Related papers: Two results on the Dunkl maximal operator

We study a two-dimensional discrete directional maximal operator along the set of the prime numbers. We show existence of a set of vectors, which are lattice points in a sufficiently large annulus, for which the $\ell^2$ norm of the…

In this paper, we study vector--valued elliptic operators of the form $\mathcal{L}f:=\mathrm{div}(Q\nabla f)-F\cdot\nabla f+\mathrm{div}(Cf)-Vf$ acting on vector-valued functions $f:\mathbb{R}^d\to\mathbb{R}^m$ and involving coupling at…

We calculate the effect of simple Hecke operators on u-expansions of higher rank Drinfeld modular forms, the eigenvalue for the Drinfeld discriminant function $\Delta_t$ and show that a certain natural class of Hecke operators is completely…

For the Dunkl operator $\Lambda_\alpha$ $(\alpha > -1/2)$ on the space of entire functions on the complex space C, the critical rate of growth for the integral means $M_p(f,r)$ of their hypercyclic functions $f$ is obtained. The rate of…

In this work we introduce two different generalizations of the Fokker-Planck equation in (1+1) dimensions by replacing the spatial derivatives in terms of generalized Dunkl-type derivatives involving reflection operators. As applications of…

In this paper, we obtained the Dunkl analogy of classical Lp Hardy inequality for $p > N + 2\gamma$ with sharp constant $\left(\frac{p-N-2\gamma}{p}\right)^{p}$, where $2\gamma$ is the degree of weight function associated with Dunkl…

In this text, we develop the theory of vectorial modular forms with values in Tate algebras introduced by the first author, in a very special case (dimension two, for a very particular representation of {\Gamma} := GL 2 (Fq[$theta$])).…

This is a companion paper to our previous one, Avatars of Stein's Theorem in the complex setting. In this previous paper, we gave a sufficient condition for an integrable function in the upper-half plane to have an integrable Bergman…

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar's constraint qualification holds, which is called the "sum…

Bourgain in his seminal paper [2] about the analysis of maximal functions associated to convex bodies, has estimated in a sharp way the $L^2$-operator norm of the maximal function associated to a kernel $K\in L^1,$ with differentiable…

We derive criteria governing two-weight estimates for multilinear fractional integrals and appropriate maximal functions. The two and one weight problems for multi(sub)linear strong fractional maximal operators are also studied; in…

We prove the boundedness of the maximal operator and Hilbert transform along certain variable parabolas in $L^p$ for $p>p_0$ with some $p_0\in (1, 2)$. Connections with the Hilbert transform along vector fields and the polynomial Carleson's…

Motivated by the recent work of Aubert-Xu and the techniques in G. Muic's article, we provide examples of computations of the Aubert-Zelevinsky duality functor for the principal and mediate series of the exceptional group $G_2$, and deduce…

We give an overview of the generalized Calder\'on-Zygmund theory for "non-integral" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted…

In this note we construct a series of singular vectors in universal affine vertex operator algebras associated to $D_{\ell}^{(1)}$ of levels $n-\ell+1$, for $n \in \Z_{>0}$. For $n=1$, we study the representation theory of the quotient…

The free Maxwell theory in D<>4 dimensions provides a physical example of a unitary, scale invariant theory which is NOT conformally invariant. The easiest way to see this is that the field strength operator F_mn is neither a primary nor a…

We construct a functional model (direct integral expansion) and study the spectra of certain periodic block-operator Jacobi matrices, in particular, of general 2D partial difference operators of the second order. We obtain the upper bound,…

In this paper, we obtain an analogue of the Serre derivation acting on the product of spaces of Drinfeld modular forms which generalizes the differential operator introduced by Gekeler in the rank two case. We further introduce a finitely…

In this paper, we consider the Schr\"odinger operators $L_k=-\Delta_k+V$, where $\Delta_k$ is the Dunkl-Laplace operator and $V$ is a non-negative potential on $R^d$. We establish that $L_k $ is essentially self-adjoint on $C_0^\infty$. In…

Consider a normal Ornstein--Uhlenbeck semigroup in $\Bbb{R}^n$, whose covariance is given by a positive definite matrix. The drift matrix is assumed to have eigenvalues only in the left half-plane. We prove that the associated maximal…