Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
bilinear maximal operator; Triebel-Lizorkin space; Besov space; Lipschitz space; $p$-quaiscontinuous; approximate differentiability
bilinear maximal operator; Triebel-Lizorkin space; Besov space; Lipschitz space; $p$-quaiscontinuous; approximate differentiability
Summary:
We study the regularity properties of bilinear maximal operator. Some new bounds and continuity for the above operators are established on the Sobolev spaces, Triebel-Lizorkin spaces and Besov spaces. In addition, the quasicontinuity and approximate differentiability of the bilinear maximal function are also obtained.
References:
[1] Carneiro, E., Moreira, D.: On the regularity of maximal operators. Proc. Am. Math. Soc. 136 (2008), 4395-4404. DOI 10.1090/S0002-9939-08-09515-4 | MR 2431055 | Zbl 1157.42003
[2] Federer, H., Ziemer, W. P.: The Lebesgue set of a function whose distribution derivatives are $p$-th power summable. Indiana Univ. Math. J. 22 (1972), 139-158. DOI 10.1512/iumj.1972.22.22013 | MR 0435361 | Zbl 0238.28015
[3] Frazier, M., Jawerth, B., Weiss, G.: Littlewood-Paley Theory and The Study of Function Spaces. Regional Conference Series in Mathematics 79. AMS, Providence (1991). DOI 10.1090/cbms/079 | MR 1107300 | Zbl 0757.42006
[4] Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften 224. Springer, Berlin (1983). DOI 10.1007/978-3-642-61798-0 | MR 0737190 | Zbl 0562.35001
[5] Grafakos, L.: Classical and Modern Fourier Analysis. Pearson, Upper Saddle River (2004). MR 2449250 | Zbl 1148.42001
[6] Hajłasz, P., Malý, J.: On approximate differentiability of the maximal function. Proc. Am. Math. Soc. 138 (2010), 165-174. DOI 10.1090/S0002-9939-09-09971-7 | MR 2550181 | Zbl 1187.42016
[7] Hajłasz, P., Onninen, J.: On boundedness of maximal functions in Sobolev spaces. Ann. Acad. Sci. Fenn., Math. 29 (2004), 167-176. MR 2041705 | Zbl 1059.46020
[8] Kilpeläinen, T., Kinnunen, J., Martio, O.: Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12 (2000), 233-247. DOI 10.1023/A:1008601220456 | MR 1752853 | Zbl 0962.46021
[9] Kinnunen, J.: The Hardy-Littlewood maximal function of a Sobolev function. Isr. J. Math. 100 (1997), 117-124. DOI 10.1007/BF02773636 | MR 1469106 | Zbl 0882.43003
[10] Kinnunen, J., Lindqvist, P.: The derivative of the maximal function. J. Reine. Angew. Math. 503 (1998), 161-167. DOI 10.1515/crll.1998.095 | MR 1650343 | Zbl 0904.42015
[11] Kinnunen, J., Saksman, E.: Regularity of the fractional maximal function. Bull. Lond. Math. Soc. 35 (2003), 529-535. DOI 10.1112/S0024609303002017 | MR 1979008 | Zbl 1021.42009
[12] Korry, S.: Boundedness of Hardy-Littlewood maximal operator in the framework of Lizorkin-Triebel spaces. Rev. Mat. Complut. 15 (2002), 401-416. DOI 10.5209/rev_REMA.2002.v15.n2.16899 | MR 1951818 | Zbl 1033.42013
[13] Korry, S.: A class of bounded operators on Sobolev spaces. Arch. Math. 82 (2004), 40-50. DOI 10.1007/s00013-003-0416-x | MR 2034469 | Zbl 1061.46034
[14] Lacey, M. T.: The bilinear maximal function map into $L^p$ for $2/3< p \leq 1$. Ann. Math. (2) 151 (2000), 35-57. DOI 10.2307/121111 | MR 1745019 | Zbl 0967.47031
[15] Liu, F., Liu, S., Zhang, X.: Regularity properties of bilinear maximal function and its fractional variant. Result. Math. 75 (2020), Article ID 88, 29 pages. DOI 10.1007/s00025-020-01215-2 | MR 4105758 | Zbl 1440.42086
[16] Liu, F., Wu, H.: On the regularity of the multisublinear maximal functions. Can. Math. Bull. 58 (2015), 808-817. DOI 10.4153/CMB-2014-070-7 | MR 3415670 | Zbl 1330.42014
[17] Liu, F., Wu, H.: On the regularity of maximal operators supported by submanifolds. J. Math. Anal. Appl. 453 (2017), 144-158. DOI 10.1016/j.jmaa.2017.03.058 | MR 3641765 | Zbl 1404.42036
[18] Luiro, H.: Continuity of the maximal operator in Sobolev spaces. Proc. Am. Math. Soc. 135 (2007), 243-251. DOI 10.1090/S0002-9939-06-08455-3 | MR 2280193 | Zbl 1136.42018
[19] Luiro, H.: On the regularity of the Hardy-Littlewood maximal operator on subdomains of $\Bbb R^n$. Proc. Edinb. Math. Soc., II. Ser. 53 (2010), 211-237. DOI 10.1017/S0013091507000867 | MR 2579688 | Zbl 1183.42025
[20] Triebel, H.: Theory of Function Spaces. Monographs in Mathematics 78. Birkhäuser, Basel (1983). DOI 10.1007/978-3-0346-0416-1 | MR 0781540 | Zbl 0546.46027
[21] Whitney, H.: On totally differentiable and smooth functions. Pac. J. Math. 1 (1951), 143-159. DOI 10.2140/pjm.1951.1.143 | MR 0043878 | Zbl 0043.05803
[22] Yabuta, K.: Triebel-Lizorkin space boundedness of Marcinkiewicz integrals associated to surfaces. Appl. Math., Ser. B (Engl. Ed.) 30 (2015), 418-446. DOI 10.1007/s11766-015-3358-8 | MR 3434042 | Zbl 1349.42037
Search
Partner of
