Schur's test, Bergman-type operators and Gleason's problem on radial-angular mixed spaces

1.   Introduction and main results

It is well known that Schur's tests give sufficient conditions for the boundedness of some integral operator $T$ from one function space to another. Thus, they become one of the most useful tools for establishing boundedness of integral operators (see, for instance, ^[1]). Recall that, in 1911, Schur's test was first given by Schur ^[2] as a discrete form. After that, Schur's test has been generalized by many authors. For instance, in 1965, Schur's test for integral operators $T$ on Lebesgue spaces $L^p(X, d\mu)$ was obtained in ^[3], where $1\le p < {\infty}$ and the space $L^p(X, d\mu)$ always denotes the set of all measurable functions $f$ on $X$ such that

In 2015, Zhao ^[4] gave a generalization of Schur's test for the boundedness of $T$ from one weighted Lebesgue spaces to another weighted Lebesgue spaces, and then applied to characterize boundedness of Forelli-Rudin operator. For the convenience of the reader, we present the following version of Schur's test from Zhao ^[4].

Theorem 1.1 (^[4]). Let $\mu$ and $\nu$ be two positive measures on the space $X$ and $K$ a nonnegative function on $X\times X$. Let $T$ be an integral operator with kernel $K$ defined by setting for any $x\in X$,

Suppose $1 < p\le q < {\infty}$ and $\frac1 p+\frac1{p'} = 1$. Let $\gamma$ and $\delta$ be two real numbers such that $\gamma+\delta = 1$. If there exist two positive functions $h_1$ and $h_2$ with two positive constants $C_1$ and $C_2$ such that

for almost all $x\in X$, and

for almost all $y\in X$, then $T: L^p(X, d\mu)\to L^q(X, d\nu)$ is bounded with

In this paper, we first extend Theorem 1.1 to the radial-angular mixed space $L_{p, q}(\Phi)$ for $1\le p, q < {\infty}$, where $\Phi$ is a generalized normal function (see Definition 1.3 below for details). In addition, the similar Schur's tests for critical points are also addressed in this paper. Let $s\in {{\mathbb R}}$ and $t > 0$. We then establish the boundedness of Bergman-type operators $P_{s, t}$ from holomorphic radial-angular mixed space $H_{p, q}(\Phi)$ to $L_{p, q}(\Phi)$ for all possible $0 < p, q < {\infty}$. At the end, we finally solve Gleason's problem on $H_{p, q}(\Phi)$ for all possible $0 < p, q < {\infty}$ as an application.

To state our main results, we need some notations.

For $z: = (z_1, \ldots, z_n)$, $\omega: = (\omega_1, \ldots, \omega_n)\in \mathbb{C}^n$, we denote the inner product of $z$ and $\omega$ by

and $|z|: = \sqrt{\langle z, z\rangle}$. Let

denote the open unit ball in complex vector space $\mathbb{C}^n$ and $\partial B$ its boundary. In addition, let $d\nu$ denote the Lebesgue measure, normalized such that $\int_B\, d\nu = 1$ and $d\sigma$ the surface measure on $\partial B$, again normalized such that $\int_{\partial B} d\sigma = 1$.

Recall that a positive continuous function $\varphi$ defined on $[0, 1)$ is said to be normal, if there exist constants $0 < a < b$, $r_0\in[0, 1)$ such that

${\rm (i)}$ $\frac{\varphi(r)}{(1-r)^a}$ is nonincreasing in $r\in[r_0, 1)$ and $\mathop{\lim}\limits_{r\to 1} \frac{\varphi(r)}{(1-r)^a} = 0$;

${\rm (ii)}$ $\frac{\varphi(r)}{(1-r)^b}$ is nondecreasing in $r\in[r_0, 1)$ and $\mathop{\lim}\limits_{r\to 1} \frac{\varphi(r)}{(1-r)^b} = {\infty}$.

Observe that the constants $a$ and $b$ are not uniquely for given normal function $\varphi$. Therefore, in what follows, let $a_{\varphi}$ denote the superemum of all possible $a$ satisfying (i) and $b_{\varphi}$ the infimum of all possible $b$ satisfying (ii). Then $a_{\varphi}$ and $b_{\varphi}$ are alwalys called characteristic exponents of $\varphi$.

We now present the notion of the generalized normal function $\Phi$ and related radial-angular mixed spaces $L_{p, q}(\Phi)$ from ^[5].

