We establish necessary and sufficient conditions for the uniform boundedness of the multi-parameter exponential sums with product Hilbert kernels $$\sum_{1\le|t_1|\le N_1,\cdots,1\le|t_k|\le N_k} \frac{e^{2\pi i P(t_1,\dots,t_k)}}{t_1\cdots t_k},$$ where $P:\mathbb{Z}^k\to\mathbb{R}$ is a polynomial of the form $P(t)=\sum_{\mathfrak{m}\in \Lambda} c_{\mathfrak{m}}\, t^{\mathfrak{m}},$ with real coefficients. The resulting bound is uniform in both the coefficients $c_{\mathfrak m}$ and the truncation parameters $N_1,\ldots,N_k$. To this end, we develop a higher-dimensional version of the multi-parameter circle method. Under the sufficient condition, we further prove $\ell^p$-boundedness of the associated discrete multiple Hilbert transform.
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 compute the large-$N$ asymptotics of the determinant of the identity plus or minus a generalized Hilbert matrix. For $N \in \mathbb{N}$ and $\delta \in \mathbb{R} \setminus \{-1, -2, \ldots\}$, let \[ H_N^\delta := \left( \frac{\sin((1+\delta)\pi)}{\pi(j+k+1+\delta)} \right)_{j,k=0}^{N-1} \] be the generalized Hilbe...
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
We examine the generalized Hilbert (matrix) operators $$ H_{g,\gamma} : (a_n) \mapsto \sum_{n=1}^{\infty} \bigg(\frac{k}{n}\bigg)^{\gamma} \frac{g_k a_n}{n+k} $$ on the $\ell^p$ spaces, $1<p<\infty$, where $g=(g_n)$ is a sequence and $-1/p<\gamma<1-1/p$. Given a partition of the natural numbers $\bigcup_j I_j = \mathbb...
Let $f$ be multiplicative, with $|f(p)|\le A$ at primes and $\sum_{n\le x}|f(n)|^2\le A^2x$ for every $x\ge1$. If $|\alpha-a/q|\le q^{-2}$, $(a,q)=1$, and $3\le R\le q\le N/R$, we prove \[ \sum_{n\le N}f(n)\operatorname{e}(n\alpha) \ll_A \frac{N}{\log N} +\frac{N}{\sqrt R}\sqrt{\log\log(3R)} \] with effective implied c...
For integers $h,k\geq 1$, let $hA$ be the $h$-fold sumset of $A$ and put $\mathcal{R}(h,k)=\{|hA|:A\subset\mathbb{Z}, |A|=k\}$. Previously, the fixed-cardinality exponent law was known only for $k\leq 3$; every fixed $k\geq 4$ remained open. We settle the problem in full by determining the complete fixed-cardinality ex...
Henry Shin· 2 citations· ⚡2
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