Related papers: Convolution equations on the Lie group (-1,1)

We study the convolution function $$ C[f(x)] := \int_1^x f(y)f({x\over y}) {{\rm d} y\over y} $$ when $f(x)$ is a suitable number-theoretic error term. Asymptotics and upper bounds for $C[f(x)]$ are derived from mean square bounds for…

This paper outlines a covariant theory of operators defined on groups and homogeneous spaces. A systematic use of groups and their representations allows to obtain results of algebraic and analytical nature. The consideration is…

To study operator algebras with symmetries in a wide sense we introduce a notion of {\em relative convolution operators} induced by a Lie algebra. Relative convolutions recover many important classes of operators, which have been already…

It is common knowledge that the Fourier transform enjoys the convolution property, i.e., it turns convolution in the time domain into multiplication in the frequency domain. It is probably less known that this property characterizes the…

Let $\mathbb{H}^{n}$ be the Heisenberg group. For $0 \leq \alpha < Q=2n+2$ and $N \in \mathbb{N}$ we consider exponent functions $p(\cdot) : \mathbb{H}^{n} \to (0, +\infty)$, which satisfies H\"older conditions, such that $\frac{Q}{Q+N} <…

Convolution of valuations was introduced by the first named author and Fu for linear spaces, and later by Alesker and the first named author for compact Lie groups. In this paper we study the convolution of invariant valuations on Lie…

If $F$ is a continuous function on the real line and $f=F'$ is its distributional derivative then the continuous primitive integral of distribution $f$ is $\int_a^bf=F(b)-F(a)$. This integral contains the Lebesgue, Henstock--Kurzweil and…

Convolution is conventionally defined as a linear operation on functions of one or more variables which commutes with shifts. Group convolution generalizes the concept to linear operations on functions of group elements representing more…

We consider the framework of an operator-valued noncommutative probability space over a unital C*-algebra B. We show how for a B-valued distribution \mu one can define convolution powers with respect to free additive convolution and with…

For the functions $f$, which can be represented in the form of the convolution $f(x)=\frac{a_{0}}{2}+\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\sum\limits_{k=1}^{\infty}e^{-\alpha k^{r}}\cos(kt-\frac{\beta\pi}{2})\varphi(x-t)dt$,…

We produce, on general homogeneous groups, an analogue of the usual H\"ormander pseudodifferential calculus on Euclidean space, at least as far as products and adjoints are concerned. In contrast to earlier works, we do not limit ourselves…

We review the properties of transversality of distributions with respect to submersions. This allows us to construct a convolution product for a large class of distributions on Lie groupoids. We get a unital involutive algebra…

We establish convolution inequalities for Besov spaces $B_{p,q}^s$ and Triebel--Lizorkin spaces $F_{p,q}^s$. As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces…

Let $G$ be a connected reductive $p$-adic group and let $\theta$ be an automorphism of $G$ of order at most two. Suppose $\pi$ is an irreducible smooth representation of $G$ that is taken to its dual by $\theta$. The space $V$ of $\pi$ then…

We study the convolution of functions of the form \[ f_\alpha (z) := \dfrac{\left( \frac{1 + z}{1 - z} \right)^\alpha - 1}{2 \alpha}, \] which map the open unit disk of the complex plane onto polygons of 2 edges when $\alpha\in(0,1)$. We…

We extend the theory of matrix weights to the variable Lebesgue spaces. The theory of matrix $\mathcal{A}_p$ weights has attracted considerable attention beginning with the work of Nazarov, Treil, and Volberg in the 1990s. We extend this…

Let G be a locally compact abelian group and let \mu be a complex valued regular Borel measure on G. In this paper we consider a generalisation of a class of Banach lattices introduced in [6]. We use Laplace transform methods to show that…

Derived from the results in [Giang et al.: \emph{Convolutions for the Fourier transforms with geometric variables and applications}, Math. Nachr. 283(12) (2010), 1758--1770], in this paper, we devoted to studying the boundedness properties…

We study a notion of VC-dimension for subsets of groups, defining this for a set $A$ to be the VC-dimension of the family $\{ (xA) \cap A : x \in A\cdot A^{-1} \}$. We show that if a finite subset $A$ of an abelian group has bounded…

We introduce the concepts of the Fourier transform and convolution generated by an arbitrary restriction of the differentiation operator in the space $L_{2}(0,b).$ In contrast to the classical convolution, the introduced convolution…