Definition 1.2. Let $\tau\in {{\mathbb R}}$ and $\varphi$ be a normal function. A continuous function $\Phi:\, [0, 1)\mapsto [0, {\infty})$ is called generalized normal function if for any $r\in[0, 1)$, $\Phi(r): = (1-r)^\tau\varphi(r)$.

Definition 1.3. Let $0 < p, q\le {\infty}$ and $\Phi$ be a generalized normal function. The radial-angular mixed space $L_{p, q}(\Phi)$ is defined to be the set of all measurable complex functions $f$ on $B$ such that

when $p\in(0, {\infty})$ and

where $M_{{\infty}}(r, f): = \mathop{\sup}\limits_{\zeta\in\partial B} |f(r\zeta)|$ and for any $q\in(0, {\infty})$,

We next state the main results about Schur's tests on radial-angular mixed spaces as follows. For any ${p(\cdot)}: = (p_1, p_2)\in(0, {\infty})^2$, in what follows, we always denote $p_-: = \min\{p_1, p_2\}$ and $p_+: = \max\{p_1, p_2\}$.

Theorem 1.4. Let $K$ a nonnegative function on $\partial B\times I\times \partial B\times I$. Let $T$ be an integral operator with kernel $K$ defined by setting for any $(x, y)\in \partial B\times I$,

Here and thereafter, $I: = [0, 1]$ and $d\lambda(r): = r^{2n-1}(1-r)^{-1}[\Phi(r)]^p\, dr$ for any $r\in I$. Suppose ${p(\cdot)}: = (p_1, p_2), \ {\vec{q}}: = (q_1, q_2)\in(1, {\infty})^2$ satisfying $1 < p_-\le p_+\le q_- < {\infty}$. Let $\gamma$ and $\delta$ be two real numbers such that $\gamma+\delta = 1$. If there exist two positive functions $h_1$ and $h_2$ defined on $\partial B\times I$ with two positive constants $C_1$ and $C_2$ such that

for almost all $(x, y)\in \partial B\times I$, and

for almost all $(s, t)\in \partial B\times I$, then $T: L_{p_1, p_2}(\Phi)\to L_{q_1, q_2}(\Phi)$ is bounded with

We also obtain the Schur's tests for the endpoint cases.

Theorem 1.5. Let $\gamma$, $\delta$ and $T$ be an integral operator with kernel $K$ defined as in Theorem 1.4. Suppose ${p(\cdot)}: = (1, p_2)$ and ${\vec{q}}: = (q_1, q_2)\in(1, {\infty})^2$ satisfying $1 < p_2\le q_- < {\infty}$. If there exist two positive functions $h_1$ and $h_2$ defined on $\partial B\times I$ with two positive constants $C_1$ and $C_2$ such that

for almost all $(x, y)\in \partial B\times I$, and

for almost all $(s, t)\in \partial B\times I$, then $T: L_{p_1, p_2}(\Phi)\to L_{q_1, q_2}(\Phi)$ is bounded with

Theorem 1.6. Let $\gamma$, $\delta$ and $T$ be an integral operator with kernel $K$ defined as in Theorem 1.4. Suppose ${p(\cdot)}: = (p_1, 1)$ and ${\vec{q}}: = (q_1, q_2)\in(1, {\infty})^2$ satisfying $1 < p_1\le q_- < {\infty}$. If there exist two positive functions $h_1$ and $h_2$ defined on $\partial B\times I$ with two positive constants $C_1$ and $C_2$ such that

for almost all $(x, y)\in \partial B\times I$, and

for almost all $(s, t)\in \partial B\times I$, then $T: L_{p_1, p_2}(\Phi)\to L_{q_1, q_2}(\Phi)$ is bounded with

Theorem 1.7. Let $\gamma$, $\delta$ and $T$ be an integral operator with kernel $K$ defined as in Theorem 1.4. Suppose ${\vec{q}}: = (q_1, q_2)\in[1, {\infty})^2$. Let $\gamma$ and $\delta$ be two real numbers such that $\gamma+\delta = 1$. If there exist two positive functions $h_1$ and $h_2$ defined on $\partial B\times I$ with two positive constants $C_1$ and $C_2$ such that

for almost all $(x, y)\in \partial B\times I$, and

for almost all $(s, t)\in \partial B\times I$, then $T: L_{1, 1}(\Phi) \to L_{q_1, q_2}(\Phi)$ is bounded with

Moreover, we find the following sufficient and necessary condition for boundedness of the integral operators $T$ from $L_{p_1, p_2}(\Phi)$ to $L_{q_1, q_2}(\Phi)$.

