On the equivalence of boundedness for multiple Hardy-Littlewood averages and related operators

Authors

  • Dah-Chin Luor

DOI:

https://doi.org/10.7146/math.scand.a-105662

Abstract

Necessary and sufficient conditions for the weight function $u$ are obtained, which provide the boundedness for a class of averaging operators from $L_p^+$ to $L_{q,u}^+$. These operators include the multiple Hardy-Littlewood averages and related maximal operators, geometric mean operators, and geometric maximal operators. We show that, under suitable conditions, the boundedness of these operators are equivalent. Our theorems extend several one-dimensional results to multi-dimensional cases and to operators with multiple kernels. We also show that in the case $p<q$, some one-dimensional results do not carry over to the multi-dimensional cases, and the boundedness of $T$ from $L_p^+$ to $L_{q,u}^+$ holds only if $u=0$ almost everywhere.

References

Andersen, K. F., Weighted inequalities for convolutions, Proc. Amer. Math. Soc. 123 (1995), no. 4, 1129–1136. https://doi.org/10.2307/2160710

Andersen, K. F. and Sawyer, E. T., Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators, Trans. Amer. Math. Soc. 308 (1988), no. 2, 547–558. https://doi.org/10.2307/2001091

Braverman, M. S., On a class of operators, J. London Math. Soc. (2) 47 (1993), no. 1, 119–128. https://doi.org/10.1112/jlms/s2-47.1.119

Cruz-Uribe, D., A new proof of the two weight norm inequality for the one-sided fractional maximal operator, Proc. Amer. Math. Soc. 125 (1997), no. 5, 1419–1424. https://doi.org/10.1090/S0002-9939-97-03838-0

Cruz-Uribe, D., The minimal operator and the geometric maximal operator in $mathbb R^n$, Studia Math. 144 (2001), no. 1, 1–37. https://doi.org/10.4064/sm144-1-1

Cruz-Uribe, D. and Neugebauer, C. J., Weighted norm inequalities for the geometric maximal operator, Publ. Mat. 42 (1998), no. 1, 239–263. https://doi.org/10.5565/PUBLMAT_42198_13

Cruz-Uribe, D., Neugebauer, C. J., and Olesen, V., Weighted norm inequalities for geometric fractional maximal operators, J. Fourier Anal. Appl. 5 (1999), no. 1, 45–66. https://doi.org/10.1007/BF01274188

Cruz-Uribe, D. V., Martell, J. M., and Pérez, C., Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, vol. 215, Birkhäuser/Springer Basel AG, Basel, 2011. https://doi.org/10.1007/978-3-0348-0072-3

Duoandikoetxea, J., Fourier analysis, Graduate Studies in Mathematics, vol. 29, American Mathematical Society, Providence, RI, 2001.

Edmunds, D. E., Kokilashvili, V., and Meskhi, A., Bounded and compact integral operators, Mathematics and its Applications, vol. 543, Kluwer Academic Publishers, Dordrecht, 2002. https://doi.org/10.1007/978-94-015-9922-1

Fu, Z. W., Liu, Z. G., and Lu, S. Z., Commutators of weighted Hardy operators in $Bbb R^n$, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3319–3328. https://doi.org/10.1090/S0002-9939-09-09824-4

García-Cuerva, J. and Rubio de Francia, J. L., Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, Notas de Matemática, 104, North-Holland Publishing Co., Amsterdam, 1985.

Grafakos, L., Modern Fourier analysis, second ed., Graduate Texts in Mathematics, vol. 250, Springer, New York, 2009. https://doi.org/10.1007/978-0-387-09434-2

Kokilashvili, V. and Meskhi, A., On one-sided potentials with multiple kernels, Integral Transforms Spec. Funct. 16 (2005), no. 8, 669–683. https://doi.org/10.1080/10652460500106105

Kokilashvili, V. and Meskhi, A., Two-weight estimates for strong fractional maximal functions and potentials with multiple kernels, J. Korean Math. Soc. 46 (2009), no. 3, 523–550. https://doi.org/10.4134/JKMS.2009.46.3.523

Kokilashvili, V., Meskhi, A., and Persson, L.-E., Weighted norm inequalities for integral transforms with product kernels, Mathematics Research Developments Series, Nova Science Publishers, Inc., New York, 2010.

Kufner, A. and Persson, L.-E., Weighted inequalities of Hardy type, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. https://doi.org/10.1142/5129

Luor, D.-C., On weighted inequalities with geometric mean operator, Bull. Aust. Math. Soc. 78 (2008), no. 3, 463–475. https://doi.org/10.1017/S0004972708000841

Luor, D.-C., Modular inequalities for the Hardy-Littlewood averages, Math. Inequal. Appl. 13 (2010), no. 3, 635–642. https://doi.org/10.7153/mia-13-45

Luor, D.-C., On the equivalence of weighted inequalities for a class of operators, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), no. 5, 1071–1081. https://doi.org/10.1017/S0308210510000776

Martín-Reyes, F. J. and Sawyer, E., Weighted inequalities for Riemann-Liouville fractional integrals of order one and greater, Proc. Amer. Math. Soc. 106 (1989), no. 3, 727–733. https://doi.org/10.2307/2047428

Martín-Reyes, F. J. and de la Torre, A., Two weight norm inequalities for fractional one-sided maximal operators, Proc. Amer. Math. Soc. 117 (1993), no. 2, 483–489. https://doi.org/10.2307/2159186

Meskhi, A., Boundedness and compactness weighted criteria for Riemann-Liouville and one-sided maximal operators, Proc. A. Razmadze Math. Inst. 117 (1998), 126–128.

