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

Abstract

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 $p \ne 2.$

1 Introduction

An operator $T:X \rightarrow Y$ 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 $\ell ^p,$ then T is said to be $\ell ^p$-singular. These notions generalize the concept of compact operators. Examples of strictly singular non-compact operators are the inclusion mappings $i_{p,q}:\ell ^p \hookrightarrow \ell ^q$, where $1\le p<q<\infty$. The following inclusions hold: $K(X) \subset S(X) \subset S_p(X),$ 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 $\ell ^p-$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 ${\mathbb {B}}^n$, extending the results previously obtained in [8, 9] for the case of the unit disk ${\mathbb {D}}$. For a holomorphic function b on ${\mathbb {B}}^n$, the operator $J_ b$ is defined as

$$\begin{aligned} J_ b f(z)=\int _{0}^{1} f(tz) Rb(tz) \frac{dt}{t},\qquad z\in {\mathbb {B}}^n\end{aligned}$$

for f holomorphic on ${\mathbb {B}}^n$. Here Rb denotes the radial derivative of b, that is,

$$\begin{aligned} Rb(z)= \sum _{k=1}^{n} z_ k \frac{\partial b}{\partial z_ k} (z),\qquad z=(z_ 1,\dots ,z_ n)\in {\mathbb {B}}^n. \end{aligned}$$

It is well-known [3, 4, 11] that $J_ b$ is bounded on the Hardy space $H^p({\mathbb {B}}^n)$ if and only if $b\in BMOA({\mathbb {B}}^n)$, the space of all holomorphic functions of bounded mean oscillation; and $J_ b$ is compact on $H^p({\mathbb {B}}^n)$ if and only if $b\in VMOA({\mathbb {B}}^n)$, the space of holomorphic functions on ${\mathbb {B}}^n$ of vanishing mean oscillation. For $0<p<\infty$, the Hardy space $H^p:=H^p({\mathbb {B}}^n)$ consists of those holomorphic functions f on ${\mathbb {B}}^n$ with

$$\begin{aligned} \Vert f\Vert _{H^p}^p=\sup _{0<r<1}\int _{{\mathbb {S}}^n} \!\! |f(r\zeta )|^p \,d\sigma (\zeta )<\infty , \end{aligned}$$

where $d\sigma$ is the surface measure on the unit sphere ${\mathbb {S}}^n:=\partial {\mathbb {B}}^n$ normalized so that $\sigma ({\mathbb {S}}^n)=1$. 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 ${\mathbb {C}}^n$, as defined here, was introduced by Z. Hu [5]. A fundamental property of the Volterra integration operator $J_ b$ is the basic identity

$$\begin{aligned} R(J_ b f)(z)=f(z)\,Rb(z),\quad z\in {\mathbb {B}}^n. \end{aligned}$$

The existence of non-compact strictly singular operators acting on the Hardy space $H^p({\mathbb {D}})$ for $p \ne 2$ can be seen by considering the inclusion mappings between the sequence spaces $\ell ^p, \, p \ne 2$, and $\ell ^2$ and utilizing the fact that $H^p({\mathbb {D}})$ contains complemented copies of $\ell ^p$ and $\ell ^2.$ The existence of such operators can be transferred to the case of the Hardy spaces $H^p({\mathbb {B}}^n), \, 1 \le p < \infty ,$ since they are all isomorphic to $H^p({\mathbb {D}})$ by the result of Wojtaszczyk [15].

We recall that, for a Banach space X, a bounded linear operator $T :X \rightarrow X$ is said to fix a copy of a given Banach space E, if there is a closed subspace $M\subset X$, linearly isomorphic to E, and $c > 0$ so that $\Vert Tx\Vert \ge c\Vert x\Vert$ for all $x \in M$ (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 $\ell ^p$ inside $H^p$.

Theorem 1.1

Let $b \in BMOA({\mathbb {B}}^n) {\setminus } VMOA({\mathbb {B}}^n)$ and $1 \le p < \infty .$ Then the operator $J_b :H^p \rightarrow H^p$ fixes an isomorphic copy of $\ell ^p$ inside $H^p.$

As a consequence, for $b\in BMOA({\mathbb {B}}^n){\setminus } VMOA({\mathbb {B}}^n)$, the operator $J_b$ is not $\ell ^p$-singular. Hence the notions of compactness, strict singularity and $\ell ^p$-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 $V:\ell ^p \rightarrow H^p$ and $U:\ell ^p \rightarrow H^p$ such that $U = J_bV$, where $V(\ell ^p) = M$ is the closed linear span of suitably chosen test functions $f_{a_k} \in H^p$ and the operator U is an isomorphism onto its range $U(\ell ^p) = J_b(M)$.

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 $\ell ^2$-singularity as the operators $J_b$.

Theorem 1.2

Let $b \in BMOA({\mathbb {B}}^n)$ and $1 \le p < \infty .$ If there exists a closed infinite-dimensional subspace $M \subset H^p$ such that $J_b:H^p \rightarrow H^p$ is bounded below on M,  then there exists a subspace $N \subset M$ isomorphic to $\ell ^p.$ In particular, the operator $J_b$ acting on $H^p$ cannot fix an isomorphic copy of $\ell ^2$, i.e. it is $\ell ^2$-singular when $p \ne 2.$

We use some standard notation. For any two points $z=(z_ 1,\dots ,z_ n)$ and $w=(w_ 1,\dots ,w_ n)$ in ${\mathbb {C}}^n$, we write $\langle z,w\rangle =z_ 1{\bar{w}}_ 1+\dots +z_ n {\bar{w}}_ n,$ and $|z|=\sqrt{\langle z,z\rangle }.$ 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 $A\lesssim B$ to mean that there exists some inessential constant $C>0$ so that $A\le C B$. The relation $A\gtrsim B$ is defined in an analogous way, and $A \asymp B$ means that both $A\lesssim B$ and $A\gtrsim B$ hold.

The paper is organized as follows. In Sect. 2, we provide an auxiliary result needed to establish Theorem 1.1 in Sect. 3; and Sect. 4 covers our second main result Theorem 1.2 for which Lemmas 4.1–4.5 are crucial tools.

2 Preliminaries

It is well known that any function in $H^p$ has radial limits

$$\begin{aligned} f(\zeta )=\lim _{r\rightarrow 1^{-}} f(r\zeta ) \end{aligned}$$

for a.e. $\zeta \in {\mathbb {S}}^n$, and $\Vert f\Vert ^p_{H^p}=\int _{{\mathbb {S}}^n}|f|^p d\sigma$. For each $a\in {\mathbb {B}}^n$, consider the test function

$$\begin{aligned} f_a(z) = \frac{(1-|a|^2)^{1/p}}{(1- \langle z, a \rangle )^{(n+1)/p}},\qquad z\in {\mathbb {B}}^n. \end{aligned}$$

It is easy to see that $f_ a \in H^p$ with $\Vert f_ a\Vert _{H^p}\asymp 1$, and its radial derivative is given by

$$\begin{aligned} Rf_ a(z)=\frac{(n+1)}{p} \,(1-|a|^2)^{1/p}\frac{\langle z,a \rangle }{(1-\langle z,a \rangle )^{\frac{n+1+p}{p}}},\qquad z\in {\mathbb {B}}^n. \end{aligned}$$

We need the following lemma regarding their values and the values of $J_b f_a$ peaking in certain subsets of ${\mathbb {S}}^n$.

Lemma 2.1

Let $b \in BMOA ({\mathbb {B}}^n), \, 1 \le p < \infty ,$ and $(a_k) \subset {\mathbb {B}}^n$ be a sequence such that $a_k \rightarrow \omega \in {\mathbb {S}}^n$. Define a non-isotropic metric ball

$$\begin{aligned} S_\varepsilon (\omega ) = \{z \in {\mathbb {S}}^n: |1 - \langle z, \omega \rangle | < \varepsilon \} \end{aligned}$$

for each $\varepsilon > 0$. Then

$$\begin{aligned} \begin{aligned}&\mathrm{(i) } \lim _{k \rightarrow \infty }\int _{{\mathbb {S}}^n {\setminus } S_\varepsilon (\omega )} |f_{a_k}|^p \, d\sigma = 0 \,\,\,\textrm{for}\,\, \textrm{each}\,\, \varepsilon>0; \\&\mathrm{(ii) } \lim _{\varepsilon \rightarrow 0}\int _{S_\varepsilon (\omega )} |f_{a_k}|^p \, d\sigma = 0 \,\,\,\textrm{for}\,\, \textrm{each}\,\, k; \\&(iii) \lim _{k \rightarrow \infty } \int _{{\mathbb {S}}^n {\setminus } S_\varepsilon (\omega )} |J_bf_{a_k}|^p \, d\sigma = 0 \,\,\,\textrm{for}\,\, \textrm{each}\,\, \varepsilon > 0; \\&(iv) \lim _{\varepsilon \rightarrow 0} \int _{S_\varepsilon (\omega )} |J_b f_{a_k}|^p \, d\sigma = 0 \,\,\,\textrm{for} \,\, \textrm{each}\,\, \mathrm{k.} \end{aligned} \end{aligned}$$

Proof

Fix $\varepsilon > 0$ and $u_0 \in {\mathbb {S}}^n {{\setminus }} S_{\varepsilon }(\omega )$. Now $|1 - \langle u_0, \omega \rangle | \ge \varepsilon$. Thus it holds that $|1 - \langle u_0, a_k\rangle | \ge \varepsilon /2$ for k large enough. So we may choose $\delta = \delta (\varepsilon ) > 0$ so that $|1 - \langle u_0, a_k\rangle | \ge \delta$ for all k large enough and all $u_0 \in {\mathbb {S}}^n {{\setminus }} S_{\varepsilon }(\omega )$ and the condition (i) follows. The proof of (ii) follows from the absolute continuity of the measures $A \mapsto \int _A |f_{a_k}|^p \, d\sigma .$

