The 17th Prairie Analysis Seminar

Mathematics DepartmentKansas State University

(Virtual event over Zoom)

November 5-6, 2021

1 Friday November 5th, 2021 2

2 Saturday November 6th, 2021 5

3 Titles and abstracts of invited talks 8

4 Titles and abstracts of contributed talks 10

Moderators and Zoom information

Event Moderator ZoomOpening and welcome Virginia Naibo https://ksu.zoom.us/j/96672828264Talks by invited speakers D. Maldonado and V. Naibo https://ksu.zoom.us/j/96672828264Parallel sessions I-A, II-A, III-A, IV-A Diego Maldonado https://ksu.zoom.us/j/92033473537Parallel sessions I-B, II-B, III-B, IV-B Virginia Naibo https://ksu.zoom.us/j/93740953837Parallel sessions I-C, II-C, III-C, IV-C Dionyssis Mantzavinos https://ksu.zoom.us/j/92091623313Parallel sessions I-D, II-D, III-D, IV-D Shuanglin Shao https://ksu.zoom.us/j/96672828264

Conference website: math.k-state.edu/pas

1 Friday November 5th, 2021All times are in Central Daylight Time (CDT)

Friday 9:50am-10:00am Opening and welcome

Friday 10:00am-11:00am Malabika Pramanik, University of British ColumbiaSets and configurations

Friday 11:00am-11:30am break

Friday 11:30am-12:30pm Parallel Sessions I

Friday 12:30pm-2:00pm Lunch break

Friday 2:00pm-3:00pm Marıa Jesus Carro, Universidad Complutense de MadridSolving Dirichlet and Neumann problems

in Lipschitz Domains at the end-point

Friday 3:00pm-3:30pm break

Friday 3:30pm-5:30pm Parallel Sessions II

Parallel Session I-A (Moderator: Maldonado; Zoom: https://ksu.zoom.us/j/92033473537)

Friday 11:30am-12pm: Emily Dautenhahn, Cornell UniversityHeat kernel estimates on manifolds with ends with mixed boundary condition

Friday 12pm-12:30pm: Chamsol Park, University of New MexicoEigenfunction Restriction Estimates on curves with nonvanishing geodesic curvatures

Parallel Session I-B (Moderator: Naibo; Zoom: https://ksu.zoom.us/j/93740953837)

Friday 11:30am-12pm: Eduard Roure Perdices, UPV/EHUSawyer-type inequalities for Lorentz spaces

Friday 12pm-12:30pm: Sergi Arias, Stockholm UniversitySome endpoint estimates for bilinear Coifman-Meyer multipliers

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Parallel Session I-C (Moderator: Mantzavinos; Zoom: https://ksu.zoom.us/j/92091623313)

Friday 11:30am-12pm: Tongou Yang, University of British ColumbiaDecoupling for smooth functions

Friday 12pm-12:30pm: Zane Li, Indiana University BloomingtonDecoupling for Cantor sets on the line and parabola

Parallel Session I-D (Moderator: Shao; Zoom: https://ksu.zoom.us/j/96672828264)

Friday 11:30am-12pm: Xueying Yu, University of WashingtonHigh-low method of NLS on the hyperbolic space

Friday 12pm-12:30pm: Abba Ramadan, University of KansasOn the standing waves of the Schrodinger equation with concentrated nonlinearity

Parallel Session II-A (Moderator: Maldonado; Zoom: https://ksu.zoom.us/j/92033473537)

Friday 3:30pm-4:00pm: Cristian Rios, University of CalgaryRegularity theory for degenerate elliptic operators

Friday 4:00pm-4:30pm: Daniel Restrepo, University of Texas at AustinThe isoperimetric problem for phase transitions

Friday 4:30pm-5:00pm: Dylan Langharst, Kent State UniversityGeneral Measure Extensions of Projection Bodies

Friday 5:00pm-5:30pm: Landon Gauthier, University of KentuckyInverse boundary value problems for polyharmonic operators with non-smooth coefficients

Parallel Session II-B (Moderator: Naibo; Zoom: https://ksu.zoom.us/j/93740953837)

Friday 3:30pm-4:00pm: John MacLellan, University of AlabamaNecessary Conditions for Two Weight Weak Type Norm Inequalities for Multilinear SingularIntegral Operators

Friday 4:00pm-4:30pm: Walton Green, Washington University in St. LouisWavelet Representation of Smooth Calderon-Zygmund Operators

Friday 4:30pm-5:00pm: Cody Stockdale, Clemson UniversityWeighted theory of compact operators

Friday 5:00pm-5:30pm: Vishwa Dewage, Louisiana State UniversityToeplitz operators with quasi-radial symbols

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Parallel Session II-C (Moderator: Mantzavinos; Zoom: https://ksu.zoom.us/j/92091623313)

Friday 3:30pm-4:00pm: Ryan Alvarado, Amherst CollegeOptimal embeddings and extensions for Triebel-Lizorkin spaces in spaces of homogeneoustype

Friday 4:00pm-4:30pm: Liding Yao, University of Wisconsin-MadisonAn in-depth look of Rychkov’s universal extension operators for Lipschitz domains

Friday 4:30pm-5:00pm: Lingxiao Zhang, University of Wisconsin-MadisonReal analytic multi-parameter singular Radon transforms

Friday 5:00pm-5:30pm: Geoffrey Bentsen, Northwestern UniversityLp-Sobolev Regularity for a Class of Local Radon-like Operators

Parallel Session II-D (Moderator: Shao; Zoom: https://ksu.zoom.us/j/96672828264)

Friday 3:30pm-4:00pm: Ryan Frier, The University of KansasA remark on the Strichartz Inequality in one dimension

Friday 4:00pm-4:30pm: Daniel Eceizabarrena, University of Massachusetts AmherstPointwise convergence over fractals for dispersive equations with homogeneous symbol

