Related papers: Some sharp Wilker type inequalities and their appl…

We find necessary and sufficient conditions for the validity of weighted Rellich and Calderon-Zygmund inequalities in L^p, 1 \leq p \leq \infty, in the whole space and in the half-space with Dirichlet boundary conditions. General operators…

We present several operator versions of the Dunkl--Williams inequality with respect to the $p$-angular distance for operators. More precisely, we show that if $A, B \in \mathbb{B}(\mathscr{H})$ such that $|A|$ and $|B|$ are invertible,…

For a real number $k$, define $\pi_k(x) = \sum_{p\le x} p^k$. When $k>0$, we prove that $$ \pi_k(x) - \pi(x^{k+1}) = \Omega_{\pm}\left(\frac{x^{\frac12+k}}{\log x} \log\log\log x\right) $$ as $x\to\infty$, and we prove a similar result when…

As main result we prove strong convergence theorems of Vilenkin-Fej\'er means when $0<p\leq 1/2.$

Let $p$ be an odd prime and let $x$ be a $p$-adic integer. In this paper, we establish supercongruences for $$ \sum_{k=0}^{p-1}\frac{\binom{x}{k}\binom{x+k}{k}(-4)^k}{(dk+1)\binom{2k}{k}}\pmod{p^2} $$ and $$…

We refine the classical Cauchy--Schwartz inequality $\|X\|_{1} \leq \|X\|_{2}$ by demonstrating that for any $p$ and $q$ with $q>p>2$, there exists a constant $C=C(p,q)$ such that $\|X\|_1 \leq 1 - C \Big{(}\|X\|_p^p -…

In this paper we obtain a Harnack type inequality for solutions to elliptic equations in divergence form with non-standard $p(x)-$type growth. A model equation is the inhomogeneous $p(x)-$laplacian. Namely, \[…

This paper addresses the regularization by sparsity constraints by means of weighted $\ell^p$ penalties for $0\leq p\leq 2$. For $1\leq p\leq 2$ special attention is payed to convergence rates in norm and to source conditions. As main…

This paper continues the investigation begun in arXiv:1906.05602 of extending the T1 theorem of David and Journ\'e, and optimal cancellation conditions, to more general weight pairs. The main additional tool developed here is a two weight…

We prove Harnack's type inequalities for bounded non-negative solutions of degenerate parabolic equations with $(p,q)$ growth $$ u_{t}-{\rm div}\left(\mid \nabla u \mid^{p-2}\nabla u + a(x,t) \mid \nabla u \mid^{q-2}\nabla u \right)=0,\quad…

A new inequality, $(x)^{p}+(1-x)^{\frac{1}{p}}\leq1$ for $p \geq 1$ and $\frac{1}{2} \geq x \geq 0$ is found and proved. The inequality looks elegant as it integrates two number pairs ($x$ and $1-x$, $p$ and $\frac{1}{p}$) whose summation…

In this paper, we prove some supercongruences via the Wilf-Zeilberger method. For instance, for any odd prime $p$ and positive integer $r$ and $\delta\in\{1,2\}$, we have \begin{align*} \sum_{n=0}^{(p^r-1)/\delta}…

In this paper we study the Sobolev inequality in the Dunkl setting using two new approaches which provide a simpler elementary proof of the classical case $p=2$, as well as an extension to the coefficient $p=1$ that was previously unknown.…

This paper studies a class of linear parabolic equations in non-divergence form in which the leading coefficients are measurable and they can be singular or degenerate as a weight belonging to the $A_{1+\frac{1}{n}}$ class of Muckenhoupt…

Motivated by recent results on the (possibly conditional) regularity for time-dependent hypoelliptic equations, we prove a parabolic version of the Poincar\'e inequality, and as a consequence, we deduce a version of the classical Moser…

Inspired by the work of Cossetti and D'Arca [CD25], we show that the general weighted $L^{p}$-Hardy type inequalities [CD25, Theorems 1.1 and 1.2] and the corresponding identities hold for all $1<p<\infty$, thus extending their results…

The paper deals about Hardy-type inequalities associated with the following higher order Poincar\'e inequality: $$ \left( \frac{N-1}{2} \right)^{2(k -l)} := \inf_{ u \in C_{c}^{\infty} \setminus \{0\}} \frac{\int_{\mathbb{H}^{N}}…

We prove an analogous Hanner's Inequality of $L^p$ spaces for positive semidefinite matrices. Let $||X||_p=\text{Tr}[(X^\ast X)^{p/2}]^{1/p}$ denote the $p$-Schatten norm of a matrix $X\in M_{n\times n}(\mathbb{C})$. We show that the…

In this paper, we prove that the double inequality \begin{equation*} 1+\alpha r'^2<\frac{\mathcal{K}_{a}(r)}{\sin(\pi a)\log(e^{R(a)/2}/r')}<1+\beta r'^2 \end{equation*} holds for all $a\in (0, 1/2]$ and $r\in (0, 1)$ if and only if…

In this paper we propose and prove some generalizations and sharpenings of certain inequalities of Wilker;'s and Shafer-Fink's type. Application of the Wu-Debnath theorem enabled us to prove some double sided inequalities.