On weak reverse integral inequalities for mean oscillations

Franciosi, Michelangelo

doi:10.1090/S0002-9939-1991-1068122-7

On weak reverse integral inequalities for mean oscillations
HTML articles powered by AMS MathViewer

by Michelangelo Franciosi PDF

Proc. Amer. Math. Soc. 113 (1991), 105-112 Request permission

Abstract:

We prove that if $f$ verifies a reverse Hölder inequality with exponent $p,1 < p < + \infty$, then ${(Mf + {f^\# })^p}$ is a ${A_1}$-weight of Muckenhoupt, where $Mf$ is the Hardy-Littlewood maximal function and ${f^\# }$ the Fefferman-Stein maximal function.

References

Colin Bennett and Robert Sharpley, Weak-type inequalities for $H^{p}$ and BMO, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 201–229. MR 545259
Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802

On Gurov-Reshetnyak classes

B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in $\textbf {R}^{n}$, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), no. 2, 257–324. MR 731786, DOI 10.5186/aasfm.1983.0806
Anna Canale, A higher integrability theorem from reverse Hölder weighted inequalities with different supports, Boll. Un. Mat. Ital. A (7) 5 (1991), no. 2, 137–146 (English, with Italian summary). MR 1120373, DOI 10.1287/ijoc.1050.0144
R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116. MR 643158, DOI 10.1080/03605308208820218
Michelangelo Franciosi, Maximum principle for second order elliptic equations and applications, J. Math. Anal. Appl. 138 (1989), no. 2, 343–348. MR 991028, DOI 10.1016/0022-247X(89)90294-1
Michelangelo Franciosi, Weighted rearrangements and higher integrability results, Studia Math. 92 (1989), no. 2, 131–139. MR 986944, DOI 10.4064/sm-92-2-131-139
Michelangelo Franciosi and Gioconda Moscariello, Higher integrability results, Manuscripta Math. 52 (1985), no. 1-3, 151–170. MR 790797, DOI 10.1007/BF01171490

A note on the maximum principle for 2nd order non variational linear elliptic equations

XXXV

F. W. Gehring, The $L^{p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. MR 402038, DOI 10.1007/BF02392268

The Hardy-Littlewood maximal theorems

Tadeusz Iwaniec, On $L^{p}$-integrability in PDEs and quasiregular mappings for large exponents, Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), no. 2, 301–322. MR 686647, DOI 10.5186/aasfm.1982.0719
Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, DOI 10.1090/S0002-9947-1972-0293384-6
Carlo Sbordone, Higher integrability from reverse integral inequalities, Methods of functional analysis and theory of elliptic equations (Naples, 1982) Liguori, Naples, 1983, pp. 347–357. MR 819021
Carlo Sbordone, On some integral inequalities and their applications to the calculus of variations, Boll. Un. Mat. Ital. C (6) 5 (1986), no. 1, 73–94 (1987). MR 897186

Some reverse integral inequalities

Ingemar Wik, Note on a theorem by Reshetnyak-Gurov, Studia Math. 86 (1987), no. 3, 287–290. MR 917054, DOI 10.4064/sm-86-3-287-290

On Muckenhoupt’s classes of weight functions