On weighted inequalities for operators of potential type

Shiying Zhao

Colloquium Mathematicae (1996)

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

Abstract

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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.

How to cite

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Zhao, 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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  2. [2] I. Genebashvili, A. Gogatishvili and V. Kokilashvili, Criteria of general weak type inequalities for integral transforms with positive kernels, Proc. Georgian Acad. Sci. (Math.) 1 (1993), 11-34. Zbl0803.42011
  3. [3] R. Macias and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 (1979), 257-270. Zbl0431.46018
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  6. [6] E. T. Sawyer, A two weight weak type inequality for fractional integrals, Trans. Amer. Math. Soc. 281 (1984), 339-345. Zbl0539.42008
  7. [7] E. T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, ibid. 308 (1988), 533-545. Zbl0665.42023
  8. [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. [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
  10. [10] I. E. Verbitsky, Weight norm inequalities for maximal operators and Pisicr’s theorem on factorization through L p ∞ , Integral Equations Oper. Theory 15 (1992), 124-153. Zbl0782.47027
  11. [11] R. L. Wheeden, A characterization of some weighted norm inequalities for the fractional maximal function, Studia Math. 107 (1993), 257-272. Zbl0809.42009

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