On an Interpolation Problem for the Classical Weighted Harmonic Bergman Space

Abstract

We provide sufficient conditions for interpolating a sequence by a function belonging to the classical weighted harmonic Bergman space defined on the unit real ball.

1 Introduction

The purpose of the article is to give an interpolation theorem in a weighted harmonic Bergman space and our result provides an answer of a question raised by Rochberg in the fourth section, comments B of [14], see below. We recall that the problem of interpolating a sequence by a function belonging to the generalized Bergman space has been studied by R. Rochberg, see Section II of [14] and let us recall this result. Let D be a symmetric homogeneous domain in ${\mathbb {C}}^n$, i.e., D could be the unit complex ball, the unit polydisc or the product of half planes for a precise definition see, e.g., [6], and $A^{p,r}(D)$ be the space of holomorphic functions on D such that $\displaystyle ||f||^p_{L_{p,r}(D)}=\int _D|f(z)|^p(B_D(z,z))^{-r}dm(z)$ is finite for $f\in A^{p,r}(D)$ where dm(z) is the Lebesgue measure in ${\mathbb {C}}^n$, $p>0$, $B_D(\cdot ,\cdot )$ represents the Bergman kernel function of D, e.g., when D is the unit complex ball $B_D(\xi ,\eta )=\displaystyle \frac{1}{Vol(D)(1-\xi {\overline{\eta }})^{n+1}}$ for $(\xi ,\eta )\in D^2$ and Vol(D) represents the volume of D, and $r>-\varepsilon _D$ where $\varepsilon _D$ is a constant relying on D. Next, Rochberg considers the $l_\infty (S)$-valued map T on $A^{p,r}(D)$ defined by $[Tf](\zeta _k)=[B_D(\zeta _k,\zeta _k)]^{-\frac{1+r}{p}}f(\zeta _k)$ for $f\in A^{p,r}(D)$ and $S=(\zeta _l)_{l\in {\mathbb {N}}}$ is a sequence of points in D, and $l_\infty (S)$ is the space of bounded sequences.

Let $d(\cdot ,\cdot )$ be the invariant distance map on D and defined by $d(a,b)=|\varphi _{a}(b)|$, the invariant distance between a and b where $\varphi _{a}(\cdot )$ is the involutive automorphism on D such that $\varphi _a(0)=a$ and $\varphi _a(a)=0$, for more properties of $\varphi _a(\cdot )$, see Theorem 2.2.2 in [16]. Then, R. Rochberg states that if $K= \displaystyle \inf _{i\ne j}d(\zeta _i,\zeta _j)$ is large enough, then T maps $A^{p,r}(D)$ onto $l_p(S)$, the space of p-Lebesgue sequences. To be precise, he states.

Theorem 1.1

(Theorem in Section II of [14]) Suppose that D is biholomorphically equivalent to a bounded symmetric domain. Suppose p, r, S are given and S satisfies $\displaystyle \inf _{i\ne j}d(\zeta _i,\zeta _j)=K>0$. Then T is a continuous map of $A^{p,r}(D)$ into $l_p(S)$. That is, there is a constant $\mathcal {K}$ such that for all $f\in A^{p,r}(D)$

$$\begin{aligned} ||Tf||_{l_p(S)}^p:=\sum _{k\in \mathbb N}|[B_D(\zeta _k,\zeta _k)]^{-\frac{1+r}{p}}f(\zeta _k)|^p\le \mathcal {K}||f||_{L_{p,r}(D)}^p. \end{aligned}$$

There is a $K_0$ so that if K is larger than $K_0$ then T is onto $l_p(S)$. In fact, there is a continuous linear map R of $l_p(S)$ into $A^{p,r}(D)$ so that $TR=I_{l_p(S)}$.

Concerning of the proof of Theorem 1.1, the author employs Schur’s lemma [7] to show that if the separation is large enough then a natural candidate for interpolation is close to the desired objective.

In the fourth section, comments B of [14], Rochberg asks on the validity of Theorem 1.1 for harmonic functions on bounded domains in ${\mathbb {R}}^n$. Therefore, Choe and Yi [2, Theorem 5.5], Tanaka [17, Theorem 1.2] resolve partially the validity of Theorem 1.1 in [14]. To be precise, they provide interpolation theorem just for the classical harmonic Bergman space and let us recall their respective results.

