相关论文: On the capacity dimensions of the Brjuno and Perez…

In this work, we prove that the complement of the Brjuno set $\mathcal{B}$ has a zero capacity with respect to the kernel $k^1_\sigma(z,\xi)=\ln^2{|z-\xi|}\left|\ln{\ln{\left(e+\frac{1}{|z-\xi|}\right)}}\right|^\sigma$ for any$\sigma > 2$.…

In this work, we prove that the complement of the Brjuno set in $\mathbb{C}^n$ has zero $C_\sigma$-capacity with respect to the kernel $k_\sigma(z,\xi)=\|z-\xi\|^{-2n+2}|\log{\|z-\xi\||^{\sigma}}$ for any $\sigma>n$. In particular, it…

We consider the Calder\'on-Zygmund kernels $K_ {\alpha,n}(x)=(x_i^{2n-1}/|x|^{2n-1+\alpha})_{i=1}^d$ in $\mathbb{R}^n$ for $0<\alpha\leq 1$ and $n\in\mathbb{N}$. We show that, on the plane, for $0<\alpha<1$, the capacity associated to the…

Analytic capacity is associated with the Cauchy kernel $1/z$ and the $L^\infty$-norm. For $n\in\mathbb{N}$, one has likewise capacities related to the kernels $K_i(x)=x_i^{2n-1}/|x|^{2n}$, $1\le i\le 2$, $x=(x_1,x_2)\in\mathbb{R}^2$. The…

We prove that the Bergman kernel of the symmetrized polydisc in dimension greater than two has zeros.

Answering a long standing question, we give an example of a Hilbert module and a nonzero bounded right linear map having a kernel with trivial orthogonal complement. In particular, this kernel is different from its own double orthogonal…

An effective formula for the Bergman kernel on $\mathbb{H}_{\gamma} = \{|z_1|^\gamma < |z_2| < 1 \}$ is obtained for rational $\gamma = \frac{m}{n} >1$. The formula depends on arithmetic properties of $\gamma$, which uncovers new symmetries…

If $Q$ is a surjection from $L^1(\mu)$, $\mu$ $\sigma$-finite, onto a Banach space containing $c_0$ then (*) $\ker Q$ is uncomplemented in its second dual. If $Q$ is a surjection from an ${\cal L}_1$-space onto a Banach space containing…

We show that, for some Cantor sets in R^d, the capacity g_s associated to the s-dimensional Riesz kernel x/|x|^{s+1} is comparable to the capacity C_{2(d-s)/3,3/2} from non linear potential theory. It is an open problem to show that, when s…

The $p-$Bergman kernel $K_p(\cdot)$ is shown to be of $C^{1,1/2}$ for $1<p<\infty$. An unexpected relation between the off-diagonal $p-$Bergman kernel $K_p(\cdot,z)$ and certain weighted $L^2$ Bergman kernel is given for $1\le p\le 2$. As…

In this paper we give an elementary proof that sets of zero $p,1$-Sobolev-Lorentz capacity are $\mathcal{H}^{n-p}$-null sets independently of non-linear potential theory. We further show that there exists a set of Sobolev-Lorentz-$(p,1)$…

Two results are proved for $\mathrm{nul} \mathbb{P}_A$, the dimension of the kernel of the Pauli operator $\mathbb{P}_A = \bigl\{\bbf{\sigma} \cdotp \bigl(\frac{1}{i} \bbf{\nabla} + \vec{A} \bigr) \bigr\} ^2 $ in $[L^2 (\mathbb{R}^3)]^2$:…

Let L be a holomorphic line bundle over a compact complex projective Hermitian manifold X. Any fixed smooth hermitian metric h on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k th tensor…

Suppose that the compact and connected Lie group G acts holomorphically on the irreducible complex projective manifold M, and that the action linearizes to the Hermitian ample line bundle L on M. Assume that 0 is a regular value of the…

We give a proof that every space of weighted square-integrable holomorphic functions admits an equivalent weight whose Bergman kernel has zeroes. Here the weights are equivalent in the sense that they determine the same space of holomorphic…

We discuss a special class of Beltrami coefficients whose associated quasiconformal mapping is bilipschitz. These are of the form the characteristic function of a planar bounded domain with smooth boundary of class C 1+epsilon times a…

In this paper we study the Bergman kernel and projection on the unbounded worm domain $$ \mathcal{W}_\infty = \big\{(z_1,z_2)\in\mathbb{C}^2 : \big|z_1-e^{i\log|z_2|^2}\big|^2<1, z_2\neq0\big\}. $$ We first show that the Bergman space of…

We calculate Bernstein-Sato b-functions for $f_{G^3}^\lambda$, a $SL_n$-invariant section of a line bundle on $SL_n/B \times SL_n/B \times \mathbb{P}^{n - 1}$ whose zero-set is the complement of the open $G$-diagonal orbit. The proof uses a…

We consider the nonexistence of minimizers for the energy containing a nonlocal perimeter with a general kernel $K$, a Riesz potential, and a background potential in $\mathbb{R}^N$ with $N\geq2$ under the volume constraint. We show that the…

We compute the Szeg\"o kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not…