Theorem 1.8. Let $T$ be an integral operator with kernel $K$ defined as in Theorem 1.4. Suppose ${p(\cdot)}: = (p_1, p_2), \ {\vec{q}}: = (q_1, q_2)\in[1, {\infty}]^2$. Then $T$ is a bounded operator from $L_{p_1, p_2}(\Phi)$ into $L_{q_1, q_2}(\Phi)$ with bound $M$ if and only if for any $u\in L_{p_1, p_2}(\Phi)$ and $\upsilon\in L_{q_1', q_2'}(\Phi)$,

On the other hand, for $s\in{{\mathbb R}}$ and $t > 0$, the Bergman-type operator $P_{s, t}$ on $L_{p, q}(\Phi)$ is defined by

The boundedness of Bergman-type operators $P_{s, t}$ has been studied extensively (see, for instance, ^[4,6,7,8]). Indeed, let $\varphi$ be a normal function and $H(B)$ denote all the holomorphic functions in $B$. The boundedness of operator $P_{s, t}$ on $L_{p, q}(\varphi)$ for $1\le p, q < {\infty}$ was studied in ^[9]. Later on, the boundedness of $P_{s, t}$ from $H_{p, q}(\varphi): = H(B)\cap L_{p, q}(\varphi)$ to $L_{p, q}(\varphi)$ for $0 < p < 1$ and $1\le q < {\infty}$ was obtained in ^[10]. The only left case $0 < q < 1$ and $0 < p < {\infty}$ was sloved by Lou ^[11] in 2007. In this paper, we extend all these known results to radial-angular mixed spaces $L_{p, q}(\Phi)$ for all possible $0 < p, q < {\infty}$.

To be exact, we establish the following boundedness of Bergman-type operators $P_{s, t}$.

Theorem 1.9. Let $0 < p, q < {\infty}$ and $\Phi(\cdot): = (1-\cdot)^\tau\varphi(\cdot)$ be a generalized normal function, where $\tau\in {{\mathbb R}}$ and $\varphi$ is a normal function with characteristic exponents $a_{\varphi}$ and $b_{\varphi}$. If

then $P_{s, t}:\, H_{p, q}(\Phi)\to L_{p, q}(\Phi)$ is a bounded operator.

Remark 1.10. For all $0 < p, \, q < {\infty}$, Theorem 1.9 gives the sufficient conditions such that the Bergman-type operator $P_{s, t}$ is bounded from $H_{p, q}(\Phi)$ to $L_{p, q}(\Phi)$. The conditions are different in the two cases: $1\le q < {\infty}$ and $0 < q < 1$.

${\rm (i)}$ When $1\le q < {\infty}$ and $0 < p < {\infty}$, the sufficient condition is $t-\tau > b_{\varphi} > a_{\varphi} > -s-\tau$. We point out that when $\tau = 0$, then $\Phi\equiv \varphi$ and the radial-angular mixed space $L_{p, q}(\Phi)$ goes back to the space $L_{p, q}(\varphi)$. Thus, in this case, Theorem 1.9 holds true for $t > b_{\varphi} > a_{\varphi} > -s$. In this sence, Theorem 1.9 extends the main theorem established in [9,Theorem A(ii)] and ^[10].

${\rm (ii)}$ When $0 < q < 1$ and $0 < p < {\infty}$, the sufficient condition is $t-\tau+n(1-1/q) > b_{\varphi} > a_{\varphi} > -s-\tau$. Similarly, we point out that, in this case, Theorem 1.9 with $\tau = 0$ goes back to [11,Theorem 1.1].

As an application, we prove that Gleason's problem in radial-angular mixed spaces $H_{p, q}(\Phi)$ for all possible $0 < p, \, q < {\infty}$ is solvable. Let $X$ be a space of holomorphic functions on a domain $\Omega$ in ${{\mathbb C}}^n$. Then Gleason's problem for $X$ and any given point $a\in \Omega$, denoted by $(\Omega, a, X)$, is the following: Given $a\in \Omega$ and $f\in X$, do there exist functions $g_1, \ldots, g_n\in X$ such that

for all $z\in \Omega$?

