On the regularity of maximal operators

Carneiro, Emanuel; Moreira, Diego

doi:10.1090/S0002-9939-08-09515-4

On the regularity of maximal operators
HTML articles powered by AMS MathViewer

by Emanuel Carneiro and Diego Moreira PDF

Proc. Amer. Math. Soc. 136 (2008), 4395-4404 Request permission

Abstract:

We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps $W^{1,p}(\mathbb {R}) \times W^{1,q}(\mathbb {R}) \to W^{1,r}(\mathbb {R})$ with $1 <p,q < \infty$ and $r\geq 1$, boundedly and continuously. The same result holds on $\mathbb {R}^n$ when $r>1$. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versions.

References

J. M. Aldaz and J. Pérez Lázaro, Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2443–2461. MR 2276629, DOI 10.1090/S0002-9947-06-04347-9
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
Piotr Hajłasz and Jani Onninen, On boundedness of maximal functions in Sobolev spaces, Ann. Acad. Sci. Fenn. Math. 29 (2004), no. 1, 167–176. MR 2041705
Juha Kinnunen, The Hardy-Littlewood maximal function of a Sobolev function, Israel J. Math. 100 (1997), 117–124. MR 1469106, DOI 10.1007/BF02773636
Juha Kinnunen and Peter Lindqvist, The derivative of the maximal function, J. Reine Angew. Math. 503 (1998), 161–167. MR 1650343
Juha Kinnunen and Eero Saksman, Regularity of the fractional maximal function, Bull. London Math. Soc. 35 (2003), no. 4, 529–535. MR 1979008, DOI 10.1112/S0024609303002017
Michael T. Lacey, The bilinear maximal functions map into $L^p$ for $2/3<p\leq 1$, Ann. of Math. (2) 151 (2000), no. 1, 35–57. MR 1745019, DOI 10.2307/121111
Hannes Luiro, Continuity of the maximal operator in Sobolev spaces, Proc. Amer. Math. Soc. 135 (2007), no. 1, 243–251. MR 2280193, DOI 10.1090/S0002-9939-06-08455-3
Diego R. Moreira and Eduardo V. Teixeira, On the behavior of weak convergence under nonlinearities and applications, Proc. Amer. Math. Soc. 133 (2005), no. 6, 1647–1656. MR 2120260, DOI 10.1090/S0002-9939-04-07876-1
Eduardo V. Teixeira, Strong solutions for differential equations in abstract spaces, J. Differential Equations 214 (2005), no. 1, 65–91. MR 2143512, DOI 10.1016/j.jde.2004.11.006