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On Grand Lebesgue spaces on sets of infinite measure
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Abstract(s)
We consider grand Lebesgue spaces on sets of infinite measure and study the dependence of these spaces on the choice of the so-called. We also consider Mikhlin and Marcinkiewicz theorems on Fourier multipliers in the setting of grand spaces. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
The theory of grand spaces is intensively developed during the last two decades. Such spaces L p)(), 1 < p < ∞, on bounded sets ⊂ Rn were introduced by T. Iwaniec and C. Sbordone [8] in connection with application to differential equations. In the last years, operators of harmonic analysis were widely studied in such spaces, see [1]–[6], [11]–[15] and the references therein. Some of these results are presented in the books [16], [17]. In all the above mentioned studies only sets of finite measure were allowed, based on the embedding L p ⊂ L p−ε. In the papers [23], [24], [26] there was suggested an approach to define grand spaces L p) a () on sets ⊆ Rn of not necessarily finite measure. In the general form given in [26], this approach is based on introducing the small power aε of a weight a into the norm of grand space, see (2.1). We call this function a, which determines the grand space L p) a (), the of this space.
The theory of grand spaces is intensively developed during the last two decades. Such spaces L p)(), 1 < p < ∞, on bounded sets ⊂ Rn were introduced by T. Iwaniec and C. Sbordone [8] in connection with application to differential equations. In the last years, operators of harmonic analysis were widely studied in such spaces, see [1]–[6], [11]–[15] and the references therein. Some of these results are presented in the books [16], [17]. In all the above mentioned studies only sets of finite measure were allowed, based on the embedding L p ⊂ L p−ε. In the papers [23], [24], [26] there was suggested an approach to define grand spaces L p) a () on sets ⊆ Rn of not necessarily finite measure. In the general form given in [26], this approach is based on introducing the small power aε of a weight a into the norm of grand space, see (2.1). We call this function a, which determines the grand space L p) a (), the of this space.
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Keywords
Lp spaces
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Publisher
Wiley-V C H Verlag Gmbh