1 Introduction
1.1 Setups and problems
Let
$\Omega $
be a domain in
$\mathbb {C}^n$
. The set of square integrable holomorphic functions on
$\Omega $
, denoted by
$A^2(\Omega )$
, forms a closed subspace of the Hilbert space
$L^2(\Omega )$
. The Bergman projection associated with
$\Omega $
is the orthogonal projection
$\textbf {P}_\Omega :L^2(\Omega )\rightarrow A^2(\Omega )$
, which can be expressed as an integral operator
$$ \begin{align*}\textbf{P}_\Omega(f)(z)=\int_{\Omega}K_\Omega(z,w)f(w)dV(w),\,\,\,\,f\in L^2(\Omega).\end{align*} $$
Here, the function
$K_\Omega :\Omega \times \Omega \rightarrow \mathbb {C}$
is the Bergman kernel of
$\Omega $
, and
$dV$
denotes the ordinary Lebesgue measure.
It is of particular interest to study the regularity of the Bergman projection on the associated
$L^p$
Sobolev spaces. We recall that the
$L^p$
Sobolev space of order k with
$1\leq p<\infty $
and
$k\in \mathbb {N}$
is the function space defined by
$$ \begin{align*} L_k^p(\Omega):=\bigg\{f\in L_{\mathrm{loc}}^1(\Omega):\|f\|_{L^p_k}:=\bigg(\int_{\Omega}\sum\limits_{|\alpha|\leq k}|D_{z,\overline{z}}^{\alpha}(f)|^p dV\bigg)^{\frac{1}{p}}<\infty\bigg\}, \end{align*} $$
where the derivatives are interpreted in the distributional sense. For a large class of smooth pseudoconvex domains, including strongly pseudoconvex domains, convex domains of finite type, domains of finite type with locally diagonalizable Levi form, and all finite type domains in
$\mathbb {C}^2$
, the regularity of the Bergman projection on
$L^p$
Sobolev spaces has been intensively studied by various authors (see [Reference Charpentier and Dupain5, Reference McNeal15–Reference Phong and Stein19] and references therein for details).
On another direction, as non-smooth cases, the
$L^p$
Sobolev regularity for the Bergman projections of Hartogs triangle and its generalizations has recently attracted particular attentions. In [Reference Chen7], Chen studied the weighted
$L^p$
Sobolev regularity of the Bergman projection on the classical Hartogs triangle. Later, Edholm and McNeal [Reference Edholm and McNeal11] showed the
$L^p$
Sobolev irregularity of the Bergman projection on two-dimensional power generalized Hartogs triangles
$$ \begin{align} H_{m,n}:=\{(z_1,z_2)\in\mathbb{C}^2:|z_1|^m<|z_2|^n<1\},\,\,\,m,n\in\mathbb{Z}^+. \end{align} $$
Subsequently, their results were extended to n-dimensional settings by Zhang [Reference Zhang22] and Liu and Tu [Reference Liu and Tu14]. Following this line, Zhang [Reference Zhang21] investigated the
$L^p_k$
irregularity of the Bergman projection on elementary Reinhardt domains defined by
$$ \begin{align} \mathcal{H}_k:=\left\{\left(z_1, \ldots, z_n\right) \in \mathbb{D}^n:\left|z_1\right|{}^{k_1}<\left|z_2\right|{}^{k_2} \cdots\left|z_n\right|{}^{k_n}\right\}\!. \end{align} $$
We also refer the readers to [Reference Chen, Jin and Yuan8] for more information on this topic.
It is worth noting that the aforementioned generalized Hartogs triangles (1.1), (1.2) and the domains in the literature above are all special Reinhardt domains with boundary singularities. A unifying result, explaining irregularity in these cases, is lacking. Therefore, inspired by this, it is reasonable and important for us to study the
$L^p$
Sobolev regularity for the Bergman projections on more general singular domains.
1.2 Monomial polyhedra
We will say that a bounded domain
$\mathcal {U}_{B}\subset \mathbb {C}^n$
with
$n\geq 2$
is a monomial polyhedron if there is a matrix
$B\in \mathbb {Z}^{n\times n}$
such that
$$ \begin{align*} \mathcal{U}_{B}:=\left\{(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}:\text{for} \, 1\leq j\leq n, |z_{1}|^{b_{1}^{j}}|z_{2}|^{b_{2}^{j}}\cdots|z_{n}|^{b_{n}^{j}}<1,\right\}\!, \end{align*} $$
where
$b^i_j\in \mathbb {Z}$
denotes the entry at the i-th row and j-th column of B. Monomial polyhedra are clearly Reinhardt and pseudoconvex. Except the trivial case
${\mathbb {D}^n=\mathcal {U}_{I_n}}$
, monomial polyhedra are singular domains with a pathological singularity at the origin. Furthermore, monomial polyhedra are an important class of singular Reinhardt domains with tractable geometry and some very interesting properties (see, for instance, [Reference Bender, Chakrabarti, Edholm and Mainkar1–Reference Chakrabarti, Tang and Zhang4]).
Another notable feature of monomial polyhedra is that they include many well-known examples of generalized Hartogs triangles. For example, when
$B=\left ( \begin {array}{@{}cc@{}} k_1 & -k_2 \\ 0 & 1 \\ \end {array} \right )$
, the monomial polyhedra are the power-generalized Hartogs triangles (1.3) studied in [Reference Edholm and McNeal9, Reference Edholm and McNeal10, Reference Khanh, Liu and Thuc13]. The monomial polyhedra also include elementary Reinhardt domains (1.2) (see [Reference Tang and Zhang20, Reference Zhang21]) and n-dimensional generalized Hartogs triangles (see [Reference Zhang22]) as special examples. It is noteworthy that monomial polyhedra significantly extend these classes of domains, we refer the readers to [Reference Bender, Chakrabarti, Edholm and Mainkar1, Section 1.4] for this topic.
It is worth mentioning that if the monomial polyhedron
$\mathcal {U}_{B}$
is bounded, then B is necessarily invertible. Moreover, we can always choose the matrix B such that
${\det \,B>0}$
and
$B^{-1}\succeq 0$
(that is, it has all nonnegative entries, see Section 2.1 for this notation), and therefore,
$\textrm {adj}\,B=(\det B)B^{-1}\succeq 0$
(see [Reference Bender, Chakrabarti, Edholm and Mainkar1, Proposition 3.2] for more details). In the rest of this article, we will always assume that
$\mathcal {U}_{B}$
is a monomial polyhedron with
$B\in \mathbb {Z}^{n\times n}$
satisfying these conditions, and set
$A:=\textrm {adj}\,B$
.
In Bender et al. [Reference Bender, Chakrabarti, Edholm and Mainkar1], by representing monomial polyhedra as quotient domains, the authors obtained the
$L^p$
regularity of the Bergman projection as follows.
Theorem 1.1 [Reference Bender, Chakrabarti, Edholm and Mainkar1, Theorem 1.2]
Let
$\textbf {P}_{\mathcal {U}_{B}}$
be the Bergman projection of monomial polyhedron
$\mathcal {U}_{B}$
. Then,
$\textbf {P}_{\mathcal {U}_{B}}$
maps
$L^p(\mathcal {U}_{B})$
to
$A^p(\mathcal {U}_{B})$
boundedly if and only if
$$ \begin{align*}p\in\left(\max\limits_{1\leq j\leq n}\frac{2{1\!1} a_j}{{1\!1} a_j+\gcd(a_j)},\min\limits_{1\leq j\leq n}\frac{2{1\!1} a_j}{{1\!1} a_j-\gcd(a_j)}\right),\end{align*} $$
where
${1\!1}:=(1,\ldots ,1)$
be the
$1\times n$
row vector, and
$a_j$
is the j-th column of the matrix A.
Recently, Chakrabarti et al. [Reference Chakrabarti, Cinzori, Gaidhane, Gregory and Wright3] computed the explicit Bergman kernels of monomial polyhedra by Bell’s transformation formula (see Theorem 2.5 in Section 2.2).
1.3 Main results
Motivated by the above, we study the
$L^p$
Sobolev regularity of the Bergman projection on monomial polyhedra. Our first main result is the following theorem, which establishes the
$L^p$
Sobolev irregularity of the Bergman projection on nontrivial monomial polyhedra.
Theorem 1.2 Let
$\mathcal {U}_{B}$
be a nontrivial monomial polyhedron (i.e.,
$\mathcal {U}_{B}\neq \mathbb {D}^n$
).
