1 Introduction
The study of Bergman spaces has long been a cornerstone of complex analysis and functional analysis, with profound connections to operator theory, partial differential equations, and geometric function theory.
For a bounded domain
$\Omega \subset {\mathbb {C}}^n$
, the p-Bergman space
${\mathcal {A}}^p(\Omega )$
consists of holomorphic functions that are
${\mathcal {L}}^p$
integrable with respect to the Lebesgue measure
These spaces generalize the classical Bergman space
${\mathcal {A}}^2(\Omega )$
, which carries a natural Hilbert space structure and is a Reproducing Kernel Hilbert Space. Recent years have witnessed significant advances in understanding p-Bergman spaces, particularly regarding the p-Bergman kernel and its properties [Reference Chen and XiongCX24, Reference Chen and ZhangCZ22, Reference Chen, Xiong and ZhangCXZ22], the duality of p-Bergman space [Reference BhatBha24, Reference Chakrabarti, Edholm and McNealCEM19], and its role in classifying complex domains [Reference Deng, Wang, Zhang and ZhouDWZZ20].
In [Reference Chen and ZhangCZ22], the authors defined the p-Bergman kernel as an extremal function satisfying
The p-Bergman kernel
$m_p(\cdot ,z)$
is the unique function that optimizes the above extremal problem. This function also mimics the properties of the Bergman kernel in the context of
${\mathcal {A}}^p(\Omega ),$
including the reproducing property of the Bergman kernel.
In [Reference BergmanBer50], the author observed that a certain set of “extremal functions” form an orthonormal basis for
${\mathcal {A}}^2(\Omega )$
. In this note, we investigate if similar “extremal functions” form a Schauder Basis for
${\mathcal {A}}^p(\Omega )$
. To define these “extremal functions,” we first define an ordering on multi-index set
${\mathbb {N}}_0^n$
Definition 1.1 We define a partial order on multi-indices as follows: For
$\alpha =(\alpha _1,\ldots , \alpha _n), \beta =(\beta _1,\ldots , \beta _n) \in {\mathbb {N}}_0^n$
, we say
$\alpha \prec \beta $
if
-
a) $|{\alpha }|<|{\beta }|,$
or -
b) $|{\alpha }|=|{\beta }|$
and there exists a
$k\, \in \{ 1,\ldots , n-1\}$
such that
$\alpha _j=\beta _j$
for
$j<k$
and
${\beta _k>\alpha _k}$
.
Note that this partial order is a well-ordering, and since each multi-index
$\beta $
has finitely many predecessors, we can find multi-indices
$\alpha ^0=(0,\ldots ,0), \alpha ^1, \ldots , \alpha ^i, \alpha ^{i+1},\ldots , \alpha ^{m}=\beta $
such that each
$\alpha ^{i+1}$
is the immediate successor of
$\alpha ^i$
.
For
$z_0\in \Omega $
, let
and let
By an argument of normality, we see that there exists a unique minimizer function, say
$m_p^\alpha (\cdot , z_0)$
. An observation by Bergman [Reference BergmanBer50] states that
$\{ m_2^\alpha (\cdot ,z)/\|{m_2^\alpha (\cdot ,z)}\|_2 :\alpha \in {\mathbb {N}}_0^n\}$
is an orthonormal basis for
${\mathcal {A}}^2(\Omega )$
.
The question was posed in [Reference Chen and ZhangCZ22] asking if
$\left\{m_p^\alpha (\cdot , z): \alpha \in {\mathbb {N}}_0^n\right\}$
forms a Schauder Basis for
${\mathcal {A}}^p(\Omega )$
. We answer this question in the case of Reinhardt domains. Let us now recall the definition of a Schauder Basis.
Definition 1.2 (Schauder Basis)
Let X be a Banach space. We say that a sequence of elements
$\{f_k\}_{k \in {\mathbb {N}}}$
is a Schauder Basis if for each
$f\in X$
, there exists a unique sequence of scalars
$\{a_k\}$
such that
$f=\sum \limits _{k\in {\mathbb {N}}} a_k(f) f_k$
, where the series converges in norm to f.
2 Extremal Schauder Basis for the p-Bergman space
We note the following Bergman inequality for the derivatives.
