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The classical Kramer sampling theorem establishes general conditions that allow the reconstruction of functions by mean of orthogonal sampling formulae. One major task in sampling theory is to find concrete, non trivial realizations of this…

We entirely compute the cohomology for a natural and large class of $\mathfrak{osp}(1|2)$ modules $M$. We study the restriction to the $\mathfrak{sl}(2)$ cohomology of $M$ and apply our results to the module $M={\mathfrak D}_{\lambda,\mu}$…

Applying methods of Real Analysis and Functional Analysis, we build two weight functions with parameters and provide two kinds of parameterized Yang-Hilbert-type integral inequalities with the best constant factors. Equivalent forms, the…

We consider various inequalities for polynomials, with an emphasis on the most fundamental inequalities of approximation theory. In the sequel a key role is played by the generalized Minkowski functional \alpha(K,x), already being used by…

We introduce affine Stanley symmetric functions for the special orthogonal groups, a class of symmetric functions that model the cohomology of the affine Grassmannian, continuing the work of Lam and Lam, Schilling, and Shimozono on the…

The inverse problem which arises in the Camassa--Holm equation is revisited for the class of discrete densities. The method of solution relies on the use of orthogonal polynomials. The explicit formulas are obtained directly from the…

Let $T$ be a linear operator that, for some $p_1\in(1,\infty)$, is bounded on $L^{p_1}(\tilde w)$ for all $\tilde w\in A_{p_1}(\mathbb R^d)$ and in addition compact on $L^{p_1}(w_1)$ for some $w_1\in A_{p_1}(\mathbb R^d)$. Then $T$ is…

In this paper, we establish various inequalities for some mappings that are linked with the illustrious Hermite-Hadamard integral inequality for mappings whose absolute values belong to the class K?;s m;1 and K?;s m;2.

We establish some weighted $L^2$ inequalities for Fourier extension operators in the setting of orthonormal systems. In the process we develop a direct approach to such inequalities based on generalised Wigner distributions, complementing…

We begin by deriving an action of the 0-Hecke algebra on standard reverse composition tableaux and use it to discover 0-Hecke modules whose quasisymmetric characteristics are the natural refinements of Schur functions known as…

We consider Lorentz-Karamata spaces, small and grand Lorentz-Karamata spaces, and the so-called $\mathcal{L}$, $\mathcal{R}$, $\mathcal{LL}$, $\mathcal{RL}$, $\mathcal{RL}$, and $\mathcal{RR}$ spaces. The quasi-norms for a function $f$ in…

We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights $w$ having finitely many zeros and singularities (i.e., points where $w$ becomes infinite) on an interval and not too ``rapidly…

We improve on several weighted inequalities of recent interest by replacing a part of the A_p bounds by weaker A_\infty estimates involving Wilson's A_\infty constant \[ [w]_{A_\infty}':=\sup_Q\frac{1}{w(Q)}\int_Q M(w\chi_Q). \] In…

This article explores weighted $(L^p, L^q)$ inequalities for the Fourier transform in rank one Riemannian symmetric spaces of noncompact type. We establish both necessary and sufficient conditions for these inequalities to hold. To prove…

We provide quantitative weighted weak type estimates for non-integral square functions in the critical case $p=2$ in terms of the $A_p$ and reverse H\"older constants associated to the weight. The method of proof uses a decoupling of the…

We obtain weighted $L^p$ inequalities for pseudo-differential operators with smooth symbols and their commutators by using a class of new weight functions which include Muckenhoupt weight functions. Our results improve essentially some…

We consider the problem of uniform interpolation of functions with values in a complex inner product space of finite dimension. This problem can be casted within a modified weighted pluripotential theoretic framework. Indeed, in the…

We prove fractional Leibniz rules and related commutator estimates in the settings of weighted and variable Lebesgue spaces. Our main tools are uniform weighted estimates for sequences of square-function-type operators and a bilinear…

Let $\mathbb{R}=(-\infty,\infty)$, and let $Q\in C^1(\mathbb{R}): \mathbb{R}\rightarrow \mathbb{R^+}=[0,\infty)$ be an even function, which is an exponent. We consider the weight $w_\rho(x)=|x|^{\rho} e^{-Q(x)}$, $\rho\geqslant 0$, $x\in…

We establish weighted inequalities for $BMO$ commutators of sublinear operators for all $0<p<\infty$. For weights $w$ satisfying the doubling condition of order $q$ with $0<q<p$ and the reverse H\"{o}lder condition, we prove that $\bullet$…