Related papers: Representation of singular integrals by dyadic ope…
In this paper the W-algebra W(2,2) and its representation theory are studied. It is proved that a simple vertex operator algebra generated by two weight 2 vectors is either a vertex operator algebra associated to a highest irreducible…
We study a family of strong fractional integral operators whose kernels have singularity on every coordinate subspace. We prove a two-weight $L^p$-$L^q$-norm inequality by allowing only one of the weights to satisfy $A_p\times…
We start with considering rank one self-adjoint perturbations $A_\alpha = A+\alpha(\,\cdot\,,\varphi)\varphi$ with cyclic vector $\varphi\in \mathcal{H}$ on a separable Hilbert space $\mathcal H$. The spectral representation of the…
We prove sharp $L^p(w)$ norm inequalities for the intrinsic square function (introduced recently by M. Wilson) in terms of the $A_p$ characteristic of $w$ for all $1<p<\infty$. This implies the same sharp inequalities for the classical…
This paper is an attempt to solve an important class of hypersingular integral equations of the second kind. To this end, we apply a new weighted and modified perturbation method which includes some special cases of the Adomian…
For the deterministic dyadic model of turbulence, there are examples of initial conditions in $l^2$ which have more than one solution. The aim of this paper is to prove that uniqueness, for all $l^2$-initial conditions, is restored when a…
We prove $L^{2}$ estimates and solvability for a variety of simply characteristic constant coefficient partial differential equations $P(D)u=f$. These estimates \[||u||_{L^2(D_{r})}\le C\sqrt{d_{r}d_{s}} ||f||_{_{L^2(D_{s})}}\] depend on…
We prove weighted $q$-variation inequalities with $2<q<\infty$ for differential and singular integral operators in higher dimensions. The vector-valued extensions of these inequalities are also given.
The work is devoted to the study of Laplace operator when the potential is a singular generalized function and plays the role of a singular perturbation of a Laplace operator. Abstract theorem obtained earlier by the authors B.N. Biyarov…
We study a class of parabolic equations in non-divergence form with measurable coefficients that exhibit singular and/or degenerate behavior governed by weights in the $A_{1+\frac{1}{n}}$-Muckenhoupt class. Under a smallness assumption on a…
Given a suitable ordering of the positive root system associated with a semisimple Lie algebra, there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra action on a Verma module…
We will present versions of the Rellich-Kondrachov theorem for pseudo-differential operators acting on localizable Hardy spaces. One of the techniques includes boundedness properties for pseudodifferential operators with symbols in the…
We give a short summary of Varopoulos' generalised Hardy-Littlewood-Sobolev inequality for self-adjoint $C_{0}$ semigroups and give a new probabilistic representation of the classical fractional integral operators on $\R^n$ as projections…
This work is motivated to study the representation theory of the non-semisimple deformed Fomin-Kirillov algebras $\mathcal{D}_4(\alpha_1, \alpha_2)$. In particular, we consider Gabriel's theorem applications in regard of constructing…
Let ${\mathsf D}$ and ${\mathsf H}$ be the self-adjoint, one-dimensional Dirac and Schr\"odinger operators in $L^{2}(\mathbb{R};\mathbb{C}^{2})$ and $L^{2}(\mathbb{R};\mathbb{C})$ respectively. It is well known that, in absence of an…
The convergence of the so-called quadratic method for computing eigenvalue enclosures of general self-adjoint operators is examined. Explicit asymptotic bounds for convergence to isolated eigenvalues are found. These bounds turn out to…
This paper provides an existence-and-uniqueness theorem characterizing the stochastic integral with respect to a Wiener process. The integral is represented as a mapping from the space of measurable and adapted pathwise locally integrable…
In this paper we study the variety of one dimensional representations of a finite $W$-algebra attached to a classical Lie algebra, giving a precise description of the dimensions of the irreducible components. We apply this to prove a…
We investigate with the help of Clifford algebraic methods the Mandelbrot set over arbitrary two-component number systems. The complex numbers are regarded as operator spinors in D\times spin(2) resp. spin(2). The thereby induced (pseudo)…
Nazarov-Treil-Volberg recently proved an elegant two-weight T1 theorem for "almost diagonal" operators that played a key role in the proof of the $A_2$ conjecture for dyadic shifts and related operators. In this paper, we obtain a…