(1) Let
$l=(l_1,\ldots ,l_n)\in \mathbb {N}^n$
, and denote
$\frac {\partial ^{|l|}}{\partial z^l}:=\frac {\partial ^{|l|}}{\partial z_1^{l_1}\ldots \partial z_n^{l_n}}$
. Then,
$\frac {\partial ^{|l|}}{\partial z^l}\circ \textbf {P}_{\mathcal {U}_{B}}$
fails to map
$C^{\infty }(\overline {\mathcal {U}_{B}})\rightarrow L^p(\mathcal {U}_{B})$
for
$$ \begin{align} p\geq \min_{1\leq j\leq n}\frac{2{1\!1} a_j}{(l+{1\!1})a_j-\gcd(a_j)}. \end{align} $$
(2) The Bergman projection
$\textbf {P}_{\mathcal {U}_{B}}$
fails to map
$L^p_k(\mathcal {U}_{B})\rightarrow L^p_k(\mathcal {U}_{B})$
for any
$k\in \mathbb {Z}^+$
and
$p\geq 2$
. In particular, the Bergman projection of any nontrivial monomial polyhedron is never bounded on any
$L^2_k$
-Sobolev space.
The next theorem gives the boundedness for the differentiated Bergman projection, which is a key tool in studying the
$L^p$
Sobolev regularity of the Bergman projection.
Theorem 1.3 Let
$\mathcal {U}_{B}$
be a nontrivial monomial polyhedron. For
$j_0=1,\ldots ,n$
, the differentiated Bergman projection
$\frac {\partial }{\partial z_{j_0}}\circ \textbf {P}_{\mathcal {U}_{B}}$
maps
$L^p_1(\mathcal {U}_{B})\rightarrow L^p(\mathcal {U}_{B})$
boundedly for
$$ \begin{align} p\in\left(\max_{1\leq j\leq n}\frac{2{1\!1} a_j}{{1\!1} a_j+a^{j_0}_j+\gcd(a_j)},\min_{1\leq j\leq n}\frac{2{1\!1} a_j}{{1\!1} a_j+a^{j_0}_j-\gcd(a_j)}\right). \end{align} $$
Finally, we obtain the
$L^p$
Sobolev regularity of the Bergman projection as follows.
Theorem 1.4 Let
$\mathcal {U}_{B}$
be a nontrivial monomial polyhedron. The Bergman projection
$\textbf {P}_{\mathcal {U}_{B}}:L^p_1(\mathcal {U}_{B})\rightarrow L^p_1(\mathcal {U}_{B})$
is bounded if
$$ \begin{align*} p\in\left(\max_{1\leq j\leq n}\frac{2{1\!1} a_j}{{1\!1} a_j+\gcd(a_j)},\min_{1\leq j,j_0\leq n}\frac{2{1\!1} a_j}{{1\!1} a_j+a^{j_0}_j-\gcd(a_j)}\right). \end{align*} $$
Remark 1.5 Combining Theorems 1.2 and 1.4 with Theorem 1.1, we see that the range on p, for boundedness on
$L^p_1(\mathcal {U}_{B})$
, is much smaller than the range for boundedness on
$L^p(\mathcal {U}_{B})=L^p_0(\mathcal {U}_{B})$
. This shows that the Bergman projection of nontrivial monomial polyhedra is very degenerate as an
$L^p$
Sobolev map.
Our main results generalize previous works on some special singular Reinhardt domains [Reference Edholm and McNeal11, Reference Liu and Tu14, Reference Zhang21, Reference Zhang22] to a much more general setting. Our strategy combines the methods of [Reference Chen7, Reference Chen, Jin and Yuan8, Reference Edholm and McNeal11] with many new ingredients. The main challenge in the proof lies in obtaining an estimate for a kind of modified Bergman kernel
$\widetilde {K}_{\mathcal {U}_{B}}(z,w)$
(see (3.7)). It should be mentioned that in the earlier works, such as [Reference Chen7, Reference Chen, Jin and Yuan8, Reference Edholm and McNeal11, Reference Liu and Tu14], the Bergman kernels of the domains can be explicitly written in a closed form, and thus the estimate for the Bergman kernel is easy to get. However, in our settings of monomial polyhedra, the Bergman kernel of
$\mathcal {U}_{B}$
is much more complicated, and so is the modified Bergman kernel
$\widetilde {K}_{\mathcal {U}_{B}}(z,w)$
. To overcome this difficulty, we adapt the techniques in [Reference Chakrabarti, Cinzori, Gaidhane, Gregory and Wright3, Reference Chakrabarti, Tang and Zhang4] to obtain an estimate for it.
The article is organized as follows. In Section 2, we review some known facts of monomial polyhedra and establish several technical lemmas for our later use. Section 3 is devoted to the proof of our main theorems. Throughout the article, we use
$\lesssim $
and
$\gtrsim $
to denote the corresponding inequalities up to a positive multiplicative constant, and we also use
$\approx $
to denote both
$\lesssim $
and
$\gtrsim $
hold.
2 Preliminaries
2.1 Some notations
Firstly, we need to introduce some notations that will be used throughout this article. For
$1\leq i,j\leq n $
and
$i,j\in \mathbb {Z}$
, we use
$b^i_j$
to denote the entry of matrix B at the intersection of the j-th column and the i-th row. We denote the i-th row vector of B by
$b^{i}$
and the j-th column vector of B by
$b_{j}$
, that is,
$b^{i}=(b_{1}^{i},\ldots ,b_{n}^{i})$
and
$b_{j}=(b_{j}^{1},\ldots ,b_{j}^{n})^{T}$
. For a matrix B with nonnegative integers,
$\gcd (b_j)$
and
$\gcd (B)$
are given as follows:
$$ \begin{align*}\gcd(b_j):=\gcd(b_{j}^{1},\ldots,b_{j}^{n}),\,\,\,\,\gcd(B):=(\gcd(b_1),\ldots,\gcd(b_n)).\end{align*} $$
For
$z=(z_1,\ldots ,z_n)$
and
$\alpha =(\alpha _1,\ldots ,\alpha _n),$
we use the convention that
$z^\alpha :=z_{1}^{\alpha _{1}}\ldots z_{n}^{\alpha _{n}}.$
For
$z=(z_1,\ldots ,z_n)$
and a matrix
$B,$
we set the matrix power
$$ \begin{align*}z^B:= (z^{b^{1}},\ldots,z^{b^{n}}).\end{align*} $$
Direct calculation shows that for
$z=(z_1,\ldots ,z_n)$
and matrices B and C, it holds that
$(z^B)^C=z^{CB}$
. It is worth mentioning that in this article, for any multi-index
$\alpha :=(\alpha _1,\ldots ,\alpha _n)$
we also regard it as a
$1\times n$
row vector. For real matrices of the same size B, C, the notations
$$ \begin{align*}B\succ C,\,\,\,B\succeq C,\,\,\,B\prec C,\,\,\,B\preceq C\end{align*} $$
stand for the natural elementwise order, e.g.,
$B\succ C$
means that
$b^i_j>c^i_j$
,
$\forall i,j$
. Also, let
${1\!1}:=(1,\ldots ,1)\in \mathbb {Z}^{1\times n}$
.
2.2 Some basic facts for monomial polyhedra
In this section, we recall some basic facts and properties of monomial polyhedra. Let
$\mathcal {U}_{B}$
be a monomial polyhedron with
$B\in \mathbb {Z}^{n\times n}$
. Then,
$\mathcal {U}_{B}$
can be written as
$$ \begin{align*} \mathcal{U}_{B}=\left\{z\in\mathbb{C}^n:z^B\,\,\mathrm{is}\,\,\mathrm{defined}\,\,\mathrm{and}\,\,(|z_1|,\ldots,|z_n|)^B\prec{1\!1}\right\}\!. \end{align*} $$
Here,
$z^B$
is defined means that
$z_j\neq 0$
for each point
$z\in \mathcal {U}_{B}$
if
$b^i_j<0$
for some i, j.
Let
$\Omega $
be a bounded Reinhardt domain. A multi-index
$\beta \in \mathbb {Z}^n$
is said to be p-allowable on
$\Omega $
if the holomorphic monomial
$z^\beta \in L^p(\Omega )$
. We denote
$$ \begin{align*}\mathscr{S}_p(\Omega):=\left\{\beta\in\mathbb{Z}^n:z^\beta\in L^p(\Omega)\right\}\!,\end{align*} $$
that is,
$\mathscr {S}_p(\Omega )$
is the set of all p-allowable multi-indices on
$\Omega $
.
