Analysis Seminar: From Subrepresentation to Sobolev: New Insights

Abstract: The classical subrepresentation formula shows that the size of a smooth function can be controlled pointwise by the Riesz potential of its gradient. The proof is a simple consequence of the fundamental theorem of calculus and the use of polar coordinates. When combined with the boundedness of the Riesz potential, this pointwise estimate gives rise to Sobolev inequalities in very general settings.
In joint work with C. Hoang and C. Pérez, we extend this inequality to a wide range of operators in harmonic analysis, from rough singular integrals to operators acting along surfaces. As a result, these operators automatically satisfy Sobolev-type mapping properties, including several new ones. We also examine the structure of the subrepresentation formula for operators and give a complete characterization in terms of an endpoint Sobolev inequality with measures.

Host: Brandon Sweeting