Internet Analysis Seminar

The topic of the Internet Analysis Seminar this year will be related to harmonic analysis associated to second-order elliptic operators. The canonical case of this arises when the operator is the standard Laplacian, and then leads to the basic material considered in standard harmonic analysis courses, such as the boundedness of the Riesz transforms on $L^p$, duality between $H^1$ and BMO and other related topics.

Changing the operator to $L=-\operatorname{div}\left(A(x)\nabla f\right)$ where $A(x)$ is a matrix with complex bounded entries that satisfies a elliptic estimate the generalizations of the standard results encountered in harmonic analysis are obtained when considering the Riesz transforms associated to the square root of the operator $L$, $\nabla L^{-\frac{1}{2}}$. Topics covered will focus on the modifications in the techniques and proofs necessary to understand how the story changes when proving the boundedness of these Riesz transforms and the $H^1-BMO$ duality statements when no longer working with the Laplacian but instead working with more general elliptic operators.