Friday 4:30pm-5:00pm: Ryan Denlinger, University of Texas at AustinScaling critical Boltzmann equation

Friday 5:00pm-5:30pm: Cole Jeznach, University of MinnesotaRegularized Distances and Geometry of Measures

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2 Saturday November 6th, 2021All times are in Central Daylight Time (CDT)

Saturday 9:00am-10:00am David Cruz-Uribe, University of AlabamaThe maximal operator on convex-set valued functions

Saturday 10:00am-10:30am break

Saturday 10:30am-12:30pm Parallel Sessions III

Saturday 12:30pm-2:00pm Lunch break

Saturday 2:00pm-3:00pm Malabika Pramanik, University of British ColumbiaSets and configurations

Saturday 3:00pm-3:30pm break

Saturday 3:30pm-4:30pm Parallel Sessions IV

Parallel Session III-A (Moderator: Maldonado; Zoom: https://ksu.zoom.us/j/92033473537)

Saturday 10:30am-11:00am: Javier Canto, BCAM - Basque Center for Applied MathematicsThe Hajlasz capacity density condition is self-improving

Saturday 11:00am-11:30am: Jean-Daniel Djida, African Institute for Mathematical Sciences -AIMS-CameroonNonlocal complement value problem for a global in time parabolic equation

Saturday 11:30am-12:00pm: Marıa Soria-Carro, The University of Texas at AustinOn a decreasing family of integro-differential operators: From fractional Laplacian to non-local Monge-Ampere

Saturday 12:00pm-12:30pm: Animesh Biswas, University of Nebraska-LincolnNonlocal Curvature with Integrable Kernel

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Parallel Session III-B (Moderator: Naibo; Zoom: https://ksu.zoom.us/j/93740953837)

Saturday 10:30am-11:00am: Duvan Cardona Sanchez, Ghent University, BelgiumSeeger-Sogge-Stein orders for the weak (1,1) boundedness of Fourier integral operators

Saturday 11:00am-11:30am: Michael Jesurum, University of Wisconsin-MadisonMaximal Operators and Fourier Restriction

Saturday 11:30am-12:00pm: Weiyan Huang, Washington University in St. LouisWeighted Inequalities for Haar multipliers

Saturday 12:00pm-12:30pm: Brandon Sweeting, University of AlabamaNew estimates for dyadic Carleson sequences

Parallel Session III-C (Moderator: Mantzavinos; Zoom: https://ksu.zoom.us/j/92091623313)

Saturday 10:30am-11:00am: Dohyun Kwon, University of Wisconsin-MadisonVolume-preserving crystalline mean curvature flow

Saturday 11:00am-11:30am: Yuxi Han, University of Wisconsin-MadisonRemarks on the vanishing viscosity process of state-constraint Hamilton-Jacobi equations

Saturday 11:30am-12:00pm: Jonathan Rehmert, Kansas State UniversityQuasisymmetric Koebe Uniformization via Transboundary Modulus

Saturday 12:00pm-12:30pm: Hyogo Shibahara, University of CincinnatiRecent developments on Gromov-Hausdorff distance with boundary

Parallel Session III-D (Moderator: Shao; Zoom: https://ksu.zoom.us/j/96672828264)

Saturday 10:30am-11:00am: Trung Truong, Kansas State UniversityImaging of bi-anisotropic periodic structures from electromagnetic near field data

Saturday 11:00am-11:30am: Thu Thi Anh Le, Kansas State UniversityImaging of 3D objects with experimental data using orthogonality sampling methods

Saturday 11:30am-12:00pm: Mohammad Shirazi, McGill UniversityApproximating Continuous Functions by Harmonic Functions on Some Bounded Domainsin Rn, n ≥ 2

Saturday 12:00pm-12:30pm: Liudmyla Kryvonos, Kent State University“Near-best” polynomial approximation of harmonic functions on compact sets in C

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Parallel Session IV-A (Moderator: Maldonado; Zoom: https://ksu.zoom.us/j/92033473537)

Saturday 3:30pm-4:00pm: Hayley Olson, University of Nebraska-LincolnAnalyzing Nonlinearities in the Nonlocal Diffusion Model

Saturday 4:00pm-4:30pm: Nicole Buczkowski, University of Nebraska-LincolnContinuous dependence for nonlocal models with respect to changes in data

Parallel Session IV-B (Moderator: Naibo; Zoom: https://ksu.zoom.us/j/93740953837)

Saturday 3:30pm-4:00pm: Wilfredo Urbina-Romero, Roosevelt UniversityGeneralized Gaussian Singular Integrals

Saturday 4:00pm-4:30pm: Jeongsu Kyeong, Temple UniversitySymmetry properties of the spectra of singular integral operators associated with the Lamesystem of elastostatics in two dimensions

Parallel Session IV-C (Moderator: Mantzavinos; Zoom: https://ksu.zoom.us/j/92091623313)

Saturday 3:30pm-4:00pm: Linhan Li, University of MinnesotaCarleson measure estimates for the Green function with applications to elliptic measures

Saturday 4:00pm-4:30pm: Joshua Flynn, University of ConnecticutSharp Hardy-Sobolev-Maz’ya, Adams and Hardy-Adams inequalities on quaternionic hy-perbolic spaces and the Cayley hyperbolic plane

Parallel Session IV-D (Moderator: Shao; Zoom: https://ksu.zoom.us/j/96672828264)

Saturday 3:30pm-4:00pm: Edward White, Florida State UniversityPolarization and Covering on sets of low smoothness

Saturday 4:00pm-4:30pm: Robert Fraser, Wichita State UniversityAlgebraic Number Fields in Harmonic Analysis

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3 Titles and abstracts of invited talksPrincipal speaker

Malabika Pramanik, University of British Columbia

Sets and configurations

Abstract: Many problems in geometric measure theory are concerned with the following question:Can large sets avoid many patterns? Stated in this level of generality, the question lacks precision,both in the quantification of size and in the specification of patterns. ”Large” could be interpretedeither as large cardinality, nontrivial Lebesgue measure, positive asymptotic or Banach density,large Hausdorff, Minkowski or Fourier dimension. ”Patterns” could be geometric in nature, forexample, arithmetic or geometric progressions, equilateral triangles, parallelograms; alternatively,they could be algebraic, such as solutions of certain equations. Regardless of the many possiblevariants of such a question, it would seem that a natural answer would be no, with any reasonabledefinition. Indeed, there is a large body of work that supports this intuition. However, there arealso many results in the literature that challenge this intuition, especially when slight variations inthe notions of size lead to very different conclusions regarding the existence of patterns.

