On small Lebesgue spaces

By C. Capone, Alberto Fiorenza. 2005

140 citations recorded by the source corpus.

We consider a generalized version of the small Lebesgue spaces, introduced in [5] as the associate spaces of the grand Lebesgue spaces. We find a simplified expression for the norm, prove relevant properties, compute the fundamental function and discuss the comparison with the Orlicz spaces.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/19e6bf98bdff72882e1cde8ac207d0de08bd282c

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Maximal functions, potentials and singular integrals in grand Morrey spaces (2011, background): “Structural properties of these spaces were investigated in the papers [3–5] etc.”
On Iwaniec-Sbordone Spaces on Sets Which May Have Infinite Measure (pp. 67-84) (2010, methodology): “Introduced in relation to problems of integrability of jacobians, these papers proved to be appropriate in various applications in partial differential equations and variational problems, they were used in the study of maximal functions, extrapolation theory etc, we refer for instance to [1, 3, 4, 7”
A direct approach to the duality of grand and small Lebesgue spaces (2009, background): “0 fgdx : f ∈M0 , ‖f‖p) ≤ 1 } They are called small Lebesgue spaces, are evidently continuously embedded in L and were first studied by the second author in [7], who found the following explicit expression of the norm (see also [4]): ‖g‖(q = inf g = ∞ ∑”
On grand and small Lebesgue and Sobolev spaces and some applications to PDE's (2018, background): “At last, it should be observed that such class of spaces, obtained in literature as endpoint of a sequence of generalizations, appeared maybe for the first time in the final remark in [23].”
Characterization of interpolation between Grand, small or classical Lebesgue spaces (2017, unknown): “sing, absolutely continuous far from the origin, concave and to satisfy ΦX(t)ΦX′(t) = t for all t∈ (0,µ(R)). Details and further material on r.i. spaces can be found in [5, Chapter 2]. Note that (see [8]) grand Lebesgue spaces Lp),α(Ω) and small Lebesgue spaces L(p,α(Ω) are r.i. spaces over Ω and ΦL”
The maximal theorem for weighted grand Lebesgue spaces (2008, background): “[7], [9], [2] and references therein), which are rearrangement-invariant Banach function spaces (see e.”
A Note on the Boundedness of the Hilbert Transform in Weighted Grand Lebesgue Spaces (2009, result): “For some properties and applications of Lp) spaces we refer to the papers [5], [7], [2], [ 1 ].”