(iii) Let now $0< \varepsilon < 1/2.$ We may assume that $b(0)=0$. We first confirm that there exists $\delta = \delta (\varepsilon ) > 0$ such that $|1 - \langle ru_0, a_k \rangle | \ge \delta$ for all k large enough and all $u_0 \in {\mathbb {S}}^n {{\setminus }} S_{\varepsilon }(\omega )$ and $0 \le r \le 1$. Fix $u_0 \in {\mathbb {S}}^n {{\setminus }} S_{\varepsilon }(\omega )$ and suppose that $0 \le r \le 1 - \varepsilon ^2.$ Then

$$\begin{aligned} |1 - \langle ru_0, \omega \rangle | \ge 1 - r|\langle u_0, \omega \rangle | \ge 1-(1-\varepsilon ^2) = \varepsilon ^2. \end{aligned}$$

Hence $|1 - \langle ru_0, a_k \rangle | \ge \frac{\varepsilon ^2}{2}$ for $k \ge k_0$ for some $k_0 = k_0(\varepsilon )$ and all $0 \le r \le 1-\varepsilon ^2$. Consider then the case $1-\varepsilon ^2 < r \le 1.$ Now

$$\begin{aligned} \begin{aligned} |1 - \langle ru_0, \omega \rangle |&= |r- \langle ru_0, \omega \rangle + 1 - r| \ge r |1-\langle u_0, \omega \rangle | - (1-r) \ge r\varepsilon - \varepsilon ^2\\&= \varepsilon (r-\varepsilon ) > \varepsilon (1/2 - \varepsilon ). \end{aligned} \end{aligned}$$

Thus it holds that

$$\begin{aligned} |1 - \langle ru_0, a_k\rangle | \ge \varepsilon /2 (1/2 - \varepsilon ) \end{aligned}$$

for all k large enough and all $1-\varepsilon ^2 < r \le 1.$ So we may choose $\delta = \delta (\varepsilon ) > 0$ so that $|1 - \langle ru_0, a_k\rangle | \ge \delta$ for all k large enough and all $u_0 \in {\mathbb {S}}^n {{\setminus }} S_{\varepsilon }(\omega )$ and $0 \le r \le 1$. For those $u_0$ and r we obtain the estimates

$$\begin{aligned} |f_{a_k}(ru_0)|^p = \frac{1-|a_k|^2}{|1- \langle ru_0, a_k \rangle |^{n+1}} \le \frac{1-|a_k|^2}{\delta ^{n+1}} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} |Rf_{a_k}(ru_0)|^p&= \left( \frac{n+1}{p}\right) ^p \,\frac{(1-|a_ k|^2)\,|\langle ru_ 0,a_ k \rangle |^p}{|1-\langle ru_ 0,a_ k \rangle |^{n+1+p}}\\&\le \frac{C r^p(1-|a_k|^2)}{|1-\langle ru_0, a_k \rangle |^{n + 1 + p}} \le \frac{Cr^p\,(1-|a_k|^2)}{\delta ^{n+1+p}}, \end{aligned} \end{aligned}$$

for all k large enough, where $C = C(n,p) > 0.$ Now, for a.e. $\zeta \in {\mathbb {S}}^n{\setminus } S_\varepsilon (\omega )$, we obtain

$$\begin{aligned} \begin{aligned} |J_bf_{a_k}(\zeta )|^p&= \lim _{r\rightarrow 1^{-}}\left| \int _0^1 f_{a_k}(t r \zeta ) Rb(t r \zeta )\frac{dt}{t}\right| ^p\\&\le C\left( |f_{a_k}(\zeta ) b(\zeta )|^p + \left( \int _0^1 |Rf_{a_k}(t\zeta )b(t\zeta )|\frac{dt}{t}\right) ^p \right) \\&\le C \left( \frac{1-|a_k|^2}{\delta ^{n+1}} |b(\zeta )|^p + \frac{1-|a_k|^2}{\delta ^{n+1+p}}\left( \int _0^1 |b(t\zeta )|\, dt\right) ^p\right) , \end{aligned} \end{aligned}$$

where constants may depend on n and p. Utilizing the well known pointwise estimate $|b(z)| \lesssim \Vert b\Vert _{BMOA} \log \frac{1}{1-|z|}$ for $z \in {\mathbb {B}}^n$, where and $\Vert b\Vert _{BMOA}$ is the $BMOA$-seminorm, we have $\int _0^1 |b(t\zeta )|\, dt \le C \Vert b\Vert _{BMOA}.$ Therefore

$$\begin{aligned} \int _{{\mathbb {S}}^n {\setminus } S_\varepsilon (\omega )}|J_bf_{a_k}(\zeta )|^p \, d\sigma (\zeta ) \lesssim \frac{1-|a_k|^2}{\delta ^{n+1}} \Vert b\Vert _{H^p}^p + \frac{1-|a_k|^2}{\delta ^{n+1+p}}\Vert b\Vert _{BMOA}^p \rightarrow 0, \end{aligned}$$

as $k \rightarrow \infty ,$ since $\Vert b\Vert _{H^p} \lesssim \Vert b\Vert _{BMOA}< \infty .$

(iv) If k is fixed, the claim follows from the absolute continuity of the measure $A \mapsto \int _{A}|J_b f_{a_k}|^p \, d\sigma$. $\square$

3 $\ell ^p-$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 $\ell ^p$. We begin with an auxiliary result.

Proposition 3.1

Let $1 \le p < \infty$ and $(a_k) \subset {\mathbb {B}}^n$ be a sequence such that $a_k \rightarrow \omega \in {\mathbb {S}}^n$. Then there exists a subsequence $(b_k)$ of $(a_k)$ such that the mapping $V:\ell ^p \rightarrow H^p$ defined as

$$\begin{aligned} V(\alpha ) = \sum _{k = 1}^\infty \alpha _k f_{b_k}, \end{aligned}$$

where $\alpha = (\alpha _k) \in \ell ^p$, is bounded.

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. $\square$

Proposition 3.2

Let $b \in BMOA ({\mathbb {B}}^n) {\setminus } VMOA ({\mathbb {B}}^n)$ and $1 \le p < \infty$. Then

$$\begin{aligned} c:= \limsup _{|a| \rightarrow 1}\Vert J_b f_a\Vert _{H^p} > 0. \end{aligned}$$

In particular, there exists a sequence $(a_k) \subset {\mathbb {B}}^n$ such that $a_k \rightarrow \omega \in {\mathbb {S}}^n$ and

$$\begin{aligned} \lim _{k \rightarrow \infty }\Vert J_b f_{a_k}\Vert _{H^p} = c. \end{aligned}$$

Proof

We may assume $b(0) = 0$. We consider first the case $p > 2$ and utilize the representation [16, Chapter 5]

$$\begin{aligned} \Vert b\Vert ^2_{ BMOA } \asymp \sup _{\Vert f\Vert _{H^p}\asymp 1}\int _{{\mathbb {B}}^n}|f(z)|^p|Rb(z)|^2(1-|z|^2) dv(z), \end{aligned}$$

where dv(z) is the normalized volume measure on ${\mathbb {B}}^n$. Now for $f \in H^p,$ by the estimates obtained in pages 144–145 in [11], we have

$$\begin{aligned} \int _{{\mathbb {B}}^n}|f(z)|^p|Rb(z)|^2(1-|z|^2)\, dv(z)\le C \Vert f\Vert _{H^p}^{p-2} \cdot \Vert J_bf\Vert _{H^p}^2,\qquad 2<p<\infty ; \end{aligned}$$

and

$$\begin{aligned} \int _{{\mathbb {B}}^n} |f(z)|^p |Rb(z)|^2 (1-|z|^2)\, dv(z)\le C \, \Vert b\Vert ^{2-p}_{ BMOA } \cdot \Vert J_b f\Vert _{H^p}^p,\qquad 1\le p \le 2. \end{aligned}$$

By replacing f with $f_a$ (note that $\Vert f_ a\Vert _{H^p}\asymp 1$), we have

$$\begin{aligned} \Vert J_bf_a\Vert _{H^p}^2 \ge C \int _{{\mathbb {B}}^n}|f_a(z)|^p|Rb(z)|^2(1-|z|^2)\, dv(z),\qquad 2< p < \infty ; \end{aligned}$$

and

$$\begin{aligned} \Vert b\Vert ^{2-p}_{ BMOA } \cdot \Vert J_b f_a\Vert ^p_{H^p} \ge C \int _{{\mathbb {B}}^n}|f_a(z)|^p|Rb(z)|^2(1-|z|^2)\, dv(z), \qquad 1 \le p \le 2. \end{aligned}$$

It is well known [16, Chapter 5] that a holomorphic function g belongs to $VMOA({\mathbb {B}}^n)$ if and only if

$$\begin{aligned} \lim _{|a| \rightarrow 1^{-}} \int _{{\mathbb {B}}^n} \frac{(1-|a|^2)}{|1-\langle z,a \rangle |^{n+1}} \,|Rg(z)|^2(1-|z|^2)\, dv(z)=0. \end{aligned}$$

Since $b \in BMOA ({\mathbb {B}}^n) {\setminus } VMOA ({\mathbb {B}}^n),$ it holds that

$$\begin{aligned} \limsup _{|a| \rightarrow 1^{-}}\int _{{\mathbb {B}}^n}|f_a(z)|^p|Rb(z)|^2(1-|z|^2)\, dv(z) > 0 \end{aligned}$$

and hence $\limsup _{|a| \rightarrow 1}\Vert J_b f_a\Vert _{H^p} >0$ for all $1 \le p < \infty .$ $\square$

As a final step towards the proof of Theorem 1.1, we construct an isomorphism from $\ell ^p$ into $H^p$ using a non-compact $J_b$ and test functions.

Proposition 3.3

Let $b \in BMOA({\mathbb {B}}^n) {\setminus } VMOA({\mathbb {B}}^n), \, 1 \le p < \infty ,$ and $(a_k) \subset {\mathbb {B}}^n$ be the sequence from Proposition 3.2. Then there exists a subsequence $(b_k)$ of $(a_k)$ such that the mapping $U:\ell ^p \rightarrow H^p,\, U(\alpha )=\sum _{k = 1}^\infty \alpha _k J_b f_{b_k},$ where $\alpha = (\alpha _k) \in \ell ^p$, is an isomorphism onto its range.