Choe and Yi state an interpolation theorem of harmonic Bergman functions on H, the upper half-space $\mathbb R^{n-1}\times {\mathbb {R}}_+$ with $n\ge 2$. First, they define that a sequence $(z_m)_{m\in {\mathbb {N}}}$ in H is $\delta$-separated for $\delta \in (0,1)$ if the pseudohyperbolic balls centred at $z_m$ and of radius $\delta$ are pairwise disjoint and consider $b^p$, the space of real-valued harmonic functions belonging to $L^p(H)$, the space of p-integrable functions on H, and consider $T_k$ the $l_p$-valued linear operator on $b^p$ defined by

$$\begin{aligned} (1):\ T_ku=\left( z_{mn}^{n/p+k}D^ku(z_m)\right) _{n\in \mathbb N}\quad \text {for}\quad u\in b^p, \end{aligned}$$

such that $D^ku(z_m)$ represents the kth-derivative of u w.r.t the last variable belonging to ${\mathbb {R}}_+$ and $(z_m)_{m\in \mathbb N}$ is a sequence in H which is called $b_p$-interpolation sequence of order k when $T_k(b_p)=l_p$. Then, they state the following interpolation theorem.

Theorem 1.2

[2, Theorem 5.5] Let $1\le p<\infty$ and $k\ge 0$ be an integer. Then there exists a positive number $\delta _0$ with the following property: Let $\{z_m\}$ be a $\delta$-separated sequence with $\delta >\delta _0$ and let $T_k:b^p\rightarrow l_p$ be the associated linear operator as in (1). Then there is a bounded linear operator $S_k:l_p\rightarrow b^p$ such that $T_kS_k$ is the identity on $l_p$. In particular, $\{z_m\}$ is a $b^p$-interpolating sequence of order k.

Later, Tanaka considers a smooth bounded domain $\Omega$ in the n-dimensional Euclidean space and defines the operator $V=V_{p,\{\lambda _j\}}$ on $b^p(\Omega )$ with value $l_p$ as $Vf=\left( [r(\lambda _k)]^{n/p}f(\lambda _k)\right) _{k\in {\mathbb {N}}}\in l_p$ for $f\in b^p(\Omega )$ such that $r(\cdot )$ stands for the boundary distance and $(\lambda _k)_{k\in {\mathbb {N}}}$ is a sequence of points in $\Omega$ and by using estimations of the harmonic Bergman kernel on $\Omega \times \Omega$ [11], he states the following interpolation theorem

Theorem 1.3

[17, Theorem 1.2] Let $1<p<\infty$. Then, we can choose a sequence $\{\lambda _j\}$ in $\Omega$ such that $V:b^p(\Omega )\rightarrow l_p$ is bounded and onto.

The purpose of the present article is to provide a positive answer of the Rochberg question mentioned above by providing an interpolation theorem in a weighted harmonic Bergman space. Precisely, we furnish sufficient conditions for interpolating a sequence by a function belonging to a weighted harmonic Bergman space on ${\mathbb {B}}^n$, the Euclidean unit ball of ${\mathbb {R}}^n$, see below Theorem 4.1. Instead of using the Bergman kernel, we employ essentially properties of the reproducing kernel for a suitable Hilbert weighted harmonic Bergman space.

1.1 The Structure of the Article

Preliminaries focus on the definition of the reproducing kernel for a suitable Hilbert weighted harmonic Bergman space, definitions of Carleson measure, interpolating and separated sequences. The third section focuses on miscellaneous lemmas helping to prove the main interpolation theorem (Theorem 4.1). The fourth section concentrates on the statement and the proof of Theorem 4.1.

2 Preliminaries

Let $(L_{p,\alpha }({\mathbb {B}}^n,d\mu _\alpha (x)), ||\cdot ||_{p,\alpha ,{\mathbb {B}}^n})$ be the space of p-integrable functions on $\Omega$, w.r.t the measure $d\mu _\alpha (x)=(1-|x|^2)^\alpha dx$ for $(\alpha ,p)\in (-1,\infty )\times [1,\infty )$, and $||\cdot ||_{p,\alpha ,\mathbb B^n}$ represents its associated norm and defined by

$$\begin{aligned} \displaystyle ||f||_{p,\alpha ,{\mathbb {B}}^n}=\left( \int _{{\mathbb {B}}^n} |f(x)|^pd\mu _\alpha (x)\right) ^{1/p},\quad \text {for}\quad f\in L_{p,\alpha }({\mathbb {B}}^n,d\mu _\alpha (x)). \end{aligned}$$

