# Analysis Seminar 2022

Current contact: Gerardo Mendoza

The seminar takes place Mondays 2:30 - 3:30 pm either in Wachman 617 or virtually via Zoom. Please see the announcment of the specific meeting. For virtual meetings, please write if you wish to join. Click on title for abstract.

• Monday February 7, 2022 at 14:30, Virtual via Zoom

Francisco Villarroya, California Polytechnic State University

We prove sufficient conditions for a Calder\'on-Zygmund operator to belong to the $p$-th Schatten-von Neumann classes $S_{p}(L^{2}(\mathbb R^{d}))$.

As in the classical $T1$ theory, the conditions are given in terms of the smoothness of the operator kernel, and the action of both the operator and its adjoint on the function $1$.

To prove membership to the Schatten class when $p>2$ we develop new bump estimates for composed compact Calder\'on-Zygmund operators, and a new extension of Carleson's Embedding Theorem.

• Monday February 14, 2022 at 14:30, Virtual via Zoom

Gabriel Araujo, University of Sao Paulo at Sao Carlos

Linear PDOs can act on various (generalized) function spaces, provided their coefficients are sufficiently regular: smooth, real-analytic and/or Gevrey spaces, as well as their generalized counterparts, just to name a few. It may then be of interest to establish properties of regularity and solvability (either local or global; several flavors of hypoellipticity; properties of the associated cohomology spaces for systems; and so on) of such PDOs in some of these spaces, sometimes providing radically different answers depending on the space under study.

Some results in the literature, however, establish relationships among such properties for whole classes of operators:

-- Greenfield (1972)   proved that   for operators with constant coefficients on tori   global hypoellipticity   implies   global analytic-hypoellipticity.

-- Dealing with   differential complexes associated to   locally integrable structures,   Caetano and Cordaro (2011)   proved that if in a given degree   the complex is locally solvable   in the smooth setup   then it is also locally solvable   in the Gevrey setup (same degree),   while Ragognette (2019),   using similar methods,   relates these with   local solvability   in the sense of   Gevrey ultradistributions.

-- Still dealing with   locally integrable structures,   Malaspina and Nicola (2014)   conjecture another connection   between smooth and Gevrey   local solvability   (a kind of converse   to the result of   Caetano and Cordaro),   which is currently open   except for a few cases.

-- In a joint work   with Cordaro (2019)   on analytic structures   we connect   (in a few particular cases)   local solvability   in the smooth sense   with a property called   semi-local analytic solvability.

Here we are interested in global properties for systems of left-invariant differential operators on compact Lie groups: regularity properties, properties on the closedness of the range and dimension of  cohomology spaces for complexes, when acting on various function spaces. Extending the methods of Greenfield and Wallach (1973) to systems, we obtain abstract characterizations for these properties and use them to derive some generalizations of Greenfield's result, as well as global versions of the result of Caetano and Cordaro for left-invariant involutive structures.

• Monday March 7, 2022 at 14:30, Virtual via Zoom

Hamid Meziani, Florida International University

For a nonsingular planar vector field $L$ with complex-valued coefficient that has local first integrals that are open maps, we consider the equation $Lu=f$ and show its solutions can be represented through a generalized Cauchy integral operator.

• Monday March 14, 2022 at 14:30, Virtual via Zoom

Clara Aldana, Universidad del Norte, Colombia

In this talk I will first talk about the isospectral problem in geometry and about isospectrality of Strum-Liouville operators on a finite interval in the simplified form of a Schrödinger operator. I will mention very interesting known results about isospectral potentials. I will introduce generalizations of the concept of isospectrality like quasi-isospectrality, and will present what we know so far about quasi-isospectral potentials. The work presented here is still on-going joint work with Camilo Perez.

• Monday March 21, 2022 at 14:30, Virtual via Zoom

Nelia Charalambous, University of Cyprus​

In this talk we will consider the essential spectrum of the Laplacian on differential forms over noncompact manifolds. We will see a brief overview of known results and discuss the main differences between the function and form spectrum. One interesting problem in the area is finding sufficient and general enough conditions on the manifold so that the essential spectrum on forms is a connected set. We will see that over asymptotically flat manifolds this is the case. The proof involves the study of the structure of the manifold at infinity via Cheeger-Fukaya-Gromov theory and Cheeger-Colding theory, combined with a generalized Weyl criterion for the computation of the spectrum. Finally, we present some recent results on the form spectrum of negatively curved manifolds.

• Monday March 28, 2022 at 14:30, Virtual via Zoom

Emanuel Carneiro, ICTP

This will be a talk at in harmonic analysis with little bits of number theory. We will discuss some of the different faces of the uncertainty principle for the Fourier transform and its recent connections to lattices and packing problems, and then slowly move towards uncharted territories. The required background in analysis will be minimal.

• Monday April 4, 2022 at 14:30, Virtual via Zoom

Tadele Mengesha, The University of Tennessee

I will present a result on Sobolev regularity of weak solutions to linear nonlocal equations. The theory we develop is concerned with obtaining higher integrability and differentiability of solutions of nonlocal equations. Under the assumption of uniform Holder continuity of coefficients, weak solutions from the energy space that correspond to highly integrable right hand side will be shown to have improved Sobolev regularity along the differentiability scale in addition to the expected integrability gain.  This result is consistent with self-improving properties of nonlocal equations that has been observed by other earlier works. To prove our result, we use a perturbation argument where optimal regularity of solutions of a simpler equation is systematically used to derive an improved regularity for the solution of the nonlocal equation.

• Monday April 11, 2022 at 14:30,

No meeting

• Monday April 18, 2022 at 14:30, Virtual via Zoom

Luis Fernando Ragognette

In collaboration with Gabriel Araújo (ICMC-USP) and Igor A. Ferra (Federal Univ. of ABC), we studied global solvability of operators on compact manifolds.

The goals of this talk are to discuss how a weak notion of global hypoellipticity implies global solvability and also to give necessary and sufficient conditions for global solvability of a class of operators of type sum of squares defined on a product compact manifolds.

• Monday April 25, 2022 at 14:30, Virtual via Zoom

Max Reinhold Jahnke, Federal University of São Carlos

It is well known that the De Rham cohomology of a compact Lie group is isomorphic to the Chevalley-Eilenberg complex. While the former is a topological invariant of the Lie group, the latter can be computed by using simple linear algebra methods. In this talk, we discuss how to obtain an injective homomorphism between the cohomology spaces associated with left-invariant involutive structures and the cohomology of a generalized Chevalley-Eilenberg complex.

We discuss some cases in which the homomorphism is surjective, such as the Dolbeault cohomology and certain elliptic and CR structures. The results provide new insights regarding the general theory of involutive structures as, for example, they reveal algebraic obstructions for solvability for the associated differential complexes.