Moduli of Holomorphic Functions and Logarithmically Convex Radial Weights

Theorem 1.1. (cf. [23, Proposition 5.4]). Let $w(t) = \frac{1}{1-t}$ , $0 \le t < 1$ . There exist functions $f_1, f_2 \in \mathcal{A}^w(\mathbb{D})$ such that $$|f_1(z)| + |f_2(z)| \ge w(|z|)$$ for all $z \in \mathbb{D}$ . ²⁰¹⁰ Mathematics Subject Classification. Primary 30H99; Secondary 32A37, 42A55, 47B38. Key words and phrases. Growth space, log-convex radial weight, circular domain. The second author was supported by the Russian Science Foundation (grant No. 14-41-00010). One may consider Theorem [1.1](#page-0-1) as a particular solution of the following approximation problem: Given a radial weight w on D, find f1, f² ∈ Hol(D) such that $$|f_1| + |f_2| \asymp w.$$ In other words, we are looking for a holomorphic mapping f : D → C 2 such that kfk ≍ w. The above problem has been solved recently for various explicit radial weights; see, for example, [\[11\]](#page-12-1), [\[12\]](#page-12-2), [\[13\]](#page-12-3), [\[17\]](#page-12-4), [\[27\]](#page-13-0). Clearly, the required property still holds if w is replaced by an equivalent radial weight. To the best of our knowledge, the largest class of weight functions w(t) is considered in [\[1\]](#page-12-5), where the direct analog of Theorem [1.1](#page-0-1) is proved under assumption that w(t) has the following doubling property: <span id="page-1-0"></span> $$(1.2) w(1-s/2) < Aw(1-s), 0 < s \le 1,$$ for some constant A > 1. Essentially the same result was independently obtained in [\[19\]](#page-12-6). As in [\[23\]](#page-12-0), the arguments in [\[1\]](#page-12-5), [\[19\]](#page-12-6) use lacunary series; see [\[14\]](#page-12-7) for a different proof. Basically, property [\(1.2\)](#page-1-0) means that w(t) grows sufficiently slowly as t → 1−. So, it is natural to ask whether a growth restriction is crucial for the corresponding results. In the present paper, we characterize those w for which the approximation problem under consideration is solvable. In particular, analogs of Theorem [1.1](#page-0-1) hold for rapidly growing radial weights. However, there are natural restrictions on the regularity of w. 1.3. Main results. Let w : [0, 1) → (0, +∞) be a weight function. Often, w is called log-convex if log w is convex. In this paper, we use a different definition related to Hadamard's three-circles theorem. Namely, w is said to be log-convex if log w(t) is a convex function of log t, 0 < t < 1. If log w(t) is a convex function of t, then w is log-convex, but the converse is not true; see Remark [6.](#page-3-0)

