Atomic decomposition and interpolation via the complex method for mixed norm Bergman spaces on tube domains over symmetric cones

TL;DR

This paper develops atomic decomposition and interpolation characterizations for mixed norm weighted Bergman spaces on tube domains over symmetric cones, using a Whitney decomposition and complex interpolation methods.

Contribution

It introduces an atomic decomposition theorem and characterizes interpolation spaces for mixed norm weighted Bergman spaces on symmetric cone tube domains.

Findings

Atomic decomposition theorem established for these spaces.

Interpolation spaces characterized via the complex method.

Provides tools for analysis on symmetric cone tube domains.

Abstract

Starting from an adapted Whitney decomposition of tube domains in $\C^n$ over irreducible symmetric cones of $\R^n,$ we prove an atomic decomposition theorem in mixed norm weighted Bergman spaces on these domains. We also characterize the interpolation space via the complex method between two mixed norm weighted Bergman spaces.

Equations

$$\overline{\Omega }=\{x^2:x\in V\}.$$

$$V=\underset{1\le i\le j\le r}{\oplus }{V}_{i,j}$$

$${\Delta }_1(x),\cdots ,{\Delta }_r(x)$$

$${\Delta }_{\mathbf{s}}(x)={\Delta }_1^{s_1-s_2}(x){\Delta }_2^{s_2-s_3}(x)\cdots {\Delta }_r^{s_r}(x),x\in \Omega ,\text{s}=(s_1,\cdots ,s_r)\in {\mathbb{C}}^r.$$

$$n_k=2(k-1)\frac{\frac{n}{r}-1}{r-1}\mathrm{and}{m}_k=2(r-k)\frac{\frac{n}{r}-1}{r-1}.$$

$$\Vert F{\Vert }_{L_{\text{s}}^{p,q}}={\left({\int }_{\Omega }\Vert F(\cdot +iy){\Vert }_p^q{\Delta }_{\text{s}-\frac{n}{r}}(y)dy\right)}^{\frac{1}{q}}<\infty$$

$$\Vert F(\cdot +iy){\Vert }_p={\left({\int }_V|F(x+iy){|}^{p}dx\right)}^{\frac{1}{p}}$$

$$P_{\text{s}}F(z)={\int }_{T_{\Omega }}{B}_{\text{s}}(z,u+iv)F(u+iv){\Delta }_{\text{s}-\frac{n}{r}}(v)dudv,$$

$$B_{\text{s}}(z,u+iv)=d_{\text{s}}{\Delta }_{-\text{s}-\frac{n}{r}}(\frac{z-u+iv}{i})$$

$$\sum _j{\left(\sum _l|{\lambda }_{l,j}{|}^p\right)}^{\frac{q}{p}}{\Delta }_{\text{s}+\frac{nq}{rp}}(y_j)<\infty ,$$

$$\sum _{l,j}{\lambda }_{l,j}{\Delta }_{\text{s}+\frac{nq}{rp}}(y_j)B_{\text{s}}(z,z_{l,j})$$

$$\Vert F{\Vert }_{A_{\text{s}}^{p,q}}^q\le C\sum _j{\left(\sum _l|{\lambda }_{l,j}{|}^p\right)}^{\frac{q}{p}}{\Delta }_{\text{s}+\frac{nq}{rp}}(y_j)$$

$$F(z)=\sum _{l,j}{\lambda }_{l,j}{\Delta }_{\text{s}+\frac{nq}{rp}}(y_j)B_{\text{s}}(z,z_{l,j}),$$

$$\sum _j{\left(\sum _l|{\lambda }_{l,j}{|}^p\right)}^{\frac{q}{p}}{\Delta }_{\text{s}+\frac{nq}{rp}}(y_j)\le C\Vert F{\Vert }_{A_{\text{s}}^{p,q}}^q$$

$$q_{\text{s}}=\underset{1\le k\le r}{\min }\left(1+\frac{s_k-\frac{n_k}{2}}{\frac{m_k}{2}}\right).$$

$$[A_{{\text{s}}_0}^{p_0,q_0},A_{{\text{s}}_1}^{p_1,q_1}{]}_{\theta }=A_{\text{s}}^{p,q}$$

$$[A_{\text{s}}^{p_0,q_0},A_{\text{s}}^{p_1,q_1}{]}_{\theta }=A_{\text{s}}^{p,q}$$

$$p_{\text{s}}=1+\underset{k}{\min }\frac{s_k+\frac{n}{r}}{(m_k-s_k{)}_{+}}$$

$$q_{\text{s}}(p)=\min \{p,p^{\prime }\}{q}_{\text{s}}.$$

$$(x|y)=\text{tr}(xy)$$

$$K=G\cap O(V),$$

$$V=\underset{1\le i\le j\le r}{\oplus }{V}_{i,j}$$

$$\left\{\begin{array}[]{ll}V_{i,i}=\mathbb{R}c_{i}\\ V_{i,j}=\{x\in V:\,\,c_{i}x=c_{j}x=\frac{1}{2}x\}\,\,\textrm{if}\,\,i<j.\end{array}\right.$$

$$A=\{P(a):a=\sum _{i=1}^r{a}_i{c}_i,a_i>0\},$$

$$G=NAK\mathrm{and}G=KAK.$$

$${\Delta }_k(nx)={\Delta }_k(x),$$

$${\Delta }_k(P(a)x)=a_1^2\cdots a_k^2{\Delta }_k(x),$$

$$\Delta (gy)=\Delta (ge)\Delta (y)=De{t}^{\frac{r}{n}}(g)\Delta (y),\forall g\in G,\forall y\in \Omega$$

$$\Omega =\{x\in {\mathbb{R}}^n:{\Delta }_k(x)>0,k=1,\cdots ,r\}.$$

Full text

1. Introduction

The context and the notations are those of [13]. Let $\Omega$ be an irreducible symmetric cone of rank r in a vector space $V$ of dimension n, endowed with an inner product $(.|.)$ for which $\Omega$ is self-dual.

We recall that $\Omega$ induces in $V$ a structure of Euclidean Jordan algebra with identity $\mathbf{e}$ such that

[TABLE]

Let $\{c_{1},\cdots,c_{r}\}$ be a fixed Jordan frame in $V$ and

[TABLE]

be its associated Peirce decomposition of $V$. We denote by

[TABLE]

the principal minors of $x\in V$ with respect to the fixed Jordan frame $\{c_{1},\cdots,c_{r}\}$. More precisely, $\Delta_{k}(x),\hskip 5.69054ptk=1,\cdots,r$ is the determinant of the projection $P_{k}x$ of $x$, in the Jordan subalgebra $V^{(k)}=\oplus_{1\leq i\leq j\leq k}V_{i,j}$. We have $\Delta_{k}(x)>0$, $k=1,\cdots,r$ when $x\in\Omega,$ and the determinant $\Delta$ of the Jordan algebra is given by $\Delta=\Delta_{r}.$ The generalized power function on $\Omega$ is defined as

[TABLE]

We adopt the following standard notations:

[TABLE]