Gleason ^[12] originally asked the question for $(B, 0, A(B))$, where $A(B)$ is the ball algebra on the unit ball $B\subset {{\mathbb C}}^n$, consisting of holomorphic functions in $B$ which are countinuous on $\bar{B}$. This problem $(B, 0, A(B))$ was solved by Leibenson. The difficulty of Gleason's problem depends on the domain $\Omega$ and the space $X$. Gleason's problem for different domains and function spaces have been studied extensively. For instance, with the help of the boundedness of Bergman-type operators, Zhu ^[13] and Choe ^[14] solved Gleason's problem for the Bergman space $L_{p, p}(\Phi)$ with $\Phi(r): = (1-r^2)^{1/p}$ and $\Phi(r): = (1-r^2)^{(1+\alpha)/p}$ respectively. Here, $1\le p < {\infty}$ and $\alpha > -1$. Hu ^[15] studied Gleason's problem for harmonic mixed norm and Bloch spaces in convex domains. In particular, for given normal function $\varphi$, Gleason's problem on mixed spaces $L_{p, q}(\varphi)$ was addressed by ^[9] for the case $1\le p, q < {\infty}$ and by ^[11] for the case $0 < p < {\infty}$, $0 < q < 1$. In this paper, we extend these conclusions and solve Gleason's problem for all $0 < p, \, q < {\infty}$ on $H_{p, q}(\Phi)$.

As an application of Theorem 1.9, we have the following result.

Theorem 1.11. Let $0 < p, \, q < {\infty}$ and $\Phi$ be a generalized normal function. Gleason's problem can be solved on $H_{p, q}(\Phi)$. More precisely, for any intrger $m\ge 1$, there exist bounded linear operators $A_{\alpha}$ on $H_{p, q}(\Phi)$ such that if $f\in H_{p, q}(\Phi)$ and $f$ and its partial derivatives of order $\le m-1$ are zero at $0$, then

for all $z\in B$. Here $\alpha: = (\alpha_1, \ldots, \alpha_n)$, $|\alpha|: = \alpha_1+\cdots+\alpha_n$ and each $\alpha_i$ is a nonnegative integer.

The rest of this paper is organized as follows. In Section 2 we will prove main results about Schur's tests, namely, Theorems 1.4, 1.5, 1.6, 1.7 and 1.8. The proofs of Theorems 1.9 and 1.11 will be given in Section 3. Section 4 is the conclusions of this paper.

Finally, we make some conventions on notation. We always denote by $C$ a positive constant which is independent of the main parameters, but it may vary from line to line. The notation $f{\lesssim} g$ means $f\le Cg$ and, if $f{\lesssim} g{\lesssim} f$, then we write $f\sim g$. If $f\le Cg$ and $g = h$ or $g\le h$, we then write $f{\lesssim} g\sim h$ or $f{\lesssim} g{\lesssim} h$, rather than $f{\lesssim} g = h$ or $f{\lesssim} g\le h$.

2.   Proofs of Schur's tests

This section is devoted to proving Theorems 1.4, 1.5, 1.6, 1.7 and 1.8. To do this, we first recall the following definition of mixed Lebesgue spaces introduced by Benedek and Panzone ^[16] in 1961.

Definition 2.1. Let ${p(\cdot)}: = (p_1, p_2)\in(0, {\infty})^2$ and $(X_i, \mu_i)$ be two totally non-trivial $\sigma$-finite measure spaces for $i\in\{1, 2\}$. The mixed Lebesgue space $L^{{p(\cdot)}}(\prod_{i = 1}^2X_i, \prod_{i = 1}^2\mu_i)$ is defined to be the set of all measurable functions $f$ on $X_1\times X_2$ such that

We now write the radial-angular mixed space $L_{p, q}(\Phi)$ with the notion of mixed Lebesgue spaces in Definition 2.1. Indeed, if $p = q$, the space $L_{p, q}(\Phi)$ is the weighted Lebesgue space; if $p\neq q$, the space $L_{p, q}(\Phi)$ becomes the weighted mixed Lebesgue space. Precisely, Let $\alpha > -1$ and $d\nu_{\alpha}: = (1-|z|^2)^{\alpha}\, d\nu(z)$. Recall that the weighted Lebesgue space $L^p_{\alpha}(B): = L^p(B, d\nu_{\alpha})$ with $p > 0$ denotes the set of all measurable complex functions $f$ on $B$ such that

Applying the integral formula in polar coordinates (see, for instance, ^[17]), we find that

Thus, $L^p_{\alpha}(B) = L_{p, p}((1-r^2)^{\frac{\alpha+1}p})$ in the sense of equivalent quasi-norms.

