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Atomic decomposition in small Lebesgue space. (English) Zbl 1486.46036

Summary: We study grand \(L^{p)}(\Omega)\) spaces, introduced by [T. Iwaniec and the second author, Arch. Ration. Mech. Anal. 119, No. 2, 129–143 (1992; Zbl 0766.46016)], and small \(L^{(q}(\Omega)\) spaces, introduced by [A. Fiorenza, Collect. Math. 51, No. 2, 131–148 (2000; Zbl 0960.46022)], and improve a result by [Fiorenza, loc. cit.] that if \(\frac1p+\frac1q=1\), \(L^{p)}\) is the dual of \(L^{(q}\), giving a constructive characterization of \(L^{(q}\) via atomic decomposition in the particular case \(p=q=2\), \(\Omega= {]}0,1[\). We notice that \(L^{(q}(]0,1[)\) is isomorphic to the dual space of \(L_b^{p)}\), the closure of \(L^\infty(]0,1[)\) in \(L^p(]0,1[)\). We also illustrate distance formula in \(L^{p)}\) to \(L^\infty\) due to [M. Carozza and the second author, Differ. Integral Equ. 10, No. 4, 599–607 (1997; Zbl 0889.35027)].
We incorporate the pair \(L_b^{p)}\), \(L^{p)}\) in the theory of o-O type pair of spaces \((E_0,E)\) according to [K.-M. Perfekt, Math. Scand. 121, No. 1, 151–160 (2017; Zbl 1434.46008); Ark. Mat. 51, No. 2, 345–361 (2013; Zbl 1283.46011)]. Description of \((L^{p)})^\star\) is also provided in terms of \((L_b^{p)})^\star\) and \((L_b^{p)})^\bot\) (see [C. Capone and M. R. Formica, J. Funct. Spaces Appl. 2012, Article ID 737534, 10 p. (2012; Zbl 1244.46011)]).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B10 Duality and reflexivity in normed linear and Banach spaces
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