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. $\square$

Proof of Theorem 1.1

By Proposition 3.1 and Proposition 3.3, we can choose a sequence $(b_k) \subset {\mathbb {B}}^n$ that induces a bounded operator $V :\ell ^p \rightarrow H^p,$ given by

$$\begin{aligned} V(\alpha ) = \sum _{k = 1}^\infty \alpha _k f_{b_k}, \end{aligned}$$

where $\alpha = (\alpha _k) \in \ell ^p,$ and an isomorphism $U:\ell ^p \rightarrow H^p,\, U = J_bV$ onto its range.

Define $M = \overline{span \{f_{b_k}\}},$ 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 $J_b|_M:M \rightarrow J_b(M)$ is an isomorphism and consequently M is isomorphic to $\ell ^p.$ In particular, the operator $J_b$ is not $\ell ^p$-singular. $\square$

4 $\ell ^2$-singularity of $J_b$

In this section, we show that if $J_b:H^p \rightarrow H^p$ is bounded below on a closed infinite-dimensional subspace M of $H^p$, then there exists a subspace $N \subset M$ isomorphic to $\ell ^p.$ In particular, this implies that $J_b:H^p \rightarrow H^p$ cannot fix an isomorphic copy of $\ell ^2$ whenever $p \ne 2.$

For $\zeta \in {\mathbb {S}}^n$, the admissible approach region $\Gamma (\zeta )$ is defined as

$$\begin{aligned} \Gamma (\zeta )=\left\{ z\in {\mathbb {B}}^n: |1-\langle z,\zeta \rangle |< 1-|z|^2 \right\} . \end{aligned}$$

If $I(z):=\{\zeta \in {\mathbb {S}}^n: z\in \Gamma (\zeta )\}$, then $\sigma (I(z))\asymp (1-|z|^2)^{n}$, and it follows from Fubini’s theorem that, for a finite positive measure $\nu$, and a positive function $\varphi$, one has

$$\begin{aligned} \int _{{\mathbb {B}}^n} \varphi (z)\,d\nu (z)\asymp \int _{{\mathbb {S}}^n} \left( \int _{\Gamma (\zeta )} \varphi (z) \frac{d\nu (z)}{(1-|z|^2)^{n}} \right) d\sigma (\zeta ). \end{aligned}$$

(4.1)

For convenience, we define the measure $\mu _b$ by

$$\begin{aligned} d\mu _b(z)=|Rb(z)|^2(1-|z|^2)dv(z), \end{aligned}$$

where dv is the normalized volume measure on ${\mathbb {B}}^n$, and set $dV_\alpha (z)=(1-|z|^2)^\alpha dv(z)$. It is well known that a holomorphic function b on ${\mathbb {B}}^n$ belongs to $BMOA({\mathbb {B}}^n)$ if and only if $\mu _ b$ is a Carleson measure; and $b\in VMOA({\mathbb {B}}^n)$ if and only if $\mu _ b$ is a vanishing Carleson measure. We recall that a positive Borel measure $\mu$ on ${\mathbb {B}}^n$ is a Carleson measure if there exists a constant $C>0$ such that

$$\begin{aligned} \mu \big (B_{\zeta }(\delta )\big )\le C \delta \, ^n \end{aligned}$$

for all $\zeta \in {\mathbb {S}}^n$ and $\delta >0$. Here $B_\zeta (\delta ) = \{z \in {\mathbb {B}}^n: |1- \langle z, \zeta \rangle | < \delta \}$. Also, $\mu$ is a vanishing Carleson measure if

$$\begin{aligned} \mu (B_\zeta (\delta ))=o(\delta ^n),\qquad \text {as }\delta \rightarrow 0. \end{aligned}$$

It is also well known that, if $\mu$ is a vanishing Carleson measure then

$$\begin{aligned} \int _{{\mathbb {B}}^n} |f_ k|^p \,d\mu \rightarrow 0 \end{aligned}$$

for any bounded sequence of functions $\{f_k\} \subset H^p$ converging to zero uniformly on compact subsets of ${\mathbb {B}}^n$, $1\le p<\infty$. Next, we establish some preliminary results en route to the proof of Theorem 1.2.

Lemma 4.1

Let $\varepsilon >0$ and $b \in H^2.$ Then there exists a compact set $K_\varepsilon \subset {\mathbb {S}}^n$ with $\sigma ({\mathbb {S}}^n{\setminus } K_\varepsilon ) < \varepsilon$ such that $\sup _{\zeta \in K_\varepsilon }\mu _b(B_\zeta (\delta ))=o(\delta ^n)$ as $\delta \rightarrow 0,$ and $\mu _{b,\varepsilon } = \chi _{\Omega _\varepsilon }|Rb|^2\, dV_1$ is a vanishing Carleson measure, where $\Omega _\varepsilon = \bigcup _{\zeta \in K_\varepsilon } \Gamma (\zeta )$.

Proof

For each $k \ge 1$, let $\nu _k$ be the projection to ${\mathbb {S}}^n$ of the measure $\mu _b$ restricted to the annulus $S_k = \{z \in {\mathbb {B}}^n: 1-1/k< |z| < 1\}.$ That is, $\nu _k$ is determined by the condition

$$\begin{aligned} \nu _k(I(\zeta , \delta ))= \mu _b\big (\{z \in S_k: |1-\langle z, \zeta \rangle |< \delta \}\big ), \end{aligned}$$

where $I(\zeta ,\delta ) = \{\xi \in {\mathbb {S}}^n: |1-\langle \xi , \zeta \rangle |<\delta \}.$ Consider the Hardy–Littlewood maximal function of $\nu _k:$

$$\begin{aligned} \nu _k^*(\zeta ) = \sup _{\delta> 0}\frac{\nu _k(I(\zeta ,\delta ))}{\sigma (I(\zeta , \delta ))} \asymp \sup _{\delta > 0}\frac{\nu _k(I(\zeta , \delta ))}{\delta ^n}. \end{aligned}$$

The maximal function theorem [16, Chapter 4] implies that it satisfies

$$\begin{aligned} \sigma (\{\zeta \in {\mathbb {S}}^n: \nu _k^*(\zeta ) >\lambda \}) \lesssim \frac{\nu _k({\mathbb {S}}^n)}{\lambda } \end{aligned}$$

(4.2)

for all $\lambda > 0$. Since $\mu _b$ is a finite measure (by the Littlewood–Paley identity we have $\mu _ b({\mathbb {B}}^n)\asymp \Vert b\Vert _{H^2}^2$), by the absolute continuity of the integral, we deduce that $\nu _k({\mathbb {S}}^n) = \mu _b(S_k) \rightarrow 0$ as $k \rightarrow \infty$. Hence $\nu _k^* \rightarrow 0$ almost everywhere on ${\mathbb {S}}^n$ as $k \rightarrow \infty$ by (4.2). Egorov’s theorem now implies that there is a set $F \subset {\mathbb {S}}^n$ with $\sigma ({\mathbb {S}}^n {\setminus } F) < \varepsilon /2$ such that $\nu _k^* \rightarrow 0$ uniformly in F as $k \rightarrow \infty$. Now for every $k \ge 1$ and $\zeta \in F$, we have

$$\begin{aligned} \sup _{0< \delta < 1/k}\frac{\mu _b(B_\zeta (\delta ))}{\delta ^n} \le \sup _{\delta > 0}\frac{\nu _k(I(\zeta ,\delta ))}{\delta ^n} \asymp \nu _k^*(\zeta ), \end{aligned}$$

where the first inequality follows from the fact that $B_\zeta (\delta ) \subset S_k$ for all $0< \delta < 1/k.$ Thereby we deduce that $\sup _{\zeta \in F} \mu _b(B_\zeta (\delta )) = o(\delta ^n$) as $\delta \rightarrow 0$. Hence, if we pick a compact subset $K_\varepsilon \subset F$ with $\sigma (F {{\setminus }} K_\varepsilon ) < \varepsilon /2,$ we get

$$\begin{aligned} \sup _{\zeta \in K_\varepsilon }\mu _b(B_\zeta (\delta ))=o(\delta ^n),\qquad \textrm{as }\delta \rightarrow 0. \end{aligned}$$

(4.3)

In order to see that $\mu _{b,\varepsilon }$ is a vanishing Carleson measure, we need to prove that $\mu _{b,\varepsilon }(B_{\zeta }(\delta ))=o(\delta ^n)$ as $\delta \rightarrow 0$ for every $\zeta \in {\mathbb {S}}^n$. From (4.3), we obtain that $\sup _{\zeta \in K_\varepsilon } \mu _{b, \varepsilon }(B_\zeta (\delta )) = o(\delta ^n$) as $\delta \rightarrow 0$. Let $\zeta \in {\mathbb {S}}^n {{\setminus }} K_\varepsilon$ and $\delta > 0$ small enough. If $B_\zeta (\delta ) \cap \Omega _\varepsilon \ne \emptyset$, then there is a point $w\in B_{\zeta }(\delta )\cap \Gamma (x)$ for some $x\in K_{\varepsilon }$. Using that $d(z,w)=|1-\langle z,w \rangle |^{1/2}$ satisfies the triangle inequality [13, Proposition 5.1.2], for $z\in B_{\zeta }(\delta )$, we have

$$\begin{aligned} \begin{aligned} |1-\langle z,x \rangle |^{1/2}&\le |1-\langle z,w \rangle |^{1/2}+|1-\langle w,x \rangle |^{1/2}\\&\le |1-\langle z,\zeta \rangle |^{1/2}+|1-\langle \zeta ,w \rangle |^{1/2}+|1-\langle w,x \rangle |^{1/2}\\&\le 2\delta ^{1/2}+(1-|w|^2)^{1/2}\\&\le 2\delta ^{1/2}+\sqrt{2}\,|1-\langle \zeta ,w \rangle |^{1/2}<4 \delta ^{1/2}. \end{aligned} \end{aligned}$$

Hence $B_{\zeta }(\delta )\subset B_ x \big (16\delta \big )$, and therefore

$$\begin{aligned} \mu _{b,\varepsilon }\big (B_{\zeta }(\delta ) \big )\le \mu _{b,\varepsilon } \big (B_ x(16\delta )\big )=o(\delta ^n),\qquad \textrm{as }\delta \rightarrow 0. \end{aligned}$$

It now follows that the measure $\mu _{b, \varepsilon }$ is a vanishing Carleson measure. $\square$

