Related papers: From dyadic $\Lambda_{\alpha}$ to $\Lambda_{\alpha…

Let $\vec{a}:=(a_1,\ldots,a_n)\in[1,\infty)^n$, $\vec{p}:=(p_1,\ldots,p_n)\in(0,\infty)^n$ and $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ be the anisotropic mixed-norm Hardy space associated with $\vec{a}$ defined via the non-tangential grand…

The one-particle density matrix $\gamma(x, y)$ for a bound state of an atom or molecule is one of the key objects in the quantum-mechanical approximation schemes. We prove the asymptotic formula $\lambda_k \sim (Ak)^{-8/3}$, $A \ge 0$, as…

We introduce the Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ for Fourier integral operators for $0<p<1$, thereby extending earlier constructions for $1\leq p\leq \infty$. We then establish various properties of these spaces,…

This document presents an elementary approach using $\varepsilon$-Poincar\'e inequality to prove generation of $L^{p}$-bounds, $p\in(1,\infty)$, for the homogeneous Landau equation with moderate soft potentials $\gamma\in[-2,0)$. The…

In this paper we introduce several quantitative methods for the lambda-calculus based on partial metrics, a well-studied variant of standard metric spaces that have been used to metrize non-Hausdorff topologies, like those arising from…

We prove $L^p$-Hardy inequalities with distance to the boundary for domains in the Heisenberg group ${\mathbb{H}}^n$, $n\geq 1$. Our results are based on a certain geometric condition. This is first implemented for the Euclidean distance in…

For $p$ prime, $A \subseteq \mathbb{Z}/p\mathbb{Z}$ and $\lambda \in \mathbb{Z}$, the sum of dilates $A + \lambda \cdot A$ is defined by \[A + \lambda \cdot A = \{a + \lambda a' : a, a' \in A\}.\] The basic problem on such sums of dilates…

We develop a new algorithm to compute a basis for $M_k(\Gamma_0(N))$, the space of weight $k$ holomorphic modular forms on $\Gamma_0(N)$, in the case when the graded algebra of modular forms over $\Gamma_0(N)$ is generated at weight two.…

We get the sharp bound for weak type $(1,1)$ inequality for $n$-dimensional Hardy operator. Moreover, the precise norms of generalized Hardy operators on the type of Campanato spaces are obtained. As applications, the corresponding norms of…

We study a class of inhomogeneous parabolic equations on the Heisenberg group $\mathbbm{H}^N$ with Hardy-type singular potentials, nonlocal memory terms, and a space-time forcing term: \begin{align} \partial_tu-\Delta_{H}u=\lambda…

We give a careful proof that a parallelized version of adiabatic quantum computation can efficiently simulate universal gate model quantum computation. The proof specifies an explicit parameter-dependent Hamiltonian $H({\lambda})$ that is…

We obtain optimal generalized versions of Hardy inequalities, which as special cases contain Hardy's inequality and Hardy's inequality involving the distance function to the boundary of $ \Omega$. In addition we obtain neccesary and…

The aim of this paper is to propose an abstract construction of spaces which keep the main properties of the (already known) Hardy spaces H^1. We construct spaces through an atomic (or molecular) decomposition. We prove some results about…

Mordechay Levin has constructed a number $\alpha$ which is normal in base 2, and such that the sequence $\left\{2^n \alpha\right\}_{n=0,1,2,\ldots}$ has very small discrepancy $D_N$. Indeed we have $N\cdot D_N = \mathcal{O} \left(\left(\log…

Let $\mathcal{U(\alpha, \lambda)}$, $0<\alpha <1$, $0 < \lambda <1$ be the class of functions $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ satisfying $$\left|\left(\frac{z}{f(z)}\right)^{1+\alpha}f'(z)-1\right|<\lambda$$ in the unit disc ${\mathbb…

In this paper we study the average $\NL_{2\alpha}$-norm over $T$-polynomials, where $\alpha$ is a positive integer. More precisely, we present an explicit formula for the average $\NL_{2\alpha}$-norm over all the polynomials of degree…

In this note, we completely characterize complex symmetric weighted composition differentiation operator on the Hardy space $H^2$ with respect to the conjugation operator $C_{\lambda,\alpha}$. Meanwhile, the normal and self-adjoint of the…

We prove a generalized Hardy-Littlewood lemma on a non-smooth domain in "$f$-norm" and give an application to a corresponding estimate for the $\dib$-Neumann problem by means of suitable weights.

We derive, via the Hardy-Littlewood method, an asymptotic formula for the number of integral zeros of a particular class of weighted quartic forms under the assumption of non-singular local solubility. Our polynomials $F({\mathbf…

Fix a space dimension $d\ge 2$, parameters $\alpha > -1$ and $\beta \ge 1$, and let $\gamma_{d,\alpha, \beta}$ be the probability measure of an isotropic random vector in $\mathbb{R}^d$ with density proportional to \begin{align*}…