The p-allowable multi-indices on monomial polyhedra
$\mathcal {U}_{B}$
are obtained in [Reference Bender, Chakrabarti, Edholm and Mainkar1] as follows.
Lemma 2.1 [Reference Bender, Chakrabarti, Edholm and Mainkar1, Proposition 5.12]
Let
$\beta \in \mathbb {Z}^n$
and
$p>0$
. Then,
$\beta \in \mathscr {S}_p(\mathcal {U}_{B})$
if and only if
$$ \begin{align*}(p\beta+2{1\!1})A\succ0.\end{align*} $$
Lemma 2.2 [Reference Bender, Chakrabarti, Edholm and Mainkar1, Proposition 5.9]
$(1)$
Let
$\beta \in \mathbb {Z}^n$
. Then,
$\beta \in \mathscr {S}_2(\mathcal {U}_{B})$
if and only if
$$ \begin{align} (\beta+{1\!1})A\succeq\gcd(A). \end{align} $$
$(2)$
For
$1 \leqslant j \leqslant n$
, let
$\Pi _j$
be the integer hypersurface determined by equality in the j-th entry of (2.1), that is,
$$ \begin{align*}\Pi_j=\left\{\beta \in \mathbb{Z}^n:(\beta+{1\!1}) a_j=\sum_{k=1}^n\left(\beta_k+1\right) a_j^k=\operatorname{gcd}\left(a_j\right)\right\}\!. \end{align*} $$
Then,
$ \Pi _j \cap \mathscr {S}_2(\mathcal {U}_{B}) \neq \varnothing. $
Let
$\mathrm {K}(B)\subset \{1,\ldots ,n\}$
be the set defined by
$$ \begin{align} \mathrm{K}(B):=\{1\leq k\leq n:b^j_k<0\,\,\mathrm{for}\,\,\mathrm{some}\,\,1\leq j\leq n\}, \end{align} $$
that is,
$\mathrm {K}(B)$
is the set of indices of those columns of B which have at least one negative entry. Define a subset
$\mathrm {L}(B)\subset \{1,\ldots ,n\}$
by
$$ \begin{align*}\mathrm{L}(B)=\left\{1 \leqslant \ell \leqslant n: a_{\ell}^k \neq 0 \text { for some } k \in \mathrm{K}(B)\right\}\!. \end{align*} $$
We also denote the domain
$$ \begin{align*}\mathbb{D}_{\mathrm{L}(B)}^n:=\left\{z \in \mathbb{D}^n: z_{\ell} \neq 0 \text { for all } \ell \in \mathrm{L}(B)\right\}\!. \end{align*} $$
It is easy for one to see that
$\mathbb {D}_{\mathrm {L}(B)}^n$
is the product domain of the unit discs
$\mathbb {D}$
and the punctured unit discs
$\mathbb {D}^*$
.
The following theorem shows that there is a proper holomorphic mapping from the product domain
$\mathbb {D}_{\mathrm {L}(B)}^n$
to the monomial polyhedron
$\mathcal {U}_{B}$
.
Theorem 2.3 [Reference Bender, Chakrabarti, Edholm and Mainkar1, Theorem 3.12]
Let
$\mathcal {U}_{B}$
be a monomial polyhedron associated with matrix
$B\in \mathbb {Z}^{n\times n}$
, and
$A=(a^i_j):=\mathrm {adj}\,B$
. Then, the mapping
$$ \begin{align*} \Phi_{A}:\,\,\,&\,\,\mathbb{D}^{n}_{\mathrm{L}(B)}\rightarrow \mathcal{U}_{B}\\ & \,\,z\,\,\,\mapsto z^{A}=(z^{a^{1}},\ldots,z^{a^{n}}) \end{align*} $$
is a proper holomorphic mapping.
The Jacobian determinant of the mapping
$\Phi _{A}$
is given in the next lemma.
Lemma 2.4 [Reference Bender, Chakrabarti, Edholm and Mainkar1, Lemma 3.8]
For any
$z\in \mathbb {D}^{n}_{\mathrm {L}(B)}$
, we have
$$ \begin{align*}\det J_{\mathbb{C}}\Phi_{A}(z)=\det\,A\cdot z^{{1\!1} A-{1\!1}}.\end{align*} $$
Given a matrix
$B\in \mathbb {Z}^{n\times n}$
and an
$n\times n$
zero matrix
$0,$
let
$$ \begin{align*} B_{+}:=\max\{B,0\},\,\,\,\,\,\,\,\,B_{-}:=\max\{-B,0\}, \end{align*} $$
that is,
$B_{+}$
is a matrix with
$(b_{+})^{i}_{j}:=\max \{b_{j}^{i},0\},$
and
$B_{-}$
is a matrix with
$(b_{-})^{i}_{j}:=\max \{-b_{j}^{i},0\}.$
It is easy for one to see that
$B=B_{+}-B_{-}.$
The following theorem gives the explicit formula for the Bergman kernel for monomial polyhedra, which will play an important role in our proof.
Theorem 2.5 [Reference Chakrabarti, Cinzori, Gaidhane, Gregory and Wright3]
Let
$\mathcal {U}_{B}$
be a monomial polyhedron with
$B\in \mathbb {Z}^{n\times n}$
, and let
$A:=(a^i_j)=\mathrm {adj}\,B$
. Denoting
$z\overline {w}:=\left (z_1\overline {w}_1, \ldots , z_n\overline {w}_n\right )$
, the Bergman kernel of
$\mathcal {U}_B$
is
$$ \begin{align*}K_{\mathcal{U}_B}(z, w)=\frac{1}{\pi^n \cdot(\operatorname{det} B)^{n-1}} \cdot \frac{\sum_{\nu \in \mathbb{N}^{1 \times n}} C_B(\nu) (z\overline{w})^\nu}{\prod_{j=1}^n\left((z\overline{w})^{\left(b_{-}\right)^j}-(z\overline{w})^{\left(b_{+}\right)^j}\right)^2}, \end{align*} $$
where
$$ \begin{align*}C_B(\nu)=\prod_{j=1}^n \mathrm{D}_{\operatorname{det} B}\left(\left(\nu-2 {1\!1} B_{-}+{1\!1}\right)a_j-1\right), \quad \nu \in \mathbb{Z}^{1 \times n} \end{align*} $$
with
$a_j \in \mathbb {Z}^{n \times 1}$
being the j-th column vector of A, and the function
$\mathrm {D}_k(r)$
is given by
$$ \begin{align*} \mathrm{D}_k(r)= \begin{cases}1+r, & 0 \leqslant r \leqslant k-1, \\ 2 k-(1+r), & k \leqslant r \leqslant 2 k-2, \\ 0, & r<0 \text { or } r>2 k-2.\end{cases} \end{align*} $$
Further, we have
$C_B(\nu )=0$
, except perhaps when
$\nu =(\nu _1,\ldots ,\nu _n)$
satisfies
$$ \begin{align} -1+\xi_j\leq\nu_j\leq2\sum_{i=1}^n\left|b^i_j\right|-1-\xi_j,\,\,\,\,1\leq j\leq n, \end{align} $$
with
$\xi _j$
being the ceiling
$\xi _j=\left \lceil \frac {1}{\det B}\cdot \sum _{i=1}^n\left |b^i_j\right |\right \rceil .$
In [Reference Chakrabarti, Tang and Zhang4], the authors obtained the optimal upper estimate for the Bergman kernel of
$\mathcal {U}_B$
as follows.
Proposition 2.6 [Reference Chakrabarti, Tang and Zhang4, Proposition 2.5]
Denote
$t:=\left (t_1, \ldots , t_n\right )$
with
$t_j:=z_j\overline {w}_j$
, and
$A:=\mathrm {adj}\,B$
. The Bergman kernel of
$\mathcal {U}_{B}$
has the following upper estimate:
$$ \begin{align*}|K_{\mathcal{U}_{B}}(z, w)|\lesssim \frac{|t^\lambda|}{\prod_{j=1}^n\left|t^{\left(b_{-}\right)^j}-t^{\left(b_{+}\right)^j}\right|{}^2},\,\,\,\,z,w\in \mathcal{U}_{B},\end{align*} $$
where
$$ \begin{align} \lambda:=2{1\!1} B_- -{1\!1}+\frac{1}{\det\,B}\gcd(A)B. \end{align} $$
2.3 Forelli–Rudin type operators on monomial polyhedra
Now, we define a class of Forelli–Rudin type operators on
$\mathcal {U}_{B}$
, which contains the Bergman projection and differentiated Bergman projection on
$\mathcal {U}_{B}$
.
