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 space $\mathcal D_{2}(\mathbb{D}^2)$ of the bidisk and is cyclic in $\mathcal D_\beta(\mathbb{D}^2)$ if and only if $\beta\leq \beta_{c}.$ We also show that the critical index satisfies $\beta_c=2-\mathrm{dim}_H(\mathcal{Z}(f)\cap \mathbb{T}^2),$ where $\mathrm{dim_H}(\mathcal{Z}(f)\cap \mathbb{T}^2)$ is the Hausdorff dimension of the zero set of the function $f$ on the two-torus $\mathbb{T}^2.$
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 mu...
Let $\mathcal{R}$ be the hyperfinite $\mathrm{II}_1$ factor. Considering the partial sum operators $(\mathcal{S}_n)_{n\geq 1}$ of the noncommutative Vilenkin-Fourier series associated with an arbitrary admissible Vilenkin group, we prove that there exists a universal constant $c>0$ such that \begin{equation*} \sup_{n\g...
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+\...
In this paper, we prove that for two cyclically ordered sequences of complex numbers of unit modulus $(\xi_j)_{j=1}^{n}$ and $(\zeta_j)_{j=1}^{n}$ the inequality: $$\bigg|\sum_{m=1}^{n}(\xi_{m+1}-\xi_m)\zeta_m\bigg|^2+ \bigg|\sum_{m=1}^{n}(\xi_{m+1}-\xi_m)\zeta_m^{-1}\bigg|^2\leqslant 4n^2\sin^2\frac{\pi}{n}$$ holds fo...
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...
Let $d\geq 2$ and $T^*_{\Omega,b}$ be the commutator of the rough maximal singular integral on $\mathbb{R}^d$ defined by $$T^*_{\Omega,b}f(x)=\sup_{\varepsilon>0}\left|\int_{|x-y|>\varepsilon}\bigl(b(x)-b(y)\bigr)\frac{\Omega(x-y)}{|x-y|^d}f(y)\,dy\right|.$$ Assume that $\Omega\in L(\log L)^2(\mathbb{S}^{d-1})$ has mea...
Sheng-Long Lin, Qing-Ying Xue· 0 citations
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