On the regularity of one-sided fractional maximal functions
-
Feng Liu
Abstract
In this paper we investigate the regularity properties of one-sided fractional maximal functions, both in continuous case and in discrete case. We prove that the one-sided fractional maximal operators
This work was supported by the NNSF of China (Grant No. 11701333, 11526122), Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents (Grant No. 2015RCJJ053), Research Award Fund for Outstanding Young Scientists of Shandong Province (Grant No. BS2015SF012) and Support Program for Outstanding Young Scientific and Technological Top-notch Talents of College of Mathematics and Systems Science (Grant No. Sxy2016K01)
Acknowledgement.
The author would like to express their deep gratitude to the referee for his/her invaluable comments and suggestions.
References
[1] ALDAZ, J. M.—PÉREZ LÁZARO, J.: Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities, Trans. Amer. Math. Soc. 359 (2007), 2443–2461.10.1090/S0002-9947-06-04347-9Suche in Google Scholar
[2] ANDERSEN, K. F.—SAWYER, E. T.: Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators, Trans. Amer. Math. Soc. 308(2) (1988), 547–558.10.1090/S0002-9947-1988-0930071-4Suche in Google Scholar
[3] BOBER, J.—CARNEIRO, E.—HUGHES, K.—PIERCE, L. B.: On a discrete version of Tanaka’s theorem for maximal functions, Proc. Amer. Math. Soc. 140 (2012), 1669–1680.10.1090/S0002-9939-2011-11008-6Suche in Google Scholar
[4] BREZIS, H.—LIEB, E.: A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.10.1007/978-3-642-55925-9_42Suche in Google Scholar
[5] CARNEIRO, E.—HUGHES, K.: On the endpoint regularity of discrete maximal operators, Math. Res. Lett. 19 (2012), 1245–1262.10.4310/MRL.2012.v19.n6.a6Suche in Google Scholar
[6] CARNEIRO, E.—-MADRID, J.: Derivative bounds for fractional maximal functions, Trans. Amer. Math. Soc. 369 (2017), 4063–4092.10.1090/tran/6844Suche in Google Scholar
[7] CARNEIRO, E.—MOREIRA, D.: On the regularity of maximal operators, Proc. Amer. Math. Soc. 136 (2008), 4395–4404.10.1090/S0002-9939-08-09515-4Suche in Google Scholar
[8] CARNEIRO, E.—SVAITER, B. F.: On the variation of maximal operators of convolution type, J. Func. Anal. 265 (2013), 837–865.10.1016/j.jfa.2013.05.012Suche in Google Scholar
[9] HAJŁASZ, P.—MALY, J.: On approximate differentiability of the maximal function, Proc. Amer. Math. Soc. 138 (2010), 165–174.10.1090/S0002-9939-09-09971-7Suche in Google Scholar
[10] HAJŁASZ, P.—ONNINEN, J.: On boundedness of maximal functions in Sobolev spaces, Ann. Acad. Sci. Fenn. Math. 29 (2004), 167–176.Suche in Google Scholar
[11] KINNUNEN, J.: The Hardy-Littlewood maximal function of a Sobolev function, Israel J. Math. 100 (1997), 117–124.10.1007/BF02773636Suche in Google Scholar
[12] KINNUNEN, J.—LINDQVIST, P.: The derivative of the maximal function, J. reine angew. Math. 503 (1998), 161–167.10.1515/crll.1998.095Suche in Google Scholar
[13] KINNUNEN, J.—SAKSMAN, E.: Regularity of the fractional maximal function, Bull. London Math. Soc. 35 (2003), 529–535.10.1112/S0024609303002017Suche in Google Scholar
[14] KURKA, O.: On the variation of the Hardy-Littlewood maximal function, Ann. Acad. Sci. Fenn. Math. 40 (2015), 109–133.10.5186/aasfm.2015.4003Suche in Google Scholar
[15] LIU, F.: A remark on the regularity of the discrete maximal operator, Bull. Austral. Math. Soc. 95 (2017), 108–120.10.1017/S0004972716000940Suche in Google Scholar
[16] LIU, F.—CHEN, T.—WU, H.: A note on the end-point regularity of the Hardy-Littlewood maximal functions, Bull. Austral. Math. Soc. 94 (2016), 121–130.10.1017/S0004972715001392Suche in Google Scholar
[17] LIU, F.—MAO, S.: On the regularity of the one-sided Hardy-Littlewood maximal functions, Czechoslovak Math. J. 67 (2017), 219–234.10.21136/CMJ.2017.0475-15Suche in Google Scholar
[18] LIU, F.—MAO, S.: Regularity of discrete multisublinear fractional maximal functions, Sci. China Math. 60 (2017), 1461–1476.10.1007/s11425-016-9011-2Suche in Google Scholar
[19] LIU, F.—WU, H.: On the regularity of the multisublinear maximal functions, Canad. Math. Bull. 58 (2015), 808–817.10.4153/CMB-2014-070-7Suche in Google Scholar
[20] LIU, F.—WU, H.: Endpoint regularity of multisublinear fractional maximal functions, Canad. Math. Bull. 60 (2017), 586–603.10.4153/CMB-2016-044-9Suche in Google Scholar
