Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 7 (2016), 1711-1736.

Parabolic weighted norm inequalities and partial differential equations

Juha Kinnunen and Olli Saari

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/apde.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We introduce a class of weights related to the regularity theory of nonlinear parabolic partial differential equations. In particular, we investigate connections of the parabolic Muckenhoupt weights to the parabolic BMO. The parabolic Muckenhoupt weights need not be doubling and they may grow arbitrarily fast in the time variable. Our main result characterizes them through weak- and strong-type weighted norm inequalities for forward-in-time maximal operators. In addition, we prove a Jones-type factorization result for the parabolic Muckenhoupt weights and a Coifman–Rochberg-type characterization of the parabolic BMO through maximal functions. Connections and applications to the doubly nonlinear parabolic PDE are also discussed.

Article information

Source
Anal. PDE, Volume 9, Number 7 (2016), 1711-1736.

Dates
Received: 15 February 2016
Revised: 20 June 2016
Accepted: 28 August 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843355

Digital Object Identifier
doi:10.2140/apde.2016.9.1711

Mathematical Reviews number (MathSciNet)
MR3570236

Zentralblatt MATH identifier
1351.42023

Subjects
Primary: 42B25: Maximal functions, Littlewood-Paley theory 42B37: Harmonic analysis and PDE [See also 35-XX] 35K55: Nonlinear parabolic equations

Keywords
parabolic BMO weighted norm inequalities parabolic PDE doubly nonlinear equations one-sided weight

Citation

Kinnunen, Juha; Saari, Olli. Parabolic weighted norm inequalities and partial differential equations. Anal. PDE 9 (2016), no. 7, 1711--1736. doi:10.2140/apde.2016.9.1711. https://projecteuclid.org/euclid.apde/1510843355


Export citation

References

  • H. Aimar, “Elliptic and parabolic BMO and Harnack's inequality”, Trans. Amer. Math. Soc. 306:1 (1988), 265–276.
  • H. Aimar and R. Crescimbeni, “On one-sided BMO and Lipschitz functions”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. $(4)$ 27:3–4 (1998), 437–456.
  • H. Aimar, L. Forzani, and F. J. Martín-Reyes, “On weighted inequalities for singular integrals”, Proc. Amer. Math. Soc. 125:7 (1997), 2057–2064.
  • L. Berkovits, “Parabolic Muckenhoupt weights in the Euclidean space”, J. Math. Anal. Appl. 379:2 (2011), 524–537.
  • R. R. Coifman and R. Rochberg, “Another characterization of BMO”, Proc. Amer. Math. Soc. 79:2 (1980), 249–254.
  • R. Coifman, P. W. Jones, and J. L. Rubio de Francia, “Constructive decomposition of BMO functions and factorization of $A\sb{p}$ weights”, Proc. Amer. Math. Soc. 87:4 (1983), 675–676.
  • D. Cruz-Uribe, C. J. Neugebauer, and V. Olesen, “The one-sided minimal operator and the one-sided reverse Hölder inequality”, Studia Math. 116:3 (1995), 255–270.
  • E. B. Fabes and N. Garofalo, “Parabolic B.M.O. and Harnack's inequality”, Proc. Amer. Math. Soc. 95:1 (1985), 63–69.
  • L. Forzani, F. J. Martín-Reyes, and S. Ombrosi, “Weighted inequalities for the two-dimensional one-sided Hardy–Littlewood maximal function”, Trans. Amer. Math. Soc. 363:4 (2011), 1699–1719.
  • J. García-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies 116, North-Holland, Amsterdam, 1985.
  • U. Gianazza and V. Vespri, “A Harnack inequality for solutions of doubly nonlinear parabolic equations”, J. Appl. Funct. Anal. 1:3 (2006), 271–284.
  • P.-A. Ivert, N. Marola, and M. Masson, “Energy estimates for variational minimizers of a parabolic doubly nonlinear equation on metric measure spaces”, Ann. Acad. Sci. Fenn. Math. 39:2 (2014), 711–719.
  • P. W. Jones, “Factorization of $A\sb{p}$ weights”, Ann. of Math. $(2)$ 111:3 (1980), 511–530.
  • J. Kinnunen and T. Kuusi, “Local behaviour of solutions to doubly nonlinear parabolic equations”, Math. Ann. 337:3 (2007), 705–728.
  • J. Kinnunen and O. Saari, “On weights satisfying parabolic Muckenhoupt conditions”, Nonlinear Anal. 131 (2016), 289–299.
  • T. Kuusi, J. Siljander, and J. M. Urbano, “Local Hölder continuity for doubly nonlinear parabolic equations”, Indiana Univ. Math. J. 61:1 (2012), 399–430.
  • A. K. Lerner and S. Ombrosi, “A boundedness criterion for general maximal operators”, Publ. Mat. 54:1 (2010), 53–71.
  • F. J. Martín-Reyes, “New proofs of weighted inequalities for the one-sided Hardy-Littlewood maximal functions”, Proc. Amer. Math. Soc. 117:3 (1993), 691–698.
  • F. J. Martín-Reyes and A. de la Torre, “Two weight norm inequalities for fractional one-sided maximal operators”, Proc. Amer. Math. Soc. 117:2 (1993), 483–489.
  • F. J. Martín-Reyes and A. de la Torre, “One-sided BMO spaces”, J. London Math. Soc. $(2)$ 49:3 (1994), 529–542.
  • F. J. Martín-Reyes, P. Ortega Salvador, and A. de la Torre, “Weighted inequalities for one-sided maximal functions”, Trans. Amer. Math. Soc. 319:2 (1990), 517–534.
  • F. J. Martín-Reyes, L. Pick, and A. de la Torre, “$A\sp +\sb \infty$ condition”, Canad. J. Math. 45:6 (1993), 1231–1244.
  • J. Moser, “On Harnack's theorem for elliptic differential equations”, Comm. Pure Appl. Math. 14:3 (1961), 577–591.
  • J. Moser, “A Harnack inequality for parabolic differential equations”, Comm. Pure Appl. Math. 17:1 (1964), 101–134.
  • J. Moser, “Correction to: `A Harnack inequality for parabolic differential equations',”, Comm. Pure Appl. Math. 20:1 (1967), 231–236.
  • S. Ombrosi, “Weak weighted inequalities for a dyadic one-sided maximal function in $\mathbb R\sp n$”, Proc. Amer. Math. Soc. 133:6 (2005), 1769–1775.
  • O. Saari, “Parabolic BMO and global integrability of supersolutions to doubly nonlinear parabolic equations”, Rev. Mat. Iberoam. 32:3 (2016), 1001–1018.
  • E. Sawyer, “Weighted inequalities for the one-sided Hardy–Littlewood maximal functions”, Trans. Amer. Math. Soc. 297:1 (1986), 53–61.
  • N. S. Trudinger, “Pointwise estimates and quasilinear parabolic equations”, Comm. Pure Appl. Math. 21:3 (1968), 205–226.
  • V. Vespri, “On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations”, Manuscripta Math. 75:1 (1992), 65–80.