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The solutions of a kind of second-order homogeneous partial differential equation are called (real kernel) alpha-harmonic functions. The alpha-harmonic functions and their first-order partial derivative functions on unit disk are estimated using the $L^p$ norm of the boundary functions of the alpha-harmonic functions. A series of inequalities are obtained. In addition, when the alpha-harmonic functions are quasiconformal, their first-order partial derivative functions are estimated by the arc length of the domain boundary and the Lipschitz constant of the boundary functions. All of the inequalities obtained in this article are optimal or asymptotically optimal., Comment: 25pages

The solutions of a kind of second-order homogeneous partial differential equation are called (real kernel) alpha-harmonic functions. In this paper, the boundary correspondence and boundary behavior of alpha-harmonic functions are studied, and the corresponding Dirichlet problem is solved. As one of its applications, an asymptotic optimal Riesz-Fejer inequality for alpha-harmonic functions is obtained. In addition, the subharmonic properties of alpha-harmonic functions is explored and an optimal radius is obtained., Comment: 21pages

In this paper, we study the relationship between the type problem and the asymptotic behaviour of the first (Dirichlet) eigenvalues $\lambda_1(B_r)$ of ``balls'' $B_r:=\{\rhor_0$ \[ r^2 \lambda_1(B_r)\ge \gamma>0, \] we obtain a sharp estimate of the volume growth: $|B_r|\ge cr^{\mu(\gamma)}.$ Moreover when $\gamma>j_0^2\approx 5.784$, where $j_0$ denotes the first positive zero of the Bessel function $J_0$, then $M$ is hyperbolic and we have a Hardy type inequality. In the case where $r_0=0$, a sharp Hardy type inequality holds. These spectral conditions are satisfied if one assumes that $\Delta\rho^2\geq2\mu(\gamma)>0$. In particular, when $\inf_M\Delta\rho^2>4$, $M$ is hyperbolic and we get a sharp Hardy type inequality. Related results for finite volume case are also studied., Comment: 20 pages. Comments are welcome! arXiv admin note: text overlap with arXiv:2401.11803

We study the density of functions which are holomorphic in a neighbourhood of the closure $\overline{\Omega}$ of a bounded non-smooth pseudoconvex domain $\Omega$, in the Bergman space $ H^2(\Omega ,\varphi)$ with a plurisubharmonic weight $\varphi$. As an application, we show that the Hartogs domain $$ \Omega _\alpha : = \{(z,w) \in D\times \C: |w|< \delta^\alpha_D(z) \}, \ \ \ \alpha>0, $$ where $D\subset \subset \C$ and $\delta_D$ denotes the boundary distance, is Bergman complete if and only if every boundary point of $D$ is non-isolated., Comment: 23pages

Multimodal neuroimaging grants a powerful in vivo window into the structure and function of the human brain. Recent methodological and conceptual advances have enabled investigations of the interplay between large-scale spatial trends, or gradients, in brain structure and function, offering a framework to unify principles of brain organization across multiple scales. Strong community enthusiasm for these techniques has been instrumental in their widespread adoption and implementation to answer key questions in neuroscience. Following a brief review of current literature on this framework, this perspective paper will highlight how pragmatic steps aiming to make gradient methods more accessible to the community propelled these techniques to the forefront of neuroscientific inquiry. More specifically, we will emphasize how interest for gradient methods was catalyzed by data sharing, open-source software development, as well as the organization of dedicated workshops led by a diverse team of early career researchers. To this end, we argue that the growing excitement for brain gradients is the result of coordinated and consistent efforts to build an inclusive community and can serve as a case in point for future innovations and conceptual advances in neuroinformatics. We close this perspective paper by discussing challenges for the continuous refinement of neuroscientific theory, methodological innovation, and real-world translation to maintain our collective progress towards integrated models of brain organization.

In this paper, we study the relationship between the type problem and the asymptotic behavior of the first eigenvalues $\lambda_1(B_r)$ of ``balls'' $B_r:=\{\rho18.624\cdots. \] Moreover, an upper bound of $\Lambda_*$ in terms of volume growth $\nu_*:=\liminf_{r\rightarrow +\infty} \frac{\log |B_r|}{\log r}$ is given as follows \[ {\Lambda_*} \lesssim \begin{cases} \nu_*^2,\ \ \ &\nu_*\gg1,\\ \nu_*\log\frac{1}{\nu_*},&1<\nu_*\ll1. \end{cases} \] The exponent $2$ for $\nu_*\gg1$ turns out to be the best possible., Comment: 21 pages. Comments welcome!

We call the solution of a kind of second order homogeneous partial differential equation as real kernel alpha-harmonic mappings. In this paper, the representation theorem, the Lipschitz continuity, the univalency and the related problems of the real kernel alpha-harmonic mappings are explored.

In this note, we extend the well-known theorems of M. Riesz and Zygmund on conjugate functions as follows. Let $\Omega$ be a domain in $\mathbb C^n$. Suppose that $f=u+iv\in \mathcal O(\Omega)$ satisfies $v(z_0)=0$ for some $z_0\in \Omega$. Then $ \|f\|_{p,z_0} \le C_p\, \|u\|_{p,z_0}$ for $11$, there exists $C_\alpha>0$ such that $ \int_{\partial \Omega_t} \frac{\exp\left(\frac{\pi}2 |f| \right)}{(1+|f|)^\alpha}\, d\omega_{z_0,t} \le C_\alpha$ for any exhaustion $\{\Omega_t\}$ of $\Omega$ with $\Omega_t\ni z_0$, where $d \omega_{z_0,t}$ is the harmonic measure of $\Omega_t$ relative to $z_0$. Analogous results for Poletsky-Stessin-Hardy spaces on hyperconvex domains are given.