Lemma 2.1 (Bergman inequality for derivatives)
Let
$f\in {\mathcal {A}}^p(\Omega )$
and let
$\alpha \in ({\mathbb {N}}_0)^n$
. For
$a\in \Omega $
, let
$r=\dfrac {\mathrm {dist}(a, b\Omega )}{\sqrt {n}}$
. Then
Proof The proof of this follows from the Cauchy inequalities (see [Reference Michael RangeRan98, Corollary I.1.7]). Consider the polydisc
$P(a, \rho )=\{z \in {\mathbb {C}}^n:| z_i-(a)_i|<\rho _i \text { for all } i=1,2,\ldots , n \}$
which is compactly contained in
$\Omega $
, whenever and for
$f\in \mathcal {O}(\Omega )$
,
$\rho _i< r$
,
where
$b_0P(z_0,\rho )=\left\{z \in {\mathbb {C}}^n| |{z_i-(z_0)_i}|=\rho _i \text { for all } i=1,2,\ldots , n \right\}$
is the distinguished boundary of the polydisc. Let
$\mathbb {T}=[0,2\pi ]^n$
denote the n-torus. Converting into polar coordinates with
$\zeta =\rho e^{i\theta }$
Integrating the above with respect to
$\rho _1,\rho _2,\ldots , \rho _n$
gives us
Therefore,
where
$1/p+1/q=1$
. The last inequality is obtained by applying Hölder’s inequality.
Lemma 2.2 Let
$\Omega $
be a bounded domain such that
$\Omega \subset B(0,R)$
and
$p\in [1,\infty )$
. Then there exists a unique minimizer
$m_p^\alpha (\cdot , z_0)$
that minimizes
Further, if
$m_p^\alpha (\cdot ,z_0)$
is that unique minimizer, it follows that
Proof The proof of existence and uniqueness is similar to the proof of existence uniqueness of
$m_p(\cdot , z_0)$
[Reference Chen and ZhangCZ22, Section 2.1, 2.2]. We’ll prove it here again for the sake of completeness.
-
i. Existence: Fix $z_0\in \Omega $
and
$\alpha \in {\mathbb {N}}_0^n$
. Let
$\left\{\varphi _k\right\}$
be a sequence of functions in
$H_p^{\alpha ,z_0}$
such that
$\|{\varphi _k}\|_p \searrow m_p^\alpha (z_0)$
as
$k\rightarrow \infty $
. Applying Lemma 2.1 with
$\alpha =0$
and observing that the
$\left\{\varphi _k\right\}$
is bounded in norm, we deduce that it is a locally bounded family. Therefore, by Montel’s theorem,
$\varphi _k$
converges locally uniformly to some holomorphic function
$m_p^\alpha (\cdot , z_0)$
and
$\partial ^\beta m_p(z_0,z_0)= \lim \limits _{k\rightarrow \infty } \partial ^\beta \varphi _k(z_0)$
for all
$\beta \in N_0^n$
. Applying Fatou’s lemma, $$ \begin{align*}\|{m_p^\alpha(\cdot, z_0)}\|_p \leq \liminf_{k\rightarrow \infty}\|{\varphi_k}\| =m_p^\alpha(z_0),\end{align*} $$proving the existence of an extremal function $m_p^\alpha (\cdot ,z_0)$
in
$ H_p^{\alpha , z_0}$
.
-
ii. Uniqueness: Let $\varphi $
be another holomorphic function in
$H_p^{\alpha ,z_0}$
such that
${\|{\varphi }\|=\|{m_p^\alpha (\cdot ,z_0)}\|=m_p^\alpha (z_0)}$
. Set
$f=\dfrac {\varphi +m_p^\alpha (\cdot ,z_0)}{2} \in H_p^{\alpha ,z_0}$
, then by convexity of the function
$t\mapsto |{t}|^p$
, we have $$ \begin{align*}[m_p^{\alpha}(z_0)]^p\leq \int_\Omega |{f(\zeta)}|^p \, dV_\zeta\leq \int_\Omega \dfrac{|{\varphi(\zeta)}|^p+ |{m_p^\alpha(\zeta,z_0)}|^p}{2}\, dV_\zeta = [m_p(z_0)]^p,\end{align*} $$that is, $\|{f}\|=m_p^\alpha (z_0)$
. Now, by strict convexity of
${\mathcal {A}}^p(\Omega )$
and the fact that
$\varphi ,m_p(\cdot ,z_0)$
, we see that
$\varphi =m_p(\cdot , z_0)$
, proving uniqueness of the minimizer.
