Riesz–Jacobi Transforms as Principal Value Integrals

  • Alejandro J. Castro
  • Adam Nowak
  • Tomasz Z. Szarek


We establish an integral representation for the Riesz transforms naturally associated with classical Jacobi expansions. We prove that the Riesz–Jacobi transforms of odd orders express as principal value integrals against kernels having non-integrable singularities on the diagonal. On the other hand, we show that the Riesz–Jacobi transforms of even orders are not singular operators. In fact they are given as usual integrals against integrable kernels plus or minus, depending on the order, the identity operator. Our analysis indicates that similar results, existing in the literature and corresponding to several other settings related to classical discrete and continuous orthogonal expansions, should be reinvestigated so as to be refined and in some cases also corrected.


Jacobi expansion Jacobi operator Riesz transform Integral representation Principal value integral 

Mathematical Subject Classification




The first-named author was partially supported by MTM2010/17974 and also by an FPU Grant from the Government of Spain. Research of the second-named author was partially supported by the National Science Centre of Poland, Project No. 2013/09/B/ST1/02057. The third-named author was partially supported by the National Science Centre of Poland, Project No. 2012/05/N/ST1/02746.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Alejandro J. Castro
    • 1
  • Adam Nowak
    • 2
  • Tomasz Z. Szarek
    • 2
  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden
  2. 2.Institute of MathematicsPolish Academy of SciencesWarszawaPoland

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