Problems of this type lie at the interface of many areas of mathematics, and draws from a diversetoolkit, spanning harmonic analysis, number theory and additive combinatorics. The two talks willprovide a survey of the vibrant research area surrounding configurations in sets and discuss someof the recent advances in the area.

Invited speakers

Marıa Jesus Carro, Universidad Complutense de Madrid

Solving Dirichlet and Neumann problems in Lipschitz Domains at the end-point

Abstract: In this talk, we shall continue the work initiated in the 80’s by B. Dahlberg, E. Fabes andC. Kenig concerning the solution of the Dirichlet and Neumann problems. In particular, it isknown that, for every Lipschitz domain Ω on the plane

Ω = x+ iy : y > ν(x),

with ν a real valued Lipschitz function, there exists 1 ≤ p0 < 2 so that the Dirichlet problem(DP) has a solution for every function f ∈ Lp(ds) and every p ∈ (p0,∞) and there exists q0 > 2so that the Neumann problem (NP) is solvable for every function f ∈ Lp(ds) and every p ∈(1, q0). Moreover, if p0 > 1, the result for the DP is false for every p ≤ p0 and the result for the NPis false for p ≥ q0.

The purpose of this talk is to present in more detail what happens at the endpoints p0 and q0. Tobe more precise, we want to find spaces X ⊂ Lp0 so that the DP is solvable for every f ∈ Xand similarly for the NP. Our results will be applied to the special case of Schwarz-ChristoffelLipschitz domains, among others, for which we explicitly compute the values of p0 and q0.

This is a joint work with Virginia Naibo and Carmen Ortiz-Caraballo.

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David V. Cruz-Uribe, OFS, The University of Alabama

he maximal operator on convex-set valued functions

Abstract: In this talk we discuss recent work on harmonic analysis on convex-set valued functions.Let Kbcs(Rd) be the collection of all bounded, convex, symmetric and closed subsets of Rd. Givenan open set Ω ⊂ Rn, we define a convex-set valued function to be a map F : Ω→ Kbcs(Rd). Sucha function is measurable if, given any open set U ⊂ Rd, F−1(U) is measurable (with respect toLebesgue measure on Rn). Using the Aumann integral, we define the integral of F on Ω to be theset ∫

Ω

F (x) dx =

∫Ω

~f(x) dx : ~f ∈ S1(Ω, F )

,

where S1(Ω, F ) is the set of integrable selector functions of F : that is, the set of vector-valuedfunctions such that

S1(Ω, F ) = ~f ∈ L1(Ω,Rd) : ~f(x) ∈ F (x), x ∈ Ω.

We define a generalization of the Hardy-Littlewood maximal operator to convex-set valued func-tions by

MF (x) = conv

(⋃Q

1

|Q|

∫Q

F (y) dy · χQ(x)

),

where conv(E) denotes the convex closure of a set E. This operator is a generalization of themaximal operator applied to vector-valued functions: given ~f : Ω→ Rd, we can associate to it theconvex-set valued function F defined by

F (x) = conv(f(x),−f(x)).

In this case we have thatsup|s| : s ∈MF (x) = M ~f(x),

where on the right-hand side we have the classical Hardy-Littlewood maximal operator. The ad-vantage of this approach is that if F is convex-set valued, then so is MF , whereas the Hardy-Littlewood maximal operator maps vector-valued functions to scalar functions.

In this talk we will discuss the foundational theory of convex-set valued functions, the maximaloperator applied to these functions, and the connection with matrix-weighted norm inequalities. Aparticular motivation for this approach is the sparse domination of vector-valued singular integralsin terms of convex-set valued functions by Nazarov, Petermichl, Treil, and Volberg. The project isjoint work with Marcin Bownik.

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4 Titles and abstracts of contributed talks

Ryan Alvarado, Amherst College (session II-C)

Optimal embeddings and extensions for Triebel-Lizorkin spaces in spaces of homogeneous type

Abstract: Embedding and extension theorems for certain classes of function spaces in Rn (such asSobolev spaces) have played a fundamental role in the area of partial differential equations. In thistalk, we will discuss some recent work which builds upon such results and identifies necessary andsufficient conditions guaranteeing that certain Sobolev-type inequalities and extension results holdfor the scale of Triebel-Lizorkin spaces (M s

p,q spaces) in the general context of spaces of homoge-neous type. An interesting facet of this work is how the range of s (the smoothness parameter) forwhich these inequalities and extension results hold is intimately linked to the geometric makeup ofthe underlying space.

Sergi Arias, Stockholm University (session I-B)

Some endpoint estimates for bilinear Coifman-Meyer multipliers

Abstract: In this talk we will review some known estimates for bilinear Coifman-Meyer multipliersand present some new results while acting on some endpoint spaces. More precisely, we willconsider the case in which one of the arguments of the operator is fixed in the space of functionswith local bounded mean oscillation, bmo, while the other one is either in a Lebesgue space or inthe Hardy space H1.

The most natural bilinear operator, the product of two functions, is a particular instance of Coifman-Meyer multipliers. As an application of the main results, we will also present some estimates onproducts of functions in local bmo with functions in either a Lebesgue space or in the local Hardyspace h1 , as well as related Kato-Ponce-type inequalities.

