|
Session 18. Harmonic analysis, orthogonal expansions and Dunkl theory
|
Dimension free \(L^p\) estimates for Riesz transforms via an \(H^{\infty}\) joint functional calculus |
Błażej Wróbel, Uniwersytet Wrocławski, Poland
|
|
|
In 1983 E. M. Stein proved dimension free \(L^p\) bounds for classical Riesz transforms on \(R^d\). Since then many authors studied the phenomenon of dimension free estimates for Riesz transforms defined in various contexts. In this talk we present a fairly general scheme for deducing the dimension free \(L^p\) boundedness of \(d\)-dimensional Riesz transforms from the \(L^p\) boundedness of one-dimensional Riesz transforms. The crucial tool we use is an \(H^{\infty}\) joint functional calculus for strongly commuting operators. The scheme is applicable to all Riesz transforms acting on 'product' spaces, e.g.: Riesz transforms connected with (classical) multi-dimensional orthogonal expansions, Riesz transforms in the 'product' (rational) Dunkl setting, and discrete Riesz transforms on products of groups having polynomial growth.
|
|
| Print version |
|