For $1\le p<\infty$ and a sequence of functions $\{f_ k\}\subset H^p$, given a subset A of ${\mathbb {S}}^n$, we consider the quantities

$$\begin{aligned} A(j,k)&=\int _{A}\left( \int _{\Gamma (\xi )}|J_b f_j|^{p-2}|f_k|^2 |Rb|^2 dV_{1-n}\right) d\sigma (\xi ); \\ A(\infty ,k)&= \sup _{\Vert f\Vert _{H^p}=1} \int _{A}\left( \int _{\Gamma (\xi )}|J_b f|^{p-2}|f_k|^2 |Rb|^2 dV_{1-n}\right) d\sigma (\xi );\\ A(k, \infty )&= \sup _{\Vert f\Vert _{H^p}=1} \int _{A}\left( \int _{\Gamma (\xi )}|J_b f_k|^{p-2}|f|^2 |Rb|^2 dV_{1-n}\right) d\sigma (\xi ). \end{aligned}$$

Lemma 4.2

Let $b \in BMOA({\mathbb {B}}^n)$, $0< \delta < 1$ and $\{f_k\}$ be a normalized sequence in $H^p$, which converges to zero uniformly on compact subsets of ${\mathbb {B}}^n.$ If $p>2$, there exists a subsequence denoted still by $\{f_k\}$, a decreasing sequence $\varepsilon _m > 0, \, \varepsilon _m \rightarrow 0,$ and compact sets $K_m\subset {\mathbb {S}}^n$ satisfying $K_m\subset K_{m+1}$ and $\sigma (E_m) < \varepsilon _m$, where $E_m = {\mathbb {S}}^n {\setminus } K_m$ such that

$$\begin{aligned}&E_m(j,k)\lesssim \delta ^2 4^{-j-k-m}\ \ \ for\ j,k<m;\\&E_m(\infty ,k)\lesssim \delta ^2 4^{-k-2m}\ \ \ for\ k<m;\\&E_m(k,\infty )\lesssim \delta ^2 4^{-k-2m}\ \ \ for\ k<m;\\&K_m(k,m)\lesssim \delta ^2 4^{-k-2m}\ \ \ for\ k\le m;\\&K_m(m,k)\lesssim \delta ^2 4^{-k-2m}\ \ \ for\ k\le m, \end{aligned}$$

for all $m\ge 1$. In particular, by defining ${\tilde{E}}_m = E_m {{\setminus }} E_{m+1}$, we have that ${\tilde{E}}_m(j,k)\lesssim \delta ^2 4^{-j-k-m}$for $k \ne m$ or $j \ne m$.

Proof

Since $b \in BMOA({\mathbb {B}}^n)$, the operator $J_ b$ is bounded on $H^p$. Also, Lemma 4.1 implies that for any $\varepsilon >0$, there exists a compact set $K_\varepsilon \subset {\mathbb {S}}^n$ with $\sigma (E_\varepsilon ) < \varepsilon$ where $E_\varepsilon = {\mathbb {S}}^n {{\setminus }} K_\varepsilon$ such that

$$\begin{aligned} \mu _{b,\varepsilon } = \chi _{\Omega _\varepsilon }|Rb|^2\, dV_1 \end{aligned}$$

(4.4)

is a vanishing Carleson measure. Note that, by (4.1) and Hölder’s inequality with exponent $p/2>1$,

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {S}}^n}&\left( \int _{\Gamma (\xi )}|J_b f_k|^{p-2} |f_m|^2 |Rb|^2 dV_{1-n}\right) d\sigma (\xi ) \asymp \int _{{\mathbb {B}}^n}|J_b f_k|^{p-2}|f_m|^2 d\mu _b\\&\le \left[ \int _{{\mathbb {B}}^n} |J_bf_k|^p d\mu _b\right] ^{\frac{p-2}{p}}\left[ \int _{{\mathbb {B}}^n} |f_m|^p d\mu _b\right] ^{\frac{2}{p}} \lesssim \Vert J_ b f_ k\Vert _{H^p}^{p-2}\cdot \Vert f_ m\Vert _{H^p}^2<\infty . \end{aligned} \end{aligned}$$

The last estimate is due to Carleson–Hörmander theorem on Carleson measures. Therefore, by absolute continuity, for all fixed $(k,m) \in {\mathbb {N}}^2$, one has

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \int _{E_\varepsilon } \left( \int _{\Gamma (\xi )}|J_b f_k|^{p-2}|f_m|^2 |Rb|^2 dV_{1-n}\right) d\sigma (\xi )=0. \end{aligned}$$

(4.5)

As a simple application of (4.1), for any positive measurable function $\varphi$, we have

$$\begin{aligned} \int _{K_\varepsilon }\left( \int _{\Gamma (\xi )}\frac{\varphi (z)dv(z)}{(1-|z|^2)^n}\right) d\sigma (\xi )\lesssim \int _{\Omega _\varepsilon } \varphi (z)dv(z). \end{aligned}$$

(4.6)

By repeating the calculation above and using this formula, we obtain

$$\begin{aligned} \lim _{(k,m)\rightarrow \infty }\int _{K_\varepsilon }\left( \int _{\Gamma (\xi )}|J_b f_k|^{p-2}|f_m|^2 |Rb|^2 dV_{1-n}\right) d\sigma (\xi )=0, \end{aligned}$$

(4.7)

which follows from the vanishing Carleson measure condition (4.4), where $(k,m)\rightarrow \infty$ means that $k+m\rightarrow \infty$.

For $f\in H^p$, the admissible maximal function $f^*(\zeta )=\sup _{z\in \Gamma (\zeta )}|f(z)|$ is bounded on $L^p({\mathbb {S}}^n)$, that is, $\Vert f^*\Vert _{L^p({\mathbb {S}}^n)}\lesssim \Vert f\Vert _{H^p}$ (see [16, Chapter 4]). Assume now that $f,g \in H^p$ are unit vectors, then

$$\begin{aligned} \begin{aligned} \int _{E_\varepsilon }&\left( \int _{\Gamma (\xi )} |J_b f|^{p-2}|g|^2 |Rb|^2 dV_{1-n}\right) d\sigma (\xi )\\&\lesssim \int _{E_\varepsilon } (J_b f)^*(\xi )^{p-2} \left( \int _{\Gamma (\xi )} |g|^2 |Rb|^2 dV_{1-n}\right) d\sigma (\xi )\\&\lesssim \left( \int _{E_\varepsilon } (J_b f)^*(\xi )^p d\sigma (\xi )\right) ^{\frac{p-2}{p}}\left( \int _{E_\varepsilon } \left( \int _{\Gamma (\xi )} |g|^2 |Rb|^2 dV_{1-n}\right) ^{\frac{p}{2}} d\sigma (\xi )\right) ^{\frac{2}{p}}. \end{aligned} \end{aligned}$$

Observe that both factors go to zero as $\sigma (E_\varepsilon )\rightarrow 0$ due to the absolute continuity of the measure, as the boundedness of $J_ b$ gives $\Vert (J_ b f)^*\Vert _{L^p({\mathbb {S}}^n)}\lesssim \Vert J_ b f \Vert _{H^p}\le \Vert J_ b\Vert$ and, by the version of Calderón area theorem for the unit ball [1, 11], we have

$$\begin{aligned} \int _{{\mathbb {S}}^n} \left( \int _{\Gamma (\xi )} |g|^2 |Rb|^2 dV_{1-n}\right) ^{\frac{p}{2}} d\sigma (\xi )\asymp \Vert J_ b g \Vert ^p_{H^p}\le \Vert J_ b\Vert ^p. \end{aligned}$$

Hence, as $\varepsilon \rightarrow 0$, we have

$$\begin{aligned} E_{\varepsilon }(\infty ,k) \lesssim \Vert J_ b \Vert ^{p-2} \left( \int _{E_\varepsilon } \left( \int _{\Gamma (\xi )} |f_ k|^2 |Rb|^2 dV_{1-n}\right) ^{\frac{p}{2}} d\sigma (\xi )\right) ^{\frac{2}{p}} \longrightarrow 0. \end{aligned}$$

(4.8)

and

$$\begin{aligned} E_{\varepsilon }(k,\infty ) \lesssim \left( \int _{E_\varepsilon } (J_b f_ k)^*(\xi )^p d\sigma (\xi )\right) ^{\frac{p-2}{p}}\, \Vert J_ b \Vert ^{2} \longrightarrow 0. \end{aligned}$$

(4.9)

We will choose a subsequence of $(f_k)$, which we will also denote as $(f_k)$, and $\varepsilon _1> \varepsilon _2 \dots >0$ in the following way. Assume that functions $f_1,\ldots , f_{m-1}$, numbers $\varepsilon _1> \ldots> \varepsilon _{m-1} > 0$ and compact sets $K_1 \subset \ldots \subset K_{m-1}$ are chosen for some $m \ge 2$. Then (4.5) together with (4.8) and (4.9) yields that there exists $\varepsilon _m < \varepsilon _{m-1}$ with $K_{m-1}\subset K_{m}$ and $\sigma (E_m) < \varepsilon _m$ such that $E_m(j,k) \lesssim \delta ^2 4^{-j-k-m}$ for every $j,k < m$, and $E_m(\infty ,k)\lesssim \delta ^2 4^{-k-2m}$, $E_m(k,\infty )\lesssim \delta ^2 4^{-k-2\,m}$ for $k<m$. After that we can use (4.7) to find $f_m$ such that $K_m(k,m) \lesssim \delta ^2 4^{-k-2\,m}$ and $K_m(m,k) \lesssim \delta ^2 4^{-k-2\,m}$ for $k\le m$. Hence for $j,k<m$, we have

$$\begin{aligned} \begin{aligned} {\tilde{E}}_m(j,k)&=\int _{{\tilde{E}}_m}\left( \int _{\Gamma (\xi )}|J_b f_j|^{p-2}|f_k|^2 |Rb|^2 dV_{1-n}\right) d\sigma (\xi ) \le E_m(j,k)\\&\lesssim \delta ^2 4^{-j-k-m}. \end{aligned} \end{aligned}$$

For $k<m$, ${\tilde{E}}_m(m,k) \le E_m(\infty ,k) \lesssim \delta ^2 4^{-k-2\,m}$ and ${\tilde{E}}_m(k,m) \le E_m(k,\infty ) \lesssim \delta ^2 4^{-k-2\,m}$.