Geoffrey Bentsen, Northwestern University (session II-C)

Lp-Sobolev Regularity for a Class of Local Radon-like Operators

Abstract: We consider local Radon-like operators which integrate over three-dimensional familiesof curves in R3. Important examples of such operators include convolution with measures sup-ported on curves and restricted X-ray transforms. We use Bourgain-Demeter decoupling and L2

estimates for frequency-localized oscillatory integral operators to prove sharp (local) Lp-Sobolevregularity for a class of Radon-like operators associated to fibered folding canonical relations.

Animesh Biswas, University of Nebraska-Lincoln (session III-A)

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Nonlocal Curvature with Integrable Kernel

Abstract: The focus of this talk will be on the recently introduced topic of nonlocal curvature,defined as

HJΩ(x) :=

∫Rn

J(x− y)(χΩc(y)− χΩ(y))dy,

where x ∈ Rn, Ω ⊂ Rn, χ is the characteristic function for a set, J is a radially symmetric,nonnegative, nonincreasing convolution kernel. Several papers have studied the case of nonlocalcurvature with nonintegrable singularity, a generalization of the classical curvature concept, whichrequires the regularity of the boundary to be above C2. Nonlocal curvature of this form appears inmany different applications, such as image processing, curvature driven motion, deformations. Ourresults offer some generalizations and extensions to the constant mean curvature problem, wherecounterparts to Alexandrov’s theorem in the nonlocal framework were established independentlyby two separate groups: Ciraolo, Figalli, Maggi, Novaga, and respectively, Cabre, Fall, Sola-Moreles, Weth. By using the concept of nonlocal curvature for integrable kernels J , as discussedby Mazon, Rossi, Toledo, we are able to lower requirements on the smoothness of the boundary.

Nicole Buczkowski, University of Nebraska-Lincoln (session IV-A)

Continuous dependence for nonlocal models with respect to changes in data

Abstract: Stability of solutions with respect to changes in data is an important feature for solutionsthat model physical phenomena. As measurements of physical parameters can never be exact,small changes in measured data will hopefully generate similarly small changes in the solutions.The exact dependence can be studied with analysis and partial differential equation tools and serveengineers, computational scientists, and also be important in other theoretical studies. Nonlocalmodels, with their capability to handle discontinuities, record long range interactions through akernel which gives additional flexibility. In this talk, we show several results on the continuousdependence (or stability) of the solution with respect to changes in (linear and nonlinear) forcingterms, boundary or collar data, as well as with respect to the kernel of the nonlocal operator.Numerical results will also be given.

Javier Canto, BCAM - Basque Center for Applied Mathematics (session III-A)

The Hajlasz capacity density condition is self-improving

Abstract: In this talk, we introduce a capacity density condition in terms of non-local Hajlaszgradients and we prove that it is self-improving. The way of doing so is by characterizing it interms of an upper bound on the Assouad codimension of the underlying set. The proof relates thecapacity density condition with boundary Poincare inequalities, adapts Keith-Zhong techniquesfor establishing local Hardy inequalities which are finally improved by Koskela-Zhong arguments.This is joint work with Antti V. Vahakangas (Jyvaskyla University).

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Duvan Cardona Sanchez, Ghent University, Belgium (session III-B)

Seeger-Sogge-Stein orders for the weak (1,1) boundedness of Fourier integral operators

Abstract: In this talk we discuss the sharpness of the Seeger-Sogge-Stein order for the weak (1,1)boundedness of Fourier integral operators. That the Fourier integral operators of order−(n− 1)/2are of weak (1,1) type was proved by Terence Tao. In this talk we discuss the sharpness of thisorder as well as its versions for canonical relations satisfying additional rank conditions. Jointwork with Michael Ruzhansky.

Emily Dautenhahn, Cornell University (session I-A)

Heat kernel estimates on manifolds with ends with mixed boundary condition

Abstract: The heat kernel is an object whose importance reaches across several fields of mathe-matics, in particular probability theory and the study of PDEs. In some spaces, we understand thebehavior of the heat kernel very well; in other spaces, its behavior is more elusive. In many of thespaces where heat kernel estimates are known, they are of the form of two-sided Gaussian bounds.In this talk, we will explore some examples where heat kernel estimates are NOT of this type. Inparticular, we will look at the heat kernel on certain Riemannian manifolds with boundary with“nice” ends; we will take Dirichlet condition along some portion of the boundary and Neumanncondition along the rest. This is joint work with Laurent Saloff-Coste.

Ryan Denlinger, University of Texas at Austin (session II-D)

Scaling critical Boltzmann equation

Abstract: We discuss recent results on scaling critical local well-posedness theories for Boltz-mann’s equation via dispersive estimates, by analogy with the cubic nonlinear Schrodinger equa-tion. Joint work with N. Pavlovic and T. Chen.

Vishwa Dewage, Louisiana State University (session II-B)

Toeplitz operators with quasi-radial symbols

Abstract: The n-dimensional Fock space is defined to be the space of holomorphic functions thatare square integrable with respect to the Gaussian measure. Using representation theory, we di-agonalize the Toeplitz operators on Fock space with essentially bounded quasi-radial symbols.

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Then we show that the commutative C∗-algebra generated by these Toeplitz operators is isometri-cally isomorphic to a space of bounded functions that are uniformly continuous with respect to thesquare root metric. This is a joint work with my advisor, Prof. Gestur Olafsson.

Jean-Daniel Djida, African Institute for Mathematical Sciences - AIMS-Cameroon (session III-A)

Nonlocal complement value problem for a global in time parabolic equation

Abstract: We investigate weak solutions to a semilinear parabolic equation involving nonlocaloperators of Levy type. In many cases, this type of equation arose when describing the chaoticdynamics of a polymer molecule in a liquid. This equation is nonlocal in time and space. It con-tains a term called the interaction potential that depends on the time integral of the solution overthe entire interval of solving the problem. The existence of a weak solution to the nonlocal com-plement value problem is proven. Furthermore, under fair conditions on the interaction potential,the uniqueness of this solution is established.