For $\textbf{{s}}=(s_{1},\cdots,s_{r})\in\mathbb{R}^{r}$ and $\rho$ real, the notation $\textbf{{s}}+\rho$ will stand for the vector whose coordinates are $s_{k}+\rho,\hskip 5.69054ptk=1,\cdots,r.$ For $1\leq p\leq\infty$ and $1\leq q<\infty,$ let $L^{p,\,\,q}_{\textbf{s}}$ denote the mixed norm Lebesgue space constisting of measurable functions $F$ on $T_{\Omega}$ such that

[TABLE]

where

[TABLE]

(with the obvious modification if $p=\infty$). The mixed norm weighted Bergman space $A_{\textbf{{s}}}^{p,q}$ is the (closed) subspace of $L^{p,\,\,q}_{\textbf{{s}}}$ consisting of holomorphic functions. Following [11], $A_{\textbf{{s}}}^{p,q}$ is non-trivial if only if $\textbf{{s}}_{k}>\frac{n_{k}}{2},\hskip 5.69054ptk=1,\cdots,r.$ When $p=q$, we write $L^{p,\,\,q}_{\textbf{{s}}}=L^{p}_{\textbf{{s}}}$ and $A^{p,\,\,q}_{\textbf{{s}}}=A^{p}_{\textbf{{s}}}$ which are respectively the usual weighted Lebesgue space and the usual weighted Bergman space. Moreover, when $p=q=2$ the orthogonal projector $P_{\textbf{{s}}}$ from the Hilbert space $L^{2}_{\textbf{{s}}}$ onto its closed subspace $A^{2}_{\textbf{{s}}}$ is called weighted Bergman projector. It is well known that $P_{\textbf{{s}}}$ is the integral operator on $L_{\textbf{{s}}}^{2}$ given by the formula

[TABLE]

where

[TABLE]

is the reproducing kernel on $A^{2}_{\textbf{{s}}},$ called weighted Bergman kernel of $T_{\Omega}$. Precisely, $\Delta_{-\textbf{{s}}-\frac{n}{r}}(\frac{x+iy}{i})$ is the holomorphic determination of the $(-\textbf{{s}}-\frac{n}{r})$-power which reduces to the function $\Delta_{-\textbf{{s}}-\frac{n}{r}}(y)$ when $x=0$.

Our first result is an atomic decomposition theorem for functions in mixed norm weighted Bergman spaces on tube domains over symmetric cones. It generalizes the result of [2] for usual weighted Bergman spaces on tube domains over symmetric cones and the result of [18] for mixed norm weighted Bergman spaces on the upper half-plane (the case $n=r=1).$

Theorem A**.**

Let s be a vector of $\mathbb{R}^{r}$ such that $s_{k}>\frac{n_{k}}{2},\hskip 2.84526ptk=1,\cdots,r.$ Assume that $P_{\textbf{{s}}}$ extends to a bounded operator on $L_{\textbf{s}}^{p,q}$. Then there is a sequence of points $\{z_{l,j}=x_{l,j}+iy_{j}\}_{l\in\mathbb{Z},\hskip 2.84526ptj\in\mathbb{N}}$ in $T_{\Omega}$ and a positive constant $C$ such that the following assertions hold.

1. (i)

For every sequence $\{\lambda_{l,j}\}_{l\in\mathbb{Z},\hskip 2.84526ptj\in\mathbb{N}}$ such that

[TABLE]

the series

[TABLE]

is convergent in $A_{\textbf{s}}^{p,q}$. Moreover, its sum $F$ satisfies the inequality

[TABLE] 2. (ii)

Every function $F\in A_{\textbf{s}}^{p,q}$ may be written as

[TABLE]

with

[TABLE]

Our second result is an interpolation theorem between mixed norm weighted Bergman spaces. It generalizes the result of [5] for usual weighted Bergman spaces. We adopt the following notation.

[TABLE]

Theorem B**.**

1. (1)

Let $\textbf{{s}}_{0},\textbf{{s}}_{1}\in\mathbb{R}^{r}$ be such that $(s_{0})_{k},(s_{1})_{k}>\frac{n}{r}-1,\quad k=1,\cdots,r$ and let $1\leq p_{0},p_{1}\leq\infty$, $1\leq q_{0},\hskip 2.84526ptq_{1}<\infty$ be such that $1\leq q_{i}<q_{{\textbf{{s}}}_{i}}$ for every $i=0,1.$ Then for every $\theta\in(0,1),$ we have

[TABLE]

*with equivalent norms, where $\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}},\frac{1}{q}=\frac{1-\theta}{q_{0}}+\frac{\theta}{q_{1}}$ and $\frac{\textbf{{s}}}{q}=\frac{(1-\theta){\textbf{{s}}}_{0}}{q_{0}}+\frac{\theta{\textbf{{s}}}_{1}}{q_{1}}$. * 2. (2)

Let $\textbf{{s}}\in\mathbb{R}^{r}$ be such that $s_{k}>\frac{n_{k}}{2},\hskip 5.69054ptk=1,\cdots,r.$ Assume that $P_{\textbf{{s}}}$ extends to a bounded operator on $L^{p_{i},q_{i}}_{\textbf{{s}}},\hskip 5.69054pti=0,1$ for $1\leq p_{0},p_{1}\leq\infty$ and $1<q_{0},q_{1}<\infty.$ Then for every $\theta\in(0,1),$ we have

[TABLE]

with equivalent norms, where $\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$ and $\frac{1}{q}=\frac{1-\theta}{q_{0}}+\frac{\theta}{q_{1}}.$ 3. (3)

Let ${\textbf{{s}}}\in\mathbb{R}^{r}$ be such that $(s)_{k}>\frac{n}{r}-1,\hskip 5.69054ptk=1,\cdots,r,$ let $1\leq p_{0}<p_{1}\leq\infty$ and let $q_{0},q_{1}$ be such that $1\leq q_{0}<q_{\textbf{{s}}}\leq q_{1}.$ We assume that $P_{\textbf{{s}}}$ extends to a bounded operator on $L^{p_{1},q_{1}}_{\textbf{{s}}}.$ Then for some values of $\theta\in(0,1)$, we have

[TABLE]

with equivalent norms, where $\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$ and $\frac{1}{q}=\frac{1-\theta}{q_{0}}+\frac{\theta}{q_{1}}.$

We recall sufficient conditions on $p,q$ and s under which the weighted Bergman projector $P_{\textbf{{s}}}$ extends to a bounded operator on $L^{p,q}_{\textbf{{s}}}.$ We adopt the following notations.

[TABLE]

and

[TABLE]

Theorem 1.1** ([12] and [17]).**

The weighted Bergman projector $P_{\textbf{{s}}}$ extends to a bounded operator on $L^{p,q}_{\textbf{{s}}}$ whenever $\frac{1}{q_{\textbf{{s}}}(p)}<\frac{1}{q}<1-\frac{1}{q_{\textbf{{s}}}(p)}$ in the following two cases:

1. (i)

$s_{j}>\frac{n_{j}}{2},\hskip 5.69054ptj=1,\cdots r$* and $1\leq p<p_{\textbf{{s}}}$ [12];* 2. (ii)

$s_{j}>\frac{n}{r}-1,\hskip 5.69054ptj=1,\cdots r$* and $1\leq p\leq\infty$ [17].*

We restrict to tube domains over Lorentz cones ($r=2$). For real $\textbf{{s}}=(s,s),$ this problem was recently completely solved on tube domains over Lorentz cones (cf. [4] for a combination of results from [8] and [1]; cf. also [7] for the unweighted case $\textbf{{s}}=(\frac{n}{r},\cdots,\frac{n}{r})$). For vectorial $\textbf{{s}}=(s_{1},s_{2}),$ Theorem 1.1 was also extended in [4] to other values $p$ and $q.$

The plan of this paper is as follows. In Section 2, we overview some preliminaries and useful results about symmetric cones and tube domains over symmetric cones. In Section 3, we study atomic decomposition of mixed norm Bergman spaces and we prove a more precise statement of Theorem A. In Section 4, we study interpolation via the complex method between mixed norm weighted Bergman spaces and we prove Theorem B. In particular we give a more precise statement of assertion (3) of this theorem (Theorem 4.6) and we ask an open question. A final remark will point out a connection between the two main theorems of the paper (Theorem A and Theorem B).