Moreover, with the notation of Definition 2.1, we can write

here and thereafter, $f_r(\cdot): = f(r\cdot)$, $I: = [0, 1]$, $d\lambda: = r^{2n-1}(1-r)^{-1}[\Phi(r)]^p\, dr$. This implies that $L_{p, q}(\Phi) = L^{(p, q)}(\partial B\times I, d\sigma\times d\lambda)$. Therefore, to show Theorems 1.4, 1.5, 1.6, 1.7 and 1.8, it suffices to prove Schur's tests for mixed Lebesgue spaces.

In what follows, let $L^{{p(\cdot)}}_{\mu}: = L^{{p(\cdot)}}(X\times X, \mu_1\times\mu_2)$ and $L^{{\vec{q}}}_{\nu}: = L^{{\vec{q}}}(X\times X, \nu_1\times\nu_2)$ for $\vec p, \vec q\in(0, {\infty})^2$, where $(X, \mu_i)$ and $(X, \nu_i)$ are measure spaces. Then we obtain the following propositions, which extend Schur's tests on weighted Lebesgue spaces given in Zhao ^[4].

Proposition 2.2. Let $\mu_i$ and $\nu_i$ be positive measures on the space $X$ for $i\in\{1, 2\}$ and $T$ an integral operator with nonnegative kernel $K$ defined by setting for any $(x, y)\in X\times X$,

Suppose ${p(\cdot)}: = (p_1, p_2), \ {\vec{q}}: = (q_1, q_2)\in(1, {\infty})^2$ satisfying $1 < p_-\le p_+\le q_- < {\infty}$. Let $\gamma$ and $\delta$ be real numbers such that $\gamma+\delta = 1$. If there exist two positive functions $h_1$ and $h_2$ defined on $X\times X$ with two positive constants $C_1$ and $C_2$ such that for almost all $(x, y)\in X\times X$,

and for almost all $(s, t)\in X\times X$,

then $T: L^{{p(\cdot)}}_{\mu}\to L^{{\vec{q}}}_{\nu}$ is bounded with $\|T\|_{L^{{p(\cdot)}}_{\mu}\to L^{{\vec{q}}}_{\nu}}\le C_1^{1/p_2'}C_2^{1/q_2}.$

Proof. If $f\in L^{{p(\cdot)}}_{\mu}$, then for almost every $(x, y)\in X\times X$, we have

Using Hölder's inequality, we have

which, together with (2.2), further implies that

In addition, from the assumption $1 < p_-\le p_+\le q_- < {\infty}$, it follows that $q_1\ge p_2$, $q_1\ge p_1$, $q_2\ge p_2$ and $q_2\ge p_1$. Thus, combining the above inequality and Minkowski's inequality, we conclude that

Now applying (2.3), we obtain

This finishes the proof of Proposition 2.2. □

Proposition 2.3. Let $\gamma$, $\delta$ and $T$ be an integral operator with kernel $K$ defined as in Proposition 2.2. Suppose ${p(\cdot)}: = (1, p_2)$ and ${\vec{q}}: = (q_1, q_2)\in(1, {\infty})^2$ satisfying $1 < p_2\le q_- < {\infty}$. If there exist two positive functions $h_1$ and $h_2$ defined on $X\times X$ with two positive constants $C_1$ and $C_2$ such that for almost all $(x, y)\in X\times X$,

and, for almost all $(s, t)\in X\times X$,

then $T: L^{{p(\cdot)}}_{{\mu}}\to L^{{\vec{q}}}_{{\nu}}$ is bounded with $\|T\|_{L^{{p(\cdot)}}_{{\mu}}\to L^{{\vec{q}}}_{{\nu}}}\le C_1^{1/p_2'}C_2^{1/q_2}.$

Proof. Let $f\in L^{{p(\cdot)}}_{{\mu}}$. From (2.4) and Hölder's inequality, we infer that, for almost every $(x, y)\in X\times X$,