Definition 2.7 Assume that
$\mathcal {K}$
is an integral operator on
$\mathcal {U}_{B}$
with kernel
$k(z,w)$
, that is,
$$ \begin{align*}\mathcal{K}f(z)=\int_{\mathcal{U}_{B}}k(z,w)f(w)dV(w),\,\,\,\,z\in \mathcal{U}_{B}.\end{align*} $$
Let
$\tau ,\sigma \in \mathbb {R}^{1\times n}$
be multi-indices. If the integral kernel
$k(z,w)$
satisfies the following estimate:
$$ \begin{align} |k(z,w)|\lesssim \frac{\left|z^\tau\right|\cdot\left|w^\sigma\right|}{\prod_{j=1}^n\left|(z\overline{w})^{\left(b_{-}\right)^j}-(z\overline{w})^{\left(b_{+}\right)^j}\right|{}^2},\,\,\,\,z,w\in \mathcal{U}_{B}, \end{align} $$
then we say that
$\mathcal {K}$
is an operator on
$\mathcal {U}_{B}$
of type-
$(\tau ,\sigma )$
.
To obtain the
$L^p$
boundedness for the operators of type-
$(\tau ,\sigma )$
, we need the Forelli–Rudin type estimate on
$\mathcal {U}_{B}$
as follows. For the Forelli–Rudin type estimates on smooth pseudoconvex domains, we refer the readers to [Reference Khanh, Liu and Thuc12] for more information.
Lemma 2.8 Let
$\tau ,\sigma \in \mathbb {R}^{1\times n}$
be multi-indices, and assume that
$k_{\tau ,\sigma }(z,w)$
is a kernel on
$\mathcal {U}_{B}$
satisfying (2.5). Let
$\alpha $
,
$\beta $
be real numbers, and
$\gamma \in \mathbb {R}^{1\times n}$
. Suppose that
$\alpha $
,
$\beta ,$
and
$\gamma $
satisfy
$\beta>-1$
,
$2\alpha -\beta -2>0,$
and
$$ \begin{align*}\left(\alpha(\sigma-2{1\!1} B)+2\beta {1\!1} B_-+\gamma+2{1\!1}\right)A\succ0,\end{align*} $$
then we have
$$ \begin{align*} \int_{\mathcal{U}_{B}}|k_{\tau,\sigma}(z,w)|^\alpha \psi(w)^\beta|w^\gamma|dV(w)\lesssim \left|z^{\alpha\tau+2(\alpha-\beta-2){1\!1} B_-}\right|\cdot\psi(z)^{-(2\alpha-\beta-2)},\,\,\,\,z\in\mathcal{U}_{B}, \end{align*} $$
where
$\psi (z):=\prod _{j=1}^n\left (\left |z^{2(b_-)^j}\right |-\left |z^{2(b_+)^j}\right |\right )$
,
$z\in \mathcal {U}_{B}.$
Proof Denote
$$ \begin{align*}F(z):=\int_{\mathcal{U}_{B}}|k_{\tau,\sigma}(z,w)|^\alpha \psi(w)^\beta|w^\gamma|dV(w),\,\,\,\,z\in \mathcal{U}_{B}.\end{align*} $$
Note that
$\alpha>0$
and
$\prod _{j=1}^n|w^{2(b_-)^j}|=|w^{2{1\!1} B_-}|$
, then we have
$$ \begin{align} F(z)&\lesssim \int_{\mathcal{U}_{B}}\frac{\left|z^{\alpha\tau}\right|\cdot\left|w^{\alpha\sigma}\right|}{\prod_{j=1}^n\left|(z\overline{w})^{\left(b_{-}\right)^j}-(z\overline{w})^{\left(b_{+}\right)^j}\right|{}^{2\alpha}}\prod_{j=1}^n\left(\left|w^{2(b_-)^j}\right|-\left|w^{2(b_+)^j}\right|\right)^{\beta}|w^{\gamma}|dV(w)\nonumber\\ &=\left|z^{\alpha\tau-2\alpha {1\!1} B_-}\right|\cdot\int_{\mathcal{U}_{B}}\left|w^{\alpha\sigma+\gamma+2(\beta-\alpha){1\!1} B_-}\right|\prod_{j=1}^n \frac{(1-|w^{b^j}|^2)^{\beta}}{|1-(z\overline{w})^{b^j}|^{2\alpha}}dV(w), \end{align} $$
where we used the fact that
$(b_+)^j-(b_-)^j=b^j$
,
$j=1,\ldots ,n$
.
We denote the matrix
$\widetilde {A}=\left (\widetilde {a}^i_j\right ):=\left (\frac {a^i_j}{\gcd (a_j)}\right )$
. From Theorem 2.3, it is easy to see that
$\Phi _{\widetilde {A}}:\mathbb {D}^{n}_{\mathrm {L}(B)}\rightarrow \mathcal {U}_{B}$
is a proper holomorphic mapping. Then, by changing the coordinates
$w=\Phi _{\widetilde {A}}(\eta )=\eta ^{\widetilde {A}}$
, we can transfer (2.6) to the integral on the polydisc
$\mathbb {D}^n$
as follows:
$$ \begin{align} (2.6)&\lesssim\left|z^{\alpha\tau-2\alpha {1\!1} B_-}\right|\nonumber\\ &\,\,\,\,\,\times\int_{\mathbb{D}^n}\prod_{j=1}^n\frac{(1-|\eta^{b^j \widetilde{A}}|^2)^\beta}{\left|1-z^{b^j}\overline{\eta}^{b^j\widetilde{A}}\right|{}^{2\alpha}}\cdot\left|\eta^{(\alpha\sigma+\gamma+2(\beta-\alpha){1\!1} B_-)\widetilde{A}}\right|\cdot|\det J_{\mathbb{C}}\Phi_{\widetilde{A}}(\eta)|^2dV(\eta)\nonumber\\ &=\left|z^{\alpha\tau-2\alpha {1\!1} B_-}\right|\cdot\int_{\mathbb{D}^n}\prod_{j=1}^n\frac{(1-|\eta^{b^j \widetilde{A}}|^2)^\beta}{\left|1-z^{b^j}\overline{\eta}^{b^j\widetilde{A}}\right|{}^{2\alpha}}\left|\eta^{(\alpha(\sigma-2{1\!1} B_-)+2\beta {1\!1} B_-+\gamma+2{1\!1})\widetilde{A}-2{1\!1}}\right|dV(\eta), \end{align} $$
where the last line follows from Lemma 2.4. From the definition of
$\widetilde {A,}$
it is easy for one to check that
$$ \begin{align*}\eta^{b^j\widetilde{A}}=\eta_j^{\frac{\det\,B}{\gcd(a_j)}},\,\,\,\,j=1,\ldots,n.\end{align*} $$
Consequently, (2.7) continues as
$$ \begin{align} F(z)\lesssim\left|z^{\alpha\tau-2\alpha{1\!1} B_-}\right|\prod_{j=1}^n\int_{\mathbb{D}}\frac{\left(1-\left|\eta_j\right|{}^{\frac{2\det\,B}{\gcd(a_j)}}\right)^\beta}{\left|1-z^{b^j}\overline{\eta}_j^{\frac{\det\,B}{\gcd(a_j)}}\right|{}^{2\alpha}}\cdot\left|\eta_j^{\left(\alpha(\sigma-2{1\!1} B_-)+2\beta{1\!1} B_-+\gamma+2{1\!1}\right)\widetilde{a}_j-2}\right|dV(\eta_j). \end{align} $$
By our assumptions, we can apply the Forelli–Rudin estimate on the unit disc
$\mathbb {D}$
(see, for instance, [Reference Chakrabarti, Tang and Zhang4, Lemma 2.9]) to obtain that
$$ \begin{align*} (2.8)&\lesssim \left|z^{\alpha\tau-2\alpha{1\!1} B_-}\right|\cdot\prod_{j=1}^n\left(1-|z^{b^j}|^2\right)^{-(2\alpha-\beta-2)}=\left|z^{\alpha\tau+2(\alpha-\beta-2){1\!1} B_-}\right|\cdot\psi(z)^{-(2\alpha-\beta-2)}, \end{align*} $$
which gives the desired result. The proof is complete.
