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Hardy-type inequalities for fractional powers of the Dunkl--Hermite operator

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We prove Hardy-type inequalities for the conformally invariant fractional powers of the Dunkl—Hermite operator. Consequently, we also obtain Hardy inequalities for the fractional harmonic oscillator as well.
The strategy is as follows: first, by introducing suitable polar coordinates, we reduce the problem to the Laguerre setting. Then, we push forward an argument developed by R. L. Frank, E. H. Lieb and R. Seiringer, initially developed in the Euclidean setting, to get a Hardy inequality for the fractional-type Laguerre operator. Such argument is based on two facts: first, to get an integral representation for the corresponding fractional operator, and second, to write a proper ground state representation.
This is joint work with \'O. Ciaurri (Universidad de La Rioja, Spain) and S. Thangavelu (Indian Institute of Science of Bangalore, India).

This talk is part of the Isaac Newton Institute Seminar Series series.

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