Abstract
In this chapter we continue our study of the Hardy-Littlewood maximal operator. In Chap. 3 we showed that the log Hölder continuity conditions LH 0 and LH ∞ are sufficient for the maximal operator to be bounded. In this chapter we will show that they are not necessary, even though they are the best possible pointwise decay conditions. To find weaker sufficient conditions we build upon the proof of Theorem 3.16, which showed that LH 0 and LH ∞ play distinct roles.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Bibliography
M. Aguilar Cañestro and P. Ortega Salvador. Weighted weak type inequalities with variable exponents for Hardy and maximal operators. Proc. Japan Acad. Ser. A Math. Sci., 82(8):126–130, 2006.
M. Aguilar Cañestro and P. Ortega Salvador. Weak-type inequalities and convergence of the ergodic averages in variable Lebesgue spaces with weights. Proc. Roy. Soc. Edinburgh Sect. A, 139(4):673–683, 2009.
M. Asif, V. Kokilashvili, and A. Meskhi. Boundedness criteria for maximal functions and potentials on the half-space in weighted Lebesgue spaces with variable exponents. Integral Transforms Spec. Funct., 20(11–12):805–819, 2009.
M. Asif and A. Meskhi. On the essential norm for the Hilbert transforms in L p(x) spaces. Georgian Math. J., 15(2):209–223, 2008.
M. Asif and A. Meskhi. Weighted estimates of a measure of noncompactness for maximal and potential operators. J. Inequal. Appl., pages Art. ID 697407, 19, 2008.
R. A. Bandaliev. The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces. Czechoslovak Math. J., 60(135)(2):327–337, 2010.
R. A. Bandaliev. The boundedness of multidimensional Hardy operators in weighted variable Lebesgue spaces. Lith. Math. J., 50(3):249–259, 2010.
R. A. Bandaliev. Embedding between variable exponent Lebesgue spaces with measures. Azerbaijan J. Math., 2(1):119–125, 2012.
E. I. Berezhnoĭ. Sharp estimates for operators on cones in ideal spaces. Trudy Mat. Inst. Steklov., 204(Issled. po Teor. Differ. Funktsii Mnogikh Peremen. i ee Prilozh. 16):3–34, 1993.
E. I. Berezhnoĭ. Two-weighted estimations for the Hardy-Littlewood maximal function in ideal Banach spaces. Proc. Amer. Math. Soc., 127(1):79–87, 1999.
M.-M. Boureanu. Existence of solutions for an elliptic equation involving the p(x)-Laplace operator. Electron. J. Differential Equations, pages No. 97, 10 pp. (electronic), 2006.
S. Boza and J. Soria. Weighted Hardy modular inequalities in variable L p spaces for decreasing functions. J. Math. Anal. Appl., 348(1):383–388, 2008.
S. Boza and J. Soria. Weighted weak modular and norm inequalities for the Hardy operator in variable L p spaces of monotone functions. Rev. Mat. Complut., 25(2):459–474, 2012.
C. Capone, D. Cruz-Uribe, and A. Fiorenza. The fractional maximal operator and fractional integrals on variable L p spaces. Rev. Mat. Iberoam., 23(3):743–770, 2007.
M. Christ and R. Fefferman. A note on weighted norm inequalities for the Hardy-Littlewood maximal operator. Proc. Amer. Math. Soc., 87(3):447–448, 1983.
D. Cruz-Uribe, L. Diening, and P. Hästö. The maximal operator on weighted variable Lebesgue spaces. Frac. Calc. Appl. Anal., 14(3):361–374, 2011.
D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer. The maximal function on variable L p spaces. Ann. Acad. Sci. Fenn. Math., 28(1):223–238, 2003. See also errata [63].
D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer. Weighted norm inequalities for the maximal operator on variable Lebesgue spaces. J. Math. Anal. Appl., 394(2):744–760, 2012.
D. Cruz-Uribe and F. Mamedov. On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces. Rev. Mat. Complut., 25(2):335–367, 2012.
D. Cruz-Uribe, J. M. Martell, and C. Pérez. Sharp weighted estimates for classical operators. Adv. Math., 229:408–441, 2011.
D. Cruz-Uribe, J. M. Martell, and C. Pérez. Weights, Extrapolation and the Theory of Rubio de Francia, volume 215 of Operator Theory: Advances and Applications. Birkhäuser/Springer Basel AG, Basel, 2011.
