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Session 18. Harmonic analysis, orthogonal expansions and Dunkl theory
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Riesz-Jacobi transforms as principal value integrals
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Tomasz Z. Szarek, Institute of Mathematics, Polish Academy of Sciences, Poland
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The talk is based on the joint work with Alejandro J. Castro and Adam Nowak
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We establish integral representations for Riesz transforms of all orders in the context
of classical Jacobi expansions. We show that in case of odd orders these operators express
as principal value integrals against kernels having non-integrable singularities on the diagonal.
On the other hand, in case of even orders Riesz-Jacobi transforms are not singular operators,
in the sense that they are given as usual integrals plus or minus, depending on the order,
the identity operator.
This extends and refines the results obtained in [1] in the ultraspherical situation, which is
a very special case of the Jacobi setting.
We present an approach that is simpler and more classic than the one elaborated in [1].
However, our arguments require some auxiliary technical results such as quite precise estimates of
derivatives of the Poisson-Jacobi kernel. The latter are obtained by means of the techniques
developed recently in [3,4].
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References
- J.J.Betancor, J.C.Fariña, L.Rodriguez-Mesa, R.Testoni, Higher order Riesz
transforms in the ultraspherical setting as principal value integral operators, Integr.
Equ. Oper. Theory 70, 2011, 511-539.
- A.J.Castro, A.Nowak, T.Z.Szarek, Riesz-Jacobi transforms as principal value
integrals, preprint 2014, 1-30.
- A.Nowak, P.Sjögren, Calder`n-Zygmund operators related to Jacobi expansions,
J. Fourier Anal. Appl. 18, 2012, 717-749.
- A.Nowak, P.Sjögren, T.Z.Szarek, Analysis related to all admissible parameters
in the Jacobi setting, preprint 2012. arXiv:1211.3270
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