Sawyer's duality principle for grand Lebesgue spaces

Abstract

The aim of this paper is to extend Sawyer's duality principle from the cone of decreasing functions of the Lebesgue space to the cone of decreasing functions of the grand Lebesgue space and to prove the boundedness of classical Hardy operators in the grand Lebesgue spaces.

Grand Lebesgue Spaces are really Banach algebras relative to the convolution on unimodular locally compact groups equipped with Haar measure

“…The Grand Lebesgue Spaces have been widely investigated, see, e.g., [6, 7, 13, 19, 31, 32, 37, 38, 42] and references therein. They play an important role in the theory of partial differential equations (PDEs) (see, e.g., [1, 15, 17, 18, 22]), in interpolation theory (see, e.g., [2, 12, 14, 16]), in the theory of probability ([10, 20, 36, 43–45]), in statistics and in the theory of random fields (see, e.g., [35], [41, Chapter 5]), in functional analysis and so on.…”

Grand Lebesgue Spaces are really Banach algebras relative to the convolution on unimodular locally compact groups

“…The Grand Lebesgue Spaces have been widely investigated, e.g. [8], [5], [4], [14], [24], [28], [30], [33] - [37], [25], [31], etc. They play an important role in the theory of Partial Differential Equations (PDEs) (see e.g.…”

Local grand Lebesgue spaces on quasi-metric measure spaces and some applications

“…An approach to aggrandize Lebesgue spaces on sets of infinite measure may be found in [3][4][5][6]. Grand spaces have been intensively studied during the last decades; see, for instance, [7][8][9][10][11][12][13]. We refer also to [14] and references therein.…”