Related papers: Weighted estimates for bilinear fractional integra…

Let $\mathcal{H}$ be a separable Hilbert space and let $A^{2}_{\varphi}(\mathcal{H})$ be the $\mathcal{H}$-valued Bergman spaces with exponential weights. In the present paper, we give the complete characterizations for the boundedness and…

In this paper, Hardy type operator $H_{\beta}$ on $\bR^{n}$ and its adjoint operator $H_{\beta}^{*}$ are investigated. We use novel methods to obtain two main results. One is that we obtain the operators $H_{\beta}$ and $H_{\beta}^{*}$…

In this paper we establish some (presumably new) interesting expressions for the composition of some well known fractional integral operators $ I^{\mu}_{a+}, D^{\mu}_{a+} $,$ I^{\gamma , \mu}_{a+}$ and also derive an integral operator…

In this paper we study the Hilbert transformations over $L^2(\mathbb{R})$ and $L^2(\mathbb{T})$ from the viewpoint of symmetry. For a linear operator over $L^2(\mathbb{R})$ commutative with the ax+b group we show that the operator is of the…

In this work we establish the following fractional Hardy's inequality $$C\int_{\mathbb{H}^n_+}\frac{|f(\xi)|^p}{x_1^{sp+\alpha}}d\xi\leq \int_{\mathbb{H}^n}\int_{\mathbb{H}^n}\frac{|f(\xi)-f(\xi')|^p}{d({\xi}^{-1}\circ…

In this paper, we characterize the sharp constant and maximizing functions for weighted Poincar\'e inequalities. These results lead to refinements of Hardy's inequality obtained by adding remainder terms involving \(L^p\) norms. We use…

Using the framework of a previous article joint with Axelsson and McIntosh, we extend to systems two results of S. Hofmann for real symmetric equations and their perturbations going back to a work of B. Dahlberg for Laplace's equation on…

We construct a family of bilinear differential operators which satisfy certain gauge properties. These operators can be naturally associated with $q$-deformations of classical integrable hierarchies. In particular, we consider the case when…

We obtain some sharp $L^p$ weighted Fourier restriction estimates of the form $\|Ef\|_{L^p(B^{n+1}(0,R),Hdx)} \lessapprox R^{\beta}\|f\|_2$, where $E$ is the Fourier extension operator over the truncated paraboloid, and $H$ is a weight…

We prove that the operator $L_0=-(1+|x|)^\beta(-\Delta)^{\alpha/2}$ with $\alpha\in(0,2)$, $d>\alpha$ and $\beta\ge0$ generates a compact semigroup or resolvent on $L^2(\R^d;(1+|x|)^{-\beta}\,dx)$, if and only if $\beta>\alpha$. When…

Let $0<\alpha<n$ and $T_{\Omega,\alpha}$ be the homogeneous fractional integral operator which is defined by \begin{equation*} T_{\Omega,\alpha}f(x):=\int_{\mathbb R^n}\frac{\Omega(x-y)}{|x-y|^{n-\alpha}}f(y)\,dy, \end{equation*} where…

In this article we consider the generalized integral operators acting on the Hilbert space $H^2$. We characterize when these operators are uniform, strong and weakly asymptotic Toeplitz and Hankel operators. Moreover we completely describe…

On the twisted Fock spaces $ \mathcal{F}^\lambda(\C^{2n}) $ we consider a family of unitary operators $\rho_\lambda(a,b) $ indexed by $ (a,b) \in \C^n \times \C^n.$ The composition formula for $ \rho_\lambda(a,b) \circ…

In this paper, we will define $a$-deformed Laguerre operators $L_{a,\alpha}$ and $a$-deformed Laguerre holomorphic semigroups on $L^2\left(\left(0,\infty\right),d\mu_{a,\alpha}\right)$. Then we give a spherical harmonic expansion, which…

For any $0<\alpha<n$, the homogeneous fractional integral operator $T_{\Omega,\alpha}$ is defined by \begin{equation*} T_{\Omega,\alpha}f(x)=\int_{\mathbb R^n}\frac{\Omega(x-y)}{|x-y|^{n-\alpha}}f(y)\,dy. \end{equation*} In this paper, we…

In this short note, we determine the spectrum of the Heisenberg oscillator which is the operator defined as $L + |z|^2$ on the 3-dimensional Heisenberg group $H_1=\mathbb C \times \mathbb R$ where $L$ stands for the positive sublaplacian…

In this paper, we study the boundedness properties of the (dyadic) maximal bilinear operator associated with rough homogeneous kernels on $\mathbb{R}$. We establish sharp $L^{p_1}(\mathbb{R}) \times L^{p_2}(\mathbb{R}) \to…

Ranges of the real-valued parameters $\alpha$, $a$, $b$, and $m$ are identified for which the operator $$\mathcal{A}_{\alpha}(a,b)f(x):=x^\alpha\left(f''(x)+\frac{a}{x}f'(x)+\frac{b}{x^2}f(x)\right), \quad x>0,$$ generates an analytic…

Let F be a real quadratic field with ring of integers O and with class number 1. Let Gamma be a congruence subgroup of GL_2 (O). We describe a technique to compute the action of the Hecke operators on the cohomology H^3 (Gamma; C). For F…

We prove $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups; these are sharp up to two endpoints. The results can be applied to improve currently known bounds on sparse…