Let $Har({\mathbb {B}}^n)$ be the set of harmonic functions on ${\mathbb {B}}^n$ and $\mathcal {A}_{Har}^{p,\alpha }({\mathbb {B}}^n)$ be the space of weighted harmonic Bergman space on ${\mathbb {B}}^n$, i.e., the intersection of $Har({\mathbb {B}}^n)$ with $L_{p,\alpha }({\mathbb {B}}^n, d\mu _\alpha (x))$. Let $x\in {\mathbb {B}}^n$, there is $b_\alpha (x,\cdot )\in \mathcal {A}_{Har}^{2,\alpha }({\mathbb {B}}^n)$ and called the reproducing kernel for the Hilbert space $\mathcal {A}_{Har}^{2,\alpha }({\mathbb {B}}^n)$ such that $\displaystyle u(x)=\int _{{\mathbb {B}}^n}u(y)b_\alpha (x,y)d\mu _\alpha (y)$ for $u\in \mathcal {A}_{Har}^{2,\alpha }({\mathbb {B}}^n)$ and $u\rightarrow u(x)$ are bounded point evaluation functionals on $\mathcal {A}_{Har}^{p,\alpha }({\mathbb {B}}^n)$ for $p\ge 1$. We recall that the expression of $b_\alpha (x,y)$ is a little bit hard to express it in a closed form, i.e., given in terms of suitable elementary functions and only involve a finite number of such functions and preferably just one, e.g., in the unweighted situation, $b_0(x,y)$ is furnished explicitly as $\displaystyle b_0(x,y)=\frac{(n-4)|x|^4|y|^4+(8x\cdot y-2n-4)|x|^2|y|^2+n}{n|{\mathbb {B}}^n|_{Leb}(1-2x\cdot y+|x|^2|y|^2)^{n/2+1}}$ where $|{\mathbb {B}}^n|_{Leb}$ stands for the Lebesgue measure of ${\mathbb {B}}^n$ [1, Theorem 8.13]. However, the reproducing kernels for $L_{2,h}(D,c_\alpha (1-|z|^2)^\alpha )$, the weighted Hilbert holomorphic Bergman space in D (the unit complex ball) and $c_\alpha$ is the normalizing constant, is expressed in a closed form as $\displaystyle \frac{1}{(1-z{\overline{w}})^{\alpha +n+1}}$ for $(z,w)\in D^2$ [5, §4.6, p. 96].

We stress that $b_\alpha (x,y)$ has been determined in several places but not in a closed form, e.g, [3, 9, 12]. E.,g. Miao provides an expression of $b_\alpha (x,y)$ for $(x,y)\in \mathbb B^n\times {\mathbb {B}}^n$ in terms of a series [12, Proposition 3]. Precisely, he states that for $\alpha >-1$, we have

$$\begin{aligned} \displaystyle b_\alpha (x,y)=\frac{2}{n|{\mathbb {B}}^n|_{Leb}\Gamma (\alpha +1)} \sum _{m=0}^\infty \frac{\Gamma (m+\frac{n}{2}+\alpha +1)}{\Gamma (m+\frac{n}{2})} Z_m(x,y), \end{aligned}$$

such that for $\xi \in {\mathbb {S}}^{n-1}$, the unit $(n-1)$-sphere, $Z_m(\xi ,\cdot )\in \mathcal {H}_m({\mathbb {S}}^{n-1})$, the space of homogeneous harmonic polynomials on ${\mathbb {S}}^{n-1}$ of degree m, i.e., for $(x=|x|\xi ,y=|y|\eta )\in {\mathbb {B}}^n\times {\mathbb {B}}^n$ for $\eta \in {\mathbb {S}}^{n-1}$, we obtain $Z_m(x,y)=|x|^m|y|^mZ_m(\xi ,\eta )$ and we call $Z_m(\xi ,\cdot )$ the zonal harmonic of degree m, $\Gamma (\cdot )$ is the standard gamma function.

Let us state some useful definitions.

