For the Dirichlet-type space $\mathcal D(\rho^{(1)},\rho^{(2)})$ on the unit bidisc $\mathbb D^2,$ where $\rho^{(1)}$ and $\rho^{(2)}$ are finite positive Borel measures on the closed unit disc $\overline{\mathbb D},$ we show that the L\'evy measure $\nu_{(\mathscr M_z,1)}$ associated with the completely alternating multisequence $\left\{\|z^\alpha\|_{\mathcal D(\rho^{(1)},\rho^{(2)})}^2 \right\}_{\alpha\in\mathbb Z_+^2}$ admits the explicit representation \begin{equation*} d\nu_{(\mathscr M_z,1)}(x)=\frac{1}{1-x_1}d((S_{*}\rho^{(1)})\times\delta_1)(x)+\frac{1}{1-x_2}d(\delta_1\times (S_{*}\rho^{(2)}))(x) \end{equation*} on $[0,1]^2\backslash\{(1,1)\},$ where $S_{*}\rho^{(i)},$ $i=1,2$, denotes the pushforward measure of $\rho^{(i)}$ by the map $S:\overline{\mathbb D}\rightarrow [0,1]$ defined by $S(z)=|z|^2,$ $z\in \overline{\mathbb D},$ and $\delta_1$ denotes the Dirac measure at 1.
Consider the Dirichlet-type spaces on the bidisc defined by $$\mathcal{D}_{\beta}(\mathbb{D}^2)=\Bigg\{f(z_1,z_2)=\sum_{k,l}a_{kl}z_1^kz_2^l\in\mathcal{O}(\mathbb{D}^2): \sum_{k,l\ge 0}|a_{kl}|^2(k+l+1)^{\beta}<+\infty\Bigg\}.$$ Given $\beta_c\in(0,2],$ we construct a function $f$ that belongs to the Dirichlet-type spa...
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on...
We study the Stokes operator with no-slip boundary conditions on the spaces $\mathrm{L}_{\sigma,n}(\Omega)$ and ${\mathrm{L}^1(\Omega,\mathbb{C}^d)}/{\nabla \mathrm{W}^{1,1}(\Omega,\mathbb{C})}$, where $\Omega\subset\mathbb{R}^d$ is an arbitrary bounded $\mathrm{C}^{1,\alpha}$-domain. We show that the Stokes operator o...
We prove the following inequality for Dirichlet polynomials: \[ \int_0^1 |f(1/2+\sigma)|\,d\sigma\lesssim \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^T |f(it)| \, dt. \] In particular, for a Dirichlet series $f(s) = \sum_{n\geq 1} a_n n^{-s}$ belonging to the Hardy space $\mathscr{H}^1$ of Dirichlet series, \[ \left|a_1+\...
We prove that the curvature equation $\sum_{j=1}^n \alpha_j E_j(\tau_{\mathcal{M}})=\sum_{j=1}^n \alpha_j E_j(\tau_{\mathcal{N}})$ yields uniqueness up to translation for any two closed $C^2_+$ hypersurfaces $\mathcal{M}, \mathcal{N}\hookrightarrow\mathbb{R}^{n+1}$ whenever $(\alpha_1,\ldots,\alpha_n)\in\mathbb{R}_{\ge...
We study the log-concavity property of first Dirichlet eigenfunctions on domains in $\mathbb{CP}^2$. For every smooth $1$-convex domain $\Omega\subset\mathbb{CP}^2$, we prove the quantitative estimate \[ \nabla^2(-\log u)>\max\left\{\psi(s),\frac85\right\}g, \text{ where } \psi(|\grad f|^2) = \frac{s}{\sqrt{1+s}}-\log(...
Yu-Sen Xia· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