Also for the case $\max \{j,k\}>m$, if $j=\max \{j,k\}>m$, we may use ${\tilde{E}}_m= E_m{{\setminus }} E_{m+1}\subset {\mathbb {S}}^n{{\setminus }} E_{m+1}=K_{m+1}\subset K_j$ to obtain

$$\begin{aligned} {\tilde{E}}_m(j,k)\le K_j(j,k)\lesssim \delta ^2 4^{-2j-k}\le \delta ^2 4^{-j-k-m}. \end{aligned}$$

Analogously, we have ${\tilde{E}}_m(j,k)\le K_k(j,k)\lesssim \delta ^2 4^{-j-k-m}$ for $k=\max \{j,k\}>m$. Thus, ${\tilde{E}}_m(j,k)\lesssim \delta ^2 4^{-j-k-m}$ for $k\ne m$ or $j\ne m$. $\square$

Next, we establish an analogous version of Lemma 4.2 in the case $1 \le p \le 2$.

Lemma 4.3

Let $1 \le p \le 2$, $b \in BMOA({\mathbb {B}}^n)$, $0< \delta < 1$, and $\{f_k\}$ be a normalized sequence in $H^p$, which converges to zero uniformly on compact subsets of ${\mathbb {B}}^n.$ Then there exists a subsequence denoted still by $\{f_k\}$, a decreasing sequence $\varepsilon _m > 0, \, \varepsilon _m \rightarrow 0,$ and compact sets $K_m\subset {\mathbb {S}}^n$ satisfying $K_m\subset K_{m+1}$ and $\sigma (E_m) < \varepsilon _m$, where $E_m = {\mathbb {S}}^n {{\setminus }} K_m$ such that

$$\begin{aligned} L_m(m) = \int _{K_m}\left( \int _{\Gamma (\xi )}|f_m|^2 |Rb|^2 dV_{1-n}\right) ^{\frac{p}{2}} d\sigma (\xi ) \end{aligned}$$

and

$$\begin{aligned} F_m(k) = \int _{E_m}\left( \int _{\Gamma (\xi )}|f_k|^2 |Rb|^2 dV_{1-n}\right) ^{\frac{p}{2}} d\sigma (\xi ) \end{aligned}$$

satisfy $F_m(k) \lesssim \delta 4^{-k-m}$ for $k < m$ and $L_m(m) \lesssim \delta 4^{-2\,m}$ for all $m\ge 1$. In particular,

$$\begin{aligned} {\tilde{F}}_m(k)=\int _{{\tilde{E}}_m}\left( \int _{\Gamma (\xi )}|f_k|^2 |Rb|^2 dV_{1-n}\right) ^{\frac{p}{2}} d\sigma (\xi ) \lesssim \delta 4^{-k-m} \end{aligned}$$

for $k \ne m$, where ${\tilde{E}}_ m=E_ m {{\setminus }} E_{m+1}$. Also, we have ${\tilde{F}}_m(m) \lesssim 1$.

Proof

As before, the operator $J_ b$ is bounded on $H^p$ and for any $\varepsilon >0$, there exists a compact set $K_\varepsilon \subset {\mathbb {S}}^n$ with $\sigma (E_\varepsilon ) < \varepsilon$ such that $\mu _{b,\varepsilon }$ is a vanishing Carleson measure. By the version of Calderón’s area theorem for the unit ball [1, 11],

$$\begin{aligned} \int _{{\mathbb {S}}^n}\left( \int _{\Gamma (\xi )}|f_k|^2 |Rb|^2 dV_{1-n}\right) ^{\frac{p}{2}}d\sigma (\xi )\asymp \Vert J_bf_k\Vert _{H^p}^{p}\lesssim 1. \end{aligned}$$

In particular, ${\tilde{F}}_m(k)\lesssim 1$. Therefore, by absolute continuity, for all k, one has

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \int _{E_\varepsilon }\left( \int _{\Gamma (\xi )}|f_k|^2 |Rb|^2 dV_{1-n}\right) ^{\frac{p}{2}}d\sigma (\xi )=0. \end{aligned}$$

(4.10)

Now, using Hölder’s inequality, the $L^p$-boundedness of the admissible maximal function and (4.6), we get

$$\begin{aligned} \begin{aligned}&\int _{K_\varepsilon }\left( \int _{\Gamma (\xi )}|f_m|^2 |Rb|^2 dV_{1-n}\right) ^{\frac{p}{2}} d\sigma (\xi )\\&\le \int _{K_\varepsilon }|f^*_m(\xi )|^{\frac{(2-p)p}{2}}\left( \int _{\Gamma (\xi )}|f_m|^p |Rb|^2 dV_{1-n}\right) ^{\frac{p}{2}}d\sigma (\xi )\\&\le \Vert f^*_m\Vert _{L^p({\mathbb {S}}^n)}^{\frac{(2-p)p}{2}}\left( \int _{K_\varepsilon }\int _{\Gamma (\xi )}|f_m|^p |Rb|^2 dV_{1-n}d\sigma (\xi )\right) ^{\frac{p}{2}}\\&\lesssim \left( \int _{\Omega _\varepsilon } |f_m|^p d\mu _b \right) ^{\frac{p}{2}}. \end{aligned} \end{aligned}$$

As $\mu _{b,\varepsilon }$ is a vanishing Carleson measure, we obtain

$$\begin{aligned} \lim _{m \rightarrow \infty }\int _{K_\varepsilon }\left( \int _{\Gamma (\xi )}|f_m|^2 |Rb|^2 dV_{1-n}\right) ^{\frac{p}{2}}d\sigma (\xi )=0. \end{aligned}$$

(4.11)

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 $\varepsilon _m > 0, \, \varepsilon _m \rightarrow 0,$ and compact sets $K_m$ satisfying $K_m\subset K_{m+1}$ and $\sigma (E_m) < \varepsilon _m$ where $E_m = {\mathbb {S}}^n {{\setminus }} K_m$ such that $F_m(k) \lesssim \delta 4^{-k-m}$ for $k < m$ and $L_m(m) \lesssim \delta 4^{-2\,m}$ for all $m\ge 1$.

More precisely, we have

$$\begin{aligned} {\tilde{F}}_m(k) = \int _{{\tilde{E}}_m}\left( \int _{\Gamma (\xi )} |f_k|^2 |Rb|^2 dV_{1-n}\right) ^{\frac{p}{2}} d\sigma (\xi ) \le F_m(k) \lesssim \delta 4^{-k-m} \end{aligned}$$

for $k < m$. Moreover, it holds that ${\tilde{E}}_m \subset K_{m+1} \subset K_k$ for $k > m$. Hence

$$\begin{aligned} \begin{aligned} {\tilde{F}}_m(k) \le L_k(k)\lesssim \delta 4^{-2k} < \delta 4^{-k-m} \end{aligned} \end{aligned}$$

for $k > m$. Therefore, it always holds that ${\tilde{F}}_m(k)\lesssim \delta 4^{-k-m}$ for $k\ne m$. $\square$

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 ${\mathbb {B}}_n$.

Lemma 4.4

Let $1 \le p < \infty$, $b \in BMOA({\mathbb {B}}^n)$, and $(f_k) \subset H^p$ be such that $\Vert f_k\Vert _{H^p} = 1$ for all k and $(f_k)$ converges to zero uniformly on compact subsets of ${\mathbb {B}}^n.$ Then there exists a subsequence $(f_{n_k})$ of $(f_k)$ such that the linear mapping $U:\ell ^p \rightarrow H^p,$ defined as

$$\begin{aligned} U(\alpha ) = \sum _{k = 1}^\infty \alpha _k J_bf_{n_k}, \end{aligned}$$

where $\alpha = (\alpha _k) \in \ell ^p$, is bounded.

Proof

We divide our proof into three cases depending on the value of p, namely cases $1 \le p \le 2,$ $2 < p \le 3$ and $3< p < \infty$. 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 $0<\delta <1$, we choose a subsequence of $\{f_k\}$ denoted still by $\{f_k\}$, a decreasing sequence $\varepsilon _m > 0, \, \varepsilon _m \rightarrow 0,$ and compact sets $K_m$ from Lemmas 4.2 and 4.3. Set ${\widetilde{E}}_0 = {\mathbb {S}}^n{\setminus } E_1=K_1$.

Let us first look at the case $1 \le p \le 2.$ Set $\alpha _ 0=0$. By the version of the area theorem for the unit ball [1, 11], we have

$$\begin{aligned} \begin{aligned}&\Vert U(\alpha )\Vert _{H^p}^p \asymp \int _{{\mathbb {S}}^n} \left( \int _{\Gamma (\zeta )}\left| \sum _{k=1}^\infty \alpha _k f_kRb\right| ^2 \, dV_{1-n}\right) ^{\frac{p}{2}}\, d\sigma (\zeta )\\&= \sum _{m=0}^\infty \int _{{\tilde{E}}_m} \left( \int _{\Gamma (\zeta )}\left| \sum _{k=1}^\infty \alpha _k f_kRb\right| ^2 \, dV_{1-n}\right) ^{\frac{p}{2}}\, d\sigma (\zeta )\\&\lesssim \sum _{m=0}^\infty \int _{{\tilde{E}}_m} \left( \int _{\Gamma (\zeta )} |\alpha _m f_mRb|^2 dV_{1-n}+\int _{\Gamma (\zeta )}\left| \sum _{\begin{array}{c} k=1\\ k\ne m \end{array}}^\infty \alpha _k f_kRb\right| ^2 \, dV_{1-n}\right) ^{\frac{p}{2}} d\sigma (\zeta ). \end{aligned} \end{aligned}$$

According to the assumption $1 \le p \le 2$, then $d(F,G)=\Vert F-G\Vert _{L^{\frac{p}{2}}}^{\frac{p}{2}}$ is a metric and hence

$$\begin{aligned} \begin{aligned} \Vert U(\alpha )\Vert _{H^p}^p&\lesssim \sum _{m=1}^\infty \int _{{\tilde{E}}_m} \left( \int _{\Gamma (\zeta )} |\alpha _m f_mRb|^2 dV_{1-n}\right) ^{\frac{p}{2}}\, d\sigma (\zeta )\\&\ \ \ +\sum _{m=0}^\infty \int _{{\tilde{E}}_m} \left( \int _{\Gamma (\zeta )}\left| \sum _{\begin{array}{c} k=1\\ k\ne m \end{array}}^\infty \alpha _k f_kRb\right| ^2 \, dV_{1-n}\right) ^{\frac{p}{2}}\, d\sigma (\zeta ). \end{aligned} \end{aligned}$$