Definition 2.1

We say that $\nu$, a positive Borel measure on ${\mathbb {B}}^n$, is a Carleson measure for $\mathcal {A}_{Har}^{p,\alpha }({\mathbb {B}}^n)$ if

$$\begin{aligned} \int _{{\mathbb {B}}^n}|f(x)|^pd\nu (x)\lesssim ||f||_{p,\alpha ,\mathbb B^n}^p, \quad \text {for all}\quad f\in \mathcal {A}_{Har}^{p,\alpha }(\mathbb B^n). \end{aligned}$$

Definition 2.2

Let $\mathcal {S}=(a_k)_{k\in {\mathbb {N}}}$ be a sequence in $\mathbb B^n$, we say that $\mathcal {S}$ is $\mathcal {A}_{Har}^{p,\alpha }({\mathbb {B}}^n)$-interpolating sequence if for any real-valued sequence $u=(u_k)_{k\in {\mathbb {N}}}\in l_{p,\mathcal {S},\alpha }$, i.e.,

$$\begin{aligned} ||u||^p_{l_{p,\mathcal {S},\alpha }}=\displaystyle \sum _{k\in \mathbb N}(1-|a_k|^2)^\alpha |u_k|^p<\infty , \end{aligned}$$

there is a function $h\in \mathcal {A}^{p,\alpha }_{Har}({\mathbb {B}}^n)$ such that $h(a_k)=u_k\cdot$

Definition 2.3

Let $\rho \in (0,1)$ and $E_\rho (x)=\{y\in \mathbb B^n:|y-x|<\rho (1-|x|)\}$ for $x\in {\mathbb {B}}^n$. We say that a sequence $(a_k)_{k\in {\mathbb {N}}}$ in ${\mathbb {B}}^n$ is $\rho$-separated if the family $\left( E_\rho (a_k)\right) _{k\in {\mathbb {N}}}$ are pairwise disjoint.

Remark 2.4

The family $\left( E_{\frac{\rho }{3}}(a_k)\right) _{k\in {\mathbb {N}}}$ covers ${\mathbb {B}}^n$, see Lemma 4 in [12].

We write $A\lesssim B$ (or equivalently $B\gtrsim A$) when A is less, up to a multiplicative positive constant, to B, and $A\approx B$ when $A\lesssim B$ and $A\gtrsim B$.

3 Miscellaneous Lemmas

This section contains three meaningful lemmas. The first one is a known result on the boundedness of the harmonic Bergman projection corresponding to $b_\alpha (x,y)$, this lemma is a corollary of [8, Theorem 1.4], the second lemma states a sufficient condition for the convergence of the series $\displaystyle \sum _{j:j\ne k}|b_\alpha (a_k,a_j)|(1-|a_j|^2)^\alpha$ for a sequence $(a_k)_{k\in {\mathbb {N}}}$ in ${\mathbb {B}}^n$, and the third one states that the set:

$$\begin{aligned} \mathcal {H}_{\alpha ,\mathcal {S}}:=\left\{ f_v(x)=\displaystyle \sum _{j\in {\mathbb {N}}}v_j\frac{b_\alpha (x,a_j)}{b_\alpha (a_j,a_j)}:v =(v_j)_{j\in {\mathbb {N}}}\in l_{p,\mathcal {S},\alpha } \text { and }x\in {\mathbb {B}}^n\right\} , \end{aligned}$$

is included in $\mathcal {A}^{p,\alpha }_{Har}({\mathbb {B}}^n)$ whenever for $p>1$ and $\alpha >-1$.

Lemma 3.1

Let $p\ge 1$ and $\alpha >-1$, then the operator $\mathcal {T}$ defined on $L_{p,\alpha }({\mathbb {B}}^n, d\mu _\alpha (x))$ by

$$\begin{aligned} \displaystyle \mathcal {T}[g](x)=\int _{{\mathbb {B}}^n} b_\alpha (x,y)g(y)d\mu _\alpha (y) \end{aligned}$$

(3.1)

is bounded from $L_{p,\alpha }({\mathbb {B}}^n, d\mu _\alpha (x))$ to $\mathcal {A}_{Har}^{p,\alpha }({\mathbb {B}}^n)$.

Lemma 3.2

Let $\alpha >-1$ and $\delta _{a_k}$ be the Dirac measure at $a_k\in {\mathbb {B}}^n$. If $\displaystyle \nu =\sum _{k\in \mathbb N}(1-|a_k|^2)^\alpha \delta _{a_k}$ is a Carleson measure for $\mathcal {A}_{Har}^{1,\alpha }({\mathbb {B}}^n)$, then the series $\displaystyle \sum _{j:j\ne k}|b_\alpha (a_k,a_j)|(1-|a_j|^2)^\alpha$ converges for all $k\in {\mathbb {N}}$.