The well-known Schur test is one of the commonly used tools for obtaining the
$L^p$
boundedness of integral operators. Here, we present a generalized version of the Schur test, which was introduced by Khanh–Liu–Thuc in [Reference Khanh, Liu and Thuc12].
Lemma 2.9 [Reference Khanh, Liu and Thuc12, Theorem 5.1]
Let
$(X, \mu )$
,
$(Y, \upsilon )$
be measure spaces with
$\sigma $
-finite, positive measures; let
$1<p\leq q<\infty $
and
$r\in \mathbb {R}$
. Let
$k: X\times Y\rightarrow \mathbb {C}$
be a measurable function. Assume that there exist positive measurable functions
$h_1, h_2$
on Y and g on X such that
$$ \begin{align*}h_1^{-1}h_2 \in L^{\infty}(Y, d\upsilon) \end{align*} $$
and the inequalities
$$ \begin{align*} \int_Y|k(x,y)|^{rp'}h_1(y)^{p'}d\upsilon(y)\leq C_1g(x)^{p'}, \end{align*} $$
$$ \begin{align*} \int_X|k(x,y)|^{(1-r)q}g(x)^{q}d\mu(x)\leq C_2h_2(y)^q \end{align*} $$
hold for almost every
$x\in (X, \mu )$
and
$y\in (Y, \upsilon )$
, where
$\frac {1}{p}+\frac {1}{p'}=1$
and
$C_1, C_2$
are positive constants. Then, the integral operator
$\mathcal {K}$
associated with the kernel
$k(x,y)$
defined by
$$ \begin{align*}\mathcal{K}(f)(x):=\int_Y k(x,y)f(y)d\upsilon(y),\,\,\,\,f\in L^p(Y, \upsilon) \end{align*} $$
is bounded from
$L^p(Y, \upsilon )$
to
$L^q(X, \mu )$
.
Now, we are ready to give the
$L^p$
boundedness for the operators on
$\mathcal {U}_{B}$
of type-
$(\tau ,\sigma )$
.
Proposition 2.10 Let
$\tau ,\sigma \in \mathbb {R}^{1\times n}$
be multi-indices, and let
$\mathcal {K}$
be an operator on
$\mathcal {U}_{B}$
of type-
$(\tau ,\sigma )$
. Assume that
$1<p<\infty $
satisfies the following two conditions:
(1)
$p(\sigma -2{1\!1} B_-+2{1\!1})A\succ 2{1\!1} A$
;
(2)
$p(\tau -2{1\!1} B_-)A+2{1\!1} A\succ 0$
.
Then, the operator
$\mathcal {K}:L^p(\mathcal {U}_{B})\rightarrow L^p(\mathcal {U}_{B})$
is bounded.
Proof Let
$p':=\frac {p}{p-1}$
be the Hölder conjugate of p. Set
$0<b<\min \left \{\frac {1}{p'},\frac {1}{q}\right \}$
and
$c:=(c_1,\ldots ,c_n)\in \mathbb {R}^{1\times n}$
to be determined with
$$ \begin{align} cA\prec \frac{1}{p'}(\sigma-2{1\!1} B_-+2{1\!1})A. \end{align} $$
We define the functions on
$\mathcal {U}_{B}$
as follows:
$$ \begin{align*} \begin{cases} h_1(z):=\left|z^{2b{1\!1} B_- -c}\right|\cdot\psi(z)^{-b};\\ h_2(z):=\left|z^{\frac{1}{p'}(\sigma-2{1\!1} B_- +2 b p{1\!1} B_-)}\right|\cdot\psi(z)^{-b};\\ g(z):=\left|z^{\frac{1}{p'}(\tau-2{1\!1} B_-+2 b p'{1\!1} B_-)}\right|\cdot\psi(z)^{-b}. \end{cases} \end{align*} $$
Taking
$$ \begin{align*}(\alpha,\beta,\gamma)=(1,-bp',(2b{1\!1} B_- -c)p')\end{align*} $$
and
$$ \begin{align*}(\alpha,\beta,\gamma)=\left(1,-bp,(\tau-2{1\!1} B_-+2 b p'{1\!1} B_-)\frac{p}{p'}\right)\end{align*} $$
in Lemma 2.8, we obtain that
$$ \begin{align*} \int_{\mathcal{U}_{B}}|k(z,w)|h_1(w)^{p'}dV(w)\lesssim g(z)^{p'},\,\,\,z\in \mathcal{U}_{B} \end{align*} $$
and
$$ \begin{align*} \int_{\mathcal{U}_{B}}|k(z,w)|g(z)^{p}dV(z)\lesssim h_2(w)^p,\,\,\,w\in \mathcal{U}_{B,} \end{align*} $$
respectively, where
$k(z,w)$
is the integral kernel of the operator
$\mathcal {K}$
.
Therefore, in order to employ Lemma 2.9 to show the boundedness of the operator
$\mathcal {K}$
, it only remains to take suitable c satisfying (3.4) such that
$$ \begin{align*}\frac{h_2(z)}{h_1(z)}=\left|z^{\frac{1}{p}(\sigma-2{1\!1} B_-)+c}\right|\in L^\infty(\mathcal{U}_{B}).\end{align*} $$
Indeed, from the conditions in the proposition, we learn that
$c=-\frac {1}{p}(\sigma -2 {1\!1} B_-)$
satisfies (3.4) and
$\frac {h_2(z)}{h_1(z)}\in L^\infty (\mathcal {U}_{B})$
. Consequently, by Lemma 2.9, it follows that
$\mathcal {K}$
is bounded on
$L^p(\mathcal {U}_{B})$
. The proof is complete.
2.4 Integration by parts
At the end of this section, we give the integration by parts on discs and annuli, which will be useful in the proof of Theorem 1.3. We first define the vector field
$$ \begin{align*}\mathcal{T}_w=\overline{w}\frac{\partial}{\partial \overline{w}}-w\frac{\partial}{\partial w}.\end{align*} $$
The vector field
$\mathcal {T}_w$
is useful in studying the Sobolev regularity and irregularity for the Bergman projections in various settings (see, for instance, [Reference Charpentier, Dupain and Mounkaila6, Reference Chen7, Reference Edholm and McNeal11]). The crucial property
$\mathcal {T}_w$
satisfies that
$\mathcal {T}_w$
annihilates
$C^1$
radial functions of
$w\in \mathbb {C}$
(see [Reference Chen7, Remark 3.3]. In particular,
$\mathcal {T}_w$
annihilates defining functions of discs and annuli centered at the origin along their boundaries. Consequently, we have the following integration by parts.
Lemma 2.11 [Reference Edholm and McNeal11, Proposition 2.11]
Let
$\Omega \subset \mathbb {C}$
be either a disc or an annulus centered at the origin. Then, if f,
$g\in L_1^1(\Omega )\cap C(\overline {\Omega })$
, then
$$ \begin{align*}\int_{\Omega}\mathcal{T}_wf\cdot gdV=-\int_{\Omega}f\cdot\mathcal{T}_w gdV.\end{align*} $$
3 The proof of the main theorems
Finally, in this section, we can give the proof of Theorems 1.2–1.4 as follows.
Proof of Theorem 1.2
Take any
$j_0\in \{1,\ldots ,n\}$
and temporarily fix it. By Lemma 2.2, we know that there exists
$\beta \in \Pi _{j_0} \cap \mathscr {S}_2(\mathcal {U}_{B})$
. Define the subsets of
$\{1,\ldots ,n\}$
as follows:
$$ \begin{align*}I_\beta:=\{i:\beta_i<0\},\,\,\,\,J_\beta:=\{j:\beta_j\geq0\},\end{align*} $$
and define the function
$$ \begin{align*}f(w):=\prod_{j\in J_\beta}w_j^{\beta_j}\cdot \prod_{i\in I_\beta}\overline{w}_i^{-\beta_i}.\end{align*} $$
It is clear that
$f\in C^\infty (\overline {\mathcal {U}_{B}})$
.