Daniel Eceizabarrena, University of Massachusetts Amherst (session II-D)

Pointwise convergence over fractals for dispersive equations with homogeneous symbol

Abstract: Let P ∈ C∞(Rn \ 0) be a real, homogeneous and non-singular symbol and the dis-persive equation

i ∂tu+ P (D)u = 0, u(x, 0) = f(x). (1)

For α ∈ [0, n], we tackle the α-almost everywhere convergence problem; that is, for which s > 0do we have

limt→0

u(x, t) = f(x), α-a.e. ∀f ∈ Hs(Rn) ? (2)

The original problem was proposed by Carleson in 1980 for the Schrodinger equation, that is, forP (ξ) = |ξ|2 and α = n.

In this work in collaboration with Felipe Ponce-Vanegas (BCAM), we prove that:

• For general P , convergence holds if s > (n− α + 1)/2.

• This is optimal: there are α ≤ n and saddle-like symbols with counterexamples with s <(n− α + 1)/2.

• If P has dispersion and α < n/2, then s > (n− α)/2, and this is optimal.

• If P (ξ) = ξk1 + . . . + ξkn, k ≥ 2 an integer and α < n, we construct counterexamples. Thisis a generalization of the recent work by An, Chu, Pierce for α = n. Main difficulties aredealing with Weil sums and computing the Hausdorff dimension of the divergence sets. Forthe latter we use a mass transference principle from Diophantine approximation.

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In the talk I will focus on the counterexamples of the last bullet point.

Joshua Flynn, University of Connecticut (session IV-C)

Sharp Hardy-Sobolev-Maz’ya, Adams and Hardy-Adams inequalities on quaternionic hyperbolicspaces and the Cayley hyperbolic plane

Abstract: The main goal of this talk is to establish the Poincare-Sobolev and Hardy-Sobolev-Maz’ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane. A crucialpart of our work is to establish appropriate factorization theorems on these spaces which are of theirindependent interests. Combining the factorization theorems and the Geller type operators with theHelgason-Fourier analysis on symmetric spaces, the precise heat and Bessel-Green-Riesz kernelestimates and the Kunze-Stein phenomenon for connected real simple groups of real rank onewith finite center, we succeed to establish the higher order Poincare-Sobolev and Hardy-Sobolev-Maz’ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane. The kernelestimates required to prove these inequalities are also sufficient for us to establish, as a byprod-uct, the Adams and Hardy-Adams inequalities on these spaces. This paper, together with ourearlier works, completes our study of the factorization theorems, higher order Poincare-Sobolev,Hardy-Sobolev-Maz’ya, Adams and Hardy-Adams inequalities on all rank one symmetric spacesof noncompact type.

Robert Fraser, Wichita State University (session IV-D)

Algebraic Number Fields in Harmonic Analysis

Abstract: We discuss applications of algebraic number fields to solve problems in Euclidean har-monic analysis. In particular, we will discuss the use of algebraic number fields to construct Salemsets in Rn. Joint work with Kyle Hambrook.

Ryan Frier, The University of Kansas (session II-D)

A remark on the Strichartz Inequality in one dimension

Abstract: In this paper, we study the extremal problem for the Strichartz inequality for the Schrodingerequation on R2. We show that the solutions to the associated Euler-Lagrange equation are exponen-tially decaying in the Fourier space and thus can be extended to be complex analytic. Consequentlywe provide a new proof to the characterization of the extremal functions: the only extremals areGaussian functions, which was investigated previously by Foschi and Hundertmark-Zharnitsky.

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Landon Gauthier, University of Kentucky (session II-A)

Inverse boundary value problems for polyharmonic operators with non-smooth coefficients

Abstract: We consider the polyharmonic with lower order coefficients of the form

(−∆)m +Q ·D + q.

The main interest is to lower the regularity of Q and q to establish uniqueness. The key componentwe used to lower the regularity is by adapting an averaging argument first introduced by Habermanand Tataru.

Walton Green, Washington University in St. Louis (session II-B)

Wavelet Representation of Smooth Calderon-Zygmund Operators

Abstract: We represent a bilinear Calderon-Zygmund operator at a given smoothness level as afinite sum of cancellative, complexity zero operators, involving smooth wavelet forms, and con-tinuous paraproduct forms. This representation results in a sparse T (1)-type bound, which inturn yields directly new sharp weighted linear and multilinear estimates on Lebesgue and Sobolevspaces. Moreover, we apply the representation theorem to study fractional differentiation of bilin-ear operators, establishing Leibniz-type rules in weighted Sobolev spaces which are new even inthe simplest case of the pointwise product.

Yuxi Han, University of Wisconsin-Madison (session III-C)

Remarks on the vanishing viscosity process of state-constraint Hamilton-Jacobi equations

Abstract: We investigate the convergence rate in the vanishing viscosity process of the solutionsto the subquadratic state-constraint Hamilton-Jacobi equations. We give two different proofs ofthe fact that, for nonnegative Lipschitz data that vanish on the boundary, the rate of convergenceis O(

√ε) in the interior. Moreover, the one-sided rate can be improved to O(ε) for nonnegative

compactly supported data and O(ε1/p) (where 1 < p < 2 is the exponent of the gradient term) fornonnegative data f ∈ C2(Ω) such that f = 0 and Df = 0 on the boundary. Our approach relies ondeep understanding of the blow-up behavior near the boundary and semiconcavity of the solutions.