Proof

Miao has shown in [13, Proposition 4] that there is a constant $\mathfrak {C}$ such that

$$\begin{aligned} |b_\alpha (x,y)|\le \mathfrak {C}(1-|x||y|)^{-n-\alpha }\quad \text {for}\quad (x,y)\in {\mathbb {B}}^{2n}. \end{aligned}$$

(3.2)

Let us show that $b_\alpha (a_k,\cdot )\in \mathcal {A}_{Har}^{1,\alpha }({\mathbb {B}}^n)$ for $a_k\in {\mathbb {B}}^n$. Therefore, by employing (3.2) and spherical coordinates, we have

$$\begin{aligned}{} & {} \int _{{\mathbb {B}}^n}|b_\alpha (a_k,x)|(1-|x|^2)^\alpha dx \le \mathfrak {C}\int _{{\mathbb {B}}^n}(1-|a_k||x|)^{-n-\alpha }(1-|x|^2)^\alpha dx\nonumber \\{} & {} \quad <\mathfrak {C}\max (2^\alpha ,1)\int _{{\mathbb {B}}^n}(1-|a_k||x|)^{-n-\alpha }(1-|x|) ^\alpha dx\end{aligned}$$

(3.3)

$$\begin{aligned}{} & {} \quad \le 2\pi \mathfrak {C}\max (2^\alpha ,1)\int _0^1(1-|a_k|r)^{-n-\alpha }(1-r)^\alpha dr\nonumber \\{} & {} \quad \le \mathfrak {C}\mathfrak {K}\max (2^\alpha ,1) (1-|a_k|)^{1-n}. \end{aligned}$$

(3.4)

To obtain (3.4), we have used a partial integration where $\mathfrak {K}$ is a positive constant and which could be finite or infinite and the fact Inequality (3.3) is strict, thus we get

$$\begin{aligned} \int _{{\mathbb {B}}^n}|b_\alpha (a_k,x)|(1-|x|^2)^\alpha dx<\infty . \end{aligned}$$

(3.5)

We stress that we can show Inequality (3.5) without using (3.2) this by observing that $\mathcal {A}_{Har}^{2,\alpha }({\mathbb {B}}^n)\subset \mathcal {A}_{Har}^{1,\alpha }({\mathbb {B}}^n)$, it is this due to the fact that the measure $d\mu _\alpha (x)$ is finite on ${\mathbb {B}}^n$ for $\alpha >-1$, and then since that $b_\alpha (a_k,\cdot )\in \mathcal {A}_{Har}^{2,\alpha }({\mathbb {B}}^n)$ then $b_\alpha (a_k,\cdot )\in \mathcal {A}_{Har}^{1,\alpha }({\mathbb {B}}^n)$.

Now, by using the fact that $\displaystyle \nu =\sum _{k\in \mathbb N}(1-|a_k|^2)^\alpha \delta _{a_k}$ is a Carleson measure for $\mathcal {A}_{Har}^{1,\alpha }({\mathbb {B}}^n)$, we get

$$\begin{aligned} \sum _{j:j\ne k}|b_\alpha (a_k,a_j)|(1-|a_j|^2)^\alpha\lesssim & {} \int _{{\mathbb {B}}^n}|b_\alpha (a_k,x)|(1-|x|^2)^\alpha dx<\infty . \end{aligned}$$

(3.6)

$\square$

Lemma 3.3

Let $p>1$, and $\alpha >-1$. Then $\mathcal {H}_{\alpha ,\mathcal {S}}$ is included in $\mathcal {A}^{p,\alpha }_{Har}({\mathbb {B}}^n)$.

Proof

Let $f_v\in \mathcal {H}_{\alpha ,\mathcal {S}}$, thus by construction it is a harmonic function and it remains to show that it belongs to $L_{p,\alpha }({\mathbb {B}}^n, d\mu _\alpha (x))$. We set up a function $g\in L_{p,\alpha }({\mathbb {B}}^n, d\mu _\alpha (x))$ and show that $|f_v(x)|\lesssim \mathcal {T}[g](x)$ for $x\in {\mathbb {B}}^n$ where $\mathcal {T}$ is the operator defined as (3.1), then we apply Lemma 3.1.

