Rigidity of Volterra-type integral operators on Hardy spaces of the unit ball

We establish that the Volterra-type integral operator $J_b$ on the Hardy spaces $H^p$ of the unit ball $\mathbb{B}_n$ exhibits a rather strong rigid behavior. More precisely, we show that the compactness, strict singularity and $\ell^p$-singularity of $J_b$ are equivalent on $H^p$ for any $1 \le p<\infty$. Moreover, we show that the operator $J_b$ acting on $H^p$ cannot fix an isomorphic copy of $\ell^2$ when $p \ne 2.$


Introduction
An operator T : X → Y between Banach spaces X and Y is strictly singular if its restriction to any infinite-dimensional subspace M of X is not a linear isomorphism onto its range, i.e. the restriction is not bounded below on M.This class of operators forms a two-sided operator ideal and was introduced by T. Kato [6] in connection with the perturbation theory of Fredholm operators.If T is not bounded below on any subspace M isomorphic to the sequence space ℓ p , then T is said to be ℓ p -singular.These notions generalize the concept of compact operators.Examples of strictly singular non-compact operators are the inclusion mappings i p,q : ℓ p ֒→ ℓ q , where 1 ≤ p < q < ∞.The following inclusions hold: K(X) ⊂ S(X) ⊂ S p (X), where K(X) is the class of compact operators on X, S(X) the class of strictly singular operators and S p (X) the class of ℓ p −singular operators on X.In general, these classes are distinct, but coincide e.g. in the case of X being a Hilbert space, see [12,Chapter 5].
The purpose of this paper is to study the strict singularity of the Volterra-type integration operator J b acting on the Hardy spaces of the unit ball B n , extending the results previously obtained in [8,9] for the case of the unit disk D. For a holomorphic function b on B n , the operator J b is defined as for f holomorphic on B n .Here Rb denotes the radial derivative of b, that is, It is well-known [3,4,11] that J b is bounded on the Hardy space H p (B n ) if and only if b ∈ BM OA(B n ), the space of all holomorphic functions of bounded mean oscillation; and J b is compact on H p (B n ) if and only if b ∈ V M OA(B n ), the space of holomorphic functions on B n of vanishing mean oscillation.For 0 < p < ∞, the Hardy space H p := H p (B n ) consists of those holomorphic functions f on B n with where dσ is the surface measure on the unit sphere S n := ∂B n normalized so that σ(S n ) = 1.
The mentioned operator J b became extremely popular in recent years, being studied in many spaces of holomorphic functions (see [2,10,11,14] and the references therein).As far as we know, the generalization of the operator J b acting on holomorphic function spaces of the unit ball of C n , as defined here, was introduced by Z. Hu [5].A fundamental property of the Volterra integration operator J b is the basic identity The existence of non-compact strictly singular operators acting on the Hardy space H p (D) for p = 2 can be seen by considering the inclusion mappings between the sequence spaces ℓ p , p = 2, and ℓ 2 and utilizing the fact that H p (D) contains complemented copies of ℓ p and ℓ 2 .The existence of such operators can be transferred to the case of the Hardy spaces H p (B n ), 1 ≤ p < ∞, since they are all isomorphic to H p (D) by the result of Wojtaszczyk [15].We recall that, for a Banach space X, a bounded linear operator T : X → X is said to fix a copy of a given Banach space E, if there is a closed subspace M ⊂ X, linearly isomorphic to E, and c > 0 so that T x ≥ c x for all x ∈ M (that is, the restriction T |M defines an isomorphism onto its range).Our first result, proved in the case of the unit disk in [8], shows that J b is compact on H p if and only if it is strictly singular; as when it is not compact it fixes an isomorphic copy of ℓ p inside H p .
Hence the notions of compactness, strict singularity and ℓ p -singularity coincide in the case of J b acting on H p .
Theorem 1.1 is established in a similar manner as in the one-dimensional case, by constructing bounded operators V : ℓ p → H p and U : ℓ p → H p such that U = J b V , where V (ℓ p ) = M is the closed linear span of suitably chosen test functions f a k ∈ H p and the operator U is an isomorphism onto its range U (ℓ p ) = J b (M ).
