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Weighted Norm Estimates for Singular Integrals with LlogL Kernels: Regularity of Weak Solutions of Some Degenerate Quasilinear Equations

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Special Functions, Partial Differential Equations, and Harmonic Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 108))

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Summary

The purpose of this paper is to summarize some known results about two different types of problems. A number of open questions related to each of the problems are indicated. The presentation is largely descriptive; no proofs are given, but they can be found in the appropriate papers listed among the references.

Dedicated to Calixto Caldéron

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References

  1. Calderón, C.P.: Differentiation through starlike sets in R m. Stud. Math. 48, 1–13 (1973)

    MATH  Google Scholar 

  2. Calderón, A.P., Zygmund, A.: On singular integrals. Am. J. Math. 78, 289–309 (1956)

    Article  MATH  Google Scholar 

  3. Chanillo, S., Watson, D.K., Wheeden, R.L.: Some integral and maximal operators related to starlike sets. Stud. Math. 107, 223–255 (1993)

    MATH  MathSciNet  Google Scholar 

  4. Christ, M., Rubio de Francia, J.L.: Weak type (1,1) bounds for rough operators, II. Invent. Math. 93, 225–237 (1988)

    Google Scholar 

  5. Cordoba, A., Fefferman, C.L.: A weighted norm inequality for singular integrals. Stud. Math. 57, 97–101 (1976)

    MATH  MathSciNet  Google Scholar 

  6. Duandikoetxea, J.: Weighted norm inequalities for homogeneous singular integrals. Trans. Am. Math. Soc. 336, 869–880 (1993)

    Article  Google Scholar 

  7. Fedii, V.S.: On a criterion for hypoellipticity. Math. USSR Sb. 14, 15–45 (1971)

    Article  Google Scholar 

  8. Gilbarg, D., Trudinger, N: Elliptic Partial Differential Equations of Second Order, revised 3rd printing. Springer, Berlin (1998)

    Google Scholar 

  9. Kurtz, D.S., Wheeden, R.L.: Results on weighted norm inequalities for multipliers. Trans. Am. Math. Soc. 255, 343–362 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kurtz, D.S., Wheeden, R.L.: A note on singular integrals with weights. Proc. Am. Math. Soc. 81, 391–397 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kusuoka, S., Stroock, D.: Applications of the Malliavin Calculus II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32, 1–76 (1985)

    MATH  MathSciNet  Google Scholar 

  12. Muckenhoupt, B., Wheeden, R.L.: Weighted norm inequalities for singular and fractional integrals. Trans. Am. Math. Soc. 161, 249–258 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  13. Pérez, C.: Weighted norm inequalities for singular integral operators. J. Lond. Math. Soc. 49, 296–308 (1994)

    Article  MATH  Google Scholar 

  14. Rios, C., Sawyer, E.T., Wheeden, R.L.: A priori estimates for infinitely degenerate quasilinear equations. Differ. Integ. Equ. 21, 131–200 (2008)

    MATH  MathSciNet  Google Scholar 

  15. Rios, C., Sawyer, E.T., Wheeden, R.L.: Hypoellipticity for infinitely degenerate quasilinear equations and the Dirichlet problem. J. d’Analyse Math. 119, 1–62 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. Rios, C., Sawyer, E.T., Wheeden, R.L.: A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations. Adv. Math. 193, 373–415 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Sawyer, E.T., Wheeden, R.L.: Hölder continuity of weak solutions to subelliptic equations with rough coefficients Mem. Am. Math. Soc. 847, p. 157 (2006)

    MathSciNet  Google Scholar 

  18. Sawyer, E.T., Wheeden, R.L.: Degenerate Sobolev spaces and regularity of subelliptic equations. Trans. Am. Math. Soc. 362, 1869–1906 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Sawyer, E.T., Wheeden, R.L.: Regularity of degenerate Monge-Ampère and prescribed Gaussian curvature equations in two dimensions Potential Anal. 24, 267–301 (2006)

    MATH  MathSciNet  Google Scholar 

  20. Treves, F.: Introduction to Pseudodifferential and Fourier Integral Operators. Plenum Press, New York (1980)

    Book  MATH  Google Scholar 

  21. Watson, D.K.: Weighted estimates for singular integrals via Fourier transform estimates. Duke Math. J. 60, 389–399 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  22. Watson, D.K.: Weighted estimates for singular integrals via Fourier transform estimates. Duke math. J. 60, 389–399 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  23. Watson, D.K., Wheeden, R.L.: Norm estimates and representations for Calderón-Zygmund operators using averages over starlike sets. Trans. Am. Math. Soc. 351, 4127–4171 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Watson, D.K., Wheeden, R.L.: Averages over starlike sets, starlike maximal functions, and homogeneous singular integrals. Trans. Am. Math. Soc. 363, 5179–5206 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Wilson, J.M.: Weighted norm inequalities for the continuous square function. Trans. Am. Math. Soc. 314, 661–692 (1989)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Hill Center Rutgers University, Piscataway, NJ, USA

    Richard L. Wheeden

Authors
  1. Richard L. Wheeden
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Correspondence to Richard L. Wheeden .

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Editors and Affiliations

  1. DePaul University, Chicago, Illinois, USA

    Constantine Georgakis

  2. Georgia Southern University, Statesboro, Georgia, USA

    Alexander M. Stokolos

  3. Roosevelt University, Chicago, Illinois, USA

    Wilfredo Urbina

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Wheeden, R.L. (2014). Weighted Norm Estimates for Singular Integrals with LlogL Kernels: Regularity of Weak Solutions of Some Degenerate Quasilinear Equations. In: Georgakis, C., Stokolos, A., Urbina, W. (eds) Special Functions, Partial Differential Equations, and Harmonic Analysis. Springer Proceedings in Mathematics & Statistics, vol 108. Springer, Cham. https://doi.org/10.1007/978-3-319-10545-1_16

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