For $\textbf{{s}}=(s,\cdots,s)$ real, Theorem A and Theorem B were presented in the PhD dissertation of the second author [14].

2. Preliminaries

Materials of this section are essentially from [13]. We give some definitions and useful results.

Let $\Omega$ be an irreducible symmetric cone of rank $r$ in a real vector space $V$ of dimension $n$ endowed with the structure of Euclidean Jordan algebra with identity e. In particular, $\Omega$ is self-dual with respect to the inner product

[TABLE]

on $V$.

2.1. Group action

Let $G(\Omega)$ be the group of linear transformations of the cone $\Omega$ and $G$ its identity component. By definition, the subgroup $G$ of $G(\Omega)$ is a semi-simple Lie group which acts transitively on $\Omega.$ This gives the identification $\Omega\sim G/K$, where $K:=\{g\in G:\,\,\,\,g\cdot\mathbf{e}=\mathbf{e}\}$ is a maximal compact subgroup of $G$. More precisely,

[TABLE]

where $O(V)$ is the orthogonal group in $V.$ Furthermore, there is a solvable subgroup $T$ of $G$ acting simply transitively on $\Omega$. That is, every $y\in\Omega$ can be written uniquely as $y=t\cdot\mathbf{e}$, for some $t\in T.$

Let $\{c_{1},\cdots,c_{r}\}$ in $\mathbb{R}^{n}$ be a fixed Jordan frame in $V$ (that is, a complete system of idempotents) and

[TABLE]

be its associated Peirce decomposition of $V$ where

[TABLE]

We have $\mathbf{e}=\sum\limits_{1\leq i\leq r}c_{i}.$ Then the solvable Lie group $T$ factors as the semidirect product $T=NA=AN$ of a nilpotent subgroup $N$ consisting of lower triangular matrices, and an abelian subgroup $A$ consisting of diagonal matrices. The latter takes the explicit form

[TABLE]

where $P$ is the quadratic representation of $\mathbb{R}^{n}$. This also leads to the Iwasawa and Cartan decompositions of the semisimple Lie group $G:$

[TABLE]

Still following [13], we shall denote by $\Delta_{1}(x),\cdots,\Delta_{r}(x)$ the principal minors of $x\in V$, with respect to the fixed Jordan frame $\{c_{1},\cdots,c_{r}\}$. These are invariant functions under the group $N$,

[TABLE]

where $n\in N$, $x\in V$, $k=1,\cdots,r$, and satisfy a homogeneity relation under $A$,

[TABLE]

if $a=a_{1}c_{1}+\cdots+a_{r}c_{r}$.

The determinant function $\Delta(y)=\Delta_{r}(y)$ is also invariant under $K$, and moreover, satisfies the formula

[TABLE]

where $Det$ is the usual determinant of linear mappings. It follows from this formula that the measure $\frac{d\xi}{\Delta^{\frac{n}{r}}(\xi)}$ is $G$-invariant in $\Omega.$

Finally, we recall the following version of Sylvester’s theorem.

[TABLE]

2.2. Geometric properties

With the identification $\Omega\sim G/K$, the cone can be regarded as a Riemannian manifold with the $G$-invariant metric defined by

[TABLE]

if $y=t\cdot\textbf{e}$ with $t\in T$ and $\xi$ and $\eta$ are tangent vectors at $y\in\Omega$. We shall denote by $d_{\Omega}$ the corresponding invariant distance, and by $B_{\delta}(\xi)$ the associated ball centered at $\xi$ with radius $\delta$. Note that for each $g\in G$, the invariance of $d_{\Omega}$ implies that $B_{\delta}(g\xi)=gB_{\delta}(\xi)$. We also note that

• •

on compact sets of $\mathbb{R}^{n}$ contained in $\Omega,$ the invariant distance $d_{\Omega}$ is equivalent to the Euclidean distance in $\mathbb{R}^{n};$

• •

the associated balls $B_{\delta}$ in $\Omega$ are relatively compact in $\Omega$.

We also need the following crucial invariance properties of $d_{\Omega}$ and $\Delta_{k}$, obtained in [2, 1].

Lemma 2.1**.**

Let $\delta_{0}>0$. Then there is a constant $\gamma 0$ depending only on $\delta_{0}$ and $\Omega$ such that for every $0<\delta\leq\delta_{0}$ and for $\xi$, $\xi^{\prime}\in\Omega$ satisfying $d_{\Omega}(\xi,\xi^{\prime})\leq\delta$ we have

[TABLE]

Lemma 2.2**.**

Let $\delta_{0}>0$ be fixed. Then there exist two constants $\eta_{1}>\eta_{2}>0$, depending only on $\delta_{0}$ and $\Omega$, such that for every $0<\delta\leq\delta_{0}$ we have

[TABLE]

The next corollary is an easy consequence of the previous lemma.

Lemma 2.3**.**

Let $\delta_{0}=1$. Then there is a positive constant $\gamma$ such that for every $\delta\in(0,1)$ such that $\eta_{1}\delta<1,$ we have

[TABLE]

for all $\xi\in\Omega$.

2.3. Gamma function in $\Omega$

The generalized gamma function in $\Omega$ is defined in terms of the generalized power functions by

[TABLE]

This integral is known to converge absolutely if and only if $\Re e\hskip 2.84526pts_{k}>\frac{n_{k}}{2},\hskip 5.69054ptk=1,\cdots,r.$ In this case,

[TABLE]

where $\Gamma$ is the classical gamma function. We shall denote $\Gamma_{\Omega}(\textbf{{s}})=\Gamma_{\Omega}(s)$ when $\textbf{{s}}=(s,\cdots,s)$. In view of [13], the Laplace transform of a generalized power function is given for all $y\in\Omega$ by

[TABLE]

for each $\textbf{{s}}\in\mathbb{C}^{r}$ such that $\Re e\hskip 2.84526pts_{k}>\frac{n_{k}}{2}$ for all $k=1,\cdots,r$. We recall that $y^{-1}=t^{*-1}\cdot\textbf{e}$ whenever $y=t\cdot\textbf{e}$ with $t\in T.$ Here $t^{\star}$ denotes the adjoint of the transformation $t\in T$ with respect to the inner product $(\cdot|\cdot).$

The power function $\Delta_{\textbf{s}}(y^{-1})$ can be expressed in terms of the rotated Jordan frame $\{c_{r},\cdots,c_{1}\}$. Indeed if we denote by $\Delta^{*}_{k}$, $k=1,\cdots,r$, the principal minors with respect to the rotated Jordan frame $\{c_{r},\cdots,c_{1}\}$ then

[TABLE]

Here $\textbf{{s}}^{*}=(s_{r},\cdots,s_{1})$.