Our second main result extends the one obtained in [9] to the setting of the unit ball.The proof requires different techniques, as some of the tools utilized in [9] are not available or useful in higher dimensions such as the Riemann mapping theorem.We utilize different equivalent norms and Carleson measures among other techniques.It is interesting to contrast this result with the one obtained in [7] for composition operators acting on the Hardy spaces of the unit disk, since composition operators do not exhibit as rigid behavior in regard to ℓ 2 -singularity as the operators J b .
If there exists a closed infinitedimensional subspace M ⊂ H p such that J b : H p → H p is bounded below on M, then there exists a subspace N ⊂ M isomorphic to ℓ p .In particular, the operator J b acting on H p cannot fix an isomorphic copy of ℓ 2 , i.e. it is ℓ 2 -singular when p = 2.
We use some standard notation.For any two points z = (z 1 , . . ., z n ) and w = (w 1 , . . ., w n ) in C n , we write z, w = z 1 w1 + • • • + z n wn , and |z| = z, z .Typically constants are used with no attempt to calculate their exact values.Given two positive quantities A and B, depending on some parameters, we write A B to mean that there exists some inessential constant C > 0 so that A ≤ CB.The relation A B is defined in an analogous way, and A ≍ B means that both A B and A B hold.
The paper is organized as follows.In Section 2, we provide an auxiliary result needed to establish Theorem 1.1 in Section 3; and Section 4 covers our second main result Theorem 1.2 for which Lemmas 4.1-4.5 are crucial tools.

Preliminaries
It is well known that any function in H p has radial limits f (ζ) = lim r→1 − f (rζ) for a.e.ζ ∈ S n , and f p H p = S n |f | p dσ.For each a ∈ B n , consider the test function It is easy to see that f a ∈ H p with f a H p ≍ 1, and its radial derivative is given by We need the following lemma regarding their values and the values of J b f a peaking in certain subsets of S n .
Lemma 2.1.Let b ∈ BMOA(B n ), 1 ≤ p < ∞, and (a k ) ⊂ B n be a sequence such that a k → ω ∈ S n .Define a non-isotropic metric ball for k large enough.So we may choose δ = δ(ε) > 0 so that |1− u 0 , a k | ≥ δ for all k large enough and all u 0 ∈ S n \ S ε (ω) and the condition (i) follows.The proof of (ii) follows from the absolute continuity of the measures A → A |f a k | p dσ.

Thus it holds that
|1 − ru 0 , a k | ≥ ε/2(1/2 − ε) for all k large enough and all 1 − ε 2 < r ≤ 1.So we may choose δ = δ(ε) > 0 so that |1 − ru 0 , a k | ≥ δ for all k large enough and all u 0 ∈ S n \ S ε (ω) and 0 ≤ r ≤ 1.For those u 0 and r we obtain the estimates and for all k large enough, where C = C(n, p) > 0. Now, for a.e.ζ ∈ S n \ S ε (ω), we obtain where constants may depend on n and p. Utilizing the well known pointwise estimate , where and b BM OA is the BMOA-seminorm, we have

ℓ p −singularity of J b
In this section, we establish the fact that a non-compact integration operator J b acting on H p fixes a copy of ℓ p .We begin with an auxiliary result.
Then there exists a subsequence (b k ) of (a k ) such that the mapping V : ℓ p → H p defined as Proof.One just needs to follow the proof given in the one-dimensional case given in [8, Proposition 3.2] using our Lemma 2.1 as a replacement of Lemma 3.1 of [8].We left the details to the interested reader.
In particular, there exists a sequence Proof.We may assume b(0) = 0. We consider first the case p > 2 and utilize the representation [16, Chapter 5] where dv(z) is the normalized volume measure on B n .Now for f ∈ H p , by the estimates obtained in pages 144-145 in [11], we have and By replacing f with f a (note that f a H p ≍ 1), we have It is well known [16,Chapter 5] that a holomorphic function g belongs to V M OA(B n ) if and only if As a final step towards the proof of Theorem 1.1, we construct an isomorphism from ℓ p into H p using a non-compact J b and test functions.and (a k ) ⊂ B n be the sequence from Proposition 3.2.Then there exists a subsequence (b k ) of (a k ) such that the mapping U : Proof.With the use of Propositions 3.1, 3.2 and Lemma 2.1, we just need to follow the argument given in the one-dimensional case, see [8,Prop. 3.5].We omit the details.