M. de Guzmán. Differentiation of Integrals in R n. Lecture Notes in Mathematics, Vol. 481. Springer-Verlag, Berlin, 1975. With appendices by Antonio Córdoba, and Robert Fefferman, and two by Roberto Moriyón.
L. Diening. Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math., 129(8):657–700, 2005.
L. Diening. Lebesgue and Sobolev Spaces with Variable Exponent. Habilitation, Universität Freiburg, 2007.
L. Diening, P. Harjulehto, P. Hästö, and M. R˚užička. Lebesgue and Sobolev spaces with Variable Exponents, volume 2017 of Lecture Notes in Mathematics. Springer, Heidelberg, 2011.
L. Diening and P. Hästö. Muckenhoupt weights in variable exponent spaces. Preprint, 2010.
L. Diening, P. Hästö, and A. Nekvinda. Open problems in variable exponent Lebesgue and Sobolev spaces. In FSDONA04 Proceedings (Drabek and Rakosnik (eds.); Milovy, Czech Republic, pages 38–58. Academy of Sciences of the Czech Republic, Prague, 2005.
J. Duoandikoetxea. Fourier Analysis, volume 29 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001.
D. E. Edmunds, A. Fiorenza, and A. Meskhi. On a measure of non-compactness for some classical operators. Acta Math. Sin. (Engl. Ser.), 22(6):1847–1862, 2006.
D. E. Edmunds, V. Kokilashvili, and A. Meskhi. On the boundedness and compactness of weighted Hardy operators in spaces L p(x). Georgian Math. J., 12(1):27–44, 2005.
D. E. Edmunds, V. Kokilashvili, and A. Meskhi. Two-weight estimates in L p(x) spaces with applications to Fourier series. Houston J. Math., 35(2):665–689, 2009.
D. E. Edmunds and J. Rákosník. Sobolev embeddings with variable exponent. Studia Math., 143(3):267–293, 2000.
A. Fiorenza. An inequality for Jensen means. Nonlinear Anal., 16(2):191–198, 1991.
A. Fiorenza and M. Krbec. A note on noneffective weights in variable Lebesgue spaces. J. Funct. Spaces Appl., pages Art. ID 853232, 5, 2012.
N. Fusco and C. Sbordone. Higher integrability of the gradient of minimizers of functionals with nonstandard growth conditions. Comm. Pure Appl. Math., 43(5):673–683, 1990.
J. Gao, P. Zhao, and Y. Zhang. Compact Sobolev embedding theorems involving symmetry and its application. NoDEA Nonlinear Differential Equations Appl., 17(2):161–180, 2010.
J. García-Cuerva and J. L. Rubio de Francia. Weighted Norm Inequalities and Related Topics, volume 116 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1985.
L. Grafakos. Modern Fourier Analysis, volume 250 of Graduate Texts in Mathematics. Springer, New York, 2nd edition, 2008.
J. E. Hutchinson and M. Meier. A remark on the nonuniqueness of tangent cones. Proc. Amer. Math. Soc., 97(1):184–185, 1986.
S. Janson. On functions with conditions on the mean oscillation. Ark. Mat., 14(2):189–196, 1976.
B. Jawerth. Weighted inequalities for maximal operators: linearization, localization and factorization. Amer. J. Math., 108(2):361–414, 1986.
F. John and L. Nirenberg. On functions of bounded mean oscillation. Comm. Pure Appl. Math., 14:415–426, 1961.
E. Kapanadze and T. Kopaliani. A note on maximal operator on \({L}^{p(t)}(\Omega )\) spaces. Georgian Math. J., 15(2):307–316, 2008.
A. Yu. Karlovich. Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. J. Integral Equations Appl., 15(3):263–320, 2003.
A. Yu. Karlovich. Remark on the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights. J. Funct. Spaces Appl., 7(3):301–311, 2009.
A. Yu. Karlovich. Singular integral operators on variable Lebesgue spaces with radial oscillating weights. In Operator algebras, operator theory and applications, volume 195 of Oper. Theory Adv. Appl., pages 185–212. Birkhäuser Verlag, Basel, 2010.
A. Yu. Karlovich and I. M. Spitkovsky. The Cauchy singular integral operator on weighted variable Lebesgue spaces. preprint, 2012.
V. Kokilashvili and A. Meskhi. Weighted criteria for generalized fractional maximal functions and potentials in Lebesgue spaces with variable exponent. Integral Transforms Spec. Funct., 18(9–10):609–628, 2007.