2.4. Bergman distance on the tube domain $T_{\Omega}$

Following [2], we define a matrix function $\{g_{j,k}\}_{1\leq j,k\leq n}$ on $T_{\Omega}$ by

[TABLE]

where $B$ is the unweighted Bergman kernel of $T_{\Omega},$ i.e $B=B_{\textbf{{s}}}$ with $\textbf{{s}}=(\frac{n}{r},\cdots\frac{n}{r})$. The map

[TABLE]

with

[TABLE]

defines a Hermitian metric on $\mathbb{C}^{n}$, called the Bergman metric. The Bergman length of a smooth path $\gamma:[0,1]\to T_{\Omega}$ is given by

[TABLE]

and the Bergman distance $d(z_{1},z_{2})$ between two points $z_{1}$, $z_{2}$ of $T_{\Omega}$ is

[TABLE]

where the infimum is taken over all smooth paths $\gamma:[0,1]\to T_{\Omega}$ such that $\gamma(0)=z_{0}$ and $\gamma(1)=z_{1}$. It is well known that the Bergman distance $d$ is equivalent to the Euclidean distance on the compact sets of $\mathbb{C}^{n}$ contained in $T_{\Omega}$ and the Bergman balls in $T_{\Omega}$ are relatively compact in $T_{\Omega}$. Next, we again denote by $\mathbb{R}^{n}$ the group of translations by vectors in $\mathbb{R}^{n}$. Then the group $\mathbb{R}^{n}\times T$ acts simply transitively on $T_{\Omega}$ and the Bergman distance $d$ is invariant under the automorphisms of $\mathbb{R}^{n}\times T$.

2.5. A Whitney decomposition of the tube domain $T_{\Omega}$

In the sequel, the Bergman ball in $T_{\Omega}$ with centre at $z$ and radius $\eta$ will be denoted $\mathbf{B}_{\eta}(z).$

Lemma 2.4**.**

There exists a constant $R>1$ such that for all $\eta\in(0,1)$ and $z_{0}=x_{0}+iy_{0}\in T_{\Omega}$, the following inclusions hold :

[TABLE]

[TABLE]

where $g$ is the element of $T$ satisfying $g\cdot e=y_{0}$.

Proof.

From the invariance under translations and automorphisms of $T$ we have that

[TABLE]

for all $x+iy\in\mathbf{B}_{\eta}(z_{0})$. We recall that the Bergman distance $d$ and the Euclidean distance $d_{Eucl}$ are equivalent on compact sets of $\mathbb{C}^{n}$ contained in $T_{\Omega}.$ So there exists a constant $R>1$ such that

[TABLE]

for all $X+iY\in\overline{\mathbf{B}_{1}(ie)}$. The proof of the lemma follows from the following equivalence

[TABLE]

and the equivalence given in Lemma 2.2 between $d_{\Omega}$ and the Euclidean distance $||\cdot||$ in $\mathbb{R}^{n}$ on compact sets of $\mathbb{R}^{n}$ contained in $\Omega.$ ∎

The starting point of our analysis is the following Whitney decomposition of the cone $\Omega$ which was obtained e.g. in [1, 2].

Lemma 2.5**.**

There is a positive integer $N$ such that given $\delta\in(0,1)$, one can find a sequence of points $\{y_{j}\}_{j=1,2,\cdots}$ in $\Omega$ with the following property:

1. (i)

the balls $B_{\frac{\delta}{2}}(y_{j})$ are pairwise disjoint; 2. (ii)

the balls $B_{\delta}(y_{j})$ cover $\Omega$; 3. (iii)

each point of $\Omega$ belongs to at most $N$ of the balls $B_{\delta}(y_{j})$

Definition 2.6**.**

The sequence $\{y_{j}\}$ is called a $\delta$-lattice of $\Omega.$

Our goal is to obtain an atomic decomposition theorem for holomorphic functions in $A_{\textbf{{s}}}^{p,q}$ spaces. To this end, we need to derive a suitable version of the classical Whitney decomposition of $\mathbb{R}^{n}$. Let $\{y_{j}\}$ be a $\delta$-lattice of $\Omega$ and let $g_{j}\in T$ be such that $g_{j}\cdot\textbf{e}=y_{j}.$ Let $R>1$ be a constant like in Lemma 2.4. We adopt the following notations:

[TABLE]

[TABLE]

where $\{x_{l,j}\}$ is a sequence in $\mathbb{R}^{n}$ to be determined.

From Lemma 2.4 we have immediately the following.

Remark 2.7**.**

For the constant $R>1$ of Lemma 2.4, the following inclusion holds

[TABLE]

Lemma 2.8**.**

Let $\delta\in(0,1).$ There exist a positive constant $R>1,$ a positive integer $N$ and a sequence of points $\{x_{l,j}\}_{l\in\mathbb{Z},\hskip 2.84526ptj\in\mathbb{N}}$ in $\mathbb{R}^{n}$ such that the following hold.

1. (i)

$\{I_{l,j}\}_{l}$* form a cover of $\mathbb{R}^{n}$;* 2. (ii)

$\{I^{\prime}_{l,j}\}_{l}$* are pairwise disjoint;* 3. (iii)

for each $j,$ every point of $\mathbb{R}^{n}$ belongs to at most $N$ balls $I_{l,j}$.

Proof.

Fix $j$ in $\mathbb{N}$ and define the collection $\mathcal{A}_{j}$ of sets in $\mathbb{R}^{n}$ by

[TABLE]

Clearly the collection $\mathcal{A}_{j}$ is non empty. Indeed the sets $\{\frac{\delta}{R}y_{j},0_{\mathbb{R}^{n}}\}$ are members of $\mathcal{A}_{j}$. Furthermore, the collection $\mathcal{A}_{j}$ is partially ordered with respect to inclusion.

Let $\mathcal{C}$ be a totally ordered subcollection of $\mathcal{A}_{j}.$ We set $F=\underset{A\in\mathcal{C}}{\cup}A.$ Given two distinct elements $x,y$ of $F,$ there are two members $A_{1}$ and $A_{2}$ of $\mathcal{C}$ such that $x\in A_{1}$ and $y\in A_{2}.$ But either $A_{1}\subset A_{2}$ or $A_{2}\subset A_{1}.$ So we have either $x,y\in A_{1}$ or $x,y\in A_{2}.$ Hence $||g_{j}^{-1}(x-y)||\geq\frac{\delta}{R}.$ This shows that $F$ is a member of $\mathcal{A}_{j}.$ In other words, the collection $\mathcal{A}_{j}$ is inductive. An application of Zorn’s lemma then gives that the collection $\mathcal{A}_{j}$ has a maximal member $E_{j}.$ We write $E_{j}=\{x_{l,j}\}_{l\in L_{j}}.$

To prove assertion (ii), consider $l$ and $k$ such that $l\neq k$ and assume that $I^{\prime}_{l,j}\cap I^{\prime}_{k,j}$ contain at least an element $x$. Then

[TABLE]

This would contradict the property that $\{x_{l,j},\,\,x_{k,j}\}$ is a subset of ${E}_{j},$ which is a member of $\mathcal{A}_{j}$.

