### Abstract

We introduce a class of integral operators related to parametric Marcinkiewicz operators. We present a multiplier formula characterizing the L^{2} boundedness of such class of operators. Also, we prove L ^{p}_{-β} (inhomogeneous Sobolev space)→L^{p} estimates provided that the kernels are in L(logL)(S^{n-1}). In fact, we show that the global parts of the introduced operators are bounded on the Lebesgue spaces L^{p}(1 < p < ∞) while the local parts are bounded on certain Sobolev spaces L^{p}_{-β}.

Original language | English |
---|---|

Pages (from-to) | 56-81 |

Number of pages | 26 |

Journal | Communications in Mathematical Analysis |

Volume | 13 |

Issue number | 2 |

Publication status | Published - 2012 |

### Fingerprint

### Keywords

- Bessel functions
- Fourier transform
- Marcinkiewicz integrals
- Rough integral operators
- Sobolev spaces
- Triebel-Lizorkin space

### ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

*Communications in Mathematical Analysis*,

*13*(2), 56-81.

**A class of marcinkiewicz type integral operators.** / Al-Salman, Ahmad.

Research output: Contribution to journal › Article

*Communications in Mathematical Analysis*, vol. 13, no. 2, pp. 56-81.

}

TY - JOUR

T1 - A class of marcinkiewicz type integral operators

AU - Al-Salman, Ahmad

PY - 2012

Y1 - 2012

N2 - We introduce a class of integral operators related to parametric Marcinkiewicz operators. We present a multiplier formula characterizing the L2 boundedness of such class of operators. Also, we prove L p-β (inhomogeneous Sobolev space)→Lp estimates provided that the kernels are in L(logL)(Sn-1). In fact, we show that the global parts of the introduced operators are bounded on the Lebesgue spaces Lp(1 < p < ∞) while the local parts are bounded on certain Sobolev spaces Lp-β.

AB - We introduce a class of integral operators related to parametric Marcinkiewicz operators. We present a multiplier formula characterizing the L2 boundedness of such class of operators. Also, we prove L p-β (inhomogeneous Sobolev space)→Lp estimates provided that the kernels are in L(logL)(Sn-1). In fact, we show that the global parts of the introduced operators are bounded on the Lebesgue spaces Lp(1 < p < ∞) while the local parts are bounded on certain Sobolev spaces Lp-β.

KW - Bessel functions

KW - Fourier transform

KW - Marcinkiewicz integrals

KW - Rough integral operators

KW - Sobolev spaces

KW - Triebel-Lizorkin space

UR - http://www.scopus.com/inward/record.url?scp=84873356159&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84873356159&partnerID=8YFLogxK

M3 - Article

VL - 13

SP - 56

EP - 81

JO - Communications in Mathematical Analysis

JF - Communications in Mathematical Analysis

SN - 1938-9787

IS - 2

ER -