Using the triangle inequalities in $L^2$ and $L^p$ respectively, one has

$$\begin{aligned} \sum _{m=0}^\infty&\int _{{\tilde{E}}_m} \left( \int _{\Gamma (\zeta )}\Big | \sum _{\begin{array}{c} k=1\\ k\ne m \end{array}}^\infty \alpha _k f_kRb\Big |^2 \, dV_{1-n}\right) ^{\frac{p}{2}}\, d\sigma (\zeta )\\&\le \sum _{m=0}^\infty \int _{{\tilde{E}}_m}\left[ \sum _{\begin{array}{c} k=1\\ k\ne m \end{array}}^\infty |\alpha _k|\left( \int _{\Gamma (\zeta )}| f_kRb|^2 \, dV_{1-n}\right) ^{\frac{1}{2}}\, \right] ^p d\sigma (\zeta )\\&\le \sum _{m=0}^\infty \left( \sum _{\begin{array}{c} k=1\\ k\ne m \end{array}}^\infty |\alpha _k|\left[ \int _{{\tilde{E}}_m}\left( \int _{\Gamma (\zeta )}| f_kRb|^2 \, dV_{1-n}\right) ^{\frac{p}{2}}\, d\sigma (\zeta ) \right] ^{\frac{1}{p}}\right) ^p\\&=\sum _{m=0}^\infty \Big (\sum _{\begin{array}{c} k=1\\ k\ne m \end{array}}^\infty |\alpha _k|\,{\tilde{F}}_m(k)^{1/p}\Big )^{p}. \end{aligned}$$

Since $K_1\subset K_k$ for any $k\ge 1$ and ${\tilde{F}}_ 0=K_ 1$, we have ${\tilde{F}}_0(k)\le L_k(k)\lesssim \delta 4^{-2k}\le \delta 4^{-k}$. Therefore, together with the estimates ${\tilde{F}}_m(m)\lesssim 1$ and ${\tilde{F}}_m(k)\lesssim \delta 4^{-k-m}$ for $k\ne m$, we have

$$\begin{aligned} \begin{aligned} \Vert U(\alpha )\Vert _{H^p}^p&\lesssim \sum _{m=1}^\infty |\alpha _{m}|^p{\tilde{F}}_m(m)+ \sum _{m=0}^\infty \Big (\sum _{\begin{array}{c} k=1\\ k\ne m \end{array}}^\infty |\alpha _k|{\tilde{F}}_m(k)^{1/p}\Big )^{p}\\&\lesssim \Vert \alpha \Vert ^p_{\ell ^p} \left( 1+\sum _{m=0}^\infty \left( \sum _{k=1}^\infty (4^{-k-m})^{\frac{1}{p}}\right) ^{p}\right) \lesssim \Vert \alpha \Vert ^p_{\ell ^p}. \end{aligned} \end{aligned}$$

Let us then consider the case $p > 2$. Since $K_1\subset K_k$ for any $k\ge 1$, we have ${\tilde{E}}_0(j,k)\le K_1(j,k)\lesssim \delta ^2 4^{-j-k}$. Our starting point will be the estimate (a consequence of the Hardy-Stein estimates together with (4.1))

$$\begin{aligned} \Vert U(\alpha )\Vert _{H^p}^p&\asymp \int _{{\mathbb {S}}_n}\left( \int _{\Gamma (\xi )}\left| \sum _{j=1}^\infty \alpha _j J_b f_j\right| ^{p-2}\left| \sum _{k=1}^\infty \alpha _k f_k\right| ^2 |Rb|^2 dV_{1-n}\right) d\sigma (\xi )\\&= \sum _{m=0}^\infty \int _{{\tilde{E}}_m}\int _{\Gamma (\xi )}\left| \sum _{j=1}^\infty \alpha _j J_b f_j\right| ^{p-2}d\mu \,d\sigma (\xi ), \end{aligned}$$

where $d\mu = \left| \sum \limits _{k=1}^\infty \alpha _k f_k\right| ^2 |Rb|^2 dV_{1-n}$.

Now assume first that $2 < p \le 3.$ Then

$$\begin{aligned} \left| \sum _{j=1}^\infty \alpha _j J_b f_j\right| ^{p-2} \le \sum _{j=1}^\infty |\alpha _j|^{p-2} |J_b f_j|^{p-2}. \end{aligned}$$

Hence

$$\begin{aligned} \begin{aligned} \Vert U(\alpha )\Vert _{H^p}^p&\asymp \sum _{m=0}^\infty \int _{{\tilde{E}}_m}\int _{\Gamma (\xi )}\left| \sum _{j=1}^\infty \alpha _j J_b f_j\right| ^{p-2}d\mu d\sigma (\xi )\\&\le \sum _{m=1}^\infty |\alpha _m|^{p-2}I(m,m)+\sum _{m=0}^\infty \sum _{\begin{array}{c} j = 1 \\ j \ne m \end{array}}^\infty |\alpha _j|^{p-2}I(m,j), \end{aligned} \end{aligned}$$

(4.12)

where

$$\begin{aligned} I(m,j):=\int _{{\tilde{E}}_m}\int _{\Gamma (\xi )}\left| J_bf_j\right| ^{p-2}d\mu \, d\sigma (\xi ). \end{aligned}$$

Applying the triangle inequality in $L^2$, one has for $m\ge 0$ and $j \ne m$

$$\begin{aligned} I(m,j)&= \int _{{\tilde{E}}_m}\int _{\Gamma (\xi )}\left| J_bf_j\right| ^{p-2}\left| \sum _{k=1}^\infty \alpha _k f_k\right| ^2 |Rb|^2 dV_{1-n}\,d\sigma (\xi )\\&\le \left( \sum _{k=1}^\infty |\alpha _k| \left( \int _{{\tilde{E}}_m}\int _{\Gamma (\xi )}\left| J_bf_j\right| ^{p-2}|f_k|^2 |Rb|^2 dV_{1-n}\,d\sigma (\xi )\right) ^{1/2} \right) ^2\\&=\left( \sum _{k=1}^\infty |\alpha _k| {\tilde{E}}_ m(j,k)^{1/2} \right) ^2\\&\lesssim \delta ^2\Vert \alpha \Vert _{\ell ^p}^2 \left( \sum _{k=1}^\infty 2^{-j-k-m} \right) ^2=\delta ^2\Vert \alpha \Vert _{\ell ^p}^2 4^{-j-m}, \end{aligned}$$

and for $m\ge 1$, using again the triangle inequality in $L^2$,

$$\begin{aligned} I(m,m)&= \int _{{\tilde{E}}_m}\int _{\Gamma (\xi )}\left| J_bf_m\right| ^{p-2}\left| \alpha _m f_m +\sum _{\begin{array}{c} k=1 \\ k \ne m \end{array}}^\infty \alpha _k f_k\right| ^2 |Rb|^2 dV_{1-n}d\sigma (\xi )\\&\le \left( |\alpha _ m| {\tilde{E}}_ m(m,m)^{1/2}+\sum _{\begin{array}{c} k=1 \\ k \ne m \end{array}}^\infty |\alpha _ k| {\tilde{E}}_ m (m,k)^{1/2}\right) ^{2}. \end{aligned}$$

By (4.1) and the Hardy-Stein estimates, we have ${\tilde{E}}_ m(m,m)\lesssim \Vert f_ m\Vert _{H^p}^p\lesssim 1$. Also, by Lemma 4.2, ${\tilde{E}}_ m (m,k)\lesssim \delta ^2 4^{-k-2\,m}$ for $k\ne m$. Hence

$$\begin{aligned} I(m,m)\lesssim \left( |\alpha _m|+\sum _{\begin{array}{c} k=1 \\ k \ne m \end{array}}^\infty \delta 2^{-k-2m}|\alpha _k|\right) ^2 \lesssim \left( |\alpha _m|+\delta \Vert \alpha \Vert _{\ell ^p}4^{-m}\right) ^2. \end{aligned}$$

So, we deduce that

$$\begin{aligned} I(m,j)\lesssim \left\{ \begin{aligned}&\delta ^2\Vert \alpha \Vert _{\ell ^p}^2 4^{-j-m}\ \ \ \text {for}\ j\ne m,\ m\ge 0\ \text {and}\ j\ge 1;\\ \\&\left( |\alpha _m|+\delta \Vert \alpha \Vert _{\ell ^p}4^{-m}\right) ^2 \ \ \ \text {for}\ j=m\ge 1. \end{aligned}\right. \end{aligned}$$

(4.13)

Hence, bearing in mind (4.12) and $|\alpha _ j|\le \Vert \alpha \Vert _{\ell ^p}$, we obtain

$$\begin{aligned} \Vert U\alpha \Vert _{H^p}^p&\lesssim \sum _{m=1}^\infty |\alpha _m|^{p-2}I(m,m)+\sum _{m=0}^\infty \sum _{\begin{array}{c} j = 1 \\ j \ne m \end{array}}^\infty |\alpha _j|^{p-2}I(m,j)\\&\lesssim \sum _{m=1}^\infty |\alpha _m|^{p-2}( |\alpha _m|+4^{-m}\Vert \alpha \Vert _{\ell ^p})^2+ \sum _{m=0}^\infty \sum _{\begin{array}{c} j = 1 \\ j \ne m \end{array}}^\infty |\alpha _j|^{p-2}\Vert \alpha \Vert _{\ell ^p}^2 4^{-j-m}\\&\lesssim \sum _{m=1}^\infty |\alpha _m|^{p-2}( |\alpha _m|^2+4^{-2m}\Vert \alpha \Vert ^2_{\ell ^p})+ \Vert \alpha \Vert _{\ell ^p}^p \lesssim \Vert \alpha \Vert ^p_{\ell ^p}. \end{aligned}$$

