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Abstract: It is a famous result  of M. Riesz that the Szego projection operator, initially defined as the orthogonal projection from the space \(L^2(\mathbb{T})\) of square integrable functions on the circle to the Hardy space \(H^2(\mathbb{D}) \),  extends 

continuously as a projection operator from \(L^p(\mathbb{T})\) onto \(H^p(\mathbb{D})\). There is a long history of similar results in the setting of Bergman spaces, and a long list of domains where an analogous statement does not hold in the Bergman setting. We try to understand the geometric distinction between the Hardy and the Bergman situations in \(L^p\), and propose a new projection operator on Reinhardt domains which is expected to have better mapping properties. We verify that the new operator satisfies \(L^p\) estimates in some situations where the Bergman projection operator does not satisfy such estimates.  This is joint work with Luke Edholm of the University of Vienna.