Then, by the assumptions $q_1\ge p_2$ and $q_2\ge p_2$, and Minkowski's inequality, we conclude that

which, together with (2.5), further implies that

This finishes the proof of Proposition 1.5. □

Proposition 2.4. Let $\gamma$, $\delta$ and $T$ be an integral operator with kernel $K$ defined as in Proposition 2.2. Suppose ${p(\cdot)}: = (p_1, 1)$ and ${\vec{q}}: = (q_1, q_2)\in(1, {\infty})^2$ satisfying $1 < p_1\le q_- < {\infty}$. If there exist two positive functions $h_1$ and $h_2$ defined on $X\times X$ with two positive constants $C_1$ and $C_2$ such that for almost all $(x, y)\in X\times X$,

and, for almost all $(s, t)\in X\times X$,

then $T: L^{{p(\cdot)}}_{{\mu}}\to L^{{\vec{q}}}_{{\nu}}$ is bounded with $\|T\|_{L^{{p(\cdot)}}_{{\mu}}\to L^{{\vec{q}}}_{{\nu}}}\le C_1^{1/p_1'}C_2^{1/q_2}.$

Proof. This proposition is a symmetric case of Proposition 2.3. Therefore, the proof is similar and hence we omit it here. □

Proposition 2.5. Let $\gamma$, $\delta$ and $T$ be an integral operator with kernel $K$ defined as in Proposition 2.2. Suppose ${\vec{q}}: = (q_1, q_2)\in[1, {\infty})^2$. If there exist two positive functions $h_1$ and $h_2$ defined on $X\times X$ with two positive constants $C_1$ and $C_2$ such that

for almost all $(x, y)\in X\times X$, and

for almost all $(s, t)\in X\times X$, then $T: L^{\vec 1}_{\mu}\to L^{{\vec{q}}}_{\nu}$ is bounded with $\|T\|_{L^{\vec 1}_{\mu}\to L^{{\vec{q}}}_{\nu}}\le C_1C_2^{1/q_2}.$

Proof. Let $f\in L^{\vec 1}_{\mu}$. Then, by (2.6), we find that, for almost every $(x, y)\in X\times X$,

Now we first show the present proposition for the case ${\vec{q}}\in(1, {\infty})^2$. In this case, applying the Minkowski's inequality, we deduce that

which, combined with (2.6), further implies that

This completes the proof in this case. For the case $q_i = 1$ for some $i\in\{1, 2\}$, instead of using Minkowski's inequality, applying Fubini's theorem will also do the job. We omit the details in this case and hence finish the proof. □

The following result shows a necessary and sufficient condition of integral operators $T$ associated with kernel $K$ as in Proposition 2.2.

Proposition 2.6. Let $T$ be a integral operator with kernel $K$ defined as in Proposition 2.2. Suppose ${p(\cdot)}: = (p_1, p_2), \ {\vec{q}}: = (q_1, q_2)\in[1, {\infty}]^2$. Then $T$ is a bounded operator from $L^{{p(\cdot)}}(\prod_{i = 1}^2X_i, \prod_{i = 1}^2\mu_i)$ into $L^{{\vec{q}}}(\prod_{i = 1}^2X_i, \prod_{i = 1}^2\nu_i)$ with bound $M$ if and only if for any $u\in L^{{p(\cdot)}}(\prod_{i = 1}^2X_i, \prod_{i = 1}^2\mu_i)$ and $\upsilon\in L^{{\vec{q}}'}(\prod_{i = 1}^2X_i, \prod_{i = 1}^2\nu_i)$,

Proof. From [16,p. 304,Theorem 2], it follows that

Thus, if (2.7) holds true, then

Conversely, when $\|T\|_{L^{{p(\cdot)}}\to L^{{\vec{q}}}}\le M$, using Hölder's inequality, we find that

This implies that (2.7) holds true and hence finishes the proof of Proposition 2.6. □

Finally, applying (2.1), we find that Theorems 1.4, 1.5, 1.6, 1.7 and 1.8 are just corollaries of Propositions 2.2, 2.3, 2.4, 2.5 and 2.6 respectively, and hence their proofs are finished.

3.   Proofs of Theorems 1.9 and 1.11

In this section, we first give the proofs of boundedness of Bergman-type operators and then, as an application, solve Gleason' problem on $H_{p, q}(\Phi)$. To achieve this, we need the following lemmas.