Note that the monomial polyhedron
$\mathcal {U}_{B}$
is a Reinhardt domain, the Bergman kernel function can be written as
$$ \begin{align} K_{\mathcal{U}_{B}}(z,w)=\sum_{s=(s_1,\ldots,s_n)\in\mathscr{S}_2(\mathcal{U}_{B}) }\frac{z^s\overline{w}^s}{\|z^s\|^2_{L^2}}. \end{align} $$
A direct calculation shows that
$$ \begin{align*} \textbf{P}_{\mathcal{U}_{B}}f(z)&=\int_{\mathcal{U}_{B}}K_{\mathcal{U}_{B}}(z,w)f(w)dV(w)\\ &=\sum_{s\in \mathscr{S}_2(\mathcal{U}_{B})}\frac{z^s}{\|z^s\|^2_{L^2}}\int_{\mathcal{U}_{B}}\prod_{j\in J_\beta}w_j^{\beta_j}\overline{w}_j^{s_j}\cdot\prod_{i\in I_\beta}\overline{w}_i^{s_i-\beta_i}dV(w)\\ &=\frac{z^\beta}{\|z^\beta\|^2_{L^2}}\int_{\mathcal{U}_{B}}\prod_{j\in J_\beta}|w_j|^{2\beta_j}dV(w)\approx z^\beta, \end{align*} $$
where we used the fact
$\beta \in \mathscr {S}_2(\mathcal {U}_{B})$
. Consequently, for any multi-indices
$l\in \mathbb {N}^n$
,
$$ \begin{align} \frac{\partial^{|l|}}{\partial z^l}\circ \textbf{P}_{\mathcal{U}_{B}}f(z)\approx z^{\beta-l}. \end{align} $$
Now assume that
$\frac {\partial ^{|l|}}{\partial z^l}\circ \textbf {P}_{\mathcal {U}_{B}}$
maps
$C^{\infty }(\overline {\mathcal {U}_{B}})\rightarrow L^p(\mathcal {U}_{B})$
. Since
$f\in C^\infty (\overline {\mathcal {U}_{B}})$
, (3.2) shows that
$z^{\beta -l}\in L^p(\mathcal {U}_{B})$
. From Lemma 2.1, it follows that
$$ \begin{align} \left(p(\beta-l)+2{1\!1}\right)A\succ0. \end{align} $$
Notice that
$\beta \in \Pi _{j_0}$
, we have
$$ \begin{align} (\beta+{1\!1})a_{j_0}=\gcd(a_{j_0}). \end{align} $$
Combining (3.3) and (3.4) yields that
$$ \begin{align*}p<\frac{2{1\!1} a_{j_0}}{(l+{1\!1})a_{j_0}-\gcd(a_{j_0})}.\end{align*} $$
This, together with the arbitrariness of
$j_0\in \{1,\ldots ,n\}$
, proves the first statement of Theorem 1.2.
Note that A is invertible and
$A\succeq 0$
. This implies that each column
$a_j$
of A contains at least one positive entry. We now show that, for a nontrivial monomial polyhedron
$\mathcal {U}_{B}$
, some column of A must have at least two positive entries. Suppose, to the contrary, that each column of A contains just a single positive entry. Then, A is a generalized permutation matrix with positive nonzero entries, which means that B must also be a generalized permutation matrix with positive nonzero entries. Consequently, by possibly reordering the nonzero entries in the matrix B, there are positive integers
$k_1,\ldots ,k_n$
such that
$$ \begin{align*}\mathcal{U}_{B}=\left\{ z\in\mathbb{C}^n:|z_j|^{k_j}<1,\,\,\,j=1,\ldots,n\right\}=\mathbb{D}^n,\end{align*} $$
which contradicts the assumption that
$\mathcal {U}_{B}$
is nontrivial.
Assume that the entries of
$a_{j_0}$
(
$1\leq j_0\leq n$
) contain at least two positive integers. Then, there exists some
$l\in \mathbb {N}^n$
such that
$la_{j_0}>\gcd (a_{j_0})$
, which means that the right-hand side of (1.3)
$<2$
. Hence,
$\textbf {P}_{\mathcal {U}_{B}}$
fails to map
$L^p_k(\mathcal {U}_{B})\rightarrow L^p_k(\mathcal {U}_{B})$
for any
$k\in \mathbb {Z}^+$
and
$p\geq 2$
. The proof is complete.
Proof of Theorem 1.3
We divide the proof into two cases.
Case 1. Assume that
$j_0\in \mathrm {K}(B)$
, where
$\mathrm {K}(B)$
is the set defined as in (2.2). From (3.1), we can immediately get that for
$1\leq j\leq n$
,
$$ \begin{align*}\frac{\partial}{\partial z_j}K_{\mathcal{U}_{B}}(z,w)=\frac{s_j}{z_j}K_{\mathcal{U}_{B}}(z,w)\,\,\,\,and\,\,\,\,\frac{\partial}{\partial \overline{w}_j}K_{\mathcal{U}_{B}}(z,w)=\frac{s_j}{\overline{w}_j}K_{\mathcal{U}_{B}}(z,w),\end{align*} $$
which follows that
$$ \begin{align} z_j\frac{\partial}{\partial z_j}K_{\mathcal{U}_{B}}(z,w)=\overline{w}_j\frac{\partial}{\partial \overline{w}_j}K_{\mathcal{U}_{B}}(z,w),\,\,\,\,j=1,\ldots,n. \end{align} $$
Note that Theorem 2.3 yields that the integrals on
$\mathcal {U}_{B}$
can be transformed to the ones on
$\mathbb {D}^{n}_{\mathrm {L}(B)}$
, which is a product domain of unit disc and punctured unit disc. Hence, we can apply Lemma 2.11 to obtain that for any
$f\in L^p_1(\mathcal {U}_{B})$
,
$$ \begin{align} \frac{\partial}{\partial z_{j_0}}\circ\textbf{P}_{\mathcal{U}_{B}}f(z)&=\int_{\mathcal{U}_{B}}\frac{\partial}{\partial z_{j_0}}K_{\mathcal{U}_{B}}(z,w)\cdot f(w)dV(w)\nonumber\\ &=\frac{1}{z_{j_0}}\int_{\mathcal{U}_{B}}\overline{w}_{j_0}\frac{\partial}{\partial \overline{w}_{j_0}}K_{\mathcal{U}_{B}}(z,w)\cdot f(w)dV(w)\nonumber\\ &=-\frac{1}{z_{j_0}}\int_{\mathcal{U}_{B}}K_{\mathcal{U}_{B}}(z,w)\mathcal{T}_{w_{j_0}}f(w)dV(w)\nonumber\\ &=\int_{\mathcal{U}_{B}}\frac{w_{j_0}}{z_{j_0}}K_{\mathcal{U}_{B}}(z,w)\frac{\partial f}{\partial w_{j_0}}(w)dV(w)\nonumber\\&\quad-\int_{\mathcal{U}_{B}}\frac{\overline{w}_{j_0}}{z_{j_0}}K_{\mathcal{U}_{B}}(z,w)\frac{\partial f}{\partial \overline{w}_{j_0}}(w)dV(w), \end{align} $$
where from the first line to the second line we used (3.5), and the derivatives
$\frac {\partial f}{\partial w_{j_0}}$
and
$\frac {\partial f}{\partial \overline {w}_{j_0}}$
should be interpreted distributionally.
From Lemma 2.6, we know that the integral operators on
$\mathcal {U}_{B}$
with kernel
$\frac {\overline {w}_{j_0}}{z_{j_0}}K_{\mathcal {U}_{B}}(z,w)$
and
$\frac {w_{j_0}}{z_{j_0}}K_{\mathcal {U}_{B}}(z,w)$
are of type-
$(\lambda -e_{j_0},\lambda +e_{j_0})$
, where
$\lambda $
is given by (2.4), and
$e_j(j=1,\ldots ,n)$
denote the
$1\times n$
row vector that the j-th entry is
$1$
, and the other entries are
$0$
. This, together with (3.6) and Proposition 2.10, implies that
$$ \begin{align*} \left\|\frac{\partial}{\partial z_{j_0}}\circ\textbf{P}_{\mathcal{U}_{B}}f \right\|_{L^p}&\leq\left\|\int_{\mathcal{U}_{B}}\frac{w_{j_0}}{z_{j_0}}K_{\mathcal{U}_{B}}(z,w)\frac{\partial f}{\partial w_{j_0}}(w)dV(w)\right\|_{L^p}\\ &\,\,\,\,\,\,\,\,\,\,\,\,+\left\|\int_{\mathcal{U}_{B}}\frac{\overline{w}_{j_0}}{z_{j_0}}K_{\mathcal{U}_{B}}(z,w)\frac{\partial f}{\partial \overline{w}_{j_0}}(w)dV(w)\right\|_{L^p}\\ &\lesssim \left\|\frac{\partial f}{\partial \overline{w}_{j_0}}\right\|_{L^p}+\left\|\frac{\partial f}{\partial w_{j_0}}\right\|_{L^p}\leq \|f\|_{L^p_1} \end{align*} $$
holds for all p that lies in the range (1.4), which yields that
$\frac {\partial }{\partial z_{j_0}}\circ \textbf {P}_{\mathcal {U}_{B}}:L^p_1(\mathcal {U}_{B})\rightarrow L^p(\mathcal {U}_{B})$
is bounded for those p. The proof of this case is complete.