For assertion (i), let us suppose $\underset{l\in L_{j}}{\cup}I_{l,j}\neq\mathbb{R}^{n}$. Then there exists $\xi_{j}\in\mathbb{R}^{n}$ such that $\xi_{j}\notin\underset{l\in L_{j}}{\cup}I_{l,j}$. Clearly the set $E_{j}\cup\{\xi_{j}\}$ is a member of $\mathcal{A}_{j}$. This would contradict the maximality of $E_{j}$ in $\mathcal{A}_{j}$. This completes the proof of assertion (i).

To prove assertion (iii), we fix $j.$ Given $x\in\mathbb{R}^{n},$ it follows from assertion (i) that there exists a subset $L_{j}(x)$ of $L_{j}$ such that

[TABLE]

We will show that there is a positive integer $N$ independent of $\delta$ such that $Card\hskip 5.69054ptL_{j}(x)\leq N$ for all $j\in\mathbb{N}$ and $x\in\mathbb{R}^{n}.$ It follows from Lemma 2.4 and Lemma 2.5 that for every $l\in L_{j},$

[TABLE]

So the balls ${\mathbf{B}}_{\frac{\delta}{2R^{2}}}(x_{l,j}+iy_{j}),\hskip 5.69054ptl\in L_{j}$ are pairwise disjoint since $R>1.$ Moreover, for every $l\in L_{j}(x),$ we have

[TABLE]

For the first inclusion, we applied the triangle inequality. We obtain

[TABLE]

We call $m$ the invariant measure on $T_{\Omega}$ given by

[TABLE]

We conclude that

[TABLE]

because

[TABLE]

We finally prove that the collection $\{I_{l,j}\}_{l,j}$ is countable. It suffices to show that for each $j,$ the collection $\{I_{l,j}\}_{l\in L_{j}}$ is countable. We fix $j.$ To every set $I^{\prime}_{l,j},$ we assign a point of $\mathbb{Q}^{n}$ belonging to $I^{\prime}_{l,j}.$ Since $\bigcap\limits_{l}I^{\prime}_{l,j}=\emptyset,$ this defines a one-to-one correspondence from the collection $\{I^{\prime}_{l,j}\}_{l\in L_{j}}$ to a subset of $\mathbb{Q}^{n}.$ This shows that the collection $\{I^{\prime}_{l,j}\}_{l\in L_{j}}$ is at most countable. Moreover the collection $\{I_{l,j}\}_{l\in L_{j}}$ which has the same cardinal as the collection $\{I^{\prime}_{l,j}\}_{l\in L_{j}}$ is infinite: the proof is elementary since $\bigcup\limits_{l\in L_{j}}I_{l,j}=\mathbb{R}^{n}$ is unbounded. The proof of the lemma is complete.

∎

Remark 2.9**.**

We just proved in Lemma 2.8 that for each $j=1,2,\cdots,$ the index set $L_{j}$ is countable. In analogy with the one-dimensional case [18], we took $L_{j}=\mathbb{Z}$ in the statement of Lemma 2.8 and in the statement of Theorem A.

2.6. A $\delta$-lattice in $T_{\Omega}$

Definition 2.10**.**

The sequence $\{z_{lj}=x_{lj}+iy_{j}\}_{l\in\mathbb{Z},j\in\mathbb{N}}$ defined in Lemma 2.8 will be called a $\delta$-lattice  in $T_{\Omega}$.

We have the following lemma.

Lemma 2.11**.**

Let $\{z_{l,j}=x_{l,j}+iy_{j}\}_{l\in\mathbb{Z},j\in\mathbb{N}}$ be a $\delta$-lattice in $T_{\Omega}$. There exists a positive constant $C=C(\delta,R)$ such that for all $l\in\mathbb{Z}$, $j\in\mathbb{N},$ the following hold.

1. (a)

[TABLE] 2. (b)

[TABLE]

Proof.

We denote $Det$ the usual determinant of an endomorphism of $\mathbb{R}^{n}.$

1. (a)

We set $u=g_{j}^{-1}(x-x_{lj}).$ Then

[TABLE]

For the second equality, we applied the formula (2.1). This proves assertion (a). 2. (b)

By assertion (iii) of Lemma 2.8, we have

[TABLE]

for every $j.$ Then

[TABLE]

We set $w=u+iv.$ By Lemma 2.4, we have the implication

[TABLE]

with $\hskip 5.69054ptg\cdot\textbf{e}=v.$ So

[TABLE]

But $d_{\Omega}(y,y_{j})<\delta$ and $d_{\Omega}(y,v)<R.$ This implies that $d(v,y_{j})<\delta+R.$ Henceforth $\Delta^{\frac{n}{r}}(v)\leq C\Delta^{\frac{n}{r}}(y_{j})$ by Lemma 2.1. This gives assertion (b).

∎

3. Atomic decomposition

3.1. The sampling theorem

We first record the following lemma (See e.g. [2]).

Lemma 3.1**.**

Let $1\leq p<\infty.$ Given $\delta\in(0,1)$, there exists a positive constant $C$ such that, for each holomorphic function $F$ in $T_{\Omega}$ we have

1. (i)

$|F(z)|^{p}\leq C\delta^{-2n}\int_{\textbf{B}_{\delta}(z)}|F(u+iv)|^{p}\frac{du\,dv}{\Delta^{\frac{2n}{r}}(v)}$; 2. (ii)

if $d(z,\zeta)<\delta$ then

[TABLE]

For the second lemma, the reader should refer to [1], Lemma 4.5.

Lemma 3.2**.**

Suppose $\delta\in(0,1)$ and $1\leq p,q<\infty$. There exists a positive constant $C$ such that

[TABLE]

for every holomorphic function $F$ on $T_{\Omega}$ and every $y\in\Omega$.

The following is our sampling theorem.

Theorem 3.3**.**

Let $\delta\in(0,1)$ satisfy the assumption of Corollary 2.3 and let $\{z_{l,j}=x_{l,j}+iy_{j}\}_{l\in\mathbb{Z},\hskip 2.84526ptj\in\mathbb{N}}$ be a $\delta$-lattice in $T_{\Omega}$. Let $1\leq p,q<\infty$ and let $\textbf{s}\in\mathbb{R}^{r}$ be such that $s_{k}>\frac{n_{k}}{2},\hskip 5.69054ptk=1,\cdots,r.$ There exists a positive constant $C_{\delta}=C_{\delta}(\textbf{{s}},p,q)$ such that for every $F\in A_{\textbf{{s}}}^{p,q}$, we have

[TABLE]

Moreover, if $\delta$ is small enough, the converse inequality

[TABLE]

is also valid.

Proof.

