Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces

In this paper, we discuss the boundedness of the fractional type Marcinkiewicz integral operators associated to surfaces, and we extend a result given by Chen et al. (J. Math. Anal. Appl. 276:691-708, 2002). They showed that under certain conditions the fractional type Marcinkiewicz integral operators are bounded from the Triebel-Lizorkin spacesḞ αRecently the second author, together with Xue and Yan, greatly weakened their assumptions. In this paper, we extend their results to the case where the operators are associated to the surfaces of the formTo prove our result, we discuss a characterization of the homogeneous Triebel-Lizorkin spaces in terms of lacunary sequences. MSC: Primary 42B20; secondary 42B25; 47G10

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“…In [2][3][4][5][6][7][8][9][10] and [15][16][17][18] boundedness of maximal operator, fractional maximal operator, Riesz potential and singular integral operators from one local Morrey-type space LM p1θ1,w1 to another one LM p2θ2,w2 has been investigated and, in particular, in [6,9] for a certain range of the parameters necessary and sufficient conditions for the operator I α to be bounded from LM p1θ1,w1 to LM p2θ2,w2 were obtained. (The definition and basic properties of these spaces are given in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Boundedness of the Riesz Potential in Local Morrey-Type Spaces

et al. 2010

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The problem of boundedness of the Riesz potential in local Morreytype 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 sharp sufficient conditions for boundedness for all admissible values The research of V. Burenkov was partially supported by the grants of the RFBR (project 09-01-00093a) and 68 V.I. Burenkov et al.of the parameters, which, for a certain range of the parameters wider than known before, coincide with the necessary ones.Keywords Riesz potential · Local and global Morrey-type spaces · Hardy operator on the cone of monotonic functions Mathematics Subject Classifications (2010) Primary 42B20 · 42B25 · 42B35

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Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces

Cited by 21 publications

References 6 publications

Wavelet characterization and modular inequalities for weighted Lebesgue spaces with variable exponent

Wavelet characterization and modular inequalities for weighted Lebesgue spaces with variable exponent

Fractional type Marcinkiewicz integral operators associated to surfaces

Boundedness of the Riesz Potential in Local Morrey-Type Spaces

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