Theorem 1.3. Let $\mathcal{D} \subset \mathbb{C}^d$ be a bounded, circular, strictly convex domain with $\mathcal{C}^2$ -boundary. If $w: [0,1) \to (0,+\infty)$ is a log-convex weight function, then there exist functions $f_m \in \mathcal{A}^w(\mathcal{D})$ , $1 \leq m \leq M = M(\mathcal{D})$ such that (1.5) $$\sum_{m=1}^{M} |f_m(z)| \ge w(r_{\mathcal{D}}(z)), \quad z \in \mathcal{D}.$$ Conversely, if (1.5) holds for some $M \in \mathbb{N}$ and some $f_1, \ldots, f_M \in \mathcal{H}ol(\mathcal{D})$ , then w(t) is equivalent to a log-convex weight function. Remark 4. Recall that, by Theorem 1.2 and Remark 3, the optimal value of M for $\mathbb{D}$ is equal to 2. It would be interesting to find the optimal value of $M(B_d)$ for the unit ball $B_d$ of $\mathbb{C}^d$ , $d \geq 2$ . - 1.4. Applications. We consider composition and multiplication operators, Volterra type operators and extended Cesàro operators on $\mathcal{A}^w(\mathcal{D})$ . - 1.5. Organization of the paper. Section 2 is devoted to the proof of Theorem 1.2. Circular domains in $\mathbb{C}^d$ are considered in Section 3. Applications are discussed in the final Section 4; main applications are related to the unit ball of $\mathbb{C}^d$ , $d \geq 1$ . Acknowledgements. The authors are grateful to Alexander Borichev for useful discussions and elucidating comments. Also, we are thankful to the anonymous referee for helpful suggestions and constructive remarks. 2. $$\mathcal{D}$$ is the unit disk <span id="page-3-2"></span>In this section, we prove Theorem [1.2.](#page-1-1) 2.1. Equivalence to a log-convex weight function is necessary. Given a continuous function u : D → (0, +∞), put $$M_u(r) = \max_{|z|=r} u(z), \quad 0 \le r < 1.$$ Now, suppose that w is a radial weight, f1, f² ∈ Hol(D) and |f1|+|f2| ≍ w. Without loss of generality, we also assume that f^j (0) 6= 0 for j = 1, 2. By Hadamard's three circles theorem, M|f1^| and M|f2^| are log-convex functions on the interval [0, 1); see, e.g., [\[8,](#page-12-9) Ch. 6, Sect. 3.13]. Hence, the sum M|f1^| + M|f2^| is also log-convex (see, for example, [\[18,](#page-12-10) p. 51]). Observe that $$M_{|f_j|} \le M_{|f_1|+|f_2|} \le M_{|f_1|} + M_{|f_2|}, \quad j = 1, 2.$$ Thus, we have the following equivalences on [0, 1): $$w \simeq M_{|f_1|+|f_2|} \simeq M_{|f_1|} + M_{|f_2|}.$$ Since the function M|f1^|+M|f2^| is log-convex, the implication [\(1.3\)](#page-1-2)⇒[\(1.4\)](#page-1-3) is proved. <span id="page-3-1"></span>Remark 5. Let w be a radial weight such that $$w \asymp |f_1| + \dots + |f_K|$$ for some f1, . . . , f^K ∈ Hol(D), K ∈ N. Repeating the arguments used above for K = 2, we deduce that w(t) is equivalent to a log-convex weight function. The rest of the section is devoted to a constructive proof of the implication [\(1.4\)](#page-1-3)⇒[\(1.3\)](#page-1-2). 2.2. Log-convex weight functions: preliminaries. Let w : [0, 1) → (0, +∞) be a log-convex weight function. Recall that, by definition, log w(t) is a convex function of log t, that is, $$\Phi(x) = \Phi_w(x) = \log w(e^x), \quad x \in (-\infty, 0),$$ is a convex function. <span id="page-3-0"></span>Remark 6. It is natural to compare the above property and the following one: log w(t) is a convex function of t. The latter property implies that Φ^w is convex, as the composition of two increasing convex functions. The reverse implication does not hold. Moreover, there exist log-convex weight functions that are not even equivalent to the exponent of a convex function. In what follows, we argue in terms of the function Φ. Observe that w(t) = exp(Φ(log t)), 0 < t < 1. To prove the implication [\(1.4\)](#page-1-3)⇒[\(1.3\)](#page-1-2), we may replace w by an equivalent weight function. So, without loss of generality, we assume that Φ is a strictly convex C 2 -function. In particular, the tangent to the graph of Φ is unique at each point (x, Φ(x)), x < 0. Also, below we repeatedly use the following property without  <span id="page-4-0"></span>Figure 1. Basic induction construction explicit reference: the slope of the tangent to the graph of $\Phi$ at $(x, \Phi(x))$ is a strictly increasing function of $x \in (-\infty, 0)$ . Finally, note that $\log v(e^x) = \log a + \beta x$ for $v(t) = at^{\beta}$ , a > 0, $\beta > 0$ . This observation allows to reduce the proof of the implication $(1.4) \Rightarrow (1.3)$ to certain manipulations with linear functions. <span id="page-4-1"></span>2.3. Basic induction construction. Fix a number $x_0 \in (-\infty, 0)$ and a parameter h > 0. By induction, we construct linear functions $\ell_k(x)$ and numbers $x_k \in (x_0, 0)$ , $k = 1, 2, \ldots$ , such that - $\ell_k(x)$ is a tangent to the graph of $\Phi(x)$ ; - $\ell_k(x)$ intersects the graph of $\Phi(x) h$ at the points whose x-coordinates are $x_{k-1}$ and $x_k, x_{k-1} < x_k$ . Note that $\ell_k(x)$ and $x_k$ are uniquely defined by the above properties. So, the induction construction proceeds. See Figure 1. Given $\ell_k(x)$ , we define parameters $a_k > 0$ and $\beta_k > 0$ by the following identity: $$\ell_k(x) = \log a_k + \beta_k x, \quad x \in \mathbb{R}.$$ Note that the sequences $\{x_k\}_{k=0}^{\infty}$ and $\{\beta_k\}_{k=1}^{\infty}$ monotonically increase, $x_k \to 0$ and $\beta_k \to \infty$ as $k \to \infty$ . Also, we use the following brief notation: $t_k = \exp(x_k)$ , $k = 0, 1, \ldots$ Hence, the positive numbers $t_k$ monotonically increase to 1 as $k \to \infty$ . Formally, the above construction works for any h > 0. In applications, we use a sufficiently large parameter h, say, h = 2, in the case of the unit disk.