From Lemma 3.1 we have

[TABLE]

It follows from the inclusion $\textbf{B}_{\frac{\delta}{2R^{2}}}(z_{l,j})\subset\left\{u+iv:\,\,u\in I^{\prime}_{l,j},\,\,v\in B_{\frac{\delta}{2R}}(y_{j})\right\}$ that

[TABLE]

From the equivalence of $\Delta(v)$ and $\Delta(y_{j})$ whenever $v\in B_{\frac{\delta}{2R}}(y_{j}),$ we obtain that

[TABLE]

Next, a successive application of Lemma 2.8, Corollary 2.3 and the non-increasing property of the function $\Omega\ni v\mapsto\|F(\cdot+iv)\|_{p}^{p}$ gives the existence of a positive constant $\gamma$ such that

[TABLE]

Finally, we obtain

[TABLE]

We define the holomorphic function $F_{\gamma}$ by

[TABLE]

By Lemma 3.2, we get

[TABLE]

It follows from (3.8), Lemma 2.5 and the equivalence of $\Delta(y)$ and $\Delta(y_{j})$ whenever $y\in B_{\frac{\delta}{2R^{2}}}(y_{j})$ that

[TABLE]

Moreover, taking $v=\gamma y$ we obtain

[TABLE]

So the estimate (3.2) is a direct consequence of (3.7) and (3.9).

Conversely, a successive application of Lemma 2.8, the triangle inequality and assertion a) of Lemma 2.11 gives

[TABLE]

for all $y\in\Omega$. In the sequel, for fixed $y\in\Omega,$ we set

[TABLE]

and we write

[TABLE]

[TABLE]

Using assertion (ii) of Lemma 3.1, we obtain easily that

[TABLE]

To prove (3.3) it suffices to establish the following inequality:

[TABLE]

To this end, first observe that by assertion (b) of Lemma 2.11, we have

[TABLE]

Now by Lemma 2.1, we have the equivalence $\Delta(v)\sim\Delta(y_{j})\sim\Delta(y)$ whenever $v\in B_{R}(y)$ and $y\in B_{\delta}(y_{j})$ with equivalence constants independent of $\delta.$ This combined with an application of assertion (iii) of Lemma 2.5 gives that

[TABLE]

Next, from the non-increasing property of the mapping $v\in\Omega\mapsto\|F(\cdot+iv)\|_{p},$ Corollary 2.3 and the $G$-invariance of the measure $\frac{dv}{\Delta^{\frac{n}{r}}(v)}$ on $\Omega,$ there exists a positive constant $\gamma$ independent of $\delta$ such that

[TABLE]

Finally, taking $t=\gamma y$ on the right hand side of the previous inequality, we obtain that

[TABLE]

∎

3.2. Proof of Theorem A

We can now prove the atomic decomposition theorem (Theorem A). Here is its more precise statement.

Theorem 3.4**.**

Let $\delta\in(0,1)$ and let $\{z_{l,j}=x_{l,j}+iy_{j}\}_{l\in\mathbb{Z},\hskip 2.84526ptj\in\mathbb{N}}$ be a $\delta$-lattice in $T_{\Omega}.$ Let s be a vector of $\mathbb{R}^{r}$ such that $s_{k}>\frac{n_{k}}{2},\hskip 2.84526ptk=1,\cdots,r.$ Assume that $P_{\textbf{{s}}}$ extends to a bounded operator on $L_{\textbf{s}}^{p,q}$. Then there exists a positive constant $C$ such that the following two assertions hold.

1. (i)

For every sequence $\{\lambda_{l,j}\}_{l\in\mathbb{Z},\hskip 2.84526ptj\in\mathbb{N}}$ such that

[TABLE]

the series

[TABLE]

is convergent in $A_{\textbf{s}}^{p,q}$. Moreover, its sum $F$ satisfies the inequality

[TABLE] 2. (ii)

For $\delta$ small enough, every function $F\in A_{\textbf{s}}^{p,q}$ may be written as

[TABLE]

with

[TABLE]

Proof of Theorem 3.4.

Let $p\in[1,\infty],\hskip 5.69054ptq\in(1,\infty),$ and call $p^{\prime}$ and $q^{\prime}$ their conjugate exponents, i.e $\frac{1}{p}+\frac{1}{p^{\prime}}=1$ and $\frac{1}{q}+\frac{1}{q^{\prime}}=1$. Let $\textbf{{s}}\in\mathbb{R}^{r}$ such that $s_{k}>\frac{n_{k}}{2},\hskip 5.69054ptk=1,\cdots,r.$ Recall that (cf. [11]) if $P_{\textbf{{s}}}:L^{p^{\prime},q^{\prime}}_{\textbf{{s}}}\to A^{p^{\prime},q^{\prime}}_{\textbf{{s}}}$ is bounded, then the dual space of $A^{p^{\prime},q^{\prime}}_{\textbf{{s}}}$ identifies with $A^{p,q}_{\textbf{{s}}}$ with respect to the pairing

[TABLE]

Denote by $l^{p,q}_{\textbf{{s}}}$ the space of complex sequences $\{\lambda_{l,j}\}_{l\in\mathbb{Z},\hskip 2.84526ptj\in\mathbb{N}}$ such that

[TABLE]

We have the duality $l^{p,q}_{\textbf{{s}}}=(l_{\textbf{{s}}}^{p^{\prime},q^{\prime}})^{\prime}$ with respect to the pairing

[TABLE]

Then from the first part of the sampling theorem, the operator

[TABLE]

is bounded. So the adjoint operator $R^{*}$ of $R$ is also a bounded operator from $l^{p,q}_{\textbf{{s}}}$ to $A^{p,q}_{\textbf{{s}}}$. Its explicit formula is

[TABLE]

This completes the proof of assertion (i).

From the second part of the sampling theorem, if $\delta$ is small enough, the adjoint operator $R^{*}:l^{p,q}_{\nu}\to A^{p,q}_{\nu}$ of $R$ is onto. Moreover, we call $\mathcal{N}$ the subspace of $l^{p,q}_{\nu}$ consisting of all sequences $\{\lambda_{l,j}\}_{l\in\mathbb{Z},\,j\in\mathbb{N}}$ such that the mapping

[TABLE]

vanishes identically. Then the linear operator

[TABLE]

is a bounded isomorphism from the Banach quotient space $l_{\nu}^{p,q}/\mathcal{N}$ to $A_{\textbf{s}}^{p,q}$. The inverse operator $\varphi^{-1}$ of $\varphi$ is continuous. This gives assertion (ii). ∎

4. Interpolation

In this section we determine the interpolation space via the complex method between two mixed norm weighted Bergman spaces.

4.1. Interpolation via the complex method between Banach spaces

Throughout this section we denote by $S$ the open strip in the complex plane defined by

[TABLE]

Its closure $\overline{S}$ is

[TABLE]

Let $X_{0}$ and $X_{1}$ be two compatible Banach spaces, i.e. they are continuously embedded in a Hausdorff topological space. Then $X_{0}+X_{1}$ becomes a Banach space with the norm

[TABLE]

We will denote by $\mathcal{F}(X_{0},X_{1})$ the space of analytic mappings

[TABLE]

with the following properties:

1. (1)

$f$ is bounded and continuous on $\overline{S};$ 2. (2)

$f$ is analytic in $S;$ 3. (3)

For $k=0,1$ the function $y\mapsto f_{k+iy}$ is bounded and continuous from the real line into $X_{k}$.

