Morrey spaces, their duals and preduals

Rosenthal, Marcel; Triebel, Hans

doi:10.1007/s13163-013-0145-z

Cited by 28 publications

(38 citation statements)

References 28 publications

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“…It was showed in Blasco et al that

h^{λ, q} false(double-struckR false)

is a Banach space, and the dual space of

h^{λ, q} false(double-struckR false)

L^{q^{'}, λ} false(double-struckR false)

; see also other studies . Now, we state the factorization of

H^{1} false(double-struckR false)

in terms of

h^{λ, q} false(double-struckR false)

and

L^{q^{'}, λ} false(double-struckR false)

as follows; for the case of factorization in terms of Calderón‐Zygmund operators T of convolution type and generalized Morrey spaces and their predual spaces on

R^{n}

, we refer the reader to Komori and Mizuhara ,.…”

Section: Introductionmentioning

confidence: 76%

“…We begin with recalling the notion of blocks in Blasco et al 31 For more properties of blocks and related spaces, see other studies. 21,[31][32][33]…”

Section: Definition 12 a Locally Integrable Real-valued Functionmentioning

confidence: 99%

“…Here, an atom a is a function in

L^{1} false(double-struckR false)

, which is supported in an interval I : = I ( x , r ), with

x \in double-struckR

and r ∈ (0, ∞ ), and satisfies that

‖ a {false‖}_{L^{\infty} (R)} \leq r^{- 1} and \int_{I} a (x) d x = 0 .

As a corollary of Theorem and the well‐known fact that

{false[H^{1} false(double-struckR false) false]}^{*} = BMO false(double-struckR false)

(see Fefferman and Stein), the second main result of this article concerns a factorization of

H^{1} false(double-struckR false)

via C Γ and

C_{normalΓ}^{*}

, here and hereafter, for any linear operator T , T ∗ denotes the adjoint operator of T on

L^{2} false(double-struckR false)

. We begin with recalling the notion of blocks in Blasco et al For more properties of blocks and related spaces, see other studies …”

Section: Introductionmentioning

confidence: 99%

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Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces

Tao

Yang

2019

Math Methods in App Sciences

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Let CΓ be the Cauchy integral operator on a Lipschitz curve Γ. In this article, the authors show that the commutator [b,CΓ] is bounded (resp, compact) on the Morrey space Lp,λfalse(double-struckRfalse) for any (or some) p ∈ (1,∞) and λ ∈ (0,1) if and only if b∈0.1emBMOfalse(double-struckRfalse) (resp, CMOfalse(double-struckRfalse)). As an application, a factorization of the classical Hardy space H1false(double-struckRfalse) in terms of CΓ and its adjoint operator is obtained.

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“…It was showed in Blasco et al that

h^{λ, q} false(double-struckR false)

is a Banach space, and the dual space of

h^{λ, q} false(double-struckR false)

L^{q^{'}, λ} false(double-struckR false)

; see also other studies . Now, we state the factorization of

H^{1} false(double-struckR false)

in terms of

h^{λ, q} false(double-struckR false)

and

L^{q^{'}, λ} false(double-struckR false)

as follows; for the case of factorization in terms of Calderón‐Zygmund operators T of convolution type and generalized Morrey spaces and their predual spaces on

R^{n}

, we refer the reader to Komori and Mizuhara ,.…”

Section: Introductionmentioning

confidence: 76%

“…We begin with recalling the notion of blocks in Blasco et al 31 For more properties of blocks and related spaces, see other studies. 21,[31][32][33]…”

Section: Definition 12 a Locally Integrable Real-valued Functionmentioning

confidence: 99%

“…Here, an atom a is a function in

L^{1} false(double-struckR false)

, which is supported in an interval I : = I ( x , r ), with

x \in double-struckR

and r ∈ (0, ∞ ), and satisfies that

‖ a {false‖}_{L^{\infty} (R)} \leq r^{- 1} and \int_{I} a (x) d x = 0 .

As a corollary of Theorem and the well‐known fact that

{false[H^{1} false(double-struckR false) false]}^{*} = BMO false(double-struckR false)

(see Fefferman and Stein), the second main result of this article concerns a factorization of

H^{1} false(double-struckR false)

via C Γ and

C_{normalΓ}^{*}

, here and hereafter, for any linear operator T , T ∗ denotes the adjoint operator of T on

L^{2} false(double-struckR false)

. We begin with recalling the notion of blocks in Blasco et al For more properties of blocks and related spaces, see other studies …”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces

Tao

Yang

2019

Math Methods in App Sciences

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“…Up to authors' knowledge, the rst such a result appears in [10, p. 132] who showed that global classical Morrey spaces L p,λ (R n ) are embedded into the weighted Lebesgue space L p (R n , w) with the weight w(x) = (1 + |x|) −µ , µ > λ. As a close result, we mention also embeddings of inhomogeneous classical Morrey spaces into power weighted Lebesgue spaces given in [17,Theorem 3.1] (see also [23,Theorem 2.8]). The results in [10] and [17] were recently extended in [2,Section 6] …”

Section: Introductionmentioning

confidence: 86%

“…As a close result, we mention also embeddings of inhomogeneous classical Morrey spaces into power weighted Lebesgue spaces given in [17,Theorem 3.1] (see also [23,Theorem 2.8]). The results in [10] and [17] were recently extended in [2,Section 6] …”

Section: Introductionmentioning

confidence: 86%

Embeddings of local generalized Morrey spaces between weighted Lebesgue spaces

Almeida

2017

Nonlinear Analysis

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Abstract. We prove that local generalized Morrey spaces are closely embedded between weighted Lebesgue spaces. We show that such embeddings are strict in all the cases under consideration by constructing counterexamples. As a consequence, continuous embeddings between generalized Morrey spaces and generalized Stummel spaces are established, as well as between Stummel classes (vanishing Stummel spaces). In particular, we obtain embeddings into a new Stummel class of functions with some vanishing property at innity. We also partially improve a known result on the coincidence of Stummel spaces with a modication of Morrey spaces where the supremum norm is replaced by an integral L p -norm.

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Preservation of certain vanishing properties of generalized Morrey spaces by some classical operators

Alabalık

Almeida

2020

Math Methods in App Sciences

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We show that certain vanishing properties defining closed subspaces of generalized Morrey spaces are preserved under the action of various classical operators of harmonic analysis, such as maximal operators, singular-type operators, Hardy operators, and fractional integral operators. Those vanishing subspaces were recently used to deal with the delicate problem on the description of the closure of nice functions in Morrey norm.

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Morrey spaces, their duals and preduals

Cited by 28 publications

References 28 publications

Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces

Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces

Embeddings of local generalized Morrey spaces between weighted Lebesgue spaces

Preservation of certain vanishing properties of generalized Morrey spaces by some classical operators

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