Karapetrović Conjecture: Hilbert Norm

• Karapetrović Conjecture is a formulation that precisely determines the operator norm of the Hilbert matrix on weighted Bergman spaces using an explicit closed-form formula.
• Recent advances by Bao–Tian–Wulan rigorously established the conjectured norm in an expanded range, employing intricate integral representations and sharp operator estimates.
• The approach utilizes Beta function identities and discriminant analysis to bridge gaps in previous results, enhancing our understanding of operator norms in analytic function spaces.

The Karapetrović conjecture describes the precise operator norm of the Hilbert matrix operator on weighted Bergman spaces $A^p_\alpha$, positing an explicit closed-form value as a function of the weight parameter $\alpha$ and integrability parameter $p$. The conjecture asserts that for the bounded range $-1<\alpha<p-2$, the operator norm is given by $\pi/\sin((\alpha+2)\pi/p)$. Recent progress, notably by Bao–Tian–Wulan, has rigorously established this norm formula throughout an optimal two-sided region that substantially subsumes all previous partial results (Bao et al., 20 Jan 2026).

1. Formulation of the Conjecture and Definitions

The weighted Bergman space $A^p_\alpha$ consists of holomorphic functions $f$ on the unit disk $\D$ such that

$\|f\|_{A^p_\alpha}^p = \int_\D |f(z)|^p\,dm_\alpha(z), \quad dm_\alpha(z) = (\alpha+1)(1-|z|^2)^\alpha\,dm(z),$

with $dm(z) = \pi^{-1}\,dx\,dy$, for parameters $\alpha>-1$, $0<p<\infty$. For a function $f(z) = \sum_{k=0}^\infty a_k z^k$, the Hilbert matrix operator $\H$ acts by

$$\H(f)(z) = \sum_{n=0}^\infty \Bigl(\sum_{k=0}^\infty \frac{a_k}{n+k+1}\Bigr) z^n,$$

whenever the double series defines an analytic function in $\D$. The operator norm is given by

$$\|\H\|_{A^p_\alpha\to A^p_\alpha} = \sup_{\|f\|_{A^p_\alpha}=1}\|\H(f)\|_{A^p_\alpha}.$$

Karapetrović (2018) conjectured that whenever $1<\alpha+2<p$,

$$\|\H\|_{A^p_\alpha\to A^p_\alpha} = \frac{\pi}{\sin((\alpha+2)\pi/p)},$$

with equivalent validity stated for the broader boundedness interval $-1<\alpha<p-2$.

2. Main Theorem and the Improved Parameter Range

Bao–Tian–Wulan established a major advance on the Karapetrović conjecture, introducing the explicit upper bound curve

$$\alpha_{\rm up}(p) = \frac{6p^3-29p^2+17p-2 + 2p\sqrt{6p^2-11p+4}}{(3p-1)^2},$$

for $p>1$. The theorem asserts that for

$0\leq\alpha\leq\alpha_{\rm up}(p),$

the Hilbert matrix operator is bounded on $A^p_\alpha$ (i.e., if $p>\alpha+2$), and the conjectured norm formula holds: $$\|\H\|_{A^p_\alpha\to A^p_\alpha} = \frac{\pi}{\sin((\alpha+2)\pi/p)}.$$ This improvement encompasses all prior sufficient conditions for $\alpha>\tfrac{1}{47}$ and $\alpha\ne1$, substantially enlarging the domain of parameter values for which the conjecture is resolved (Bao et al., 20 Jan 2026).

3. Proof Structure and Key Analytical Techniques

The resolution proceeds through the following key elements:

1. Lower Bound via Test Functions: Karapetrović demonstrated that evaluating $\H$ on the family $f_r(z)=(1-rz)^{-(\alpha+2)/p}$ yields

$$\|\H\|_{A^p_\alpha\to A^p_\alpha} \geq B\!\bigl(\tfrac{\alpha+2}{p},1-\tfrac{\alpha+2}{p}\bigr) = \frac{\pi}{\sin((\alpha+2)\pi/p)},$$

leveraging the Beta function identity.

