Hausdorff operators on Hardy type spaces

Abstract

Скопина Мария Александровна -доктор физико-математических наук, Санкт-Петербургский государственный университет (г. Санкт-Петербург); Региональный научно-образовательный математический центр Южного федерального университета (г. Ростов-на-Дону).

Approximation of Hausdorff operators

“…and, as is shown first in [8] (see also [14] or [10]), such an operator is bounded in L 1 (R) whenever ϕ ∈ L 1 (R).…”

“…The number of publications is growing considerably; to add some most notable, we mention [1,5,9,12,13,15]. There are two survey papers: [4] and [10]). In the latter, as well as in [11], numerous open problems are given.…”

Hausdorff Operators on Some Spaces of Holomorphic Functions on the Unit Disc

“…In view of (17) to prove ( 14) it remains to check this equality on a dense subset of L p (dA α ). We shall use (16) to check (14) on the subset of bounded functions…”

“…The one-dimensional Hausdorff operators were introduced by Hardy [11, Chapter XI]. The modern theory of Hausdorff operators was inspired by [15]; see surveys [16], [17]. Hardy introduced his operators as a continuous analog of Hausdorff means; similarly, to get a mean of a holomorphic function f on D it is natural to integrate f over all natural holomorphic mixing of the disc, i. e., over the elements of the group Aut(D) of Möbius transformations of D (see theorems 4 and 5 below).…”

Boundedness of Hausdorff operators on Hardy spaces over homogeneous spaces of Lie groups

“…The multidimensional case was studied in [5]. For more details of the development of the theory of Hausdorff operators up to 2014 see [6], and [7].…”