Finally, we consider the case $p > 3$. As before, we have

$$\begin{aligned} \Vert U(\alpha )\Vert _{H^p}^p \asymp \sum _{m=0}^\infty \int _{{\tilde{E}}_m}\int _{\Gamma (\xi )}\left| \sum _{j=1}^\infty \alpha _j J_b f_j\right| ^{p-2}d\mu d\sigma (\xi ). \end{aligned}$$

As $p>3$, we can use the triangle inequality in $L^{p-2}$ in order to obtain

$$\begin{aligned} \Vert U(\alpha )\Vert _{H^p}^p&\lesssim \sum _{m = 0}^\infty \left( \sum _{j = 1}^\infty |\alpha _j| \left( \int _{{\widetilde{E}}_m}\int _{\Gamma (\xi )}|J_bf_j|^{p-2}d\mu d\sigma (\xi )\right) ^{\frac{1}{p-2}}\right) ^{p-2}\\&=\sum _{m = 0}^\infty \left( \sum _{j = 1}^\infty |\alpha _j| I(m,j)^{\frac{1}{p-2}}\right) ^{p-2}. \end{aligned}$$

Observe that the estimates obtained in (4.13) for I(m, j) are valid for all $p>2$. Hence

$$\begin{aligned} \Vert U(\alpha )\Vert _{H^p}^p&\lesssim \sum _{m =1}^\infty |\alpha _m|^{p-2} I(m,m) +\sum _{m = 0}^\infty \left( \sum _{\begin{array}{c} j = 1 \\ j \ne m \end{array}}^\infty |\alpha _j| I(m,j)^{\frac{1}{p-2}}\right) ^{p-2}\\&\lesssim \sum _{m =1}^\infty |\alpha _m|^{p-2}\left( |\alpha _m|+\Vert \alpha \Vert _{\ell ^p}4^{-m}\right) ^2\\&+\Vert \alpha \Vert _{\ell ^p}^p \sum _{m = 0}^\infty \left( \sum _{\begin{array}{c} j = 1 \\ j \ne m \end{array}}^\infty (4^{-j-m})^{\frac{1}{p-2}}\right) ^{p-2}\\&\lesssim \Vert \alpha \Vert _{\ell ^p}^p. \end{aligned}$$

Hence U is bounded from $\ell ^p$ into $H^p$ for $1 \le p <\infty .$ $\square$

Lemma 4.5

Let $1 \le p < \infty$, $b \in BMOA({\mathbb {B}}^n)$, and $(f_k) \subset H^p$ be such that $\Vert f_k\Vert _{H^p} = 1$ for all k and $(f_k)$ converges to zero uniformly on compact subsets of ${\mathbb {B}}^n$. Assume also that $\inf _k \Vert J_bf_k\Vert _{H^p} \asymp 1.$ Then there exists a subsequence, still denoted by $(f_k)$, such that the linear mapping

$$\begin{aligned} U:\ell ^p \rightarrow H^p, \, U(\alpha ) = \sum _{k = 1}^\infty \alpha _k J_bf_k, \end{aligned}$$

where $\alpha = (\alpha _k) \in \ell ^p$, is bounded below.

Proof

The proof is also divided into three cases depending on the value of p, namely cases $1 \le p \le 2,$ $2 < p \le 3$ and $3< p < \infty$. For $0<\delta <1$, which will be determined later, we choose a subsequence of $\{f_k\}$ denoted still by $\{f_k\}$, a decreasing sequence $\varepsilon _m > 0, \, \varepsilon _m \rightarrow 0,$ 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

$$\begin{aligned} \Vert U(\alpha )\Vert _{H^p}^p&\asymp \int _{{\mathbb {S}}_n}\left( \int _{\Gamma (\xi )}\left| \sum _{j=1}^\infty \alpha _j J_b f_j\right| ^{p-2}\left| \sum _{k=1}^\infty \alpha _k f_k\right| ^2 |Rb|^2 dV_{1-n}\right) d\sigma (\xi )\\&\ge \sum _{m=1}^\infty \int _{{\tilde{E}}_m}\int _{\Gamma (\xi )}\left| \sum _{j=1}^\infty \alpha _j J_b f_j\right| ^{p-2}d\mu d\sigma (\xi ), \end{aligned}$$

where $d\mu = \left| \sum \limits _{k=1}^\infty \alpha _k f_k\right| ^2 |Rb|^2 dV_{1-n}$.

For the case $p > 3$, using the standard estimate $(a+b)^q \le 2^{q-1}(a^q+b^q),$ where $a, b \ge 0$ and $q \ge 1$, we obtain

$$\begin{aligned}&\Vert U(\alpha )\Vert _{H^p}^p \gtrsim \sum _{m=1}^\infty \int _{{\tilde{E}}_m}\int _{\Gamma (\xi )}\Big | \alpha _m J_bf_m+\sum _{\begin{array}{c} j = 1 \\ j \ne m \end{array}}^\infty \alpha _j J_bf_j\Big |^{p-2}d\mu d\sigma (\xi )\\&\ge \sum _{m=1}^\infty \int _{{\tilde{E}}_m} \int _{\Gamma (\xi )}\left( 2^{3-p}\Big |\alpha _m J_bf_m\Big |^{p-2}-\Big |\sum _{\begin{array}{c} j = 1 \\ j \ne m \end{array}}^\infty \alpha _j J_bf_j\Big |^{p-2}\right) d\mu \,d\sigma (\xi )\\&\ge 2^{3-p}\sum _{m=1}^\infty |\alpha _m|^{p-2} I(m,m) -\sum _{m =1}^\infty \Big (\sum _{\begin{array}{c} j = 1 \\ j \ne m \end{array}}^\infty |\alpha _j| I(m,j)^{\frac{1}{p-2}}\Big )^{p-2}. \end{aligned}$$

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

$$\begin{aligned} \sum _{m =1}^\infty \Big (\sum _{\begin{array}{c} j = 1 \\ j \ne m \end{array}}^\infty |\alpha _j| I(m,j)^{\frac{1}{p-2}}\Big )^{p-2}\le C_ 1 \delta ^2 \Vert \alpha \Vert ^p_{\ell ^p} \end{aligned}$$

for some positive constant $C_ 1$. Hence,

$$\begin{aligned} \Vert U(\alpha )\Vert _{H^p}^p \gtrsim 2^{3-p}\sum _{m=1}^\infty |\alpha _m|^{p-2} I(m,m) -C_ 1 \delta ^2 \Vert \alpha \Vert ^p_{\ell ^p}. \end{aligned}$$

(4.14)

Now, the trivial estimate (a consequence of the inequality $|A-B|^2 \le 2 (|A|^2+|B|^2)$)

$$\begin{aligned} \big | \sum _{k=1}^{\infty } \alpha _ k \,f_ k \big |^2 \ge \frac{1}{2} \,|\alpha _ m \,f_ m |^2 -\big | \sum _{k\ne m} \alpha _ k \,f_ k |^2, \end{aligned}$$

gives

$$\begin{aligned} \begin{aligned} I(m,m)&= \int _{{\tilde{E}}_m}\int _{\Gamma (\xi )}\left| J_bf_m\right| ^{p-2}|\sum _{k = 1}^\infty \alpha _k f_k|^2 |Rb|^2 dV_{1-n}\,d\sigma (\xi )\\&\ge \frac{1}{2}\,|\alpha _ m|^2\, {\tilde{E}}_ m (m,m)-B(m), \end{aligned} \end{aligned}$$

with

$$\begin{aligned} B(m):=\int _{{\tilde{E}}_m}\int _{\Gamma (\zeta )}\left| J_bf_m\right| ^{p-2}\big |\sum _{\begin{array}{c} k = 1 \\ k \ne m \end{array}}^\infty \alpha _k f_k \big |^2 |Rb|^2 dV_{1-n}\,d\sigma (\zeta ). \end{aligned}$$

As before, we use the triangle inequality in $L^2$ and the estimates ${\tilde{E}}_m(m,k)\lesssim \delta ^2 4^{-k-2\,m}$ for $k\ne m$ to obtain

$$\begin{aligned} B(m)\le \Big (\sum \limits _{\begin{array}{c} k = 1 \\ k \ne m \end{array}}^\infty |\alpha _ k|\,{\tilde{E}}_m(m,k)^{\frac{1}{2}}\Big )^{2} \le C_ 2 \delta ^2 \, \Vert \alpha \Vert ^2_{\ell ^p}\,4^{-2m} \end{aligned}$$

for some positive constant $C_ 2$. As ${\tilde{E}}_m(m,m) \asymp \Vert J_ b f_ m \Vert _{H^p}^p \ge d^p$, where $d=\inf _{k \in {\mathbb {N}}} \Vert J_b f_k\Vert _{H^p}$, we have that

$$\begin{aligned} I(m,m)\ge \frac{C_ 3}{2}\, |\alpha _ m|^2 \,d^p -C_ 2 \delta ^2 \, \Vert \alpha \Vert ^2_{\ell ^p}\,4^{-2m}, \end{aligned}$$

where $C_ 3$ is another positive constant. Putting the previous estimates in (4.14), we obtain

$$\begin{aligned} \begin{aligned} \Vert U(\alpha )\Vert _{H^p}^p&\gtrsim 2^{2-p}C_ 3d^p\sum _{m=1}^\infty |\alpha _m|^{p} -C_ 2 2^{3-p}\delta ^2 \, \Vert \alpha \Vert ^2_{\ell ^p}\,\sum _{m=1}^{\infty } |\alpha _ m|^{p-2}\,4^{-2m}\\&-C_ 1 \delta ^2 \Vert \alpha \Vert ^p_{\ell ^p}\\&\ge 2^{2-p}C_ 3d^p\Vert \alpha \Vert ^p_{\ell ^p} -C_ 2 2^{3-p}\delta ^2 \, \Vert \alpha \Vert ^p_{\ell ^p} -C_ 1 \delta ^2 \Vert \alpha \Vert ^p_{\ell ^p}\\&\ge 2^{1-p}C_ 3d^p\Vert \alpha \Vert ^p_{\ell ^p} \end{aligned} \end{aligned}$$

after taking $\delta >0$ small enough. Hence U is bounded below.