The space $\mathcal{F}(X_{0},X_{1})$ is a Banach space with the following norm:

[TABLE]

If $\theta\in(0,1)$, the complex interpolation space $[X_{0},X_{1}]_{\theta}$ is the subspace of $\mathcal{F}(X_{0},X_{1})$ consisting of holomorphic functions $g$ on $T_{\Omega}$ such that $f_{\theta}=g$ for some $f\in\mathcal{F}(X_{0},X_{1})$. The space $[X_{0},X_{1}]_{\theta}$ is a Banach space with the following norm:

[TABLE]

Referring to [6] and [19] (cf. also [22]), the complex method of interpolation spaces is functorial in the following sense: if $Y_{0}$ and $Y_{1}$ denote two other compatible Banach spaces of measurable functions on $T_{\Omega},$ then if

[TABLE]

is a linear operator with the property that $T$ maps $X_{0}$ boundedly into $Y_{0}$ and $T$ maps $X_{1}$ boundedly into $Y_{1}$, then $T$ maps $[X_{0},X_{1}]_{\theta}$ boundedly into $[Y_{0},Y_{1}]_{\theta}$, for each $\theta\in(0,1)$. See [6] for more information about complex interpolation.

A classical example of interpolation via the complex method concerns $L^{p,q}$ spaces with a change of measures. We state it in our setting of a tube domain $T_{\Omega}$ over a symmetric cone $\Omega.$

Theorem 4.1**.**

[9, 20]** Let $1\leq p_{0},p_{1},q_{0},q_{1}\leq\infty.$ Given two positive measurable functions (weights) $\omega_{0},\hskip 5.69054pt\omega_{1}$ on $\Omega,$ then for every $\theta\in(0,1),$ we have

[TABLE]

[TABLE]

with equal norms, provided that

[TABLE]

[TABLE]

[TABLE]

We finally record the Wolff reiteration theorem [21, 15] .

Theorem 4.2**.**

Let $A_{1},A_{2},A_{3},A_{4}$ be compatible Banach spaces. Suppose $[A_{1},A_{3}]_{\theta}=A_{2}$ and $[A_{2},A_{4}]_{\varphi}=A_{3}.$ Then

[TABLE]

with $\xi=\frac{\theta\varphi}{1-\theta+\theta\varphi},\psi=\frac{\varphi}{1-\theta+\theta\varphi}.$

4.2. Preliminary results on tube domains over symmetric cones

We recall the following notations given in the introduction:

[TABLE]

and

[TABLE]

for every $k=1,\cdots,r.$ We recall the following two results ([17, 4]).

Lemma 4.3**.**

Let $\textbf{{s}},\textbf{{t}}\in\mathbb{R}^{n}$ be such that $s_{k},t_{k}>\frac{n_{k}}{2},\hskip 5.69054ptk=1,\cdots,r.$ Then the subspace $A^{2,2}_{\textbf{{t}}}\cap A^{p,q}_{\textbf{{s}}}$ is dense in the weighted Bergman space $A^{p,q}_{\textbf{{s}}}$ for all $1\leq p\leq\infty$ and $1\leq q<\infty$

Corollary 4.4**.**

Let $\textbf{{s}}\in\mathbb{R}^{n}$ be such that $s_{k}>\frac{n_{k}}{2},\hskip 5.69054ptk=1,\cdots,r.$ Assume that $\textbf{{t}}\in\mathbb{R}^{n}$ and $1\leq p,q<\infty$ are such that $P_{\textbf{{t}}}$ extends to a bounded operator on $L_{\textbf{{s}}}^{p,q}.$ Then $P_{\textbf{{t}}}$ is the identity on $A_{\textbf{{s}}}^{p,q};$ in particular $P_{\textbf{{t}}}(L_{\textbf{{s}}}^{p,q})=A_{\textbf{{s}}}^{p,q}.$

The following theorem was proved in [4].

Theorem 4.5**.**

Let $\textbf{{s}},\textbf{{t}}\in\mathbb{R}^{r}$ and $1\leq p,q\leq\infty$. Then the positive Bergman operator $P^{+}_{\textbf{{t}}}$ defined by

[TABLE]

is bounded on $L_{\textbf{{s}}}^{p,q}$ when $t_{k}>\frac{n}{r}-1,\hskip 5.69054ptk=1,\cdots,r$ and

[TABLE]

[TABLE]

In this case, $P_{\textbf{{t}}}$ extends to a bounded operator from $L_{\textbf{{s}}}^{p,q}$ onto $A_{\textbf{{s}}}^{p,q}.$

Proof.

For $q>1,$ the first part of this theorem is just the case $\alpha=0$ in Theorem 3.8 of [4] for symmetric cones. The case $q=1$ is an easy exercise (cf. e.g. Theorem II.7 of [3]). The proof of the surjectivity of $P_{\textbf{{t}}}$ uses the previous corollary. ∎

4.3. Proof of Theorem B

(1) We adopt the following notations:

[TABLE]

and

[TABLE]

It suffices to show the existence of a positive constant $C$ such that the following two estimates are valid.

[TABLE]

[TABLE]

We first the estimate (4.1). By Theorem 4.1, we have

[TABLE]

with equivalent norms, provided that

[TABLE]

[TABLE]

[TABLE]

In particular, for every $g\in L^{p,q}_{\textbf{{s}}},$ we have

[TABLE]

By Theorem 4.5, for t large (i.e. each $t_{k},\hskip 5.69054ptk=1,\cdots,r$ is large), the weighted Bergman projector $P_{\textbf{{t}}}$ extends to a bounded operator from $L^{p_{i},q_{i}}_{\textbf{{s}}_{i}}$ onto $A^{p_{i},q_{i}}_{\textbf{{s}}_{i}},\hskip 5.69054pti=0,1$ and hence from $L^{p,q}_{\textbf{{s}}}$ onto $A^{p,q}_{\textbf{{s}}}.$ Then by Corollary 4.4, for every $g\in A^{p_{i},q_{i}}_{\textbf{{s}}_{i}},\hskip 5.69054pti=0,1$ and for every $g\in A^{p,q}_{\textbf{{s}}},$ we have $P_{\textbf{{t}}}g=g.$

Now let $g\in A^{p,q}_{\textbf{{s}}}.$ For $f\in\mathcal{F}(L^{p_{0},q_{0}}_{\textbf{{s}}_{0}},L^{p_{1},q_{1}}_{\textbf{{s}}_{1}}),$ we define the mapping

[TABLE]

by $(P_{\textbf{{t}}}\circ f)_{\zeta}=P_{\textbf{{t}}}\circ f_{\zeta}.$ Then $P_{\textbf{{t}}}\circ f\in\mathcal{F}(A^{p_{0},q_{0}}_{\textbf{{s}}_{0}},A^{p_{1},q_{1}}_{\textbf{{s}}_{1}})$ and if $f_{\theta}=g,$ we have $(P_{\textbf{{t}}}\circ f)_{\theta}=P_{\textbf{{t}}}\circ f_{\theta}=P_{\textbf{{t}}}g=g.$ So

[TABLE]

for every $f\in\mathcal{F}(L^{p_{0},q_{0}}_{\textbf{{s}}_{0}},L^{p_{1},q_{1}}_{\textbf{{s}}_{1}})$ such that $f_{\theta}=g.$ By Theorem 4.5, we get

[TABLE]

This proves the estimate (4.1).