Lemma 3.1. ([10,Lemma 4]) Let $0 < p\le 1$, $1\le q < {\infty}$, $t+s > 0$ and $f\in H(B)$. Then there exists a positive constant $C$ such that, for any $0\le r < 1$,

Lemma 3.2. ([9,Lemma 2.1]) Let $0 < q < 1$ and $t+s > n(1-1/q)$. Then, for any measurable function $f$ on $B$, there exists a positive constant $C$ such that, for any $0\le r < 1$,

Lemma 3.3. ([19,Lemma 6]) Let $s_1 > s_2 > 0$ and $0\le r < 1$. Then there exists a positive constant $C$ such that

Lemma 3.4. ([11,Lemma 2.1]) Let $0 < p \le 1$, $\alpha, \beta > 0$, $0\le r < 1$ and $f:\, [0, 1)\to [0, {\infty})$ be increasing. Then there exists a positive constant $C$ such that

Lemma 3.5. ([9,Lemma 2.3]) Let $0 < p < {\infty}$, $0\le r < 1$ and $\Phi(\cdot): = (1-\cdot)^\tau\varphi(\cdot)$ be a generalized normal function, where $\tau\in {{\mathbb R}}$ and $\varphi$ is a normal function with characteristic exponents $a_{\varphi}$ and $b_{\varphi}$. If $s-\tau+t > b_{\varphi} > a_{\varphi} > s-\tau$, then there exists a positive constant $C$ such that

We now show Theorem 1.9.

Proof of Theorem 1.9. We prove Theorem 1.9 by separately considering $1\le q < {\infty}$ in Step 1 and $0 < q < 1$ in Step 2.

Step 1) In this step we show that if $1\le q < {\infty}$, $0 < p < {\infty}$ and $t-\tau > b_{\varphi} > a_{\varphi} > -s-\tau$, then $P_{s, t}$ is bounded from $H_{p, q}(\Phi)$ to $L_{p, q}(\Phi)$. To achieve this, We further deal with the two cases $0 < p\le 1$ and $1 < p < {\infty}$ separately.

Case i) $0 < p\le 1$ and $1\le q < {\infty}$. In this case, let $f\in H_{p, q}(\Phi)$. From Lemma 3.1, Tonelli's theorem, Lemma 3.5 and the fact that $t-\tau > b_{\varphi} > a_{\varphi} > -s-\tau$, we infer that

Letting $\rho = r^{\frac1p}$, by the facts that $0 < p < 1$ and the monotonicity of $M_q(\rho, f)$ on $\rho$ for holomorphic function $f$, we know that $M_q(r^{\frac1p}, f)\le M_q(r, f)$ and $(1-r^{\frac1p})^{-1}\le (1-r)^{-1}$. In addition, since $\Phi$ is the generalized normal function, we have

Thus, $[\Phi(r^{\frac1p})]^p{\lesssim} [\Phi(r)]^p$ and hence

This finishes the proof of Theorem 1.9 in the case $0 < p\le 1$ and $1\le q < {\infty}$.

Case ii) $1 < p < {\infty}$ and $1\le q < {\infty}$. In this case, let $f\in L_{p, q}(\Phi)$ and $t = t_1+t_2 = t_3+t_4$ such that $t_1 > 0$, $a_{\varphi}+t_1+\tau > t_3$, $t_3+s > t_1$ and $t_2-\tau > b_{\varphi}$. We point out that these assumptions on $t_i$ are reasonable. Indeed, taking a sufficiently small $\epsilon > 0$, let

and $t_4: = (1+\epsilon)b_{\varphi}-(1-\epsilon)a_{\varphi}.$ Then one can verify $t_1$, $t_2$, $t_3$ and $t_4$ satisfy all the above conditions from $t-\tau > b_{\varphi} > a_{\varphi} > -s-\tau$. With these assumptions, applying Lemma 3.1, Hölder's inequality and Lemma 3.3, we find that

This, together with Tonelli's theorem and Lemma 3.5, further implies that

which completes the proof of Theorem 1.9 in the case $1 < p < {\infty}$ and $1\le q < {\infty}$. Thus, we finish the proof of Step 1.

Step 2) In this step we show that if $0 < q < 1$, $0 < p < {\infty}$ and $t-\tau+n(1-1/q) > b_{\varphi} > a_{\varphi} > -s-\tau$, then $P_{s, t}$ is bounded from $H_{p, q}(\Phi)$ to $L_{p, q}(\Phi)$. For this purpose, We considering the following two cases $p\le q$ and $p > q$ separately.

