Advertisement

\(L^p\)-Boundedness of Two Singular Integral Operators of Convolution Type

  • Sten KaijserEmail author
  • John Musonda
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 179)

Abstract

We investigate boundedness properties of two singular integral operators defined on \(L^p\)-spaces \((1<p<\infty )\) on the real line, both as convolution operators on \(L^p(\mathbb R)\) and on the spaces \(L^p(\omega )\), where \(\omega (x)=1/(2\cosh \frac{\pi }{2}x)\). It is proved that both operators are bounded on these spaces and estimates of the norms are obtained. This is achieved by first proving boundedness for \(p=2\) and weak boundedness for \(p=1\), and then using interpolation to obtain boundedness for \(1<p\le 2\). To obtain boundedness also for \(2\le p<\infty \), we use duality in the translation invariant case, while the weighted case is partly based on the expositions on the conjugate function operator in (M. Riesz, Mathematische Zeitschrift, 27, 218–244, 1928) [7].

Keywords

Convolution operators Sech (function) Hilbert transform Hardy space Weak type estimates 

Notes

Acknowledgement

This research was supported by the International Science Program (Uppsala University and Sida foundation) and the Research environment MAM (Mathematics and Applied Mathematics), School of Education, Culture and Communication, Mälardalen University.

References

  1. 1.
    Araaya, T.K.: The Meixner-Pollaczek polynomials and a system of orthogonal polynomials in a strip. J. Comput. Appl. Math. 170, 241–254 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Kaijser, S.: Några nya ortogonala polynom. Normat. 47(4), 156–165 (1999)MathSciNetGoogle Scholar
  3. 3.
    Meixner, J.: Orthogonale polynomsysteme mit einer besonderen gestalt der erzeugenden funktion. J. Lond. Math. Soc. 9, 6–13 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Musonda, J.: Three systems of orthogonal polynomials and associated operators. U.U.D.M. project report. (2012:8)Google Scholar
  5. 5.
    Musonda, J., Kaijser, S.: Three Systems of Orthogonal Polynomials and \(L^2\)-Boundedness of Two Associated Operators(2015). http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-29566
  6. 6.
    Pollaczek, F.: Sur une famille de polynomes orthogonaux qui contient les polynomes d’Hermite et de Laguerre comme cas limites. C. R. Acad. Sci. Paris. 230, 1563–1565 (1950)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Riesz, M.: Sur les fonctions conjuguées. Mathematische Zeitschrift. 27, 218–244 (1928)CrossRefGoogle Scholar
  8. 8.
    Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces, pp. 205–209. Princeton University Press, Princeton, N.J. (1971)zbMATHGoogle Scholar
  9. 9.
    Thorin, O.: An extension of a convexity theorem due to M. Riesz, Kungl. Fysiogr. Sällsk. i Lund Förh. 8, 166–170 (1938)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden
  2. 2.Division of Applied Mathematics, School of Education, Culture and CommunicationMälardalen UniversityVästeråsSweden

Personalised recommendations