Lemma 2.1. Let the numbers $x_k$ , k = 0, 1, ..., and the linear functions $\ell_k$ , k = 1, 2, ..., be those introduced in subsection 2.3. Then <span id="page-4-2"></span>(2.1) $$\ell_k(x) \ge \ell_{k+1}(x) + h$$ for all $x, x_0 \le x \le x_{k-1}, k \ge 1$ ; <span id="page-4-3"></span> $$(2.2) \ell_{k+1}(x) > \ell_k(x) + h for all x, x_{k+1} < x < 0, k > 1.$$

Lemma 2.2. Let the numbers $x_k$ , $t_k$ , k = 0, 1, ..., and the linear functions $\ell_k$ , $k=1,2,\ldots$ , be those introduced in subsection 2.3. Assume that $h\geq 2$ . Then, for $k = 1, 2, \dots,$ - $\begin{array}{ll} \text{(i)} & a_k t^{\beta_k} \leq w(t), \quad t \in [t_0,1); \\ \text{(ii)} & e^{-h} w(t) \leq a_k t^{\beta_k}, \quad t \in [t_{k-1},t_k]; \\ \text{(iii)} & \sum_{m \geq 1, \ |m-k| \geq 2} a_m t^{\beta_m} < \frac{1}{2} a_k t^{\beta_k}, \quad t \in [t_{k-1},t_k]. \end{array}$

Theorem 3.1. Let $\mathcal{D} \subset \mathbb{C}^d$ be a bounded, circular, strictly convex domain with $\mathcal{C}^2$ -boundary. There exist $\delta = \delta(\mathcal{D}) \in (0,1)$ and $Q = Q(\mathcal{D}) \in \mathbb{N}$ with the following properties: for every $n \in \mathbb{N}$ , there exist homogeneous holomorphic polynomials $W_q[n]$ of degree $n, 1 \leq q \leq Q$ , such that <span id="page-8-2"></span> $$(3.1) ||W_q[n]||_{L^{\infty}(\partial \mathcal{D})} \le 1;$$ <span id="page-8-3"></span>(3.2) $$\max_{1 \le q \le Q} |W_q[n](\zeta)| \ge \delta \quad \text{for all } \zeta \in \partial \mathcal{D}.$$ As observed in [10], to prove Theorem 3.1, it suffices to repeat mutatis mutandis the arguments used in [16, Theorem 2.6]. 3.3. Auxiliary estimates. We need a modification of property (iii) from Lemma 2.2.

Lemma 3.2. Let $\delta \in (0,1)$ . Then there exists $h(\delta) \geq 2$ with the following property: Let the numbers $x_k$ , $t_k$ , $k = 0, 1, \ldots$ , and the linear functions $\ell_k$ , $k = 1, 2, \ldots$ , be those introduced in subsection 2.3. Assume that $h \geq h(\delta)$ . Then (iii_$$\delta$$) $\sum_{m\geq 1, |m-k|\geq 2} a_m t^{\beta_m} < \frac{\delta}{2} a_k t^{\beta_k}, \quad t \in [t_{k-1}, t_k],$ for $k = 1, 2, \dots$

Corollary 4.1. Suppose that $g \in \mathcal{H}ol(\mathcal{D}), \ \varphi : \mathcal{D} \to \mathcal{D}$ is a holomorphic mapping and $\mathcal{Y}(\mathcal{D})$ is a lattice. Then the weighted composition operator $C^g_{\omega}$ maps $\mathcal{A}^w(\mathcal{D})$ into $\mathcal{Y}(\mathcal{D})$ if and only if $$|g(z)|w(r_{\mathcal{D}}(\varphi(z))) \in \mathcal{Y}(\mathcal{D}).$$ 4.2. Integral operators. Assume that $\mathcal{D}$ is the unit ball $B_d$ of $\mathbb{C}^d$ . Given $g \in$ $\mathcal{H}ol(B_d)$ and a holomorphic mapping $\varphi: B_d \to B_d$ , the Volterra type operator $V_{\alpha}^g: \mathcal{H}ol(B_d) \to \mathcal{H}ol(B_d)$ is defined as $$(V_{\varphi}^g f)(z) = \int_0^1 f(\varphi(tz)) \frac{\mathcal{R}g(tz)}{t} dt, \quad f \in \mathcal{H}ol(B_d), \ z \in B_d,$$ where $$\mathcal{R}g(z) = \sum_{j=1}^{d} z_j \frac{\partial g}{\partial z_j}(z), \quad z \in B_d,$$ is the radial derivative of g. If $\varphi(z) \equiv z$ , then $V_{\varphi}^{g}$ is denoted by $J_{g}$ and it is called an extended Cesàro operator (see [15]). In fact, if d = 1, then we have <span id="page-10-0"></span> $$(J_g f)(z) = \int_0^z f(w)g'(w) dw, \quad f \in \mathcal{H}ol(\mathbb{D}), \ z \in \mathbb{D}.$$ The above operator was introduced by Pommerenke [22] as a natural generalization of the classical Cesàro operator. For d=1, various properties of the operator $J_q$ are discussed in surveys [3], [26]. Direct calculations show that (4.1) $$\mathcal{R}V_{\omega}^{g}f(z) = f(\varphi(z))\mathcal{R}g(z), \quad z \in B_{d},$$ for all $f, g \in \mathcal{H}ol(B_d)$ (cf. [15]). Applying (4.1) and Corollary 4.1, we obtain the following fact. <span id="page-10-2"></span>Corollary 4.2. Let $\mathcal{Y}(B_d)$ be a lattice on $B_d$ . Then the operator $\mathcal{R}V_{\mathcal{G}}^g$ maps $\mathcal{A}^w(B_d)$ into $\mathcal{Y}(B_d)$ if and only if $|\mathcal{R}g(z)|w(|\varphi(z)|) \in \mathcal{Y}(B_d)$ . Given a space $X \subset \mathcal{H}ol(B_d)$ , a typical problem is to characterize $g \in \mathcal{H}ol(B_d)$ such that $J_g$ is a bounded operator on X. Often the characterizing property is the following one: $$\sup_{z \in B_d} |\mathcal{R}g(z)|(1-|z|) < \infty,$$ that is, g is in the Bloch space $\mathcal{B}(B_d)$ (see, e.g., [4], [15]). Hence, it is interesting to find those X for which the answer is different. To give such examples, consider the following exponential weight functions: $$w_{\alpha}(r) = \exp\left(\frac{1}{(1-r)^{\alpha}}\right), \quad 0 \le r < 1, \ \alpha > 0.$$ Note that the above weight functions are not doubling.