V. Kokilashvili and A. Meskhi. Two-weighted inequalities for Hardy-Littlewood functions and singular integrals in L p(⋅) spaces. J. Math. Sci., 173(6):656–673, 2011.
V. Kokilashvili, N. Samko, and S. Samko. The maximal operator in variable spaces \({L}^{p(\cdot )}(\Omega,\rho )\) with oscillating weights. Georgian Math. J., 13(1):109–125, 2006.
V. Kokilashvili, N. Samko, and S. Samko. The maximal operator in weighted variable spaces L p(⋅). J. Funct. Spaces Appl., 5(3):299–317, 2007.
V. Kokilashvili, N. Samko, and S. Samko. Singular operators in variable spaces \({L}^{p(\cdot )}(\Omega,\rho )\) with oscillating weights. Math. Nachr., 280(9–10):1145–1156, 2007.
V. Kokilashvili and S. Samko. Maximal and fractional operators in weighted L p(x) spaces. Proc. A. Razmadze Math. Inst., 129:145–149, 2002.
V. Kokilashvili and S. Samko. On Sobolev theorem for Riesz-type potentials in Lebesgue spaces with variable exponent. Z. Anal. Anwendungen, 22(4):899–910, 2003.
V. Kokilashvili and S. Samko. Singular integrals in weighted Lebesgue spaces with variable exponent. Georgian Math. J., 10(1):145–156, 2003.
V. Kokilashvili and S. Samko. Maximal and fractional operators in weighted L p(x) spaces. Rev. Mat. Iberoamericana, 20(2):493–515, 2004.
V. Kokilashvili and S. Samko. Operators of harmonic analysis in weighted spaces with non-standard growth. J. Math. Anal. Appl., 352(1):15–34, 2009.
T. Kopaliani. On some structural properties of Banach function spaces and boundedness of certain integral operators. Czechoslovak Math. J., 54(129)(3):791–805, 2004.
T. Kopaliani. Infimal convolution and Muckenhoupt A p(⋅) condition in variable L p spaces. Arch. Math. (Basel), 89(2):185–192, 2007.
T. Kopaliani. On the Muckenchaupt condition in variable Lebesgue spaces. Proc. A. Razmadze Math. Inst., 148:29–33, 2008.
T. Kopaliani. A characterization of some weighted norm inequalities for maximal operators. Z. Anal. Anwend., 29(4):401–412, 2010.
F. Leonetti. Weak differentiability for solutions to nonlinear elliptic systems with p, q-growth conditions. Ann. Mat. Pura Appl. (4), 162:349–366, 1992.
A. K. Lerner. On modular inequalities in variable L p spaces. Arch. Math. (Basel), 85(6):538–543, 2005.
A. K. Lerner. Some remarks on the Hardy-Littlewood maximal function on variable L p spaces. Math. Z., 251(3):509–521, 2005.
A. K. Lerner. On some questions related to the maximal operator on variable L p spaces. Trans. Amer. Math. Soc., 362(8):4229–4242, 2010.
A. K. Lerner and S. Ombrosi. A boundedness criterion for general maximal operators. Publ. Mat., 54(1):53–71, 2010.
A. K. Lerner, S. Ombrosi, and C. Pérez. Sharp A 1 bounds for Calderón-Zygmund operators and the relationship with a problem of Muckenhoupt and Wheeden. Int. Math. Res. Not. IMRN, (6):Art. ID rnm161, 11, 2008.
E. H. Lieb and M. Loss. Analysis, volume 14 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2001.
K. Lindberg. On subspaces of Orlicz sequence spaces. Studia Math., 45:119–146, 1973.
L. Maligranda. Hidegoro Nakano (1909–1974) – on the centenary of his birth. In M. Kato, L. Maligranda, and T. Suzuki, editors, Banach and Function Spaces III: Proceedings of the International Symposium on Banach and Function Spaces 2009, pages 99–171. Yokohama Publishers, 2011.
F. Mamedov and A. Harman. On a weighted inequality of Hardy type in spaces L p(⋅). J. Math. Anal. Appl., 353(2):521–530, 2009.
F. Mamedov and A. Harman. On a Hardy type general weighted inequality in spaces L p(⋅). Integral Equations Operator Theory, 66(4):565–592, 2010.
