Sharp stability of convex functionals on weighted Bergman spaces with applications

Theorem 1. Let $G: [0,1] \to \mathbb{R}$ be a continuous and convex non-linear function such that G(0) = 0. Then, for every $f \in \mathcal{A}^2_{\alpha}(\mathbb{D})$ there exists a constant $c_G$ depending on G and $\alpha > -1$ for which $$\int_{\mathbb{D}} G(|f(z)|^2 (1-|z|^2)^{\alpha+2}) d\mu(z) \leqslant \int_{\mathbb{D}} G((1-|z|^2)^{\alpha+2}) d\mu(z) - c_G \min_{|c|=||f||_{\mathcal{A}^2_{\alpha}}} \frac{||f-cf_w||_{\mathcal{A}^2_{\alpha}}^2}{||f||_{\mathcal{A}^2_{\alpha}}^2}.$$ In particular, taking $G(t) = t^{\frac{p}{2}}, p > 2$ we obtain the following corollary: Corollary 2. There exists a computable constant $c_{p,\alpha}$ for p > 2 and $\alpha > -1$ such that $$||f||_{\mathcal{A}_{\beta}^{p}}^{p} \leq ||f||_{\mathcal{A}_{\alpha}^{2}}^{p} \left(1 - c_{p,\alpha} \min_{|c| = ||f||_{\mathcal{A}_{\alpha}^{2}}} \frac{||f - cf_{w}||_{\mathcal{A}_{\alpha}^{2}}^{2}}{||f||_{\mathcal{A}_{\alpha}^{2}}^{2}}\right)^{\frac{p}{2}}$$ $$\leq ||f||_{\mathcal{A}_{\alpha}^{2}}^{p} - c_{p,\alpha}^{\frac{p}{2}} \min_{|c| = ||f||_{\mathcal{A}_{\alpha}^{2}}} \frac{||f - cf_{w}||_{\mathcal{A}_{\alpha}^{2}}^{p}}{||f||_{\mathcal{A}_{\alpha}^{2}}^{p}},$$ where $\beta + 2 = \frac{p(\alpha+2)}{2}$ . We can prove Theorem 1 using a more direct approach which works in a slightly general situation. Namely, we have the following theorem:

Theorem 3. Let A be a positive operator with the unit trace norm and G be a continuous convex, non-linear function. Then, for some computable absolute constant $c_A > 0$ , we have $$\int_{\mathbb{D}} G(\langle \kappa_z, A\kappa_z \rangle) d\mu(z) \leqslant \int_{\mathbb{D}} G((1-|z|^2)^{\alpha+2}) d\mu(z) - c_A \inf_{\xi \in \mathbb{D}} \|A - \langle \cdot, \kappa_\xi \rangle \kappa_\xi \|_{\mathcal{S}_1},$$ where $S_1$ denotes the trace norm and $\kappa_{\xi}$ normalized Bergman reproducing kernel in $\mathcal{A}^2_{\alpha}(\mathbb{D})$ . We turn now our attention to higher dimensional weighted Bergman spaces. Let $\mathcal{A}^p_{\alpha}(\mathbb{B}_n)$ be the space of the holomorphic functions in the unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ such that $$||f||_{\mathcal{A}_{\alpha}^{p}(\mathbb{B}_{n})} := \left(\frac{\Gamma(\alpha+n+1)}{n!\Gamma(\alpha+1)} \int_{\mathbb{B}_{n}} |f(z)|^{p} (1-|z|^{2})^{\alpha+n+1} d\mu_{n}(z)\right)^{\frac{1}{p}} < +\infty,$$ where $\alpha > -1$ , $d\mu_n(z) = \frac{dV(z)}{(1-|z|^2)^{n+1}}$ and dV(z) is the usual Lebesgue volume measure. In [35] it is proved that if $$\int_{\mathbb{B}_n} G(|f(z)|^p (1-|z|^2)^{\alpha+n+1}) d\mu_n(z) \leqslant \int_{\mathbb{B}_n} G((1-|z|^2)^{\alpha+n+1}) d\mu_n(z)$$ is true for all convex functions G, then the extremizers are exactly multiples of $(1-\langle z,w\rangle)^{-\frac{2(\alpha+n+1)}{p}}$ and the quantities $\int_{\mathbb{B}_n}G(|f(z)|^p(1-|z|^2)^{\alpha+n+1})d\mu_n(z)$ and $\int_{\mathbb{B}_n}G((1-|z|^2)^{\alpha+n+1})d\mu_n(z)$ are close only if f is close to the set of extremizers. Here, we will get the optimal stability result:

Theorem 5. Let $G:[0,1] \to \mathbb{R}$ be a continuous and strictly increasing or convex and increasing function such that G(0) = 0. Then, for every $f \in H^2$ , there exists a constant $c_G$ depending on G for which $$\int_{\mathbb{D}} G(|f(z)|^2 (1-|z|^2)) d\mu(z) \le \int_{\mathbb{D}} G((1-|z|^2)) d\mu(z) - c_G(1-T),$$ where $T \leq 1$ is the constant such that $$|f(z)|^2(1-|z|^2) \leqslant T||f||_{H^2}^2.$$ The paper is organized as follows. The second section describes the fundamental relations between the main notions in the context of the functionals on the weighted Bergman spaces that are induced by convex functions. The proof of stability through majorization theory and stability estimates of local inequalities with applications to Fock and Hardy spaces (using the technique from [\[14\]](#page-29-9)) are given in the third section. Two proofs of the general result for the positive operator with finite trace norm using the methods from [\[14\]](#page-29-9) and [\[35\]](#page-30-15) are the main content of the fourth section. In the same section, we give the wavelet transform interpretation. The fifth section is devoted to the proof of the stability of the convex functionals on A^p α (Bn) assuming that the analog of Kulikov's inequality in a higher-dimensional setting holds for every convex function. In the last, sixth section we prove the result for Hardy space setting and functionals given by strictly increasing functions.

Lemma 8. Let us denote $\rho(t) = \mu(\{z \in \mathbb{D} : u(z) > t\}), t > 0$ . Then the following is true: a) $$\Delta \log u(z) \geqslant -4(\alpha+2),$$ b) The function $\rho(t)$ is non-increasing and absolutely continuous on compact subsets of $(0, +\infty)$ and there holds the differential inequality $$\rho'(t) \leqslant -\frac{\rho(t) + \pi}{(\alpha + 2)t}$$ c) $T = \max_{z \in \mathbb{D}} u(z) \leqslant 1$ and T = 1 if and only if $A = \langle \cdot, \kappa_{z_0} \rangle \kappa_{z_0}$ for some $z_0$ . For T = 1 we have $\rho(t) = \rho_0(t) = \pi \left(t^{-\frac{1}{\alpha+2}} - 1\right)$ while in the case T < 1 there exists a unique $t$ such that $\rho(t) > \rho_0(t)$ for $0 < t < t$ and $\rho(t) < \rho_0(t)$ for $t^* < t < T$ .

Lemma 9. For every $t_0 > 0$ there exists $T_0 \in (t_0, 1)$ and the absolute constant C such that $$\rho(t) \leqslant \pi \left(1 + Ct_0^{-3} \delta^2\right) \left(\left(\frac{T}{t}\right)^{\frac{1}{\alpha+2}} - 1\right) =: \tilde{\rho}(t), \quad t \in (t_0, T),$$ $$if \ f \in \mathcal{A}_{\alpha}^2 \ and \ \|f\|_{\mathcal{A}_{\alpha}^2} = 1. \ Here \ \delta = \sqrt{\frac{1-T}{T}}.$$

Lemma 12. For every $\alpha > -1$ , $f \in \mathcal{A}^2_{\alpha}(\mathbb{B}_n)$ and $t_0 \in (0,1)$ , there exist $T_0 \in (t_0,1)$ and the absolute constant C' depending only on $t_0$ and n such that $$\rho(t) \leqslant \frac{\omega_n}{2n} \left( 1 + C' \delta^2 \right) \left( \left( \frac{T}{t} \right)^{\frac{1}{\alpha + n + 1}} - 1 \right)^n =: \tilde{\rho}(t), \quad t \in (t_0, T),$$ \nif $f \in \mathcal{A}_{\alpha}^2(\mathbb{B}_n)$ and $||f||_{\mathcal{A}_{\alpha}^2}(\mathbb{B}_n) = 1$ and $\delta = \sqrt{\frac{1 - T}{T}}.$ Since ρ⁰ is again a convex and a decreasing function, we have the analogs of [\(22\)](#page-24-1) and, therefore, [\(21\)](#page-24-0) hold for some τ¹ ⩽ t ⩽ τ² < τ³ ⩽ T for T close to 1. Along with Theorem 5.2. from [\[35\]](#page-30-15), we finish a proof of Theorem .