Weiyan Huang, Washington University in St. Louis (session III-B)

Weighted Inequalities for Haar multipliers

Abstract: Harmonic analysis studies the boundedness of operators in Lebesgue spaces. Dyadicarguments play an important role in this topic. In R, the dyadic grid D consists of intervals of the

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form I = [m2−k, (m + 1)2−k), where m, k are integers. Each dyadic interval I is associated witha Haar function hI , and the collection of Haar functions hII∈D forms an orthonormal basis ofL2(dx). A Haar multiplier is an operator T who has the form Tf(x) =

∑I∈D c(x, I)〈f, hI〉hI(x),

where c(x, I) is a function on R×D. In 1998, Katz and Pereyra studied a class of Haar multipliersT tw whose coeficients are given by c(x, I) = wt(x)

(mIw)t, where w is a weight on R and mIw denotes its

average on I . The authors proved the sufficient and necessary conditions on the weight w for T twto be bounded on Lp(dx). In this talk, we will talk about the weighted version of this result; thatis, we will see some sufficient and necessary conditions for T tw to be bounded on Lp(vdx) wherew, v are weights on R. We will also talk about some analogues of such Haar multipliers in spacesof homogeneous type and analogous results.

Michael Jesurum, University of Wisconsin-Madison (session III-B)

Maximal Operators and Fourier Restriction

Abstract: Several recent developments in Fourier restriction theory have involved using certainmaximal restriction theorems to give a pointwise interpretation of the restriction operator. We givean overview of these results and discuss the relationship between maximal restriction theorems andclassical restriction theorems.

Cole Jeznach, University of Minnesota (session II-D)

Regularized Distances and Geometry of Measures

Abstract: In joint work with Max Engelstein and Svitlana Mayboroda we generalize the notion ofthe regularized distance function

Dµ,α(x) = (

∫|x− y|−d−α dµ(y))−1/α

to functions with more general integrands. We provide a large class of integrands for which thecorresponding distance functions contain geometric information about µ. In particular, we produceexamples that are in some sense far from the original kernel |x − y|−d−α but still characterize thegeometry of µ since they have nice symmetries with respect to flat sets. In co-dimension 1, theseexamples are explicit, but in higher co-dimensions, our proof of existence of such examples isnon-constructive, and thus we have no additional information about their structure.

Liudmyla Kryvonos, Kent State University (session III-D)

“Near-best” polynomial approximation of harmonic functions on compact sets in C

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Abstract: For a function u that is continuous on a compact set K and harmonic in its interior weconstruct a sequence of polynomial approximants that give almost optimal rate of approximationon K, and in the interior points of K converge faster than on the whole of K. We also show thatthe geometric convergence insideK is possible for sets whose boundary is an analytically boundedJordan domain.

Dohyun Kwon, University of Wisconsin-Madison (session III-C)

Volume-preserving crystalline mean curvature flow

Abstract: We consider the global existence of volume-preserving crystalline mean curvature flowin a non-convex setting. We show that a natural geometric property, associated with reflectionsymmetries of the Wulff shape, is preserved with the flow. Using this geometric property, weaddress the global existence and regularity of the flow for smooth anisotropies. For the non-smoothcase, we establish global existence results for the types of anisotropies known to be globally well-posed. This is a joint work with Inwon Kim (UCLA) and Norbert Pozar (Kanazawa University).

Jeongsu Kyeong, Temple University (session IV-B)

Symmetry properties of the spectra of singular integral operators associated with the Lame systemof elastostatics in two dimensions

Abstract: In this talk, I will discuss Mellin Transform techniques for studying spectral properties ofsingular integral operators associated with the Lame system on infinite sectors in two dimensionsand investigate symmetry properties.

Dylan Langharst, Kent State University (session II-A)

General Measure Extensions of Projection Bodies

Abstract: The inequalities of Petty and Zhang are affine isoperimetric-type inequalities providingsharp bounds for voln−1

n (K)voln(ΠK), where ΠK is a projection body of a convex body K.In this talk, we shall discuss a number of generalizations of Zhang’s inequality to the setting ofarbitrary measures. Additionally, we shall discuss extensions of the projection body operator Πto the setting of arbitrary measures and functions, while providing associated inequalities for thisoperator; in particular, Zhang-type inequalities.

Thu Thi Anh Le, Kansas State University (session III-D)

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Imaging of 3D objects with experimental data using orthogonality sampling methods

Abstract: We consider the electromagnetic inverse scattering problem that aims to reconstruct thelocation and shape of an unknown object from the electromagnetic field scattered by that object. Ithas applications in radar and nondestructive testing. In this talk, we investigate a modified versionof the Orthogonality Sampling Method (OSM) for Maxwell’s equations. This modification allowsthe method to work with more types of polarization associated with the data. Numerical resultstesting against 3D experimental data from the Fresnel institute will be presented. The results showthat the modified OSM performs better than its original version in real data verification. This isjoint work with Dinh-Liem Nguyen, Hayden Schmidt, and Trung Truong.

Linhan Li, University of Minnesota (session IV-C)

Carleson measure estimates for the Green function with applications to elliptic measures

Abstract: We are interested in the relations between an elliptic operator on a domain, the geometryof the domain, and the boundary behavior of the Green function. In joint works with Guy Davidand Svitlana Mayboroda, we show that if the coefficients of the operator satisfy a quadratic Car-leson condition, then the Green function on the half-space is almost affine, in the sense that thenormalized difference between the Green function with a sufficiently far away pole and a suitableaffine function at every scale satisfies a Carleson measure estimate. We demonstrate with coun-terexamples that our results are optimal, in the sense that the class of the operators consideredare essentially the best possible. Analogous results can be derived on sets with lower-dimensionalboundaries. In this talk, I will also talk about applications of these quantitative results to ellipticmeasures and characterizations of uniform rectifiable sets of higher co-dimension.

