Advertisement

Rough maximal bilinear singular integrals

  • Eva Buriánková
  • Petr HonzíkEmail author
Article
  • 1 Downloads

Abstract

We study the rough maximal bilinear singular integral
$$\begin{aligned} T^{*}_\varOmega (f,g)(x)=\! \sup _{\varepsilon >0}\left| \int _{\mathbb {R}^{n}\setminus B\left( 0,\varepsilon \right) }\! \int _{\mathbb {R}^{n}\setminus B\left( 0,\varepsilon \right) }\!\frac{ \varOmega ((y,z)/|(y,z)|)}{ |(y,z)|^{2n}}f(x-y)g(x-z) dydz\right| , \end{aligned}$$
where \(\varOmega \) is a function in \(L^\infty (\mathbb S^{2n-1})\) with vanishing integral. We prove it is bounded from \(L^p\times L^q\rightarrow L^r,\) where \(1<p,q<\infty \) and \(1/r=1/p+1/q.\) We also discuss results for \(\varOmega \in L^s(\mathbb S^{2n-1}),\)\(1<s<\infty \).

Keywords

Singular integrals Bilinear operators Maximal operators Fourier multipliers 

Mathematics Subject Classification

42B20 42B99 

Notes

References

  1. 1.
    Calderón, A.P., Zygmund, A.: On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Calderón, A.P., Zygmund, A.: On singular integrals. Am. J. Math. 78, 289–309 (1956)CrossRefzbMATHGoogle Scholar
  3. 3.
    Coifman, R.R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Am. Math. Soc. 212, 315–331 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41, 909–996 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Duoandikoetxea, J.: Fourier Analysis, Graduate Studies in Mathematics, vol. 29. American Mathematical Society, Providence (2001)Google Scholar
  6. 6.
    Grafakos, L.: Modern Fourier Analysis, Graduate Texts in Mathematics, vol. 250, 3rd edn. Springer, New York (2014)Google Scholar
  7. 7.
    Grafakos, L., Honzik, P., He, D.: Rough bilinear singular integrals. Adv. Math. 326, 54–78 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Grafakos, L., Torres, R.: Maximal operator and weighted norm inequalities for multilinear singular integrals. Indiana Univ. Math. J. 51, 1261–1276 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grafakos, L., Torres, R.H.: Multilinear Calderón–Zygmund theory. Adv. Math. 165, 124–164 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Meyer, Y.: Wavelets and Operators, Cambridge Studies in Advanced Mathematics, vol. 37. Cambridge University Press, Cambridge (1992)Google Scholar
  11. 11.
    Triebel, H.: Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration, EMS Tracts in Mathematics, vol. 11. European Mathematical Society (EMS), Zürich (2010)CrossRefzbMATHGoogle Scholar

Copyright information

© Universitat de Barcelona 2019

Authors and Affiliations

  1. 1.Charles UniversityPrague 8Czech Republic

Personalised recommendations