Case 2. Assume that
$j_0\notin \mathrm {K}(B)$
. Denote the modified kernel
$\widetilde {K}_{\mathcal {U}_{B}}(z,w)$
by
$$ \begin{align} \widetilde{K}_{\mathcal{U}_{B}}(z,w):=K_{\mathcal{U}_{B}}(z,w)-\sum_{\nu\in\mathscr{S}_2(\mathcal{U}_{B}),\nu_{j_0}=0 }\frac{z^\nu \overline{w}^\nu}{\|z^\nu\|_{L^2}^2}. \end{align} $$
Now, we need the following claim, which gives the estimate for the new kernel
$\widetilde {K}_{\mathcal {U}_{B}}(z,w)$
.
Claim The integral operator on
$\mathcal {U}_{B}$
with kernel
$\frac {w_{j_0}}{z_{j_0}}\widetilde {K}_{\mathcal {U}_{B}}(z,w)$
is of type-
$(\lambda -e_{j_0}, \lambda +e_{j_0})$
.
Proof of the Claim
Denote
$z^{(j_0)}:=(z_1,\ldots ,z_{j_0-1},0,z_{j_0+1},\ldots ,z_n)$
and
$w^{(j_0)}:=(w_1,\ldots ,w_{j_0-1},0,w_{j_0+1},\ldots ,w_n)$
. By Theorem 2.5, we know that
$$ \begin{align} \widetilde{K}_{\mathcal{U}_{B}}(z,w)&=K_{\mathcal{U}_{B}}(z,w)-K_{\mathcal{U}_{B}}\left(z^{(j_0)},w^{(j_0)}\right)\nonumber\\ &\approx \frac{\sum\limits_{\nu \in \mathbb{N}^{1 \times n}} C_B(\nu) (z\overline{w})^\nu}{\prod\limits_{j=1}^n\left((z\overline{w})^{\left(b_{-}\right)^j}-(z\overline{w})^{\left(b_{+}\right)^j}\right)^2}-\frac{\sum\limits_{\nu \in \mathbb{N}^{1 \times n},\nu_{j_0}=0} C_B(\nu) \left(z^{(j_0)}\overline{w^{(j_0)}}\right)^\nu}{\prod\limits_{j=1}^n\left((z^{(j_0)}\overline{w^{(j_0)}})^{\left(b_{-}\right)^j}-(z^{(j_0)}\overline{w^{(j_0)}})^{\left(b_{+}\right)^j}\right)^2}. \end{align} $$
A direct calculation shows that
$$ \begin{align*}\prod\limits_{j=1}^n\left((z^{(j_0)}\overline{w^{(j_0)}})^{\left(b_{-}\right)^j}-(z^{(j_0)}\overline{w^{(j_0)}})^{\left(b_{+}\right)^j}\right)^2=\prod_{j=1}^n\left(z^{(j_0)}\overline{w^{(j_0)}}\right)^{2\left(b_{-}\right)^j}\prod_{j=1}^n\left(1-\left(z^{(j_0)}\overline{w^{(j_0)}}\right)^{b^j}\right)^2.\end{align*} $$
Note that
$j_0\notin \mathrm {K}(B)$
, we define two subsets of
$\{1,\ldots ,n\}$
as follows:
$$ \begin{align*}I:=\{j:b^j_{j_0}=0\},\,\,\,\,J=\{1,\ldots,n\} \setminus I=\{j:b^j_{j_0}>0\}.\end{align*} $$
It is clear that
$\left (z^{(j_0)}\overline {w^{(j_0)}}\right )^{b^j}=0$
for
$j\in J$
. Thus, we have
$$ \begin{align*}\prod_{j=1}^n\left(1-\left(z^{(j_0)}\overline{w^{(j_0)}}\right)^{b^j}\right)^2=\prod_{j\in I}\left(1-\left(z\overline{w}\right)^{b^j}\right)^2.\end{align*} $$
It is easy for one to see that
$(b_-)^j_{j_0}=0$
for
$j=1,\ldots ,n$
, which follows that
$$ \begin{align*}\prod_{j=1}^n\left(z^{(j_0)}\overline{w^{(j_0)}}\right)^{2\left(b_{-}\right)^j}=\prod_{j=1}^n\left(z\overline{w}\right)^{2(b_-)^j}\!.\end{align*} $$
Combining the above equations gives that
$$ \begin{align} \prod\limits_{j=1}^n\left((z^{(j_0)}\overline{w^{(j_0)}})^{\left(b_{-}\right)^j}-(z^{(j_0)}\overline{w^{(j_0)}})^{\left(b_{+}\right)^j}\right)^2=\prod_{j=1}^n\left(z\overline{w}\right)^{2(b_-)^j}\cdot\prod_{j\in I}\left(1-\left(z\overline{w}\right)^{b^j}\right)^2. \end{align} $$
Taking (3.9) into (3.8) shows that the modified kernel
$\widetilde {K}_{\mathcal {U}_{B}}(z,w)$
can be written as
$$ \begin{align} \widetilde{K}_{\mathcal{U}_{B}}(z,w)=\frac{\sum\limits_{\nu \in \mathbb{N}^{1 \times n}} C_B(\nu) (z\overline{w})^\nu-\sum\limits_{\nu \in \mathbb{N}^{1 \times n},\nu_{j_0}=0} C_B(\nu) (z\overline{w})^\nu\cdot \prod\limits_{j\in J}\left(1-\left(z\overline{w}\right)^{b^j}\right)^2}{\prod\limits_{j=1}^n\left(z\overline{w}\right)^{2(b_-)^j}\cdot\prod\limits_{j=1}^n\left(1-\left(z\overline{w}\right)^{b^j}\right)^2}. \end{align} $$
A straightforward calculation implies that
$$ \begin{align*}\prod\limits_{j\in J}\left(1-\left(z\overline{w}\right)^{b^j}\right)^2=1-2\sum_{j\in J}\left(z\overline{w}\right)^{b^j}+P(z,w),\end{align*} $$
where
$P(z,w)$
is a homogeneous polynomial with respect to
$\left (z\overline {w}\right )^{b^1},\ldots ,\left (z\overline {w}\right )^{b^n}$
. Therefore,
$$ \begin{align*} \mathrm{the}\,\,\mathrm{numerator}\,\,\mathrm{of}\,\,(3.10)&=\sum\limits_{\nu \in \mathbb{N}^{1 \times n},\nu_{j_0}\geq1} C_B(\nu) (z\overline{w})^\nu+2\sum_{j\in J}\sum\limits_{\nu \in \mathbb{N}^{1 \times n},\nu_{j_0}=0}C_B(\nu) (z\overline{w})^{\nu+b^j}\\ &\,\,\,\,\,\,\,\,\,\,\,\,-P(z,w)\sum\limits_{\nu \in \mathbb{N}^{1 \times n},\nu_{j_0}=0}C_B(\nu) (z\overline{w})^{\nu}. \end{align*} $$
Notice that
$P(z,w)$
is bounded on
$\mathcal {U}_{B}$
. Hence, we have
$$ \begin{align}\left|\frac{w_{j_0}}{z_{j_0}}\widetilde{K}_{\mathcal{U}_{B}}(z,w)\right|&\lesssim \frac{\left|\frac{w_{j_0}}{z_{j_0}}\right|\sum\limits_{\nu \in \mathbb{N}^{1 \times n},\nu_{j_0}\geq1} C_B(\nu) |(z\overline{w})^\nu|}{\prod_{j=1}^n\limits\left|(z\overline{w})^{\left(b_{-}\right)^j}-(z\overline{w})^{\left(b_{+}\right)^j}\right|^2}\nonumber\\ &\,\,\,\,\,\,\,\,\,\,\,+\frac{\left|\frac{w_{j_0}}{z_{j_0}}\right|\sum\limits_{j\in J}\sum\limits_{\nu \in \mathbb{N}^{1 \times n},\nu_{j_0}=0}C_B(\nu) |(z\overline{w})^{\nu+b^j}|}{\prod_{j=1}^n\limits\left|(z\overline{w})^{\left(b_{-}\right)^j}-(z\overline{w})^{\left(b_{+}\right)^j}\right|^2}\nonumber\\ &:=M(z,w)+N(z,w). \end{align} $$
We first estimate
$M(z,w)$
. It is easy to see that
