 < q ≤

Theorem 5.4. Let 1 < q ≤ Aq = {g ∈



hp (Ω) which are totally

1≤p
n−1 . Then the set n−2



(n−1)/(n−2)

1≤p
hp (Ω) : g ∈ / hλ (Ω, U )for any open set U with U ∩ ∂Ω = ∅ and any defining function λ of Ω}

is dense and Gδ in the space



hp (Ω).

1≤p
Remark 5.5. According to Theorem 5.3, the functions in the space



hp (Ω) are generically totally

1≤p
unbounded in Ω, despite the fact that all these functions have non-tangential limits almost everywhere at

K. Kioulafa / J. Math. Anal. Appl. 484 (2020) 123697

18

the points of the boundary of Ω (by Fatou’s theorem). Similar remarks can be made for Theorems 2.1 and 4.5. Acknowledgments I would like to thank T. Hatziafratis and V. Nestoridis for helpful discussions. References [1] S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory, Springer, 2001. [2] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 2015. [3] T. Hatziafratis, K. Kioulafa, V. Nestoridis, On Bergman type spaces of holomorphic functions and the density, in these spaces, of certain classes of singular functions, Complex Var. Elliptic Equ. 63 (7–8) (2018) 1011–1032. [4] G.M. Henkin, J. Leiterer, Theory of Functions on Complex Manifolds, Birkhäuser, 1984. [5] S.G. Krantz, Function Theory of Several Complex Variables, 2nd ed., AMS Chelsea Publishing, 2000. [6] V. Nestoridis, Domains of holomorphy, Ann. Math. Québec 42 (1) (2018) 101–105. [7] V. Nestoridis, A.G. Siskakis, A. Stavrianidi, S. Vlachos, Generic non-extendability and total unboundedness in function spaces, J. Math. Anal. Appl. 475 (2) (2019) 1720–1731. [8] R.M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer, 1986. [9] W. Rudin, Function Theory in the Unit Ball of C n , Springer, 1980. [10] M. Siskaki, Boundedness of derivatives and anti-derivatives of holomorphic functions as a rare phenomenon, J. Math. Anal. Appl. 426 (2) (2018) 1073–1086. [11] E.M. Stein, Boundary Behavior Holomorphic Functions of Several Complex Variables, Princeton University Press, 1972. [12] E.L. Stout, H p -functions on strictly pseudoconvex domains, Amer. J. Math. 98 (1976) 821–852. [13] K.H. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer, 2005.