Zane Li, Indiana University Bloomington (session I-C)

Decoupling for Cantor sets on the line and parabola

Abstract: Motivated by Biggs’ study of Vinogradov’s Mean Value Theorem over ellipsephic sets,we study decoupling for Cantor sets on the the line and parabola. Our parabola decoupling resultimproves upon what one would obtain if one directly applied the Bourgain-Demeter parabola de-coupling theorem. The key feature is that we make use of the sparsity of the Fourier supports. Thisis joint work with Alan Chang, Jaume de Dios Pont, Rachel Greenfeld, Asgar Jamneshan, and JoseMadrid.

John MacLellan, University of Alabama (session II-B)

Necessary Conditions for Two Weight Weak Type Norm Inequalities for Multilinear Singular Inte-gral Operators

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Abstract: In this talk we will discuss necessary conditions for a multilinear Calderon Zygmundoperator to satisfy two weight weak type norm inequalities. We generalize results from our recentpaper, and earlier work by Lacey, Sawyer, and Uriarte-Tuero, and by Stein in the linear case. Weprove results assuming a non-degeneracy condition similar to that introduced by Stein in the linearcase, both assuming doubling conditions and without them, only assuming that our measures donot have common point masses.

Hayley Olson, University of Nebraska-Lincoln (session IV-A)

Analyzing Nonlinearities in the Nonlocal Diffusion Model

Abstract: Nonlocal calculus operators are capable of modeling phenomena that classical differen-tial operators cannot capture. This includes anomalous diffusion – such as super- and subdiffusionprocesses – present in models of subsurface diffusion and turbulence, for example. Here, we inves-tigate the introduction of nonlinearities to the nonlocal variation of a diffusion model. Consideringthe model in a nonlocal setting is useful because these nonlocal integral operators act on an interac-tion horizon which allows them to have solutions with lower requirements on regularity. This caninclude functions which are not differentiable and even discontinuous. Provided results includeconvergence results for certain classes of nonlinearities to show they behave similarly to classicaloperators when the interaction horizon shrinks down to the point of interest.

Chamsol Park, University of New Mexico (session I-A)

Eigenfunction Restriction Estimates on curves with nonvanishing geodesic curvatures

Abstract: There have been ways to study the eigenfunctions of the Laplace-Beltrami operator oncompact Riemannian manifolds (without boundary). One way to study the eigenfunctions is toconsider the Lp estimates of the eigenfunctions restricted to submanifolds in the given manifolds.Burq, Gerard, and Tzvetkov, and Hu studied many types of these restricted estimates, one of whichwas the L2 restriction estimates on curves with nonvanishing geodesic curvatures. In this talk,we would like to talk about their logarithmic improved analogues, in the presence of nonpositivesectional curvatures in the given compact Riemannian manifolds.

Abba Ramadan, University of Kansas (session I-D)

On the standing waves of the Schrodinger equation with concentrated nonlinearity

Abstract: We study the concentrated NLS on Rn, with power non-linearities, driven by the frac-tional Laplacian, (−∆)s, s > n

2. We construct the solitary waves explicitly, in an optimal range of

the parameters, so that they belong to the natural energy space Hs. Next, we provide a complete

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classification of their spectral stability. Finally, we show that the waves are non-degenerate andconsequently orbitally stable, whenever they are spectrally stable.

Incidentally, our construction shows that the soliton profiles for the concentrated NLS are in factexact minimizers of the Sobolev embedding Hs(Rn) → L∞(Rn), which provides an alternativecalculation and justification of the sharp constants in these inequalities.

Jonathan Rehmert, Kansas State University (session III-C)

Quasisymmetric Koebe Uniformization via Transboundary Modulus

Abstract: A circle domain is an open and connected subset of the plane whose complementarycomponents are points or round disks. Given a metric space, X, which is homeomorphic to a count-ably connected circle domain (and satisfies some mild conditions), we obtain a characterization ofwhen X is quasisymmetric to a circle domain. This characterization is in terms of Schramm’stransboundary modulus and Heinonen-Koskela’s Loewner property and is inspired by Bonk’s cel-ebrated uniformization result for planar carpets. This is joint work with Hrant Hakobyan.

Daniel Restrepo, University of Texas at Austin (session II-A)

The isoperimetric problem for phase transitions

Abstract : In this talk we will consider an approximation to the classical isoperimetric problem (ala De Giorgi) via the Allen-Cahn energy,i.e,

ACε(u) =ε∫Rn

|∇u|2 +1

ε

∫Rn

W (u).

subject to a suitable volume constraint.

We will show the existence, uniqueness and convergence of minimizers of this problem. Also, wewill discuss a quantitative stability result for this problem, i.e., we will show that the differenceof the energy between any function v ∈ W 1,2(Rn) and the energy of the minimizer controls thedistance between v and the minimizer.

This is a joint work with Francesco Maggi.

Cristian Rios, University of Calgary (session II-A)

Regularity theory for degenerate elliptic operators

Abstract: We explore De Giorgi/Moser regularity techniques for operators with infinitely degen-erate ellipticity. We recall the classical approach, and then discuss recent developments under theassumptions of Orlicz spaces Sobolev embeddings.

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Eduard Roure Perdices, UPV/EHU (session I-B)

Sawyer-type inequalities for Lorentz spaces

Abstract: In this talk, we will present a new mixed restricted weak type inequality for the Hardy-Littlewood maximal operator M, along with some of its applications and extensions to the multi-variable setting. This is joint work with Carlos Perez.

Hyogo Shibahara, University of Cincinnati (session III-C)

Recent developments on Gromov-Hausdorff distance with boundary

Abstract: We will discuss the recent developments on Gromov-Hausdorff distance with boundary.This distance, unlike Gromov-Hausdorff distance, captures the difference between a noncompletemetric space and its completion. We will further discuss the compactness theorem through A(c)property, tangents with boundary and the class of uniform spaces with diameter bound from below.