Therefore, for $x\in {\mathbb {B}}^n$, let us consider the following function

$$\begin{aligned} \displaystyle g(x)=\sum _{j\in \mathbb N}(1-|a_j|^2)^{\frac{\alpha }{p}}|v_j|||\mathcal {X}_ {E_{\frac{\rho }{3}}(a_j)}||^{-1}_{p,\alpha ,\mathbb B^n}\mathcal {X}_{E_{\frac{\rho }{3}}(a_j)}(x), \end{aligned}$$

such that $\mathcal {X}_{E_{\frac{\rho }{3}}(a_j)}$ stands for the characteristic function of $E_{\frac{\rho }{3}}(a_j)$. Hence, by using the fact that $\left( E_{\frac{\rho }{3}}(a_k)\right) _{k\in {\mathbb {N}}}$ are pairwise disjoint (Definition 2.3) and Remark 2.4 we have

$$\begin{aligned} ||g||_{p,\alpha ,{\mathbb {B}}^n}^p\le \displaystyle \sum _{j\in \mathbb N}(1-|a_j|^2)^\alpha |v_j|^p=||v||_{l_{p,\mathcal {S},\alpha }}^p<\infty . \end{aligned}$$

Thus $g\in L_{p,\alpha }({\mathbb {B}}^n, d\mu _\alpha (x))$ and by applying Lemma 3.1, we have that $\mathcal {T}$ is a bounded operator.

Also, Miao has shown the following estimate which is extracted from Proposition 4 in [13]

$$\begin{aligned} b_\alpha (x,x)\approx (1-|x|)^{-n-\alpha }\quad \text {for}\quad x\in \mathbb B^n.\end{aligned}$$

(3.7)

Whence, we apply (3.7) for $a_k\in {\mathbb {B}}^n$ and we get $\displaystyle \frac{1}{|b_\alpha (a_j,a_j)|}\lesssim (1-|a_j|^2)^{\alpha +n}$, so by definition of $f_v$, we have

$$\begin{aligned} |f_v(x)|\lesssim \displaystyle \sum _{j\in {\mathbb {N}}}|v_j||b_\alpha (x,a_j)|(1-|a_j|^2)^{\alpha +n}. \end{aligned}$$

Thanks to the harmonicity properties, we have that $b_\alpha (x,a_j)$ is equal to its mean value on $E_{\frac{\rho }{3}}(a_j)$, and then we have

$$\begin{aligned} |b_\alpha (x,a_j)|\le \frac{1}{|E_{\frac{\rho }{3}}(a_j)|_{Leb}} \int _{E_{\frac{\rho }{3}}(a_j)}|b_\alpha (x,y)|dy. \end{aligned}$$

Therefore, we have

$$\begin{aligned} |f_v(x)|\lesssim \displaystyle \sum _{j\in {\mathbb {N}}}|v_j| \frac{1}{|E_{\frac{\rho }{3}}(a_j)|_{Leb}}\int _{E_{\frac{\rho }{3}}(a_j)}|b_\alpha (x,y)| (1-|a_j|^2)^{\alpha +n}dy. \end{aligned}$$

(3.8)

We recall that $0<\rho <1$, thus by employing a straightforward calculus, we have

$$\begin{aligned} (1-|a_j|)\approx (1-|y|)\approx (1-|y|^2)\quad \text {for}\quad y\in E_{\frac{\rho }{3}}(a_j). \end{aligned}$$

Consequently, we integrate $(1-|a_j|)^\alpha \approx (1-|y|^2)^\alpha$ on $E_{\frac{\rho }{3}}(a_j)$ w.r.t. the variable y and we get $\displaystyle (1-|a_j|)^{\alpha }\approx \frac{1}{|E_{\frac{\rho }{3}} (a_j)|_{Leb}}\int _{E_{\frac{\rho }{3}}(a_j)}(1-|y|^2)^\alpha dy$ and by applying the fact that $|E_{\frac{\rho }{3}}(a_j)|_{Leb}\lesssim 1$, we have

$$\begin{aligned} \displaystyle \left( 1-|a_j|\right) ^{-\alpha /p}\lesssim & {} \left( \int _{E_{\frac{\rho }{3}}(a_j)}(1-|y|^2)^\alpha dy\right) ^{-1/p}. \end{aligned}$$

(3.9)

