Related papers: On the Forelli-Rudin projection theorem

We prove $L^p-L^{p^\prime}$ boundedness of spectral projections and the resolvent of the Laplace-Beltrami operator on Damek-Ricci spaces with the explicit norms in terms of the spectral parameter. To prove these results we established…

We prove two-sided inequalities between the integral moduli of smoothness of a function on $\mathbb{R}^d/\mathbb{T}^d$ and the weighted tail-type integrals of its Fourier transform/series. Sharpness of obtained results in particular is…

We show that the Lusin area integral or the square function on the unit ball of $\C^n$, regarded as an operator in weighted space $L^2(w)$ has a linear bound in terms of the invariant $A_2$ characteristic of the weight. We show a…

Semyanistyi's fractional integrals have come to analysis from integral geometry. They take functions on $R^n$ to functions on hyperplanes, commute with rotations, and have a nice behavior with respect to dilations. We obtain sharp…

We give a simple, short and self-contained presentation of Bourgain's discretised projection theorem from 2010, which is a fundamental tool in many recent breakthroughs in geometric measure theory, harmonic analysis, and homogeneous…

We introduce and study, in the framework of a theory of quantum Cartan domains, a q-analogue of the Berezin transform on the unit ball. We construct q-analogues of weighted Bergman spaces, Toeplitz operators and covariant symbol calculus.…

Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal operation, sometimes even defined via differentiation. Using the formalism of geometric algebra and geometric calculus in which the Grassmann…

The aim of my thesis is to discuss, develop and apply the newest developments of this fascinating theory connected to modern harmonic analysis. In particular, we investigate some strong convergence result of partial sums of Vilenkin-Fourier…

In this work we obtain weighted boundedness results for singular integral operators with kernels exhibiting exponential decay. We also show that the classes of weights are characterized by a suitable maximal operator. Additionally, we study…

We deal with the boundedness properties of higher order commutators related to some generalizations of the multilinear fractional integral operator of order $m$, $I_\alpha ^m$, from a product of weighted Lebesgue spaces into adequate…

We consider Marstrand type projection theorems for closest-point projections in the normed space $\mathbb{R}^2$. We prove that if a norm on $\mathbb{R}^2$ is regular enough, then the analogues of the well-known statements from the Euclidean…

Let $A_{\alpha}^{p}(\mathbb{B}^n;\mathbb{C}^d)$ be the weighted Bergman space on the unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$ of functions taking values in $\mathbb{C}^d$. For $1<p<\infty$ let $\mathcal{T}_{p,\alpha}$ be the algebra…

In this article, we introduce the fractional maximal operator on the Hyperbolic space, a non-doubling measure space, and study the weighted boundedness. Motivated in the weighted boundedness of Hardy-Littlewood maximal studied by Antezana…

We study the problem estimation of classical Lebesgue-Riesz and Grand Lebesgue Norm for the fractional integrals and derivatives for the functions from the classical Lebesgue-Riesz spaces as well as from the modified Besov's spaces.

We study composition operators acting on the weighted Bergman spaces on the bidisc, i.e. $C_{\Phi}:A^2_{\beta}(\mathbb{D}^2)\to A^2_{\beta}(\mathbb{D}^2)$ where $\Phi$ is induced by rational inner functions (RIFs) or a RIF and a smooth…

A well-known result by Lindenstrauss is that any two-dimensional normed space can be isometrically imbedded into $L_1(0,1)$. We provide an explicit form of a such an imbedding. The proof is elementary and self-contained. Applications are…

Let $\mathbb{B}^d$ be the unit ball on the complex space $\mathbb{C}^d$ with normalized Lebesgue measure $dv.$ For $\alpha\in\mathbb{R},$ denote $k_\alpha(z,w)=\frac{1}{(1-\langle z,w\rangle)^\alpha},$ the Bergman-type integral operator…

We prove new Pitt inequalities for the Fourier transforms with radial and non-radial weights using weighted restriction inequalities for the Fourier transform on the sphere. We also prove new Riemann-Lebesgue estimates and versions of the…

We introduce the mixed Bourgain-Morrey spaces and obtain their preduals. The boundedness of Hardy-Littlewood maximal operator, iterated maximal operator, fractional integral operator, singular integral operator on these spaces is proved.…

We develop a new formulation of well localized operators as well as a new proof for the necessary and sufficient conditions to characterize their boundedness between $L^2(\mathbb{R}^n,u)$ and $L^2(\mathbb{R}^n,v)$ for general Radon measures…