$$ \begin{align} \left|\frac{w_{j_0}}{z_{j_0}}\right|\sum\limits_{\nu \in \mathbb{N}^{1 \times n},\nu_{j_0}\geq1} C_B(\nu) |(z\overline{w})^\nu|=|w_{j_0}|^2\sum\limits_{\nu \in \mathbb{N}^{1 \times n},\nu_{j_0}\geq1} C_B(\nu) |(z\overline{w})^{\nu-e_{j_0}}|. \end{align} $$
For any
$\nu \in \mathbb {N}^{1 \times n}$
satisfying
$\nu _{j_0}\geq 1$
and
$C_B(\nu )\neq 0$
, from Theorem 2.5, we know that
$$ \begin{align*}(\nu-2{1\!1} B_-+{1\!1})A-{1\!1}\succeq0,\end{align*} $$
which is equivalent to
$$ \begin{align} (\nu-e_{j_0})A\succeq \gcd(A)+(2{1\!1} B_- -{1\!1})A-a^{j_0}. \end{align} $$
Consequently, by [Reference Chakrabarti, Tang and Zhang4, Lemma 2.3] we obtain that for any
$z,w\in \mathcal {U}_{B}$
,
$$ \begin{align*} \left|(z\overline{w})^{\nu-e_{j_0}}\right|&=\left|(z\overline{w})^{\frac{1}{\det B}(\nu-e_{j_0})AB}\right|\\ &\leq \left|(z\overline{w})^{\frac{1}{\det B}\left(\gcd(A)+(2{1\!1} B_- -{1\!1})A-a^{j_0}\right)B}\right|=\left|(z\overline{w})^{\lambda-e_{j_0}}\right|. \end{align*} $$
Note that Theorem 2.5 implies that
$C_B(\nu )$
is bounded, and the summation in (3.12) is finite due to (2.3). Thus, we can obtain that
$$ \begin{align*}M(z,w)=\frac{|w_{j_0}|^2\sum\limits_{\nu \in \mathbb{N}^{1 \times n},\nu_{j_0}\geq1} C_B(\nu) |(z\overline{w})^{\nu-e_{j_0}}|}{\prod_{j=1}^n\limits\left|(z\overline{w})^{\left(b_{-}\right)^j}-(z\overline{w})^{\left(b_{+}\right)^j}\right|^2}\lesssim \frac{\left|z^{\lambda-e_{j_0}}\right| \left|w^{\lambda+e_{j_0}}\right|}{\prod_{j=1}^n\limits\left|(z\overline{w})^{\left(b_{-}\right)^j}-(z\overline{w})^{\left(b_{+}\right)^j}\right|^2}, \end{align*} $$
that is, the integral operator with kernel
$M(z,w)$
is of type-
$(\lambda -e_{j_0},\lambda +e_{j_0})$
.
Now, we estimate
$N(z,w)$
, the second term in (3.11). Take any
$j\in J$
and temporarily fix it. For any
$\nu \in \mathbb {N}^{1\times n}$
with
$\nu _{j_0}=0$
and
$C_B(\nu )\neq 0$
, it is easily seen that
$\nu +b^j-e_{j_0}\in \mathbb {N}^{1\times n}$
. An argument analogous to that in (3.13) gives that
$$ \begin{align*}(\nu-e_{j_0}+b^j)A\succeq \gcd(A)+(2{1\!1} B_--{1\!1})A-a^{j_0}+b^j A. \end{align*} $$
This shows that for any
$z,w\in \mathcal {U}_{B}$
, it holds that
$$ \begin{align*} \left|\frac{w_{j_0}}{z_{j_0}}\right|\left|(z\overline{w})^{\nu+b^j}\right|&=|w_{j_0}|^2\left|(z\overline{w})^{\frac{1}{\det B}(\nu+b^j-e_{j_0})AB}\right|\\ &\leq|w_{j_0}|^2\left|(z\overline{w})^{\frac{1}{\det B}(\gcd(A)+(2{1\!1} B_--{1\!1})A-a^{j_0}+b^j A)B}\right|\\ &=|w_{j_0}|^2\left|(z\overline{w})^{\lambda-e_{j_0}}\right|\left|(z\overline{w})^{b^j}\right|<\left|z^{\lambda-e_{j_0}}\right| \left|w^{\lambda+e_{j_0}}\right|, \end{align*} $$
from which it follows directly that
$$ \begin{align*}N(z,w)\lesssim \frac{\left|z^{\lambda-e_{j_0}}\right| \left|w^{\lambda+e_{j_0}}\right|}{\prod_{j=1}^n\limits\left|(z\overline{w})^{\left(b_{-}\right)^j}-(z\overline{w})^{\left(b_{+}\right)^j}\right|^2}, \end{align*} $$
that is, the integral operator with kernel
$N(z,w)$
is of type-
$(\lambda -e_{j_0},\lambda +e_{j_0})$
.
Combining (3.11) with the above estimates for
$M(z,w)$
and
$N(z,w)$
, we know that the proof of the claim is complete.
Now, we come back to the proof of Case 2. Since
$\sum _{\nu \in \mathscr {S}_2(\mathcal {U}_{B}),\nu _{j_0}=0 }\frac {z^\nu \overline {w}^\nu }{\|z^\nu \|_{L^2}^2}$
is independent of
$w_{j_0}$
and
$\overline {w}_{j_0}$
, we have that
$$ \begin{align*}\mathcal{T}_{w_{j_0}}\left(\widetilde{K}_{\mathcal{U}_{B}}(z,w)\right)=\mathcal{T}_{w_{j_0}}\left(K_{\mathcal{U}_{B}}(z,w)\right)\!.\end{align*} $$
Through a similar process as that in (3.6), we get that
$$ \begin{align*}\frac{\partial}{\partial z_{j_0}}\circ\textbf{P}_{\mathcal{U}_{B}}f(z)=\int_{\mathcal{U}_{B}}\frac{w_{j_0}}{z_{j_0}}\widetilde{K}_{\mathcal{U}_{B}}(z,w)\frac{\partial f}{\partial w_{j_0}}(w)dV(w)-\int_{\mathcal{U}_{B}}\frac{\overline{w}_{j_0}}{z_{j_0}}\widetilde{K}_{\mathcal{U}_{B}}(z,w)\frac{\partial f}{\partial \overline{w}_{j_0}}(w)dV(w).\end{align*} $$
From the previous claim, we learn that the integral operators with kernels
$\frac {w_{j_0}}{z_{j_0}}\widetilde {K}_{\mathcal {U}_{B}}(z,w)$
and
$\frac {\overline {w}_{j_0}}{z_{j_0}}\widetilde {K}_{\mathcal {U}_{B}}(z,w)$
are of type-
$(\lambda -e_{j_0},\lambda +e_{j_0})$
. Making use of Proposition 2.10 and we obtain
$$ \begin{align*}\left\|\frac{\partial}{\partial z_{j_0}}\circ\textbf{P}_{\mathcal{U}_{B}}f \right\|_{L^p} \lesssim \left\|\frac{\partial f}{\partial \overline{w}_{j_0}}\right\|_{L^p}+\left\|\frac{\partial f}{\partial w_{j_0}}\right\|_{L^p}\lesssim \|f\|_{L^p_1}\end{align*} $$
holds for p that lies in the range (1.4), which follows that
$\frac {\partial }{\partial z_{j_0}}\circ \textbf {P}_{\mathcal {U}_{B}}:L^p_1(\mathcal {U}_{B})\rightarrow L^p(\mathcal {U}_{B})$
is bounded. The proof is complete.
Acknowledgements
The author thanks the referee for carefully reading the article and providing corrections and suggestions for improvements. The author would like to thank Prof. Feng Rong, Prof. Zhenhan Tu, and Prof. Zehua Zhou for helpful comments and suggestions to this article.