Proof of Theorem 1.1.By Proposition 3.1 and Proposition 3.3, we can choose a sequence (b k ) ⊂ B n that induces a bounded operator V : ℓ p → H p , given by where α = (α k ) ∈ ℓ p , and an isomorphism U : ℓ p → H p , U = J b V onto its range.
Define M = span{f b k }, where the closure is taken in H p .Since U is bounded below, we have that the restriction J b | M is also bounded below.Thus J b | M : M → J b (M ) is an isomorphism and consequently M is isomorphic to ℓ p .In particular, the operator J b is not ℓ p -singular.

ℓ 2 -singularity of J b
In this section, we show that if J b : H p → H p is bounded below on a closed infinitedimensional subspace M of H p , then there exists a subspace N ⊂ M isomorphic to ℓ p .In particular, this implies that J b : H p → H p cannot fix an isomorphic copy of ℓ 2 whenever p = 2.
For ζ ∈ S n , the admissible approach region Γ(ζ) is defined as ) n , and it follows from Fubini's theorem that, for a finite positive measure ν, and a positive function ϕ, one has (4.1) For convenience, we define the measure µ b by where dv is the normalized volume measure on B n , and set dV α (z) = (1 − |z| It is also well known that, if µ is a vanishing Carleson measure then for any bounded sequence of functions {f k } ⊂ H p converging to zero uniformly on compact subsets of B n , 1 ≤ p < ∞.Next, we establish some preliminary results en route to the proof of Theorem 1.2.
Lemma 4.1.Let ε > 0 and b ∈ H 2 .Then there exists a compact set Proof.For each k ≥ 1, let ν k be the projection to S n of the measure µ b restricted to the annulus where The maximal function theorem [16,Chapter 4] implies that it satisfies Egorov's theorem now implies that there is a set F ⊂ S n with σ(S n \ F ) < ε/2 such that ν * k → 0 uniformly in F as k → ∞.Now for every k ≥ 1 and ζ ∈ F , we have where the first inequality follows from the fact that B ζ (δ) ⊂ S k for all 0 < δ < 1/k.Thereby we deduce that In order to see that µ b,ε is a vanishing Carleson measure, we need to prove that µ b,ε , and therefore It now follows that the measure µ b,ε is a vanishing Carleson measure.
For 1 ≤ p < ∞ and a sequence of functions {f k } ⊂ H p , given a subset A of S n , we consider the quantities and {f k } be a normalized sequence in H p , which converges to zero uniformly on compact subsets of B n .If p > 2, there exists a subsequence denoted still by {f k }, a decreasing sequence ε m > 0, ε m → 0, and compact sets for all m ≥ 1.In particular, by defining Ẽm = E m \ E m+1 , we have that Ẽm (j, k) δ 2 4 −j−k−m for k = m or j = m.
Proof.Since b ∈ BM OA(B n ), the operator J b is bounded on H p .Also, Lemma 4.1 implies that for any ε > 0, there exists a compact set is a vanishing Carleson measure.Note that, by (4.1) and Hölder's inequality with exponent p/2 > 1, The last estimate is due to Carleson-Hörmander theorem on Carleson measures.Therefore, by absolute continuity, for all fixed (k, m) ∈ N 2 , one has As a simple application of (4.1), for any positive measurable function ϕ, we have By repeating the calculation above and using this formula, we obtain For f ∈ H p , the admissible maximal function [16,Chapter 4]).Assume now that f, g ∈ H p are unit vectors, then .