2. Integral Representation of $\H$: The Hilbert operator admits the representation

$\H(f)(z) = \int_0^1 T_t(f)(z)\,dt,$

where $T_t$ is a weighted composition operator:

$$T_t(f)(z) = \omega_t(z)\,f(\phi_t(z)),\;\; \omega_t(z) = (1-(1-t)z)^{-1},\;\; \phi_t(z) = \frac{t}{1-(1-t)z}.$$

1. Upper Bound via Integral Inequalities: After transforming to polar coordinates and using sharp operator-theoretic and real-variable estimates, a two-step comparison relates $\|T_t(f)\|_{A^p_\alpha}$ to the Bergman norm, introducing a remainder involving an auxiliary function $F_{p,\alpha}(r)$.
2. Monotonicity Lemma and Quadratic Discriminant: The crucial step involves a monotonicity lemma, establishing that $F_{p,\alpha}(r)\le0$ for $0<r\le1$ provided

$$\alpha_{\rm low}(p)\le \alpha \le \alpha_{\rm up}(p),$$

where $\alpha_{\rm low}(p)$ mirrors the formula for $\alpha_{\rm up}(p)$ with the opposite sign before the square root. This is proven by showing the related quadratic $k(r)$ has non-positive discriminant, ensuring non-positivity for the corrective term.

1. Norm Identification: The simultaneous attainment of the matching upper and lower bounds yields the sharp norm.

4. Key Formulas and Supporting Estimates

Several foundational identities are explicit in the proof process:

• Beta function–sine connection: For $p>\alpha+2>0$,

$$\int_0^1 t^{\frac{\alpha+2}p-1}(1-t)^{1-\frac{\alpha+2}p}dt = B\!\bigl(\tfrac{\alpha+2}{p},1-\tfrac{\alpha+2}{p}\bigr) = \frac{\pi}{\sin((\alpha+2)\pi/p)}.$$

• Difference-of-powers estimate: For $0<\beta<1$, $x,y>0$,

$x^\beta-y^\beta\leq \beta y^{\beta-1}(x-y).$

• Quadratic discriminant for monotonicity: Non-positivity of the discriminant for

$k(r)=A r^2+B r+C,$

with explicit $A,B,C$ depending on $p$ and $\alpha$, is equivalent to the necessary range for $\alpha$.

5. Relation to Previous Work and Status of the Conjecture

Partial results predating Bao–Tian–Wulan's theorem include:

• For $\alpha\ge0$, $p\ge2(\alpha+2)$, the norm formula holds (Karapetrović, 2018).
• Further verified in the subrange $\alpha>0$, $$\alpha+2+\sqrt{\alpha^2+\tfrac72\alpha+3}\leq p<2(\alpha+2)$$ (Lindström–Miihkinen–Wikman, 2021).
• Other specialized regimes analyzed by Dai (2024), Dmitrović–Karapetrović (2023), and similar works.

The curve $\alpha_{\rm up}(p)$ falls strictly below all previously known upper bounds for $\alpha$ when $\alpha>1/47$ and $\alpha\ne1$, establishing the norm in almost all of the previously inaccessible parameter space for $p<2(\alpha+2)$. The conjecture remains open only within the narrow triangle defined by $-1<\alpha<0$, $\alpha+2<p<2(\alpha+2)$ (Bao et al., 20 Jan 2026).

6. Summary Table of Parameter Ranges

Author(s)	Proven Range for $\alpha$	Norm Formula Proven
Karapetrović (2018)	$\alpha\geq0,\;p\geq2(\alpha+2)$	$\pi/\sin((\alpha+2)\pi/p)$
Lindström–Miihkinen–Wikman (2021)	$$\alpha+2+\sqrt{\alpha^2+\frac72\alpha+3}\leq p<2(\alpha+2)$$	$\pi/\sin((\alpha+2)\pi/p)$
Bao–Tian–Wulan (2025)	$0\leq\alpha\leq\alpha_{\rm up}(p)$	$\pi/\sin((\alpha+2)\pi/p)$

The table encapsulates the evolutionary expansion of the region for which the Karapetrović conjecture is settled.

7. Open Cases and Impact

The Karapetrović conjecture is now resolved for all $(p,\alpha)$ except the region $-1<\alpha<0$, $\alpha+2<p<2(\alpha+2)$. The proof techniques, particularly the integral-operator representation and the two-sided discriminant analysis, signal broader applicability in studying composition-type operators and sharp norm formulae in analytic function spaces. Given the explicit nature of the resulting norm, further advances in the remaining open region may require new extremal function constructions or refined convexity arguments. The results close the most significant existing gaps and complete the picture for positive weights and the majority of the parameter space (Bao et al., 20 Jan 2026).