We present an equality between the Bergman metric and Carath\'{e}odry metric.

The application of sodium laser guide star is the key difference between modern adaptive optics system and traditional adaptive optics system. Especially in system like multi-conjugate adaptive optics, sodium laser guide star asterism which is formed by several laser guide stars in certain pattern is required to probe more atmospheric turbulence in different directions. To achieve this, a sodium laser guide star asterism launching platform is required. In this paper, we will introduce the sodium laser guide star asterism launching platform built and tested on the 1.8m telescope of the Gaomeigu Observatory. The platform has two functions: one is to compare the performance of sodium laser guide stars generated by different lasers at the same place; the other is to generate sodium laser guide star asterism with adjustable shape. The field test results at the beginning of 2021 verify the important role of the platform, which is also the first time to realize sodium laser guide star asterism in China.

We show that the $p-$Bergman kernel $K_p(z)$ on a bounded domain $\Omega$ is of locally $C^{1,1}$ for $p\geq1$.The proof is based on the locally Lipschitz continuity of the off-diagonal $p-$Bergman kernel $K_p(\zeta,z)$ for fixed $\zeta\in \Omega$. Global irregularity of $K_p(\zeta,z)$ is presented for some smooth strongly pseudoconvex domains when $p\gg 1$. As an application of the local $C^{1,1}-$regularity, an upper estimate for the Levi form of $\log K_p(z)$ for $1

The $p-$Bergman kernel $K_p(\cdot)$ is shown to be of $C^{1,1/2}$ for $1

We obtain a quantitative estimate of Bergman distance when $\Omega \subset \mathbb{C}^n$ is a bounded domain with log-hyperconvexity index $\alpha_l(\Omega)>\frac{n-1+\sqrt{(n-1)(n+3)}}{2}$, as well as the $A^2(\log A)^q$-integrability of the Bergman kernel $K_{\Omega}(\cdot, w)$ when $\alpha_l(\Omega)>0$.

We prove an $H^2-$Corona theorem with estimate $C(\delta)=C\delta^{-1-q}|\log \delta|$ for $\delta\ll 1$ on delta-regular domains, where $q=\min\{n,m-1\}$ and $m$ is the number of generators. This class of domains includes smooth bounded domains with defining functions that are plurisubharmonic on boundaries and pseudoconvex domains of D'Angelo finite type., Comment: Several mistakes are corrected; Lemma 4.2 is added to explain the density

In this note, we prove an $L^2$ Hartogs-type extension theorem for unbounded domains.

We obtain a psh Hopf lemma for domains satisfying certain cusp conditions by using a sharp estimate for the Green function of a planar cusp along the axis. As an application, we obtain a negative psh exhaustion function with certain global growth estimate on a pseudoconvex domain with H\"{o}lder boundary.

In this paper we attempt to develop a general $p-$Bergman theory on bounded domains in $\mathbb C^n$. To indicate the basic difference between $L^p$ and $L^2$ cases, we show that the $p-$Bergman kernel $K_p(z)$ is not real-analytic on some bounded complete Reinhardt domains when $p\ge 4$ is an even number. By the calculus of variations we get a fundamental reproducing formula. This together with certain techniques from nonlinear analysis of the $p-$Laplacian yield a number of results, e.g., the off-diagonal $p-$Bergman kernel $K_p(z,\cdot)$ is H\"older continuous of order $\frac12$ for $p>1$ and of order $\frac1{2(n+2)}$ for $p=1$. We also show that the $p-$Bergman metric $B_p(z;X)$ tends to the Carath\'eodory metric $C(z;X)$ as $p\rightarrow \infty$ and the generalized Levi form $i\partial\bar{\partial}\log K_p(z;X)$ is no less than $B_p(z;X)^2$ for $p\ge 2$ and $ C(z;X)^2$ for $p\le 2.$ Stability of $K_p(z,w)$ or $B_p(z;X)$ as $p$ varies, boundary behavior of $K_p(z)$, as well as basic facts on the $p-$Bergman prjection, are also investigated., Comment: Final version; a concluding remark is added in section 3, Adv. Math. (2022)

In the past few years, Chinese astronomical community is actively testing astronomical sites for several new optical/infrared ground-based telescopes. These site testing campaigns conducted were mainly focused on fundamental performances of the site, such as cloud coverage, seeing, temperature, etc. With increasing interests in sodium laser guide star adaptive optics for these new telescopes in the Chinese astronomical community, it is interesting to investigate the performance of the laser guide star at these sites, especially considering that the sodium laser guide star's on-sky performance is significantly influenced by sites' local performances, such as geomagnetic field, sodium layer dynamics, density of air molecule, etc. In this paper, we studied sodium laser guide star's performance of a 20W class Quasi-CW pulsed laser developed by TIPC with numerical simulation for five selected sites in China.

Using the logarithmic capacity, we give quantitative estimates of the Green function, as well as lower bounds of the Bergman kernel for bounded pseudoconvex domains in $\mathbb C^n$ and the Bergman distance for bounded planar domains. In particular, it is shown that the Bergman kernel satisfies $K_\Omega(z)\gtrsim \delta_\Omega(z)^{-2}$ for any bounded pseudoconvex domain with $C^0-$boundary. An application to holomorphic motions is given., Comment: Items (a), (c) in Theorem 1.2 and Proposition 1.3 are new

In this article, we investigate the (big) Hankel operators $H_f$ on Hardy spaces of strongly pseudoconvex domains with smooth boundaries in $\mathbb{C}^n$. We also give a necessary and sufficient condition for boundedness of the Hankel operator $H_f$ on the Hardy space of the unit disc, which is new in the setting of one variable., Comment: 23 pages

In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions of completely monotone sequences., Comment: 15 pages