Gianmarco Brocchi
Assistant Professor of mathematics at the University of Iceland.
Title: Progress on the Kate square root estimate
Abstract:
The Kato square root estimate is a $L^2$ inequality concerning perturbations of the Laplacian. While the one-dimensional case was established in the 1980s by Coifman, McIntosh, and Meyer, the higher-dimensional extension — where the perturbation takes the form of a matrix-valued function $A$ in the divergence-form operator $-\mathrm{div}(A \nabla)$ — remained open for two more decades.
In this talk, I will introduce the Kato square root estimate and describe the first-order method, a technique that reduces the second-order operator $-\mathrm{div}(A \nabla)$ to a first-order, bisectorial operator $DB$. This method exploits a connection between harmonic and holomorphic extensions and allows us to rewrite the original estimate as a question about the boundedness of the holomorphic functional calculus for $D B$.
I will also present recent results in the theory, including extensions to Riemannian manifolds and to operators with degenerate coefficients, where the matrix $A(x)$ may lack uniform bounds or accretivity and can exhibit singular behaviour. What types of singularities can be handled? On which classes of manifolds? And in Euclidean space, can one treat anisotropic singularities, namely those that vary with direction?
New results are part of ongoing joint work with Andreas Rosén.