Let $A^2(\mathbb D)$ be the unweighted Bergman space and write $Q_m(S)=T_{B_m(S)}$ for the map induced by the $m$th higher-order Berezin transform. Su\'arez asked in 2005 whether $Q_m(S)$ converges to $S$ in operator norm for every $S$ in the full Bergman Toeplitz algebra. We answer this question negatively in a strong form: there is a compact operator $S$ such that \[ \sup_{m\geq 0}\|Q_m(S)\|=\infty. \] In particular, along a strictly increasing sequence $(m_n)$ one has \[ \|Q_{m_n}(S)-S\|\longrightarrow\infty. \] The obstruction is a moving matrix edge. The proof uses a moving family of rank-one test operators, each supported in the $m$th matrix column. Near the corresponding edge, these test operators are sent to weighted Hankel matrices, and the limiting coefficients form an explicit Pascal kernel. We prove a standalone Pascal--Hankel theorem showing that the associated weighted Hankel transformation fails to map $\ell^2$ boundedly into trace class. Finite-section convergence, trace duality, and the Uniform Boundedness Principle then transfer this instability to the exact Bergman maps.
Let $K$ be a number field and $S$ a finite set of non-archimedean places. Write $\mathcal{O}_S$ for the ring of $S$-integers of $K$ and $\mathcal{O}_S^\times$ for its unit group. Let $\pi : X \rightarrow \mathbb{P}^1$ be a morphism of (irreducible) curves defined over $K$, and denote by $\operatorname{Red}(\pi)$ the se...
We prove the Zygmund conjecture in all parameters: the maximal operator associated with products of cubes with side lengths $(s_1,\ldots,s_{m-1},\phi(s_1,\ldots,s_{m-1}))$, where $\phi$ is any positive coordinatewise nondecreasing function, satisfies the weak $L(\log L)^{m-2}$ estimate for every $m\geq3$. The estimate...
For $m\geq 2$, let $c_p(m)$ be the all-dimensional best constant in $$ \left\|\sum_{k=1}^m A_k\right\|_p \leq c_p(m)\left\|\sum_{k=1}^m |A_k|\right\|_p. $$ Tang and Zhang conjectured an explicit formula for every finite $p>1$. We disprove the conjecture with two explicit real $2\times 2$ rank-one matrices at $p=3/2$. T...
What becomes of Grothendieck's inequality when its Hilbert-space vectors are replaced by vectors in $\ell_p^r$ and $\ell_{p'}^r$? For an $m\times n$ matrix, we show that the universal dimensional loss is governed not by $r$ alone, but by the effective dimension \[ d=\min\{r,m,n\}. \] More precisely, the optimal constan...
In this paper we provide a positive version of Miljutin's theorem. Namely, we show that for any uncountable compact metric spaces $K$ and $L$ there exists a positive isomorphism $T:C(K)\longrightarrow C(L)$. Moreover, the inverse of $T$ is a regular operator, and $T$ may be chosen so that $\|T\|_\mathrm{r} \cdot \|T^{-...
Javier Laguna-Ricart, Gonzalo Martínez-Cervantes, Jakub Rondoš et al.· 0 citations
For $P=X_1^2+\cdots + X_n^2$, let $T_t^Pf(x)$ denote the solution to the linear Schr\"{o}dinger equation at time $t$. In 1980, Carleson asked for the minimal regularity of an initial data function $f\in H^s(\mathbb{R}^n)$ that guarantees pointwise convergence of $T_t^Pf(x)$ to $f(x)$ as $t\rightarrow 0$. This was resol...
Re-Na Chu· 0 citations
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