[21] LIU, F.—WU, H.: On the regularity of maximal operators supported by submanifolds, J. Math. Anal. Appl. 453 (2017), 144–158.10.1016/j.jmaa.2017.03.058Suche in Google Scholar
[22] LUIRO, H.: Continuity of the maixmal operator in Sobolev spaces, Proc. Amer. Math. Soc. 135 (2007),243–251.10.1090/S0002-9939-06-08455-3Suche in Google Scholar
[23] LUIRO, H.: On the regularity of the Hardy-Littlewood maximal operator on subdomains of ℝd, Proc. Edinburgh Math. Soc. 53 (2010), 211–237.10.1017/S0013091507000867Suche in Google Scholar
[24] LUIRO, H.: The variation of the maximal function of a radial function, arXiv:1702.00669v1.10.4310/ARKIV.2018.v56.n1.a9Suche in Google Scholar
[25] MADRID, J.: Sharp inequalities for the variation of the discrete maximal function, Bull. Austral. Math. Soc. 95 (2017), 94–107.10.1017/S0004972716000903Suche in Google Scholar
[26] PIERCE, L. B.: On discrete fractional integral operators and mean values of Weyl sums, Bull. London Math. Soc. 43 (2011), 597–612.10.1112/blms/bdq127Suche in Google Scholar
[27] TANAKA, H.: A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function, Bull. Austral. Math. Soc. 65 (2002), 253–258.10.1017/S0004972700020293Suche in Google Scholar
[28] TEMUR, F.: On regularity of the discrete Hardy-Littlewood maximal function, arXiv:1303.3993v1.Suche in Google Scholar
© 2018 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Prof. RNDr. Beloslav Riečan, DrSc., Dr.h.c. *NOV. 10, 1936 – †AUG. 13, 2018
- On the n × n × n Rubik's Cube
- Congruences involving alternating harmonic sums modulo pαqβ
- On the proximity of large primes
- On the factorizations of cubic polynomials with the same discriminant modulo a prime
- Some results on abstract convexity of functions
- Remarks on b-Metric and metric-preserving functions
- Generalization of Ostrowski inequality for convex functions
- On microscopic sets and Fubini Property in all directions
- New subfamily of meromorphic multivalent starlike functions in circular domain involving q-differential operator
- Application of tan(φ(ξ)/2)-expansion method to burgers and foam drainage equations
- Approximate sentinels for diffusion phenomena with pollution
- Remarks on a semilinear system in ℝn motivated by difference equations
- Oscillation tests for difference equations with several non-monotone deviating arguments
- On the regularity of one-sided fractional maximal functions
- A note on a Banach’s fixed point theorem in b-rectangular metric space and b-metric space
- A note about Volterra operator
- Some reverse and numerical radius inequalities
- A class of four-dimensional CR submanifolds in six dimensional nearly Kähler manifolds
- Sequential decreasing strong size properties
- Approximation of Information Divergences for Statistical Learning with Applications
- Wigner's theorem for an infinite set
- A note on automorphisms of lie ideals in prime rings
Artikel in diesem Heft
- Prof. RNDr. Beloslav Riečan, DrSc., Dr.h.c. *NOV. 10, 1936 – †AUG. 13, 2018
- On the n × n × n Rubik's Cube
- Congruences involving alternating harmonic sums modulo pαqβ
- On the proximity of large primes
- On the factorizations of cubic polynomials with the same discriminant modulo a prime
- Some results on abstract convexity of functions
- Remarks on b-Metric and metric-preserving functions
- Generalization of Ostrowski inequality for convex functions
- On microscopic sets and Fubini Property in all directions
- New subfamily of meromorphic multivalent starlike functions in circular domain involving q-differential operator
- Application of tan(φ(ξ)/2)-expansion method to burgers and foam drainage equations
- Approximate sentinels for diffusion phenomena with pollution
- Remarks on a semilinear system in ℝn motivated by difference equations
- Oscillation tests for difference equations with several non-monotone deviating arguments
- On the regularity of one-sided fractional maximal functions
- A note on a Banach’s fixed point theorem in b-rectangular metric space and b-metric space
- A note about Volterra operator
- Some reverse and numerical radius inequalities
- A class of four-dimensional CR submanifolds in six dimensional nearly Kähler manifolds
- Sequential decreasing strong size properties
- Approximation of Information Divergences for Statistical Learning with Applications
- Wigner's theorem for an infinite set
- A note on automorphisms of lie ideals in prime rings
