We prove Kahane's conjecture: if a continuous unimodular function $F$ on the circle satisfies $\|F^n\|_{A(\mathbb{T})}=o(\log|n|)$ as $|n|\to\infty$, $n\in\mathbb{Z}$, then $F(x)=e^{i(mx+b)}$ for some $m\in\mathbb{Z}$ and $b\in\mathbb{R}$. The main step is a logarithmic lower bound for averages: for every nonconstant real-valued $\varphi\in A(\mathbb{T})$, $\frac{1}{T}\int_a^{a+T}\|e^{it\varphi}\|_{A(\mathbb{T})}dt \ge c_\varphi\log T \quad(a\in\mathbb{R},\ T\ge T_\varphi)$, where $c_\varphi>0$ and $T_\varphi>1$ are independent of $a$. The proof uses exact sampling on finite subgroups of the circle, uniformly bounded localized modulations, and the weighted non-harmonic Littlewood inequality of Jaming--Kellay--Saba. A suitable cutoff and a small linear perturbation of the phase ensure that a fixed positive proportion of the coefficients at the beginning of the frequency ordering have modulus one. Passing to integer powers and removing the degree of the circle map yield Kahane's conjecture without any additional smoothness assumptions. We also give equivalent phase, convolution, and operator formulations, and derive consequences for homomorphisms of weighted Wiener algebras and for higher-dimensional tori.
We study the near $L^1$ behavior of the maximally quadratically modulated Hilbert transform \[ \mathcal{C}_2f(x) := \sup_{\lambda} \left|\operatorname{p.v.} \int_{\mathbb{R}}f(x-y)e^{2 \pi i \lambda y^2} \frac{\mathrm{d} y }{y} \right| \] and its lacunary counterpart obtained by restricting $\lambda$ to $2^{\mathbb{Z}}...
A. Fragkos, Ben Krause, Michael T. Lacey· 0 citations
Let $w \colon (0,1) \to \mathbb{R}_{+}$ be a weight. We prove that for an arbitrary Schauder basis $\{e^{i \lambda_n t}\}_{n \in \mathbb{Z}}$ in $L^2(0,1)$ and an arbitrary lacunary defect set $A \subset \mathbb{Z}$ the system $\{w(t) r_n(t)\}_{n \in \mathbb{Z} \setminus A}$ is always complete and never minimal for any...
Let $A\in\mathbb{C}^{n\times n}$ be positive definite Hermitian, where $n\ge 2$, and define $\|X\|_A^2=\mathrm{tr}(AX^*X)$ for all $X\in\mathbb{C}^{n\times n}$. We determine the smallest constant $C_A$ for which $$ \|XY-YX\|_{\mathrm{F}}^2 \le C_A\|X\|_A^2\|Y\|_A^2 $$ holds for all rank-one $X,Y\in\mathbb{C}^{n\times n...
Let $f:\mathbb{N}\to{z\in\mathbb{C}:|z|\leq1}$ be multiplicative, and suppose that $\alpha$ has a rational approximation $a/q$ with $|\alpha-a/q|\leq 1/(qx)$ and $q\leq x/((\log x)^2(\log\log x)^3)$. We prove that $\left|\sum_{n\leq x}f(n)e(n\alpha)\right|\leq \left(\frac4{\sqrt6}+o(1)\right)\frac{x}{\sqrt q} +O\left(\...
Let $m$ denote the normalized Haar measure on the unit circle $\mathbb{T}$. For $0<r<\infty$, define $\lVert f\rVert_{r}:=\left(\int_{\mathbb{T}}|f|^{r}\,\mathrm{d} m\right)^{1/r}$, with $\lVert f\rVert_{0}$ and $\lVert f\rVert_{\infty}$ interpreted as the geometric mean and the supremum norm, respectively. Set $Q_n(z)...
Let $\mathcal R$ be a normalized root system in $\mathbb R^N$ with a nonnegative multiplicity function $k$, and let $\mathcal F$ be the associated Dunkl transform. We prove a H\"ormander multiplier theorem on the Hardy spaces $H^p_{\mathrm{Dunkl}}$, $0<p\le1$, defined by conical Littlewood--Paley square functions. Let...
Jacek Dziubański, Agnieszka Hejna-Łyżwa· 0 citations
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