Therefore, by employing (3.8) and (3.9), we get

$$\begin{aligned} |f_v(x)|\lesssim & {} \sum _{j\in {\mathbb {N}}}|v_j|\frac{1}{|E_{\frac{\rho }{3}} (a_j)|_{Leb}}\int _{E_{\frac{\rho }{3}}(a_j)}|b_\alpha (x,y)|(1-|a_j|^2) ^{\alpha +n}dy\\\lesssim & {} \sum _{j\in {\mathbb {N}}}(1-|a_j|)^{\frac{\alpha }{p}}|v_j| \int _{E_{\frac{\rho }{3}}(a_j)}|b_\alpha (x,y)|(1-|a_j|^2)^{\alpha -\frac{\alpha }{p}}dy\\\lesssim & {} \sum _{j\in {\mathbb {N}}}(1-|a_j|^2)^{\frac{\alpha }{p}}|v_j| \int _{E_{\frac{\rho }{3}}(a_j)}|b_\alpha (x,y)|(1-|y|^2)^\alpha (1-|a_j|^2)^{ -\frac{\alpha }{p}}dy\\\lesssim & {} \sum _{j\in {\mathbb {N}}}(1-|a_j|^2)^{\frac{\alpha }{p}}|v_j| \int _{E_{\frac{\rho }{3}}(a_j)}|b_\alpha (x,y)|(1-|y|^2)^\alpha (1-|a_j|) ^{-\frac{\alpha }{p}}dy\\\lesssim & {} \sum _{j\in {\mathbb {N}}}(1-|a_j|^2)^{\frac{\alpha }{p}}|v_j| \int _{E_{\frac{\rho }{3}}(a_j)} |b_\alpha (x,y)|(1-|y|^2)^\alpha dy\\{} & {} \left( \int _{E_{\frac{\rho }{3}}(a_j)}(1-|y|^2)^{\alpha }dy\right) ^{-1/p}\\= & {} \sum _{j\in \mathbb N}(1-|a_j|^2)^{\frac{\alpha }{p}}|v_j|\int _{E_{\frac{\rho }{3}}(a_j)}|b_\alpha (x,y)|(1-|y|^2)^\alpha dy||\mathcal {X}_{E_{\frac{\rho }{3}}(a_j)}||^{-1}_{p,\alpha ,\mathbb B^n}\\= & {} \mathcal {T}[g](x). \end{aligned}$$

Then by applying Lemma 3.1, we obtain $f_v\in \mathcal {A}_{Har}^{p,\alpha }({\mathbb {B}}^n)$. $\square$

In the next section, we state our main result (Theorem 4.1) which states sufficient conditions for interpolating a sequence by a function in $\mathcal {A}_{Har}^{p,\alpha }(\mathbb B^n)$ for $p> 1$ and $\alpha >-1$.

4 Statement and Proof of the Main Result

Theorem 4.1

Let $p>1$, and $\alpha >-1$. We suppose that $\mathcal {S}=(a_k)_{k\in {\mathbb {N}}}$ is a $\rho$-separated sequence in ${\mathbb {B}}^n$ and the measure $\displaystyle \nu =\sum _{k\in \mathbb N}(1-|a_k|^2)^\alpha \delta _{a_k}$ is a Carleson measure for $\mathcal {A}_{Har}^{1,\alpha }({\mathbb {B}}^n)$. Furthermore, we suppose that $\mathcal {S}$ satisfies

$$\begin{aligned} \displaystyle \sum _{j:j\ne k}|b_\alpha (a_k,a_j)|(1-|a_j|^2)^\alpha <1,\quad \text {for all}\quad k\in {\mathbb {N}}. \end{aligned}$$

(4.1)

Then $\mathcal {S}$ is $\mathcal {A}_{Har}^{p,\alpha }(\mathbb B^n)$-interpolating sequence.

Remark 4.2

The fact that we have assumed that $\nu$ is a Carleson measure for $\mathcal {A}_{Har}^{p,\alpha }({\mathbb {B}}^n)$ then left hand side of Inequality (4.1) is finite, see Lemma 3.2, and for this reason we have supposed that the sequence $\mathcal {S}$ satisfies (4.1). We observe that within the framework of a weighted holomorphic Bergman space, Jevtić, Massaneda, and Thomas state an analogue result of Theorem 4.1, see Theorem 4.5 in [10]. The case of an interpolating sequence for holomorphic Bergman space of infinite order has been treated by the present author [4].

