Norm inequalities in some subspaces of Morrey space

Abstract

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Estimates for Fractional Integral Operators and Linear Commutators on Certain Weighted Amalgam Spaces

“…Since w is in the class A 1,q , we get w q ∈ A 1 ⊂ A ∞ by Lemma 7(ii). Moreover, since 1/α − 1 − 1/s < 0, then we apply inequality (16) to obtain that…”

“…(iii) if p = α and s = ∞, then ðL p , L s Þ α ðR n Þ reduces to the usual Lebesgue space L p ðR n Þ In [14] (see also [15,16]), Feuto considered a weighted version of the amalgam space ðL p , L s Þ α ðwÞ. A nonnegative measurable function w defined on R n is called a weight if it is locally integrable.…”

“…(ii) if p = α and s = ∞, then ðL p , L s Þ α ðwÞ reduces to the weighted Lebesgue space L p ðwÞ and ðWL p , L s Þ α ðwÞ reduces to the weighted weak Lebesgue space WL p ðwÞ Recently, many works in classical harmonic analysis have been devoted to norm inequalities involving several integral operators in the setting of weighted amalgam spaces (see [14][15][16]19] and [20]). These results obtained are extensions of well-known analogues in the weighted Lebesgue spaces.…”

Norm Inequalities in Generalized Morrey Spaces

“…The proof is partially inspired by [5]. The same arguments are used in [3] (see also [6]), to prove norm inequalities involving Riesz potentials and integral operators satisfying the hypothesis of Theorem 2.1 of [5] in the context of (L q , L p ) α (R n ) spaces.…”

Pre-Dual of Fofana’s Spaces

“…Many classical results for Lebesgue and the classical Morrey spaces have been extended to the setting of the spaces (L q , ℓ p ) α (R d ) (see [18,19,20,14,15,16,27,5,29,4]). Although the dual space of Lebesgue spaces L α (R d ) (1 ≤ α < ∞) and amalgam spaces (L q , ℓ p )(R d ) (1 ≤ q, p < ∞) are well known (L α ′ and (L q ′ , ℓ p ′ ) respectively with 1 p ′ + 1 p = 1), the one of (L q , ℓ p ) α (R d ) is still unknown.…”