Mohammad Shirazi, McGill University (session III-D)

Approximating Continuous Functions by Harmonic Functions on Some Bounded Domains in Rn, n ≥2

Abstract: I shall present some results obtained recently (with Paul. M. Gauthier) regarding ap-proximating (in the Carleman sense) continuous functions by harmonic functions on some specialdomains U of Rn, n ≥ 2. These domains are bounded between two hypersurfaces that are definedby graphs of two continuous functions on the projection U ′ of U to Rn−1. It will be shown that theapproximation gets better as one moves to the boundary via a subset F whose projection on Rn−1

is an Fσ polar set.

Marıa Soria-Carro, The University of Texas at Austin (session III-A)

On a decreasing family of integro-differential operators: From fractional Laplacian to nonlocalMonge-Ampere

Abstract: Integro-differential operators arise naturally in the study of stochastic processes withjumps. A classical example is the fractional Laplacian. In 2015, Caffarelli and Silvestre introduceda nonlocal analog of the Monge-Ampere operator as an infimum of integro-differential operatorsover a family of kernels whose level sets have the same measure as the level sets of the kernel ofthe fractional Laplacian. Motivated by their work, we study a new family of ordered operators that

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arise from imposing weaker geometric conditions on the kernels. The key tools in the analysisof these operators are Ryff’s theorem, from the theory of rearrangements, and Brenier–McCann’stheorem, from optimal transport. This is a work in progress with my advisor, Luis Caffarelli, formy Ph.D. dissertation.

Cody Stockdale, Clemson University (session II-B)

Weighted theory of compact operators

Abstract: The boundedness properties of singular integral operators are of central importance inanalysis. Within the last decade, optimal bounds for general Calderon-Zygmund operators actingon weighted Lebesgue spaces in terms of Muckenhoupt weight characteristics have been obtained.In addition to this theory concerning boundedness, a theory for compactness of Calderon-Zygmundoperators has recently been established. The first goal of this talk is to present the extension ofcompact Calderon-Zygmund theory to weighted spaces using sparse domination techniques. Asimilar line of research concerns the weighted boundedness of the Bergman projection in termsof Bekolle-Bonami weights, and compactness in this setting can be understood within the studyof Toeplitz operators. We also discuss the weighted theory of Toeplitz operators on the Bergmanspace.

Brandon Sweeting, University of Alabama (session III-B)

New estimates for dyadic Carleson sequences

Abstract: We obtain new sharp estimates for a family of Carleson sequences related to dyadic A2

weights. The proof uses Bellman functions, but unlike the typical situation found in literature, inour setting these functions do not have infinitesimal extremal splits and do no arise as solutions ofa PDE. This is joint work with Leonid Slavin.

Trung Truong, Kansas State University (session III-D)

Imaging of bi-anisotropic periodic structures from electromagnetic near field data

Abstract: Inverse scattering problems for three-dimensional bi-anisotropic periodic structures gov-erned by the full Maxwell equation have various applications, especially in the study of photoniccrystals. However, most of the existing works are concerned with the case of isotropic structures.Also, the presence of three matrix-valued coefficients that characterize physical properties of thebi-anisotropic periodic scatterers makes it challenging to reconstruct all the information about thestructures. Moreover, inverse problems involving periodic structures are known to be highly ill-posed. In this talk, we provide a rigorous justification of the Factorization method for the shape

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reconstruction of such structures as well as numerical examples to show efficiency of the methodas a numerical tool to solve the inverse problem.

Wilfredo Urbina-Romero, Roosevelt University (session IV-B)

Generalized Gaussian Singular Integrals

Abstract: We will discuss a general class of singular integrals with respect to the Gaussian measureand their boundedness properties.

Edward White, Florida State University (session IV-D)

Polarization and Covering on sets of low smoothness

Abstract: In this talk I will discuss some recent results on the asymptotic properties of point con-figurations that achieve optimal covering of sets lacking smoothness.

Tongou Yang, University of British Columbia (session I-C)

Decoupling for smooth functions

Abstract: For each d ≥ 0, we prove uniform decoupling inequalities in R3 for all bivariate polyno-mials of degree at most d with bounded coefficients. As an immediate consequence, we fully solvea conjecture of Bourgain-Demeter-Kemp.

Liding Yao, University of Wisconsin-Madison (session II-C)

Title: An in-depth look of Rychkov’s universal extension operators for Lipschitz domains

Abstract: Given a bounded Lipschitz domain Ω ⊂ Rn, Rychkov showed that there is a linearextension operator E for Ω which is bounded in Besov and Triebel-Lizorkin spaces. We introducea class of operators that generalize E which are more versatile for applications. We also derivesome quantitative blow-up estimates of the extended function and all its derivatives in Ω

cup to

boundary. This is a joint work with Ziming Shi.

Xueying Yu, University of Washington (session I-D)

High-low method of NLS on the hyperbolic space

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Abstract: We prove global existence and scattering for the defocusing cubic nonlinear Schrodingerequation on two dimensional hyperbolic space with subcritical initial data, using a high-low fre-quency decomposition method. This is a joint work with Gigliola Staffilani.

Lingxiao Zhang, University of Wisconsin-Madison (session II-C)

Real analytic multi-parameter singular Radon transforms

Abstract: We study operators of the form

Tf(x) = ψ(x)

∫f(γt(x))K(t) dt,

where γt(x) is a real analytic function of (t, x) mapping from a neighborhood of (0, 0) in RN ×Rn into Rn satisfying γ0(x) ≡ x, ψ(x) ∈ C∞c (Rn), and K(t) is a “multi-parameter singularkernel” with compact support in RN ; for example when K(t) is a product singular kernel. Thecelebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth γt(x), inthe single-parameter case when K(t) is a Calderon-Zygmund kernel. Street and Stein generalizedtheir work to the multi-parameter case, and gave sufficient conditions for the Lp-boundedness ofsuch operators. We show that when γt(x) is real analytic, the sufficient conditions of Street andStein are also necessary for the Lp-boundedness of T , for all such kernels K.

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