Let now $2 < p \le 3.$ The preceding estimates together with (4.13) imply that

$$\begin{aligned}&\Vert U\alpha \Vert _{H^p}^p \gtrsim \sum _{m=1}^\infty \int _{{\tilde{E}}_m}\int _{\Gamma (\xi )}\left| \sum _{j=1}^\infty \alpha _j J_b f_j\right| ^{p-2}d\mu d\sigma (\xi )\\&\ge \sum _{m=1}^\infty |\alpha _m|^{p-2} I(m,m) -\sum _{m =1}^\infty \sum _{\begin{array}{c} j = 1 \\ j \ne m \end{array}}^\infty |\alpha _j|^{p-2} I(m,j)\\&\ge \sum _{m=1}^\infty \left( \frac{C_ 3\,d^p}{2}|\alpha _m|^p-C_ 2 \delta ^2 \Vert \alpha \Vert _{\ell ^p}^p 4^{-2m}\right) -C_ 4\delta ^2\Vert \alpha \Vert _{\ell ^p}^p\sum _{m =1}^\infty \sum _{\begin{array}{c} j = 1 \\ j \ne m \end{array}}^{\infty } 4^{-j-m}\\&\gtrsim d^p\,\Vert \alpha \Vert _{\ell ^p}^p, \end{aligned}$$

whenever $\delta$ is small enough.

Finally, it remains to deal with the case $1 \le p \le 2.$ By the area theorem, we have

$$\begin{aligned} \begin{aligned} \Vert U\alpha \Vert _{H^p}^p&\asymp \int _{{\mathbb {S}}_n} \left( \int _{\Gamma (\xi )}\left| \sum _{k = 1}^\infty \alpha _k f_k \right| ^2 |Rb|^2\,dV_{1-n}\right) ^{\frac{p}{2}}\, d\sigma (\xi )\\&\ge \sum _{m=1}^{\infty } \int _{{\tilde{E}}_ m} \left( \int _{\Gamma (\xi )}\left| \sum _{k = 1}^\infty \alpha _k f_k \right| ^2 |Rb|^2\,dV_{1-n}\right) ^{\frac{p}{2}}\, d\sigma (\xi ). \end{aligned} \end{aligned}$$

Applying the $L^2$ triangle inequality, we get

$$\begin{aligned}&\left( \int _{\Gamma (\xi )}\left| \sum _{k = 1}^\infty \alpha _k f_k \right| ^2 |Rb|^2\,dV_{1-n}\right) ^{\frac{p}{2}}\\ \ge&\left| \left( \int _{\Gamma (\xi )}|\alpha _m f_m|^2 |Rb|^2 dV_{1-n}\right) ^{1/2}-\left( \int _{\Gamma (\xi )}\Big |\sum _{\begin{array}{c} k =1 \\ k \ne m \end{array}}^{\infty } \alpha _k f_k\Big |^2 |Rb|^2 dV_{1-n}\right) ^{1/2}\right| ^p . \end{aligned}$$

Then, use the $L^p$ triangle inequality to obtain

$$\begin{aligned}&\int _{{\tilde{E}}_ m} \left( \int _{\Gamma (\xi )}\left| \sum _{k = 1}^\infty \alpha _k f_k \right| ^2 |Rb|^2\,dV_{1-n}\right) ^{\frac{p}{2}}\, d\sigma (\xi )\\&\ge \left| \alpha _m {\tilde{F}}_m(m)^{1/p}-\left( \int _{{\tilde{E}}_ m}\left( \int _{\Gamma (\xi )}\Big |\sum _{\begin{array}{c} k =1 \\ k \ne m \end{array}}^{\infty } \alpha _k f_k\Big |^2 |Rb|^2 dV_{1-n}\right) ^{p/2}d\sigma (\xi )\right) ^{1/p}\right| ^p. \end{aligned}$$

This, together with the $\ell ^p$ triangle inequality, gives

$$\begin{aligned} \begin{aligned} \Vert U\alpha \Vert _{H^p}^p&\gtrsim \sum _{m=1}^{\infty }\left| \alpha _m {\tilde{F}}_m(m)^{1/p}-A(m)^{1/p}\right| ^p\\&\ge \left| \left( \sum _{m=1}^\infty |\alpha _m|^p {\tilde{F}}_m(m)\right) ^{1/p} -\left( \sum _{m=1}^\infty A(m)\right) ^{1/p}\right| ^p, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} A(m):=\int _{{\tilde{E}}_ m}\left( \int _{\Gamma (\xi )}\Big |\sum _{\begin{array}{c} k =1 \\ k \ne m \end{array}}^{\infty } \alpha _k f_k\Big |^2 |Rb|^2 dV_{1-n}\right) ^{p/2}d\sigma (\xi ). \end{aligned}$$

By Lemma 4.3 we have ${\tilde{F}}_ m(k)\le C_ 5 \delta 4^{-k-m}$ for $k\ne m$. Thus, by the estimates obtained in the proof of the case $1\le p\le 2$ in Lemma 4.4, we have

$$\begin{aligned} \begin{aligned} \sum _{m=1}^{\infty } A(m)&\le \sum _{m=1}^\infty \Big (\sum _{\begin{array}{c} k=1\\ k\ne m \end{array}}^\infty |\alpha _k|\,{\tilde{F}}_m(k)^{1/p}\Big )^{p} \\&\le C_ 5 \delta \sum _{m=1}^\infty \left( \sum _{k =1}^\infty ( 4^{-k-m})^{1/p}\right) ^p \Vert \alpha \Vert _{\ell ^p} \le C_ 6 \delta \,\Vert \alpha \Vert ^p_{\ell ^p}. \end{aligned} \end{aligned}$$

By the area theorem, we have ${\tilde{F}}_ m(m)\asymp \Vert J_ b f_ m\Vert _{H^p}^p$. Hence, there is a positive constant $C_ 7$ such that ${\tilde{F}}_ m(m)\ge C_ 7 d^p$, where $d=\inf _{k \in {\mathbb {N}}} \Vert J_b f_k\Vert _{H^p}$. Therefore we obtain the desired lower bound

$$\begin{aligned} \Vert U\alpha \Vert _{H^p}&\gtrsim C_ 7^{1/p} d\,\Vert \alpha \Vert _{\ell ^p} - \big (C_ 6 \delta \big )^{1/p} \Vert \alpha \Vert _{\ell ^p} \gtrsim d \,\Vert \alpha \Vert _{\ell ^p}, \end{aligned}$$

whenever $\delta$ is small enough. $\square$

Now we are ready to prove our second main result.

Proof of Theorem 1.2

Let us recall that the Cauchy–Szegő kernel

$$\begin{aligned} K_z(\xi )=(1-\langle \xi ,z\rangle )^{-n}=\sum _{j=0}^\infty c_j \langle \xi ,z\rangle ^j. \end{aligned}$$

reproduces $H^p$ for $1<p<\infty$, and that, by the Forelli–Rudin estimates, its $H^{p'}$ norm is dominated by $C_p (1-|z|^2)^{-n/p}$. Given a multi-index $\alpha =(\alpha _1,\alpha _2,...,\alpha _n)$, let us consider the functional $\lambda _\alpha$ defined by

$$\begin{aligned} \lambda _\alpha (f)=\int _{{\mathbb {S}}^n}f(\xi )\overline{\xi ^\alpha }d\sigma (\xi ). \end{aligned}$$

Observe that for a finite collection of multi-indices $\alpha$, the intersection of their kernels has finite codimension in the infinite-dimensional space M. Therefore, as in page 5 of [9], we can pick a normalized sequence $(g_m)\subset M$ such that

$$\begin{aligned} g_m \in \bigcap _{|\alpha |\le m}\ker \lambda _\alpha . \end{aligned}$$

Such $g_m$ is orthogonal to $\langle \xi ,z\rangle ^k$ for all $0\le k \le m$ and $z \in {\mathbb {B}}^n$, and it follows that

$$\begin{aligned} K_z^{m+1}(\xi )=\sum _{j=m+1}^\infty c_j \langle \xi ,z\rangle ^j=(1-\langle \xi ,z\rangle )^{-n}-\sum _{j=0}^m c_j \langle \xi ,z\rangle ^j \end{aligned}$$

reproduces $g_m$. We can write

$$\begin{aligned} K_z^m(\xi )=\langle \xi ,z\rangle ^{m+1}p_m(\xi ,z)K_z(\xi ), \end{aligned}$$

where $p_m(\xi ,z)$ is a polynomial holomorphic in $\xi$ and anti-holomorphic in z. Now, if $0\le |z|\le r<1$, then the $H^{p'}$ norm of $K_z^m$ is dominated by $C_p r^{m+1} \Vert p_m\Vert _{\infty }(1-r^2)^{-n/p}$. By using, say, the binomial theorem to estimate $p_m$, we see that $\lim _{m\rightarrow \infty }r^{m+1}\Vert p_m\Vert _\infty =0$. By Hölder’s inequality and the reproducing formula with $K_z^m$, this shows that $g_m\rightarrow 0$ uniformly on compact subsets of ${\mathbb {B}}_n$.

By Lemmas 4.4 and 4.5, we can find a subsequence $(g_{m_k})$ such that the operator $U:\ell ^p \rightarrow H^p$ given by  

$$\begin{aligned} U\alpha = \sum _{k=1}^\infty \alpha _k J_b g_{m_k} \end{aligned}$$

is an isomorphism onto its range $J_bN$, where $N = \overline{span \{g_{m_k}}\}$. Then the subspace N is isomorphic to $\ell ^p$ and in particular the operator $J_b$ is $\ell ^2$-singular for $p \ne 2$. $\square$

Remark 4.6

(1) Due to the fact that the sequence spaces $\ell ^p$ and $\ell ^q$ are totally incomparable for $p\ne q$ whenever $1\le p,q<\infty$, Theorem 1.2 also implies that $J_b\in S_{q}(H^p)$ for $1\le p,q<\infty$, $p \ne q$, and $b\in BMOA({\mathbb {B}}^n)$.

(2) It is known that the standard Bergman spaces $A_{\alpha }^{p}$, $\alpha >-1$, are isomorphic to $\ell ^p$. If $1\le p<\infty$, the strict singularity of the operator $J_b$ on $A_{\alpha }^{p}$ coincides with the compactness, since all strictly singular operators on $\ell ^p$ are compact. In particular, we have $J_b\in S_{q}(A_{\alpha }^{p})$ for $1\le p,q<\infty$, $p \ne q$, and b being in the Bloch space.