# The one-sided minimal operator and the one-sided reverse Holder inequality

Studia Mathematica (1995)

• Volume: 116, Issue: 3, page 255-270
• ISSN: 0039-3223

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## Abstract

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We introduce the one-sided minimal operator, ${m}^{+}f$, which is analogous to the one-sided maximal operator. We determine the weight classes which govern its two-weight, strong and weak-type norm inequalities, and show that these two classes are the same. Then in the one-weight case we use this class to introduce a new one-sided reverse Hölder inequality which has several applications to one-sided $\left({A}_{p}^{+}\right)$ weights.

## How to cite

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Cruz-Uribe, David, et al. "The one-sided minimal operator and the one-sided reverse Holder inequality." Studia Mathematica 116.3 (1995): 255-270. <http://eudml.org/doc/216232>.

@article{Cruz1995,
abstract = {We introduce the one-sided minimal operator, $m^+f$, which is analogous to the one-sided maximal operator. We determine the weight classes which govern its two-weight, strong and weak-type norm inequalities, and show that these two classes are the same. Then in the one-weight case we use this class to introduce a new one-sided reverse Hölder inequality which has several applications to one-sided $(A^+_p)$ weights.},
author = {Cruz-Uribe, David, SFO, Neugebauer, C., Olesen, V.},
journal = {Studia Mathematica},
keywords = {one-sided (A\_p) weights; reverse Hölder inequality; minimal function; one-sided reverse Hölder inequality; condition ; one-sided minimal operator; weighted norm inequalities; one-sided maximal operator},
language = {eng},
number = {3},
pages = {255-270},
title = {The one-sided minimal operator and the one-sided reverse Holder inequality},
url = {http://eudml.org/doc/216232},
volume = {116},
year = {1995},
}

TY - JOUR
AU - Cruz-Uribe, David
AU - SFO
AU - Neugebauer, C.
AU - Olesen, V.
TI - The one-sided minimal operator and the one-sided reverse Holder inequality
JO - Studia Mathematica
PY - 1995
VL - 116
IS - 3
SP - 255
EP - 270
AB - We introduce the one-sided minimal operator, $m^+f$, which is analogous to the one-sided maximal operator. We determine the weight classes which govern its two-weight, strong and weak-type norm inequalities, and show that these two classes are the same. Then in the one-weight case we use this class to introduce a new one-sided reverse Hölder inequality which has several applications to one-sided $(A^+_p)$ weights.
LA - eng
KW - one-sided (A_p) weights; reverse Hölder inequality; minimal function; one-sided reverse Hölder inequality; condition ; one-sided minimal operator; weighted norm inequalities; one-sided maximal operator
UR - http://eudml.org/doc/216232
ER -

## References

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1. [1] D. Cruz-Uribe, SFO, and C. J. Neugebauer, The structure of the reverse Hölder classes, Trans. Amer. Math. Soc. 347 (1995), 2941-2960. Zbl0851.42016
2. [2] D. Cruz-Uribe, SFO, C. J. Neugebauer and V. Olesen, Norm inequalities for the minimal and maximal operator, and differentiation of the integral, preprint. Zbl0903.42007
3. [3] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud. 116, North-Holland, Amsterdam, 1985.
4. [4] G. H. Hardy and J. E. Littlewood, A maximal theorem with function-theoretic applications, Acta Math. 54 (1930), 81-116. Zbl56.0264.02
5. [5] F. J. Martín-Reyes, New proofs of weighted inequalities for the one-sided Hardy-Littlewood maximal functions, Proc. Amer. Math. Soc. 117 (1993), 691-698. Zbl0771.42011
6. [6] F. J. Martín-Reyes, P. Ortega Salvador and A. de la Torre, Weighted inequalities for one-sided maximal functions, Trans. Amer. Math. Soc. 319 (1990), 517-534. Zbl0696.42013
7. [7] F. J. Martín-Reyes, L. Pick and A. de la Torre, $\left({A}_{\infty }^{+}\right)$ condition, Canad. J. Math. 45 (1993), 1231-1244. Zbl0797.42012
8. [8] F. J. Martín-Reyes and A. de la Torre, Two weight norm inequalities for fractional one-sided maximal operators, Proc. Amer. Math. Soc. 117 (1993), 483-489. Zbl0769.42010
9. [9] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. Zbl0236.26016
10. [10] E. Sawyer, Weighted inequalities for the one sided Hardy-Littlewood maximal functions, ibid. 297 (1986), 53-61. Zbl0627.42009
11. [11] J. O. Strömberg and R. L. Wheeden, Fractional integrals on weighted ${H}^{p}$ and ${L}^{p}$ spaces, ibid. 287 (1985), 293-321.

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