Related papers: Sparse bounds on variational norms along monomial …

In this paper, we prove a problem proposed by Bre\v{s}ar: for any graphs $G$ and $H$, $\Gamma(G\square H)\ge\Gamma(G)\Gamma(H)+ \min\{|V(G)|-\Gamma(G),|V(H)|-\Gamma(H)\}$, where $\Gamma(G)$ denotes the upper domination number of $G$.

We exhibit a class of "relatively curved" $\vec{\gamma}(t) := (\gamma_1(t),\dots,\gamma_n(t))$, so that the pertaining multi-linear maximal function satisfies the sharp range of H\"{o}lder exponents, \[ \left\| \sup_{r > 0} \ \frac{1}{r}…

This paper refines the main results from our previous study on sparse bounds of generalized commutators of multilinear fractional singular integral operators in \cite{CenSong2412}. The key improvements are: 1. We replace pointwise…

We establish a uniform domination of the family of trilinear multiplier forms with singularity over a one-dimensional subspace by positive sparse forms involving $L^p$-averages. This class includes the adjoint forms to the bilinear Hilbert…

Consider a $p$-dimensional population ${\mathbf x} \in\mathbb{R}^p$ with iid coordinates in the domain of attraction of a stable distribution with index $\alpha\in (0,2)$. Since the variance of ${\mathbf x}$ is infinite, the sample…

Let $ T_{P } f (x) = \int e ^{i P (y)} K (y) f (x-y) \; dy $, where $ K (y)$ is a smooth Calder\'on-Zygmund kernel on $ \mathbb R ^{n}$, and $ P$ be a polynomial. We show that there is a sparse bound for the bilinear form $ \langle T_P f, g…

We study sparse domination for operators defined with respect to an atomic filtration on a space equipped with a general measure $\mu$. In the case of Haar shifts, $L^p$-boundedness is known to require a weak regularity condition, which we…

Let ${\mathcal F}$ be a germ of holomorphic foliation defined in a neighborhood of the origin of ${\mathbb C}^{2}$ that has a germ of irreducible holomorphic invariant curve $\gamma$. We provide a lower bound for the vanishing multiplicity…

We consider sparse inhomogeneous Erd\H{o}s-R\'enyi random graph ensembles where edges are connected independently with probability $p_{ij}$. We assume that $p_{ij}= \varepsilon_N f(w_i, w_j)$ where $(w_i)_{i\ge 1}$ is a sequence of…

For polynomial $ P (x,y)$, and any Calder\'{o}n-Zygmund kernel, $K$, the operator below satisfies a $ (1,r)$ sparse bound, for $ 1< r \leq 2$. $$ \sup _{\epsilon >0} \Bigl\lvert \int_{|y| > \epsilon} f (x-y) e ^{2 \pi i P (x,y) } K(y) \; dy…

Inspired by the potential of improving tractability via gap- or above-guarantee parametrisations, we investigate the complexity of Dominating Set when given a suitable lower-bound witness. Concretely, we consider being provided with a…

Let n denote the number of variables and m the number of equations in a sparse polynomial system over the binary field. We study the inconsistency probability of randomly generated sparse polynomial systems over the binary field, where each…

In this paper, we study the behaviour of the generalized power domination number of a graph by small changes on the graph, namely edge and vertex deletion and edge contraction. We prove optimal bounds for $\gamma\_{p,k}(G-e)$,…

We study an inverse problem on a finite connected graph G = (X, E), on whose vertices a conductivity {\gamma} is defined. Our data consists in a sequence of partial observations of a fractional random walk on G. The observations are partial…

In dimension $d\geq 3$, a variational principle for the size of the pure point spectrum of (discrete) Schr\"odinger operators $H(\mathfrak{e},V)$ on the hypercubic lattice $\mathbb{Z}^{d}$, with dispersion relation $\mathfrak{e}$ and…

In this note, we show that if $T$ is a multilinear singular integral operator associated with a kernel satisfies the so-called multilinear $L^{r}$-H\"ormander condition, then $T$ can be dominated by multilinear sparse operators.

For the ``Hopf bundle'' $S^1\to S^{2n,1} \to {\mathbb H}^n$, horizontal lifts of simple closed curves are studied. Let $\gamma$ be a piecewise smooth, simple closed curve on a complete totally geodesic surface $S$ in the base space. Then…

Let $\gamma(G)$ and $\gamma_t(G)$ denote the domination number and the total domination number, respectively, of a graph $G$ with no isolated vertices. It is well-known that $\gamma_t(G) \leq 2\gamma(G)$. We provide a characterization of a…

In this paper, for general plane curves $\gamma$ satisfying some suitable smoothness and curvature conditions, we obtain the single annulus $L^p(\mathbb{R}^2)$-boundedness of the Hilbert transforms $H^\infty_{U,\gamma}$ along the variable…

We give a convergence proof for the approximation by sparse collocation of Hilbert-space-valued functions depending on countably many Gaussian random variables. Such functions appear as solutions of elliptic PDEs with lognormal diffusion…