-
iii. Finally, to obtain the estimates, we note that $\Omega $
is a bounded domain, therefore the normalized monomial
$\dfrac {(z-z_0)^\alpha }{\alpha !} \in H_p^{\alpha ,z_0} $
which shows that $$ \begin{align*} m_p^\alpha(z_0)\leq \dfrac{(2R)^{|{\alpha}|}}{\alpha!}\cdot (\mathrm{vol}(\Omega))^{1/p} \text{ and } |{\partial^\beta m_p^{\alpha}(z_0,z_0)}|\lesssim \dfrac{(2R)^{|{\alpha}|}}{\alpha!}\cdot (\beta+2)! \end{align*} $$
Lemma 2.3 For
$\alpha \in {\mathbb {N}}_0^n$
, there exists a bounded linear functional
$a_\alpha :{\mathcal {A}}^p(\Omega )\rightarrow {\mathbb {C}}$
such that
Proof Fix
$z_0\in \Omega $
and
$\alpha \in {\mathbb {N}}_0^n$
and we can find finite multi-indices
$\alpha ^0=(0,\ldots ,0)$
,
$\alpha ^1$
,
$\ldots $
,
$\alpha ^i$
,
$\alpha ^{i+1}$
,
$\ldots $
,
$\alpha ^{k}=\alpha $
such that each
$\alpha ^{i+1}$
is an immediate successor of
$\alpha ^i$
. Assume that for
$f\in {\mathcal {A}}^p(\Omega )$
, we have the formal summation
Differentiate the above series with respect to
$\partial ^{\alpha _0}$
,
$\partial ^{\alpha _1}$
,
$\ldots $
,
$\partial ^{\alpha _k}$
and substitute
$z=z_0$
, we get the following set of equations:
Thus, solving for
$a_{\alpha _k}(f)$
translates to solving the above system of linear equations, that is, finding the inverse of a lower triangular matrix as below:
If we write the above
$(k+1)\times (k+1)$
square matrix as
$I+N$
, where I is the identity matrix and N is the above matrix with diagonal entries replaced by
$0$
. Since, all eigenvalues of N are
$0$
, it follows that
$N^{k+1}=0$
implying
that is,
$I+N$
is invertible, and the inverse is given by
If each term of
$(I+N)^{-1}$
is bounded by M, then
Further, from the above matrix, we note that
2.1 Reinhardt domain
A domain
$ \Omega \subset {\mathbb {C}}^n$
is said to be a Reinhardt domain if
A Reinhardt domain
$\Omega $
is said to be a complete Reinhardt domain if
For a bounded Reinhardt domain
$\Omega $
, the shadow of
$\Omega $
, denoted
$|{\Omega }|$
, is
When
$\Omega $
is a Reinhardt domain, the normalized monomial
$e_\alpha (\zeta ):=\dfrac {\zeta ^\alpha }{\alpha !}$
functions play an important role in the study of p-Bergman spaces. Chakrabarti and Edholm [Reference Chakrabarti and EdholmCE24, Theorem 2.16] show that these normalized monomials form a Schauder Basis for
${\mathcal {A}}^p(\Omega )$
. More precisely, let
$S(\Omega ,{\mathcal {A}}^p)$
denote the set of all
${\mathcal {A}}^p$
allowable multi-indices, that is,
then
$\displaystyle {\left\{e_\alpha : {\alpha \in S(\Omega , {\mathcal {A}}^p)}\right\}}$
is a Schauder Basis for
${\mathcal {A}}^p(\Omega )$
. Note that for a bounded complete Reinhardt domain
$\Omega $
,
$\displaystyle {S(\Omega ,{\mathcal {A}}^p)={\mathbb {N}}_0^n}$
and thus
$\left\{e_\alpha : \alpha \in {\mathbb {N}}_0^n\right\}$
is a Schauder Basis for
${\mathcal {A}}^p(\Omega )$
.
Lemma 2.4 Let
$\Omega $
be a bounded complete Reinhardt domain and
$z_0=0$
, then
forms a Schauder Basis for
${\mathcal {A}}^p(\Omega )$
.
Proof Recall that
$H_p^{\alpha , 0} = \left \{ f \in \mathcal {A}^p(\Omega ) : \partial ^\alpha f(0) = 1,\, \partial ^\beta f(0) = 0 \text { for all } \beta \prec \alpha \right \}.$
Since
$\Omega $
is a complete Reinhardt domain, any function
$f \in H_p^{\alpha , 0}$
can be written via its power series:
where
$e_\alpha (w) = \frac {w^\alpha }{\alpha !}$
, and h is holomorphic with
$\partial ^\beta h(0) = 0$
for all
$\beta \preceq \alpha $
.