F. Mamedov and Y. Zeren. On a two-weighted estimation of the maximal operator in the Lebesgue space with variable exponent. Ann. Mat. Pura Appl., 190(2):263–275, 2011.
F. Mamedov and Y. Zeren. On equivalent conditions for the general weighted Hardy type inequality in the space L p(⋅). Z. Anal. Anwend., 31(1):55–74, 2012.
P. Marcellini and G. Papi. Nonlinear elliptic systems with general growth. J. Differential Equations, 221(2):412–443, 2006.
R. A. Mashiyev, B. Çekiç, F. Mamedov, and S. Ogras. Hardy’s inequality in power-type weighted L p(⋅)(0, ∞) spaces. J. Math. Anal. Appl., 334(1):289–298, 2007.
A. Meskhi. Measure of Non-compactness for Integral Operators in Weighted Lebesgue Spaces. Nova Science Publishers, Hauppauge, New York, 2009.
B. Muckenhoupt. Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc., 165:207–226, 1972.
B. Muckenhoupt and R. L. Wheeden. Weighted norm inequalities for fractional integrals. Trans. Amer. Math. Soc., 192:261–274, 1974.
E. Nakai and K. Yabuta. Pointwise multipliers for functions of bounded mean oscillation. J. Math. Soc. Japan, 37(2):207–218, 1985.
H. Nakano. Modulared sequence spaces. Proc. Japan Acad., 27:508–512, 1951.
A. Nekvinda. Hardy-Littlewood maximal operator on \({L}^{p(x)}({\mathbb{R}}^{n})\). Math. Inequal. Appl., 7(2):255–265, 2004.
A. Nekvinda. Maximal operator on variable Lebesgue spaces for almost monotone radial exponent. J. Math. Anal. Appl., 337(2):1345–1365, 2008.
C. J. Neugebauer. Weighted variable L p integral inequalities for the maximal operator on non-increasing functions. Studia Math., 192(1):51–60, 2009.
L. Pick and M. Růžička. An example of a space L p(x) on which the Hardy-Littlewood maximal operator is not bounded. Expo. Math., 19(4):369–371, 2001.
W. Rudin. Real and Complex Analysis. McGraw-Hill Book Co., New York, third edition, 1987.
N. Samko, S. Samko, and B. Vakulov. Weighted Sobolev theorem in Lebesgue spaces with variable exponent. J. Math. Anal. Appl., 335(1):560–583, 2007.
N. Samko, S. Samko, and B. Vakulov. Weighted Sobolev theorem in Lebesgue spaces with variable exponent: corrigendum [mr2340340]. Armen. J. Math., 3(2):92–97, 2010.
S. Samko. Hardy-Littlewood-Stein-Weiss inequality in the Lebesgue spaces with variable exponent. Fract. Calc. Appl. Anal., 6(4):421–440, 2003.
S. Samko, E. Shargorodsky, and B. Vakulov. Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators. II. J. Math. Anal. Appl., 325(1):745–751, 2007.
S. Samko and B. Vakulov. Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators. J. Math. Anal. Appl., 310(1):229–246, 2005.
D. Sarason. Functions of vanishing mean oscillation. Trans. Amer. Math. Soc., 207:391–405, 1975.
C. Sbordone and I. Wik. Maximal functions and related weight classes. Publ. Mat., 38(1):127–155, 1994.
G. Sinnamon. Four questions related to Hardy’s inequality. In Function spaces and applications (Delhi, 1997), pages 255–266. Narosa, New Delhi, 2000.
S. Spanne. Some function spaces defined using the mean oscillation over cubes. Ann. Scuola Norm. Sup. Pisa (3), 19:593–608, 1965.
D. A. Stegenga. Bounded Toeplitz operators on H 1 and applications of the duality between H 1 and the functions of bounded mean oscillation. Amer. J. Math., 98(3):573–589, 1976.
E. M. Stein. Note on the class L log L. Studia Math., 32:305–310, 1969.
G. Talenti. Boundedness of minimizers. Hokkaido Math. J., 19(2):259–279, 1990.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Basel
About this chapter
Cite this chapter
Cruz-Uribe, D.V., Fiorenza, A. (2013). Beyond Log-Hölder Continuity. In: Variable Lebesgue Spaces. Applied and Numerical Harmonic Analysis. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0548-3_4
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0548-3_4
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0547-6
Online ISBN: 978-3-0348-0548-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)