Proof of Theorem 4.1

To prove that $\mathcal {S}$ is $\mathcal {A}_{Har}^{p,\alpha }(\mathbb B^n)$-interpolating sequence is to come back to show that the operator $R_{\mathcal {S}}:\mathcal {A}_{Har}^{p,\alpha }(\mathbb B^n)\rightarrow l_{p,\mathcal {S},\alpha }$, defined by $f_v\mapsto (f_v(a_k))_{k\in {\mathbb {N}}}$ with $v=(v_j)_{j\in {\mathbb {N}}}\in l_{p,S,\alpha }$ and such that $f_v(a_k)=\displaystyle \sum _{j\in \mathbb N}v_j\frac{b_\alpha (a_k,a_j)}{b_\alpha (a_j,a_j)}$ is an onto map and which is sufficient to proof that $l_{p,\mathcal {S},\alpha }$-norm of $R_{\mathcal {S}}f_v-v$ is strictly less to the $l_{p,\mathcal {S},\alpha }$-norm of v.

By using (3.7) and the fact that $(1-|a_j|^2)^n=(1-|a_j|)^n(1+|a_j|)^n\le 2^n$, we get

$$\begin{aligned} \left| \frac{b_\alpha (a_k,a_j)}{b_\alpha (a_j,a_j)}\right|\lesssim & {} (1-|a_j|^2)^{\alpha +n}|b_\alpha (a_k,a_j)|\nonumber \\\lesssim & {} (1-|a_j|^2)^{\alpha }|b_\alpha (a_k,a_j)|. \end{aligned}$$

(4.2)

By employing, successively, Inequality (4.2), the definition of the operator $R_\mathcal {S}$, Hölder inequality with the conjugate exponents p, q, and Inequality (4.1), we have

$$\begin{aligned}{} & {} ||R_{\mathcal {S}}f_v-v||^p_{l_{p,\mathcal {S},\alpha }} =\sum _{k\in \mathbb N}(1-|a_k|^2)^\alpha |f_v(a_k)-v_k|^p\\{} & {} \quad \le \sum _{k\in {\mathbb {N}}}(1-|a_k|^2)^{\alpha }\left( \sum _{j\in \mathbb N}|v_j|\left| \frac{b_\alpha (a_k,a_j)}{b_\alpha (a_j,a_j)} -\delta _{kj}\right| \right) ^p\\{} & {} \quad =\sum _{k\in {\mathbb {N}}}(1-|a_k|^2)^{\alpha }\left( \sum _{j:j\ne k}| v_j|\left| \frac{b_\alpha (a_k,a_j)}{b_\alpha (a_j,a_j)}\right| \right) ^p\\{} & {} \quad \lesssim \sum _{k\in \mathbb N}(1-|a_k|^2)^{\alpha }\left( \sum _{j:j\ne k}|v_j| (1-|a_j|^2)^\alpha |b_\alpha (a_j,a_k)|\right) ^p \\{} & {} \quad =\sum _{k\in {\mathbb {N}}}(1-|a_k|^2)^{\alpha }\left( \sum _{j:j\ne k}|v_j|(1-|a_j|^2)^{\frac{\alpha }{p}}|b_\alpha (a_j,a_k)|^{\frac{1}{p}} (1-|a_j|^2)^{\frac{\alpha }{q}}|b_\alpha (a_j,a_k)|^{\frac{1}{q}}\right) ^p\\{} & {} \quad \lesssim \sum _{k\in {\mathbb {N}}}(1-|a_k|^2)^{\alpha }\left[ \sum _{j:j\ne k}|v_j|^p(1-|a_j|^2)^\alpha |b_\alpha (a_j,a_k)|\right. \\{} & {} \qquad \left. \left( \sum _{j:j\ne k}(1-|a_j|^2)^\alpha |b_\alpha (a_j,a_k)|\right) ^{p/q}\right] \\{} & {} \quad \lesssim \sum _{k\in \mathbb N}(1-|a_k|^2)^{\alpha }\left[ \sum _{j:j\ne k}|v_j|^p(1-|a_j|^2)^\alpha |b_\alpha (a_j,a_k)|\right] \\{} & {} \quad =\sum _{j\in \mathbb N}(1-|a_j|^2)^\alpha |v_j|^p\left[ \sum _{k:k\ne j}(1-|a_k|^2)^\alpha |b_\alpha (a_j,a_k)|\right] \\{} & {} \quad <\sum _{j\in \mathbb N}(1-|a_j|^2)^\alpha |v_j|^p=||v||^p_{l_{p,\mathcal {S},\alpha }}. \end{aligned}$$

Consequently, by applying Theorem 18.3 in [15], the operator $R_{\mathcal {S}}$ is invertible which implies that it is an onto map. The proof of Theorem 4.1 is complete.

Remark 4.3

The fact that the statement of Lemma 3.1 covers the case $p=1$, might indicate that that a version of Theorem 4.1 for this case is also true.