Defining Hausdorff operators on Euclidean spaces

Abstract

After 2000, an interest in the Hausdorff operators grew, first of all in the sense of a diversity of spaces on which these operators were considered. We try to introduce a 'correct' definition of the Hausdorff operator on Euclidean spaces. This is supplemented by a variety of known and possible applications.

Matrix dilation and Hausdorff operators on modulation spaces

“…In Zhao et al, 15 we consider this problem for the case that $$$ A(y) $$$ is a diagonal matrix with the entries $$$ 1/\mid y\mid $$$. Recently, by a similar method in Zhao et al, 15 Karapetyants–Liflyand 18 make a partial extension to the general Hausdorff operator $$$ {H}_{\Phi, A} $$$. However, in fact, the method in previous studies 15,18 is not really suited for the general Hausdorff operator, which also leads some gaps in Karapetyants and Liflyand 18, Theorem 6 .…”

“…Recently, by a similar method in Zhao et al, 15 Karapetyants–Liflyand 18 make a partial extension to the general Hausdorff operator $$$ {H}_{\Phi, A} $$$. However, in fact, the method in previous studies 15,18 is not really suited for the general Hausdorff operator, which also leads some gaps in Karapetyants and Liflyand 18, Theorem 6 . Here, we try to give a reasonable definition of Hausdorff operator such that the map is continuous from $$$ \mathcal{S}\left({\mathbb{R}}^d\right) $$$ into $$$ {\mathcal{S}}^{\prime}\left({\mathbb{R}}^d\right) $$$.…”

Matrix dilation and Hausdorff operators on modulation spaces

“…In [19], we consider this problem for the case that A(y) is a diagonal matrix with the entries 1/|y|. Recently, by a similar method in [19], Karapetyants-Liflyand [11] make a partial extension to the general Hausdorff operator H Φ,A . However, in fact the method in [19,11] is not really suited for the general Hausdorff operator, which also leads some gaps in [11,Theorem 6].…”

“…Recently, by a similar method in [19], Karapetyants-Liflyand [11] make a partial extension to the general Hausdorff operator H Φ,A . However, in fact the method in [19,11] is not really suited for the general Hausdorff operator, which also leads some gaps in [11,Theorem 6]. Here, we try to give a reasonable definition of Hausdorff operator such that the map is continuous from…”

Homogeneous operators and homogeneous integral operators

“…We note that recently the study of special classes of operators, more precisely, integral operators with one or another property of invariance or symmetry, has acquired significant attention. In this regard, we would like to mention the studies of the class of so-called Hausdorff operators, see [17,18] and [19]. This class of operators differs essentially from operators with homogeneous kernels in a multidimensional situation, but the two classes are the same in the one-dimensional case.…”