Observe that both factors go to zero as σ(E ε ) → 0 due to the absolute continuity of the measure, as the boundedness of , by the version of Calderón area theorem for the unit ball [1,11], we have Hence, as ε → 0, we have We will choose a subsequence of (f k ), which we will also denote as (f k ), and ε 1 > ε 2 • • • > 0 in the following way.Assume that functions f 1 , . . ., f m−1 , numbers ε 1 > . . .> ε m−1 > 0 and compact sets K 1 ⊂ . . .⊂ K m−1 are chosen for some m ≥ 2. Then (4.5) together with (4.8) and (4.9) yields that there exists ε Also for the case max{j, k} > m, if j = max{j, k} > m, we may use Analogously, we have Ẽm (j, k) Next, we establish an analogous version of Lemma 4.2 in the case 1 ≤ p ≤ 2. Lemma 4.3.Let 1 ≤ p ≤ 2, b ∈ BM OA(B n ), 0 < δ < 1, and {f k } be a normalized sequence in H p , which converges to zero uniformly on compact subsets of B n .Then there exists a subsequence denoted still by {f k }, a decreasing sequence ε m > 0, ε m → 0, and compact sets and In particular, for k = m, where Ẽm = E m \ E m+1 .Also, we have Fm (m) 1.
Proof.As before, the operator J b is bounded on H p and for any ε > 0, there exists a compact set K ε ⊂ S n with σ(E ε ) < ε such that µ b,ε is a vanishing Carleson measure.By the version of Calderón's area theorem for the unit ball [1,11], In particular, Fm (k) 1.Therefore, by absolute continuity, for all k, one has (4.10) Now, using Hölder's inequality, the L p -boundedness of the admissible maximal function and (4.6), we get As µ b,ε is a vanishing Carleson measure, we obtain As in the proof of Lemma 4.2, we may use (4.10) and (4.11) inductively to find a subsequence denoted still by {f k }, a decreasing sequence ε m > 0, ε m → 0, and compact sets More precisely, we have for k > m.Therefore, it always holds that Fm (k) δ4 −k−m for k = m.
We now construct a bounded linear operator acting on H p in terms of the operator J b and a normalized sequence of functions converging uniformly to zero on compact subsets of B n .and (f k ) ⊂ H p be such that f k H p = 1 for all k and (f k ) converges to zero uniformly on compact subsets of B n .Then there exists a subsequence (f n k ) of (f k ) such that the linear mapping U : ℓ p → H p , defined as Proof.We divide our proof into three cases depending on the value of p, namely cases 1 ≤ p ≤ 2, 2 < p ≤ 3 and 3 < p < ∞.This results from the use of norms, which are equivalent to the standard H p norm and the choice of a particular norm depends on the value of p.For 0 < δ < 1, we choose a subsequence of {f k } denoted still by {f k }, a decreasing sequence ε m > 0, ε m → 0, and compact sets K m from Lemmas 4.2 and 4.3.Set Let us first look at the case 1 ≤ p ≤ 2. Set α 0 = 0.By the version of the area theorem for the unit ball [1,11], we have to the assumption is a metric and hence Using the triangle inequalities in L 2 and L p respectively, one has Let us then consider the case p > 2. Since K 1 ⊂ K k for any k ≥ 1, we have Ẽ0 (j, k) ≤ K 1 (j, k) δ 2 4 −j−k .Our starting point will be the estimate (a consequence of the Hardy-Stein estimates together with (4.1)) where dµ Hence where Applying the triangle inequality in L 2 , one has for m ≥ 0 and j = m and for m ≥ 1, using again the triangle inequality in L 2 , So, we deduce that Hence, bearing in mind (4.12) and |α j | ≤ α ℓ p , we obtain Finally, we consider the case p > 3.As before, we have As p > 3, we can use the triangle inequality in L p−2 in order to obtain Observe that the estimates obtained in (4.13) for I(m, j) are valid for all p > 2. Hence and (f k ) ⊂ H p be such that f k H p = 1 for all k and (f k ) to zero uniformly on compact subsets of B n .Assume also that inf k J b f k p ≍ 1.Then there exists a subsequence, still denoted by (f k ), such that the linear mapping Proof.The proof is also divided into three cases depending on the value of p, namely cases 1 ≤ p ≤ 2, 2 < p ≤ 3 and 3 < p < ∞.For 0 < δ < 1, which will be determined later, we choose a subsequence of {f k } denoted still by {f k }, a decreasing sequence ε m > 0, ε m → 0, and compact sets K m from Lemmas 4.2 and 4.3.We proceed to show that U is bounded below.