Let
$h \in \mathcal {A}^p(\Omega )$
satisfy
$\partial ^\beta h(0) = 0$
for all
$\beta \preceq \alpha $
. Consider the one-parameter family
$\displaystyle {\{e_\alpha + t h\}_{t \in \mathbb {R}} \subset H_p^{\alpha , 0}}$
. Computing the derivative of the
${\mathcal {L}}^p$
norm to the p-th power at
$t=0$
gives
since h is orthogonal to
$e_\alpha $
by hypothesis on h. In fact,
where
$\zeta _i=r_ie^{i\tau _i}$
are the polar coordinates. Now we note that
$\displaystyle {\int _{\mathbb {T}^n} e^{i\alpha \cdot \tau } \cdot \overline {e^{i\eta \cdot \tau }}\, d\tau }=0$
, whenever
$\alpha \neq \eta $
, proving orthogonality between h and
$e_\alpha $
.
Therefore, for any
$f \in H_p^{\alpha , 0}$
, the convex function
$t \mapsto \|e_\alpha + t h\|_p$
attains its minimum at
$t = 0$
. By the uniqueness of the extremal function, we conclude that
$m_p^\alpha (w, 0) = e_\alpha (w).$
Finally, the set
$\{ m_p^{\alpha }(\cdot , 0) : \alpha \in \mathbb {N}_0^n \}$
forms a Schauder Basis for
$\mathcal {A}^p(\Omega )$
; that is, for any
$f \in \mathcal {A}^p(\Omega )$
,
where the convergence is in
${\mathcal {L}}^p(\Omega )$
, and
$a_{\alpha , 0}$
is a bounded linear functional.
Corollary 2.5 For a Unit Ball
${\mathbb {B}}^n$
,
$\left \{m_p^\alpha (\cdot , 0): \alpha \in {\mathbb {N}}_0^n \right \}$
is a Schauder Basis for
${\mathcal {A}}^p({\mathbb {B}}^n)$
.
Theorem 2.6 Let
$\Omega ={\mathbb {B}}^n$
denote the Unit Ball and
$z_0\in \Omega $
. Let
$\phi _0$
be an automorphism of the Unit Ball that maps
$0$
to
$z_0$
and set
$c_0=J_{\phi _0}(0)^{-2/p}$
. Then
forms a Schauder Basis for
${\mathcal {A}}^p(\Omega )$
.
Proof Let
$f\in H_p^{\alpha ,z_0}$
, then it follows that
$T_f:=c_0 f\circ \phi _0 \cdot J_{\phi _0}^{2/p} \in H_p^{\alpha ,0}$
, where
$c_0=J_{\phi _0}(0)^{-2/p}$
and by change of variables formula,
Thus,
that is,
Further, by change of variables,
$f\in {\mathcal {A}}^p(\Omega )$
if and only if
$f(\phi _0)\cdot J_{\phi _0}^{2/p} \in {\mathcal {A}}^p(\Omega )$
and by the previous result,
which shows that
$a_\alpha ^{z_0}(f)=a_\alpha ^{0}(T_f)$
:
By change of variables,
proving
$\left\{m_{p}^{\alpha }(\cdot , z_0): \alpha \in {\mathbb {N}}_0^+\right\}$
forms a Schauder Basis for
${\mathcal {A}}^p(\Omega )$
.
Corollary 2.7 Similarly, if
$\Omega $
is a simply-connected complete Reinhardt domain which has a transitive automorphism group, then for all
$z_0\in \Omega $
,
$\left\{m_{p}^{\alpha }(\cdot , z_0): \alpha \in {\mathbb {N}}_0^n\right\}$
forms a Schauder Basis for
${\mathcal {A}}^p(\Omega )$
.
Also, we have proved the following result.
Corollary 2.8 Let
$\Omega $
be a bounded domain and let
$z,w \in \Omega $
such that there exists an
${\mathcal {A}}^p$
preserving biholomorphism (see [Reference BhatBha23, Definition 5.1])
$F:\Omega \rightarrow \Omega $
such that
$F(z)=w$
. Then
$\left\{m_p^\alpha (\cdot , z):\alpha \in {\mathbb {N}}_0^n\right\}$
is a Schauder Basis for
${\mathcal {A}}^p(\Omega )$
if and only if
$\left\{m_p^\alpha (\cdot ,w):\alpha \in {\mathbb {N}}_0^n\right\}$
is a Schauder Basis for
${\mathcal {A}}^p(\Omega )$
.
Acknowledgments
The results of this article are part of the author’s Ph.D. dissertation at Texas A&M University. The author thanks his advisor, Prof. Harold Boas, for providing valuable advice and feedback during the preparation of this note. Finally, the author gratefully acknowledges the anonymous reviewer for identifying an error in the proof of a key lemma and for providing numerous suggestions that significantly improved the clarity of this article.
