Grand Lebesgue spaces with respect to measurable functions

Capone, Claudia; Formica, Maria Rosaria; Giova, Raffaella

doi:10.1016/j.na.2013.02.021

Cited by 78 publications

(39 citation statements)

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“…Obviously, when λ = 0, the generalized grand Morrey space L p),0 ϕ (X ) coincides with the generalized grand Lebesgue space considered in [6]. On the other hand, when p = ∞, it is easy to see that L…”

Section: Introductionmentioning

confidence: 84%

“…It is known that the grand Lebesgue spaces were introduced by Iwaniec and Sbordone in [18] in 1992 to study the integrability of the Jacobian determinant of an order preserving mapping from a bounded domain Ω ⊂ R n to R n . From then on, the grand Lebesgue spaces and their generalized versions have attracted a lot of attention and found several applications in various areas of analysis, such as partial differential equations and harmonic analysis; see, for example, [36], [15], [19], [13], [7], [10], [11], [12], [9], [8], [6], [3] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces

Liu

Yuan

2017

Czech.Math.J.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces

Liu

Yuan

2017

Czech.Math.J.

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“…In other words, Theorem 1 shows that the class of fully measurable Lebesgue spaces introduced in [1], where both and Ω coincide with the interval = (0,1) ⊆ R, coincides in the case of bounded exponents with the class of spaces introduced in [3]. As a consequence, it does make no difference whether one defines the fully measurable Lebesgue spaces…”

Section: Journal Of Function Spacesmentioning

confidence: 99%

Identification of Fully Measurable Grand Lebesgue Spaces

Anatriello

Chill

Fiorenza

2017

Journal of Function Spaces

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We consider the Banach function spaces, called fully measurable grand Lebesgue spaces, associated with the function norm ( ) = ess sup ∈ ( ) ( ) ( ), where ( ) denotes the norm of the Lebesgue space of exponent ( ), and (⋅) and (⋅) are measurable functions over a measure space ( , ]), ( ) ∈ [1, ∞], and ( ) ∈ (0, 1] almost everywhere. We prove that every such space can be expressed equivalently replacing (⋅) and (⋅) with functions defined everywhere on the interval (0, 1), decreasing and increasing, respectively (hence the full measurability assumption in the definition does not give an effective generalization with respect to the pointwise monotone assumption and the essential supremum can be replaced with the simple supremum). In particular, we show that, in the case of bounded (⋅), the class of fully measurable Lebesgue spaces coincides with the class of generalized grand Lebesgue spaces introduced by Capone, Formica, and Giova.Let (Ω, ) be a finite measure space, and let M(Ω) be the set of all measurable functions on Ω with values in [−∞, +∞], M + (Ω) be the subset of the nonnegative functions, and M 0 (Ω) be the subset of the real valued functions.Let ( , ]) be a measure space, and define M( ) similarly as above.whereThen [⋅], (⋅) is a Banach function norm, and the associated Banach function spaceis called fully measurable grand Lebesgue space. This definition is one of the most abstract generalizations of the original grand Lebesgue spaces originated in the paper by Iwaniec and Sbordone [2]. Given (⋅) ∈ M( ) as above, let us setso that 1 ≤ − ≤ + ≤ ∞. In the following, we exclude the noninteresting case of constant (⋅), that is, the case − = + š 0 , because in such case we know (see [1]) that [⋅], (⋅) (Ω) = 0 (Ω).

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“…The computation of the Boyd indices of generalized grand Lebesgue spaces L p),δ introduced in [7] is a problem of interest in Function Spaces Theory and in Harmonic analysis (see e.g., [17]). …”

Section: Introductionmentioning

confidence: 99%

“…In [7], it is proved that L p),δ (0, 1) are r.i. Banach function spaces. When δ(ε) = ε, the spaces L p),δ reduce to grand Lebesgue spaces L p) introduced in [25] by Iwaniec-Sbordone in connection with the integrability of the Jacobian.…”

Section: Introductionmentioning

confidence: 99%