Weighted inequalities for potential operators on differential forms.

*(English)*Zbl 1193.47054The author extends a weak-type, two-weight inequality for potential operators [see D. Cruz-Uribe and C. Pérez, “Two-weight, weak-type norm inequalities for fractional integrals, Calderón-Zygmund operators and commutators”, Indiana Univ. Math. J. 49, No. 2, 697–721 (2000; Zbl 1033.42009)] to differential forms defined on an open set in \(\mathbb{R}^{n}\), and also proves a strong-type two-weight inequality for the solutions of the nonhomogeneous \(A\)-harmonic equation [for the scalar case, see E. Sawyer and R. L. Wheeden, “Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces”, Am. J. Math. 114, No. 4, 813–874 (1992; Zbl 0783.42011)].

Reviewer: Mihai Pascu (Bucureşti)

##### MSC:

47G40 | Potential operators |

42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |

58A10 | Differential forms in global analysis |

26D10 | Inequalities involving derivatives and differential and integral operators |

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\textit{H. Bi}, J. Inequal. Appl. 2010, Article ID 713625, 13 p. (2010; Zbl 1193.47054)

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##### References:

[1] | Agarwal RP, Ding S, Nolder C: Inequalities for Differential Forms. Springer, New York, NY, USA; 2009:xvi+387. · Zbl 1184.53001 |

[2] | do Carmo MP: Differential Forms and Applications, Universitext. Springer, Berlin, Germany; 1994:x+118. |

[3] | Warner FW: Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics. Volume 94. Springer, New York, NY, USA; 1983:ix+272. |

[4] | Martell, JM, Fractional integrals, potential operators and two-weight, weak type norm inequalities on spaces of homogeneous type, Journal of Mathematical Analysis and Applications, 294, 223-236, (2004) · Zbl 1072.42013 |

[5] | Cruz-Uribe, D; Pérez, C, Two-weight, weak-type norm inequalities for fractional integrals, Calderón-Zygmund operators and commutators, Indiana University Mathematics Journal, 49, 697-721, (2000) · Zbl 1033.42009 |

[6] | Nolder, CA, Hardy-Littlewood theorems for [inlineequation not available: see fulltext.]-harmonic tensors, Illinois Journal of Mathematics, 43, 613-632, (1999) · Zbl 0957.35046 |

[7] | Sawyer, E; Wheeden, RL, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, American Journal of Mathematics, 114, 813-874, (1992) · Zbl 0783.42011 |

[8] | Ding, S, Norm estimates for the maximal operator and Green’s operator, Dynamics of Continuous, Discrete & Impulsive Systems. Series A, 16, 72-78, (2009) · Zbl 1182.47018 |

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