Case iii) $0 < q < 1$ and $p\le q$. In this case, let $f\in H_{p, q}(\Phi)$ and $g(z): = z^{2n-1}f(z)$. Applying Lemmas 3.2 and 3.4, Tonelli's theorem, Lemma 3.5 and the fact that $t > n(1/q-1)+b_{\varphi}+\tau$, we infer that

Letting $\rho = r^{\frac1p}$, then by $0 < p\le q < 1$ and an argument similar to that used in the proof of Case i), we conclude that

This finishes the proof of Theorem 1.9 in the present case.

Case iv) $0 < q < 1$ and $q < p < {\infty}$. In this case, let $f\in L_{p, q}(\Phi)$, $g(z): = z^{2n-1}f(z)$, $\eta: = p/q > 1$ and $q(t+1)-1 = L_1+L_2 = L_3+L_4$, where $L_1 > 0$, $L_3 > L_1$, $L_2/q+(n-1)(1-1/q)-\tau > b_{\varphi}$ and $a_{\varphi} > (L_3-L_1)/q-s-\tau$. Observe that these conditions on $L_i$ are satisfied by taking a sufficiently small $\epsilon > 0$ and setting

and

With these assumptions, applying Lemma 3.2, Hölder's inequality, Lemma 3.3, Tonelli's theorem and Lemma 3.5, we find that

which completes the proof in this case and hence of Step 2. Combining this and Step 1, we obtain Theorem 1.9. □

Remark 3.6. Let $s\in{{\mathbb R}}$ and $t > 0$. Define $\widetilde{P}_{s, t}$ on $L_{p, q}(\Phi)$ by

Then the proof of Theorem 1.9 above actually show that, for any $f\in H_{p, q}(\Phi)$,

This is important in the proof of Theorem 1.11.

Applying Theorem 1.9, we next prove Theorem 1.11.

Proof of Theorem 1.11. We may assume that $m = 1$. The general case follows from induction. In this case, note that $f(0) = 0$. By Leibenson's formula, we find that, for any $f\in H_{p, q}(\Phi)$ and $z\in B$,

For any $z\in B$, let $A_k f(z): = \int_0^1\frac{\partial f}{\partial z_k}(rz)\, dr$. Then $A_k$ is obviously linear. Therefore, to finish the proof, it remains to show that $A_k$ is bounded on $H_{p, q}(\Phi)$. To this end, let $f_r(z): = f(rz)$ for any $0 < r < 1$ and $z\in B$. Then we have $P_{0, t}f_r = f_r$ (see ^[9]). Letting $r\to 1^-$, by Theorem 1.9, we conclude that $P_{0, t}f = f$, namely, for any $z\in B$,

By differential under the integral, we have

Observe that

is a polynomial in $z$ and $\overline{w}$. Therefore, $|A_kf(z)|{\lesssim} |\widetilde{P}_{0, t}f(z)|$. From this and Remark 3.6, we infer that $A_k$ is bounded on $H_{p, q}(\Phi)$ and hence finish the proof of Theorem 1.11. □

4.   Conclusions

Let $0 < p, q < \infty$, $\Phi$ be a generalized normal function and $L_{p, q}(\Phi)$ the radial-angular mixed space. In this paper, we have generalized the classical Schur's test to radial-angular mixed spaces setting and then found the sufficient and necessary condition for the boundedness of integral operators from $L_{p_1, p_2}(\Phi)$ to $L_{q_1, q_2}(\Phi)$ for $1\leq p_i, q_i\leq \infty$ with $i\in\{1, 2\}$. Furthermore, we have also established the boundedness of Bergman-type operators $P_{s, t}$ on holomorphic radial-angular mixed space $H_{p, q}(\Phi)$ for all possible $0 < p, q < \infty$. As an application, we finally solved Gleason's problem on $H_{p, q}(\Phi)$ for all possible $0 < p, q < \infty$.

Use of AI tools declaration

The authors have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Long Huang is supported by the National Natural Science Foundation of China (Grant No. 12201139) and Guangdong Basic and Applied Basic Research Foundation (Grant No. 2021A1515110905). Xiaofeng Wang is supported by the National Natural Science Foundation of China (Grant No. 11971125).

Conflict of interest

The authors declare there is no conflicts of interest.