Meskhi, A., Solution of some weight problems for the Riemann-Liouville and Weyl operators, Georgian Math. J. 5 (1998), no. 6, 565–574. https://doi.org/10.1023/B:GEOR.0000008132.69276.cc

Meskhi, A., Criteria for the boundedness and compactness of integral transforms with positive kernels, Proc. Edinb. Math. Soc. (2) 44 (2001), no. 2, 267–284. https://doi.org/10.1017/S0013091599000747

Meskhi, A., A note on two-weight inequalities for multiple Hardy-type operators, J. Funct. Spaces Appl. 3 (2005), no. 3, 223–237. https://doi.org/10.1155/2005/361878

Nassyrova, M., Persson, L.-E., and Stepanov, V., On weighted inequalities with geometric mean operator generated by the Hardy-type integral transform, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), no. 4, Article 48, 16pp.

Opic, B. and Gurka, P., Weighted inequalities for geometric means, Proc. Amer. Math. Soc. 120 (1994), no. 3, 771–779. https://doi.org/10.2307/2160469

Opic, B. and Kufner, A., Hardy-type inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Scientific & Technical, Harlow, 1990.

Ortega Salvador, P. and Ramírez Torreblanca, C., Weighted inequalities for the one-sided geometric maximal operators, Math. Nachr. 284 (2011), no. 11-12, 1515–1522. https://doi.org/10.1002/mana.200910028

Osekowski, A., Sharp inequalities for geometric maximal operators associated with general measures, Anal. Math. 39 (2013), no. 4, 287–296. https://doi.org/10.1007/s10476-013-0404-8

Persson, L.-E. and Stepanov, V. D., Weighted integral inequalities with the geometric mean operator, J. Inequal. Appl. 7 (2002), no. 5, 727–746. https://doi.org/10.1155/S1025583402000371

Persson, L.-E. and Ushakova, E. P., Some multi-dimensional Hardy type integral inequalities, J. Math. Inequal. 1 (2007), no. 3, 301–319. https://doi.org/10.7153/jmi-01-27

Pick, L. and Opic, B., On the geometric mean operator, J. Math. Anal. Appl. 183 (1994), no. 3, 652–662. https://doi.org/10.1006/jmaa.1994.1172

Prokhorov, D. V., On the boundedness and compactness of a class of integral operators, J. London Math. Soc. (2) 61 (2000), no. 2, 617–628. https://doi.org/10.1112/S002461079900856X

Prokhorov, D. V. and Stepanov, V. D., Weighted estimates for Riemann-Liouville operators and their applications, Tr. Mat. Inst. Steklova 243 (2003), no. Funkts. Prostran., Priblizh., Differ. Uravn., 289–312.

Sawyer, E., Weighted inequalities for the two-dimensional Hardy operator, Studia Math. 82 (1985), no. 1, 1–16. https://doi.org/10.4064/sm-82-1-1-16

Sinnamon, G., A weighted gradient inequality, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), no. 3-4, 329–335. https://doi.org/10.1017/S0308210500018606

Sinnamon, G., One-dimensional Hardy-type inequalities in many dimensions, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 4, 833–848. https://doi.org/10.1017/S0308210500021818

Tang, C. and Zhai, Z., Generalized Poincaré embeddings and weighted Hardy operator on $Q^alpha ,q_p$ spaces, J. Math. Anal. Appl. 371 (2010), no. 2, 665–676. https://doi.org/10.1016/j.jmaa.2010.05.063

Tang, C. and Zhou, R., Boundedness of weighted Hardy operator and its adjoint on Triebel-Lizorkin-type spaces, J. Funct. Spaces Appl. (2012), Art. ID 610649, 9pp. https://doi.org/10.1155/2012/610649

Wedestig, A., Weighted inequalities for the Sawyer two-dimensional Hardy operator and its limiting geometric mean operator, J. Inequal. Appl. (2005), no. 4, 387–394. https://doi.org/10.1155/JIA.2005.387

Xiao, J., $L^p$ and BMO bounds of weighted Hardy-Littlewood averages, J. Math. Anal. Appl. 262 (2001), no. 2, 660–666. https://doi.org/10.1006/jmaa.2001.7594

Yin, X. R. and Muckenhoupt, B., Weighted inequalities for the maximal geometric mean operator, Proc. Amer. Math. Soc. 124 (1996), no. 1, 75–81. https://doi.org/10.1090/S0002-9939-96-03252-2

Published

2018-09-05

How to Cite

Luor, D.-C. (2018). On the equivalence of boundedness for multiple Hardy-Littlewood averages and related operators. MATHEMATICA SCANDINAVICA, 123(2), 273–296. https://doi.org/10.7146/math.scand.a-105662

Issue

Section

Articles