We next prove the estimate (4.2). Let $g\in[A^{p_{0},q_{0}}_{\textbf{{s}}_{0}},A^{p_{1},q_{1}}_{\textbf{{s}}_{1}}]_{\theta}.$ We first suppose that $||g||^{anal}_{\theta}=0,$ i.e. $g=0$ in the Banach space $\left[A^{p_{0},q_{0}}_{\textbf{{s}}_{0}},A^{p_{1},q_{1}}_{\textbf{{s}}_{1}}\right]_{\theta}.$ We notice that

[TABLE]

for all $\varphi\in\mathcal{F}\left(A^{p_{0},q_{0}}_{\textbf{{s}}_{0}},A^{p_{1},q_{1}}_{\textbf{{s}}_{1}}\right).$ This implies that

[TABLE]

and hence $||g||_{\left[L^{p_{0},q_{0}}_{\textbf{{s}}_{0}},L^{p_{1},q_{1}}_{\textbf{{s}}_{1}}\right]_{\theta}}=0.$ By the estimate (4.3), we obtain $||g||_{L^{p,q}_{\textbf{{s}}}}=0.$

We next suppose that $0<||g||^{anal}_{\theta}<\infty.$ There exists $\varphi\in\mathcal{F}(L^{p_{0},q_{0}}_{\textbf{{s}}_{0}},L^{p_{1},q_{1}}_{\textbf{{s}}_{1}})$ such that $g=f_{\theta}$ and $||\varphi||_{\mathcal{F}\left(A^{p_{0},q_{0}}_{\textbf{{s}}_{0}},A^{p_{1},q_{1}}_{\textbf{{s}}_{1}}\right)}\leq 2||g||^{anal}_{\theta}.$ By (4.3) and (4.4), we obtain:

[TABLE]

This proves the estimate (4.2).

(2) In this assertion, we have $\textbf{{s}}_{1}=\textbf{{s}}_{2}=\textbf{{s}}.$ The weighted Bergman projector $P_{\textbf{{s}}}$ extends to a bounded operator from $L^{p_{i},q_{i}}_{\textbf{{s}}}$ onto $A^{p_{i},q_{i}}_{\textbf{{s}}},\hskip 5.69054pti=0,1$ and hence from $L^{p,q}_{\textbf{{s}}}$ onto $A^{p,q}_{\textbf{{s}}}.$ Then by Corollary 4.4, for every $g\in A^{p,q}_{\textbf{{s}}},$ we have $P_{\textbf{{s}}}g=g.$ The proof of assertion (2) is the same as the proof of assertion (1) with $\textbf{{t}}=\textbf{{s}}$ in the present case. More precisely, for the proof of the estimate (4.1), we replace the mapping

[TABLE]

with $f\in\mathcal{F}(L^{p_{0},q_{0}}_{\textbf{{s}}_{0}},L^{p_{1},q_{1}}_{\textbf{{s}}_{1}}),$ by the mapping

[TABLE]

with $f\in\mathcal{F}(L^{p_{0},q_{0}}_{\textbf{{s}}},L^{p_{1},q_{1}}_{\textbf{{s}}}).$ The proof of the estimate (4.2) remains the same. (3) We are going to prove the following more precise statement.

Theorem 4.6**.**

Let $\textbf{{s}}\in\mathbb{R}^{r}$ be such that $s_{k}>\frac{n}{r}-1,\hskip 5.69054ptk=1,\cdots,r.$ Let $1\leq p_{0},p_{1}\leq\infty$ and let $q_{0},q_{1}$ be such that $1\leq q_{0}<q_{\textbf{{s}}}\leq q_{1}<\infty.$ Assume that $P_{\textbf{{s}}}$ extends to a bounded operator on $L^{p_{1},q_{1}}_{\textbf{{s}}}.$ Let $\theta,\varphi\in(0,1)$ be related by the equation

[TABLE]

*and assume that *

[TABLE]

Then for $\xi=\frac{\theta\varphi}{1-\theta+\theta\varphi},\hskip 5.69054pt\psi=\frac{\varphi}{1-\theta+\theta\varphi},$ we have

[TABLE]

with equivalent norms, with

[TABLE]

and

[TABLE]

Proof.

We apply the Wolff reiteration theorem (Theorem 4.2) with $A_{1}=A^{p_{0},q_{0}}_{\textbf{{s}}},\hskip 5.69054ptA_{2}=A^{p_{2},2}_{\textbf{{s}}},\hskip 5.69054ptA_{3}=A^{p_{3},q_{3}}_{\textbf{{s}}}$ and $A_{4}=A^{p_{4},q_{4}}_{\textbf{{s}}}.$ On the one hand, we observe that $q_{\textbf{{s}}}>2$ and hence the couple $(p_{2},2)$ satisfies the condition

[TABLE]

of Theorem 1.1. So $P_{s}$ extends to a bounded operator on $L^{p_{2},2}_{\textbf{{s}}}$ as well as we assumed that $P_{\textbf{{s}}}$ extends to a bounded operator on $L^{p_{1},q_{1}}_{\textbf{{s}}}.$ We next apply assertion (2) of Theorem B to get the identity $[A_{2},A_{4}]_{\varphi}=A_{3}$ with $p_{3}$ and $q_{3}$ defined by the system $(\star\star\star\star).$

On the other hand, the condition $(\star\star)$ and the definition of $q_{3}$ given by the second equality of $(\star\star\star\star)$ imply that $1<q_{3}<q_{\textbf{{s}}}.$ We recall that $1<q_{0}<q_{\textbf{{s}}}.$ Then by assertion (1) of Theorem B, we obtain the identity $[A_{1},A_{3}]_{\theta}=A_{2}$ with

[TABLE]

The latter identity and the second identity of $(\star\star\star\star)$ give the relation $(\star).$ The former identity and the first identity of $(\star\star\star\star)$ give the relation $(\star\star\star).$

∎

Question. Can Theorem 4.6 and consequently Theorem B be extended to other values of the interpolation parameters $p_{0},q_{0},\textbf{{s}}_{0},p_{1},q_{1},\textbf{{s}}_{1}$?

Final remark. We recall that $g\in[A_{{\textbf{{s}}}_{0}}^{p_{0},q_{0}},A_{{\textbf{{s}}}_{1}}^{p_{1},q_{1}}]_{\theta}$ if there exists a mapping $f\in\mathcal{F}(A_{{\textbf{{s}}}_{0}}^{p_{0},q_{0}},A_{{\textbf{{s}}}_{1}}^{p_{1},q_{1}})$ such that $f_{\theta}=g.$ For ${\textbf{{s}}}_{0}={\textbf{{s}}}_{1}$ real and $p_{i}=q_{i},\hskip 5.69054pti=1,...,r,$ an explicit construction was presented in [5] for such a mapping $f$ in terms of an analytic family of operators and the atomic decomposition of the relevant (usual) Bergman spaces and this construction was generalized in [14] to mixed norm Bergman spaces associated to the same scalar parameter $\textbf{{s}}=(\nu,\cdots,\nu)$. It may be interesting to extend this construction to mixed norm Bergman spaces $A_{{\textbf{{s}}}_{0}}^{p_{0},q_{0}},A_{{\textbf{{s}}}_{1}}^{p_{1},q_{1}}$ associated to more general vectors ${\textbf{{s}}}_{0},{\textbf{{s}}}_{1}\in\mathbb{R}^{r}.$