# On weighted inequalities for operators of potential type

Colloquium Mathematicae (1996)

- Volume: 69, Issue: 1, page 95-115
- ISSN: 0010-1354

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topZhao, Shiying. "On weighted inequalities for operators of potential type." Colloquium Mathematicae 69.1 (1996): 95-115. <http://eudml.org/doc/210331>.

@article{Zhao1996,

abstract = {In this paper, we discuss a class of weighted inequalities for operators of potential type on homogeneous spaces. We give sufficient conditions for the weak and strong type weighted inequalities sup\_\{λ>0\} λ|\{x ∈ X : |T(fdσ)(x)|>λ \}|\_\{ω\}^\{1/q\} ≤ C (∫\_\{X\} |f|^\{p\}dσ)^\{1/p\} and (∫\_\{X\} |T(fdσ)|^\{q\}dω )^\{1/q\} ≤ C (∫\_X |f|^\{p\}dσ )^\{1/p\} in the cases of 0 < q < p ≤ ∞ and 1 ≤ q < p < ∞, respectively, where T is an operator of potential type, and ω and σ are Borel measures on the homogeneous space X. We show that under certain restrictions on the measures those sufficient conditions are also necessary. A consequence is given for the fractional integrals in Euclidean spaces.},

author = {Zhao, Shiying},

journal = {Colloquium Mathematicae},

keywords = {fractional maximal functions; operators of potential type; weights; norm inequalities; homogeneous spaces; weak and strong type weighted inequalities; fractional integrals},

language = {eng},

number = {1},

pages = {95-115},

title = {On weighted inequalities for operators of potential type},

url = {http://eudml.org/doc/210331},

volume = {69},

year = {1996},

}

TY - JOUR

AU - Zhao, Shiying

TI - On weighted inequalities for operators of potential type

JO - Colloquium Mathematicae

PY - 1996

VL - 69

IS - 1

SP - 95

EP - 115

AB - In this paper, we discuss a class of weighted inequalities for operators of potential type on homogeneous spaces. We give sufficient conditions for the weak and strong type weighted inequalities sup_{λ>0} λ|{x ∈ X : |T(fdσ)(x)|>λ }|_{ω}^{1/q} ≤ C (∫_{X} |f|^{p}dσ)^{1/p} and (∫_{X} |T(fdσ)|^{q}dω )^{1/q} ≤ C (∫_X |f|^{p}dσ )^{1/p} in the cases of 0 < q < p ≤ ∞ and 1 ≤ q < p < ∞, respectively, where T is an operator of potential type, and ω and σ are Borel measures on the homogeneous space X. We show that under certain restrictions on the measures those sufficient conditions are also necessary. A consequence is given for the fractional integrals in Euclidean spaces.

LA - eng

KW - fractional maximal functions; operators of potential type; weights; norm inequalities; homogeneous spaces; weak and strong type weighted inequalities; fractional integrals

UR - http://eudml.org/doc/210331

ER -

## References

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- [8] E. T. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), 813-874. Zbl0783.42011
- [9] E. T. Sawyer, R. L. Wheeden and S. Zhao, Weighted norm inequalities for operators of potential type and fractional maximal functions, Potential Anal., to appear. Zbl0873.42012
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