We consider first the case p > 2. As done before, we have For the case p > 3, using the standard estimate (a + b) q ≤ 2 q−1 (a q + b q ), where a, b ≥ 0 and q ≥ 1, we obtain The last inequality is a consequence of the triangle inequality in L p−2 (see the proof of the case p > 3 in Lemma 4.4).Also, the quantities I(m, j) are the ones appearing in Lemma 4.4.By the estimates in (4.13), we have As before, we use the triangle inequality in L 2 and the estimates Ẽm (m, k) δ , where C 3 is another positive constant.Putting the previous estimates in (4.14), we obtain after taking δ > 0 small enough.Hence U is bounded below.
Let now 2 < p ≤ 3. The preceding estimates together with (4.13) imply that whenever δ is small enough.
Finally, it remains to deal with the case 1 ≤ p ≤ 2. By the area theorem, we have Applying the L 2 triangle inequality, we get Proof of Theorem 1.2.We observe by the boundedness of point evaluations on H p that a normalized basis (g n ) of M is locally bounded (consider e.g.closed balls B(0, 1−1/n), n = 1, 2, . . ., and recall that Then by Montel's theorem (g n ) is a normal family and hence there is a subsequence (g n k ) converging to some holomorphic function g uniformly on compact subsets of B n .Also, g ∈ H p , by the uniform convergence on rS n for every 0 < r < 1 and the definition of H p -norm.Now take f k = gn k −g gn k −g H p to obtain a normalized sequence (f k ) converging to zero uniformly on compact subsets of B n .Now by Lemmas 4.4 and 4.5, we can find a subsequence (f n k ) such that the operator U : ℓ p → H p given by U α = ∞ k=1 α k J b f n k is an isomorphism onto its range J b N , where N = span{f n k }.Then the subspace N is isomorphic to ℓ p and in particular the operator J b is ℓ 2 -singular for p = 2. Remark 4.6.(1) Due to the fact that the sequence spaces ℓ p and ℓ q are totally incomparable for p = q whenever 1 ≤ p, q < ∞, Theorem 1.2 also implies that J b ∈ S q (H p ) for 1 ≤ p, q < ∞, p = q, and b ∈ BM OA(B n ).
(2) It is known that the standard Bergman spaces A p α , α > −1, are isomorphic to ℓ p .If 1 ≤ p < ∞, the strict singularity of the operator J b on A p α coincides with the compactness, since all strictly singular operators on ℓ p are compact.In particular, we have J b ∈ S q (A p α ) for 1 ≤ p, q < ∞, p = q, and b being in the Bloch space.
Since µ b is a finite measure (by the Littlewood-Paley identity we have µ b (B n ) ≍ b 2 H 2 ), by the absolute continuity of the integral, we deduce that ν k (S n ) = µ b (S k ) → 0 as k → ∞.Hence ν * k → 0 almost everywhere on S n as k → ∞ by (4.2).

=m α k f k 2 |Rb| 2 dV 1 ( 4
Fm (m) 1/p − A(m) 1/p p By Lemma 4.3 we have Fm (k) ≤ C 5 δ4 −k−m for k = m.Thus, by the estimates obtained in the proof of the case 1 ≤ p ≤ 2 in Lemma 4.4, we have −k−m ) 1/p p α ℓ p ≤ C 6 δ α p ℓ p .By the area theorem, we have Fm (m) ≍ J b f m p H p .Hence, there is a positive constant C 7 such that Fm (m) ≥ C 7 d p , where d = inf k∈N J b f k H p .Therefore we obtain the desired lower boundU α H p C 1/p 7 d α ℓ p − C 6 δ 1/p α ℓ p d α ℓ p ,whenever δ is small enough.Now we are ready to prove our second main result.
2 ) α dv(z).It is well known that a holomorphic function b on B n belongs to BM OA(B n ) if and only if µ b is a Carleson measure; and b ∈ V M OA(B n ) if and only if µ b is a vanishing Carleson measure.We recall that a positive Borel measure µ on B n is a Carleson measure if there exists a constant C > 0 such that 2satisfies the triangle inequality [13, Proposition 5.1.2],for z ∈ B ζ (δ), we have