Corollary 4.3. Let $\alpha > 0$ and let $g \in \mathcal{H}ol(B_d)$ . The operator $J_g : \mathcal{A}^{w_{\alpha}}(B_d) \to \mathcal{A}^{w_{\alpha}}(B_d)$ is bounded if and only if $$\sup_{z \in B_d} |g(z)|(1-|z|)^{\alpha} < \infty.$$

Corollary 4.4. Let $d \geq 1$ and let $v : [0,1) \to (0,+\infty)$ be an arbitrary weight function. Suppose that $g \in \mathcal{H}ol(B_d)$ , $\varphi : B_d \to B_d$ is a holomorphic mapping and $\mathcal{Y}(B_d)$ is a lattice. Then the weighted composition operator $C_{\varphi}^g$ maps $\mathcal{A}^v(B_d)$ into $\mathcal{Y}(B_d)$ if and only if $|g(z)|\widetilde{v}(|\varphi(z)|) \in \mathcal{Y}(B_d)$ .

Definition 1. Given a radial weight v on $B_d$ , $d \ge 1$ , the associated weight $\tilde{v}_d$ is defined by $$\widetilde{v}_d(z) = \sup\{|f(z)|: f \in \mathcal{H}ol(B_d), |f| \le v \text{ on } B_d\}.$$ As observed in [5], $\tilde{v}_1$ is a radial weight, so the associated weight function $\tilde{v}_1$ : $[0,1) \to (0,+\infty)$ is correctly defined. Moreover, $\tilde{v}_1$ is known to be log-convex (see [6]). Now, assume that $d \geq 2$ . If $f \in \mathcal{H}ol(B_d)$ and $|f| \leq v$ , then every slice-function $f_{\zeta}(\lambda) = f(\lambda \zeta)$ , $\zeta \in \partial B_d$ , $\lambda \in \mathbb{D}$ , is in $\mathcal{H}ol(\mathbb{D})$ and $|f_{\zeta}(\lambda)| \leq v(|\lambda|)$ , $\lambda \in \mathbb{D}$ . Thus, on the one hand, $\widetilde{v}_d(z) \leq \widetilde{v}_1(|z|)$ , $z \in B_d$ . On the other hand, if $f \in \mathcal{H}ol(\mathbb{D})$ and $|f| \leq v$ on $\mathbb{D}$ , then $F(z_1, \ldots, z_d) := f(z_1) \in \mathcal{H}ol(B_d)$ and $|F| \leq v$ on $B_d$ ; hence, $\widetilde{v}_d(z) \geq \widetilde{v}_1(|z|)$ . So $\widetilde{v}_d$ is a radial weight and the identity $\widetilde{v}_d = \widetilde{v}_1 := \widetilde{v}$ holds for the associated weight functions. Clearly, $\mathcal{A}^v(B_d) = \mathcal{A}^{\widetilde{v}}(B_d)$ , $d \geq 1$ , isometrically. Hence, given an arbitrary weight function $v:[0,1)\to (0,+\infty)$ , the study of $\mathcal{A}^v(B_d)$ reduces to that of $\mathcal{A}^w(B_d)$ , where $w=\widetilde{v}$ is a log-convex weight function. For example, an extension of Corollary 4.1 has the following form: