Hardy‐Littlewood Maximal Operator, Singular Integral Operators and the Riesz Potentials on Generalized Morrey Spaces

Nakai, Eiichi

doi:10.1002/mana.19941660108

Cited by 375 publications

(276 citation statements)

References 13 publications

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“…The spaces GM pθ,r −λ were used by G. Lu [4] for studying embedding theorems for vector fields of Hörmander type. T. Mizuhara [5] and E. Nakai [7] studied the boundedness of various integral operators in the spaces GM p∞,w .…”

Section: W(r)rmentioning

confidence: 99%

Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces

2004

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Abstract. The problem of boundedness of the Hardy-Littewood maximal operator in local and global Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted L p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions are also necessary.For x ∈ R n and r > 0, let B(x, r) denote the open ball centred at x of radius r. Definition 1. Let 0 < p, θ ≤ ∞ and let w be a non-negative measurable function on (0, ∞). We denote by LM pθ,w and GM pθ,w the local and global Morrey-type spaces respectively, defined to be the spaces of all functions f ∈ L loc p (R n ) with finite quasinormsrespectively.Lemma 1. Let 0 < p, θ ≤ ∞ and let w be a non-negative measurable function on (0, ∞).

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Section: W(r)rmentioning

confidence: 99%

Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces

2004

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“…In [15], [16], [28] and [30] there were obtained sufficient conditions on functions ω 1 and ω 2 for the boundedness of the singular operator…”

Section: Preliminaries On Morrey Spaces With Constant Exponents Pmentioning

confidence: 99%

“…The following statement, containing the results in [28], [30] was proved in [15] (see also [16]). Note that Theorems 3.6 and 3.7 do not impose condition (3.2).…”

Section: Preliminaries On Morrey Spaces With Constant Exponents Pmentioning

confidence: 99%

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Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces

Guliyev¹,

Hasanov²,

Samko³

2010

MATH. SCAND.

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We consider generalized Morrey spaces M p(·),ω ( ) with variable exponent p(x) and a general function ω(x, r) defining the Morrey-type norm. In case of bounded sets ⊂ R n we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev-Adams type, also of variable order. The conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ω(x, r), which do not assume any assumption on monotonicity of ω(x, r) in r.

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“…[2, Theorem 3.1.4 (b)]) has been extended to various function spaces. For Morrey spaces, Sobolev's inequality was studied in [1], [27], [5], [25], etc., for Morrey spaces of variable exponent in [3], [13], [14], [22], [23], etc., for grand Morrey spaces in [21] and [17], and also for grand Morrey spaces of variable exponent in [11]. Recently, Sobolev's inequality has been extended by the authors [19] to an inequality for general potentials of functions in Musielak-Orlicz-Morrey spaces.…”

Section: Introductionmentioning

confidence: 99%

Sobolev and Trudinger type inequalities on grand Musielak-Orlicz-Morrey spaces

Maeda¹,

Mizuta²,

Ohno³

et al. 2015

Ann. Acad. Sci. Fenn. Math.

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Hardy‐Littlewood Maximal Operator, Singular Integral Operators and the Riesz Potentials on Generalized Morrey Spaces

Cited by 375 publications

References 13 publications

Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces

Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces

Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces

Sobolev and Trudinger type inequalities on grand Musielak-Orlicz-Morrey spaces

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