Related papers: Discrete Fractional Integration Operators Along th…

Cilleruelo conjectured that if $f\in\mathbb{Z}[x]$ is an irreducible polynomial of degree $d\ge 2$ then, $\log \operatorname{lcm} \{f(n)\mid n<x\} \sim (d-1)x\log x.$ In this article, we investigate the analogue of prime arguments, namely,…

Let p be a prime = 3 (mod 4). A number of elegant number-theoretical properties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2\pi/p) and C(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} cot(n^2\pi/p) are proved. For example, T(p) equals p times…

Let $\mathcal K$ be a flag kernel on a homogeneous nilpotent Lie group $G$. We prove that operators $T$ of the form $T(f)= f*\mathcal K$ form an algebra under composition, and that such operators are bounded on $L^{p}(G)$ for $1<p<\infty$.

We introduce a notion of fractional (noninteger order) derivative on an arbitrary nonempty closed subset of the real numbers (on a time scale). Main properties of the new operator are proved and several illustrative examples given.

We construct a Schwartz function $\varphi$ such that for every exponentially small perturbation of integers $\Lambda$, the set of translates $\{\varphi(t-\lambda), \lambda\in\Lambda\}$ spans the space $L^p(R)$, for every $p > 1$. This…

If $f\in \{f\in L^p(\mathbb{R}): f(x)=\int_{-\pi}^{\pi}e^{ix\xi}d\beta(\xi), \beta\in B.V.([-\pi,\pi]) \}$, then $f$ is determined by its samples on the integers by taking an appropriate limit. Specifically, $\| f - L_{\phi_\alpha}f…

For a proper function $f$ on the plane, we study the operator \[ Tf(x,y) = \lim_{\varepsilon\to 0} \int_\varepsilon^1 f(x-t,y-t^k) \frac{e^{2\pi i \gamma(t)}}{\psi(t)} dt, \] where $k\ge1$ and $\psi$ and $\gamma$ are functions defined near…

We establish pointwise ergodic theorems for operators of Radon type along subsets of prime numbers of the form $\big\{\{ \varphi_1(n)\} < \psi(n)\big\}$. We achieve this by proving $\ell^p(\mathbb{Z})$ boundedness of $r$-variations, where…

In this paper, we study the $L^p$ boundedness of a class of oscillating multiplier operator for the Dunkl transform, $T_{m_\alpha}=\mathcal{F}_k^{-1}(m_{\alpha}\mathcal{F}_k(f))$ with $m(\xi)=|\xi|^{-\alpha}e^{\pm i|\xi|}\phi(\xi)$. We…

We determine exactly when two classes of integral operators are bounded on weighted $L^p$ spaces over the Siegel upper half-space.

We introduce sufficient conditions on discrete singular integral operators for their maximal truncations to satisfy a sparse bound. The latter imply a range of quantitative weighted inequalities, which are new. As an application, we prove…

We consider Hadamard fractional derivatives and integrals of variable fractional order. A new type of fractional operator, which we call the Hadamard-Marchaud fractional derivative, is also considered. The objective is to represent these…

Let $f$ be a normalized, ordinary newform of weight $\ge 2$. For each prime $\mathfrak{p}$ of $F=\mathbb{Q}(a_n)_{n\in \mathbb{N}}$, there is an associated $\mathfrak{p}$-adic $L$-function $\mathcal{L}_\mathfrak{p}(f)\in \Lambda \otimes…

We prove maximum and comparison principles for fractional discrete derivatives in the integers. Regularity results when the space is a mesh of length $h$, and approximation theorems to the continuous fractional derivatives are shown. When…

We study the structure of strictly singular non-compact operators between $L_p$ spaces. Answering a question raised in [Adv. Math. 316 (2017), 667-690], it is shown that there exist operators $T$, for which the set of points…

We describe the idempotent Fourier multipliers that act contractively on $H^p$ spaces of the $d$-dimensional torus $\mathbb{T}^d$ for $d\geq 1$ and $1\leq p \leq \infty$. When $p$ is not an even integer, such multipliers are just…

In this paper, we study the $\ell^p$-maximal regularity for the fractional difference equation with finite delay: \begin{equation*} \ \ \ \ \ \ \ \ \left\{\begin{array}{cc} \Delta^{\alpha}u(n)=Au(n)+\gamma u(n-\lambda)+f(n), \ n\in \mathbb…

We consider the problem of sums of dilates in groups of prime order. We show that given $A\subset \Z{p}$ of sufficiently small density then $$\big| \lambda_{1}A+\lambda_{2}A+...+ \lambda_{k}A \big|…

We study a family of convolution operators whose symbols and kernels have singularity on the light-cone in $\mathbb{R}^{n+1}$. First, we prove a desired ${\bf L}^p\to {\bf L}^q$ norm inequality which has been left open. Moreover, we obtain…

We define fractional derivatives $\pppa$ in Sobolev spaces based on $L^p(0,T)$ by an operator theory, and characterize the domain of $\pppa$ in subspaces of the Sobolev-Slobodecki spaces $W^{\alpha,p}(0,T)$. Moreover we define $\pppa u$ for…