Related papers: On Stein's Conjecture on the Polynomial Carleson O…

We use explicit class field theory of rational function fields to prove a dynamical criterion for a polynomial of the form $x^{p^r}+ax+b$ over a field of characteristic $p$ to have dynamical Galois group as large as possible. When $p=2$ and…

We characterize the Carleson measures $\mu$ on the unit disk for which the image of the Hardy space $H^p$ under the corresponding embedding operator is closed in $L^p(\mu)$. In fact, a more general result involving $(p,q)$-Carleson measures…

Given a Calder\'{o}n--Zygmund (C--Z for short) operator $T$, which satisfies H\"ormander condition, we prove that: if $T$ maps all the characteristic atoms to $WL^{1}$, then $T$ is continuous from $L^{p}$ to $L^{p}(1<p<\infty)$. So the…

Let $L$ be a Schr\"odinger operator of the form $L=-\Delta+V$ acting on $L^2(\mathbb R^n)$ where the nonnegative potential $V$ belongs to the reverse H\"older class $B_q$ for some $q\geq n.$ In this article we will show that a function…

We impose standard $ T1 $-type assumptions on a Calder\'on-Zygmund operator $ T $, and deduce that for bounded compactly supported functions $ f, g $ there is a sparse bilinear form $ \Lambda $ so that $$ \lvert \langle T f, g \rangle\rvert…

We study a class of Schr\"{o}dinger-Poisson systems with $(p,q)$-Laplacian. Using fixed point theory, we obtain a new existence result for nontrivial solutions. The main novelty of the paper is the combination of a double phase operator and…

For a Calderon-Zygmund operator T on d-dimensional space, that has a sufficiently smooth kernel, we prove that for any 1< p \le 2, and weight w in A_p, that the maximal truncations T_* of T map L^p(w) to weak-L^p(w), with norm bounded by…

Let $p$ be an odd prime, and let $\sum_{n=0}^{\infty} a_{n}X^{n}\in\mathbb{F}_p[[X]]$ be the reduction modulo $p$ of the Artin-Hasse exponential. We obtain a polynomial expression for $a_{kp}$ in terms of those $a_{rp}$ with $r<k$, for even…

Given a piecewise linear (PL) function $p$ defined on an open subset of $\R^n$, one may construct by elementary means a unique polyhedron with multiplicities $\D(p)$ in the cotangent bundle $\R^n\times \R^{n*}$ representing the graph of the…

Let $\tau_\nu$ ($\nu \in \mathbb{Z}$) be a character of $K=S(U(n)\times U(n))$, and $SU(n,n)\times_K\mathbb{C}$ the associated homogeneous line bundle over $\mathcal{D}=\{Z\in M(n,\mathbb{C}): I-ZZ^* > 0\}$. Let $\mathcal{H}_\nu$ be the Hua…

We prove a formula for a characteristic polynomial of an operator expressed as a polynomial of rank 1 operators. The formula uses a discrete analog of path integration and implies a generalization of the Forman-Kenyon's formula [4,6] for a…

A convex-polynomial is a convex combination of the monomials $\{1, x, x^2, \ldots\}$. This paper establishes that the convex-polynomials on $\mathbb R$ are dense in $L^p(\mu)$ and weak$^*$ dense in $L^\infty(\mu)$, precisely when…

We prove a compact $T(1)$ theorem, involving quantitative estimates, analogous to the quantitative classical $T(1)$ theorem due to Stein. We also discuss the $C_c^\infty$-to-$CMO$ mapping properties of non-compact Calder\'on-Zygmund…

In this paper, we investigate the boundedness of bilinear Calder\'on-Zygmund operators $T$ from ${L^{p_1}\left(w_1\right)} \times {L^{p_2}\left(w_2\right)}$ to ${L^{p,\infty}\left(v_{\vec{w}}\right)}$ with the stopping time method, where $1…

The celebrated union-closed conjecture is concerned with the cardinalities of various subsets of the Boolean $d$-cube. The cardinality of such a set is equivalent, up to a constant, to its measure under the uniform distribution, so we can…

We show that for a Steinhaus random multiplicative function $f:\mathbb{N}\to\mathbb{D}$ and any polynomial $P(x)\in\mathbb{Z}[x]$ of $\text{deg}\ P\ge 2$ which is not of the form $w(x+c)^{d}$ for some $w\in \mathbb{Z}$, $c\in \mathbb{Q}$,…

Given a closed, convex cone $K\subseteq \mathbb{R}^n$, a multivariate polynomial $f\in\mathbb{C}[\mathbf{z}]$ is called $K$-stable if the imaginary parts of its roots are not contained in the relative interior of $K$. If $K$ is the…

A theorem of A. and C. R\'enyi on periodic entire functions states that an entire function $f(z) $ must be periodic if $ P(f(z)) $ is periodic, where $ P(z) $ is a non-constant polynomial. By extending this theorem, we can answer some open…

For $d \geq 2, \ D \geq 1$, let $\mathscr{P}_{d,D}$ denote the set of all degree $d$ polynomials in $D$ dimensions with real coefficients without linear terms. We prove that for any Calder\'{o}n-Zygmund kernel, $K$, the maximally modulated…

Let $c_{kl} \in W^{2,\infty}(\mathbb{R}^d, \mathbb{C})$ for all $k,l \in \{1, \ldots, d\}$. We consider the divergence form operator $A = - \sum_{k,l=1}^d \partial_l (c_{kl} \, \partial_k) $in $L_2(\mathbb{R}^d)$ when the coefficient matrix…