Research

Publications

(Click a title to view the respective abstract.)

Caffarelli-Kohn-Nirenberg identities, inequalities and their stabilities.

We set up a one-parameter family of inequalities that contains both the Hardy inequalities (when the parameter is 1) and the Caffarelli-Kohn-Nirenberg inequalities (when the parameter is optimal). Moreover, we study these results with the exact remainders to provide direct understandings to the sharp constants, as well as the existence and non-existence of the optimizers of the Hardy inequalities and Caffarelli-Kohn-Nirenberg inequalities. As an application of our identities, we establish some sharp versions with optimal constants and theirs attainability of the stability of the Heisenberg Uncertainty Principle and several stability results of the Caffarelli-Kohn-Nirenberg inequalities.

Cazacu, C., Flynn, J., Lam, N., & Lu, G. (2023).
Journal de Mathématiques Pures et Appliquées.
arXiv:2211.14622
Sharp Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane.

Though Adams and Hardy-Adams inequalities can be extended to general symmetric spaces of noncompact type fairly straightforwardly by following closely the systematic approach developed in our early works on real and complex hyperbolic spaces, higher order Poincaré-Sobolev and Hardy-Sobolev-Maz’ya inequalities are more difficult to establish. The main purpose of this goal is to establish the Poincaré-Sobolev and Hardy-Sobolev-Maz’ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane. A crucial part of our work is to establish appropriate factorization theorems on these spaces which are of their independent interests. To this end, we need to identify and introduce the “Quaternionic Geller’s operators” and “Octonionic Geller’s operators” which have been absent on these spaces. Combining the factorization theorems and the Geller type operators with the Helgason-Fourier analysis on symmetric spaces, the precise heat and Bessel-Green-Riesz kernel estimates and the Kunze-Stein phenomenon for connected real simple groups of real rank one with finite center, we succeed to establish the higher order Poincaré-Sobolev and Hardy-Sobolev-Maz’ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane. The kernel estimates required to prove these inequalities are also sufficient for us to establish, as a byproduct, the Adams and Hardy-Adams inequalities on these spaces. This paper, together with our earlier works, completes our study of the factorization theorems, higher order Poincaré-Sobolev, Hardy-Sobolev-Maz’ya, Adams and Hardy-Adams inequalities on all rank one symmetric spaces of noncompact type.

Flynn, J., Lu, G., & Yang, Q. (2023).
Revista Matemática Iberoamericana.
arXiv:2106.06055
Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields and second order derivatives.

The present work has as a first goal to extend the previous results in \cite{CFL20} to weighted uncertainty principles with nontrivial radially symmetric weights applied to curl-free vector fields. Part of these new inequalities generalize the family of Caffarelli-Kohn-Nirenberg (CKN) inequalities studied by Catrina and Costa in \cite{CC} from scalar fields to curl-free vector fields. We will apply a new representation of curl-free vector fields developed by Hamamoto in \cite{HT21}. The newly obtained results are also sharp and minimizers are completely described. Secondly, we prove new sharp second order interpolation functional inequalities for scalar fields with radial weights generalizing the previous results in \cite{CFL20}. We apply new factorization methods being inspired by our recent work \cite{CFL21}. The main novelty in this case is that we are able to find a new independent family of minimizers based on the solutions of Kummer’s differential equations. We point out that the two types of weighted inequalities under consideration (first order inequalities for curl-free vector fields vs. second order inequalities for scalar fields) represent independent families of inequalities unless the weights are trivial.

Cazacu, C., Flynn, J., & Lam, N. (2023).
Calculus of Variations and Partial Differential Equations
arXiv:2111.15067
Hardy’s Identities and Inequalities on Cartan-Hadamard Manifolds.

We study the Hardy identities and inequalities on Cartan-Hadamard manifolds using the notion of a Bessel pair. These Hardy identities offer significantly more information on the existence/nonexistence of the extremal functions of the Hardy inequalities. These Hardy inequalities are in the spirit of Brezis-Vázquez in the Euclidean spaces. As direct consequences, we establish several Hardy type inequalities that provide substantial improvements as well as simple understandings to many known Hardy inequalities and Hardy-Poincaré-Sobolev type inequalities on hyperbolic spaces in the literature.

Flynn, J., Lam, N., Lu, G., & Mazumdar, S. (2023).
The Journal of Geometric Analysis
arXiv:2103.12788
Hardy-Poincaré-Sobolev type inequalities on hyperbolic spaces and related Riemannian manifolds

We establish sharpened forms of the Hardy type identities and inequalities which are substantial improvements of Hardy inequalities for the operators $-\Delta_{\mathbb{H}}- \frac{(N-1)^2}{4}$ on the hyperbolic spaces $\mathbb{H}^{N}$ . More precisely, we study the refined Hardy-Poincaré-Sobolev type inequalities in the spirit of Brezis-Vázquez [17] and Brezis-Marcus [15] on hyperbolic spaces, spherically symmetric Riemannian manifolds and more general Riemannian manifolds. Spherically symmetric manifolds are also of significant importance in physics and general relativity. Our approaches are to first establish the Hardy-Poincaré-Sobolev type identities using the notion of Bessel pairs on hyperbolic spaces and related Riemannian manifolds. Using a Hardy identity on upper half space, we also establish a sharp Sobolev inequality on a novel example of noncomplete and non-extendable Riemannian manifold with positive Ricci curvature. In particular, in dimension 3, our example shows that the completeness assumption of the manifolds in the rigidity result of Ledoux [47] cannot be removed.

Flynn, J., Lam, N., & Lu, G. (2022).
Journal of Functional Analysis
Sharp second order uncertainty principles.

We study sharp second order inequalities of Caffarelli-Kohn-Nirenberg type in the euclidian space $\mathbb{R}^{N}$ , where $N$ denotes the dimension. This analysis is equivalent to the study of uncertainty principles for special classes of vector fields. In particular, we show that when switching from scalar fields $u: \mathbb{R}^n \rightarrow \mathbb{C}$ to vector fields of the form $u = \nabla U$ ( $U$ being a scalar field) the best constant in the Heisenberg Uncertainty Principle (HUP) increases from $\frac{N^{2}}{4}$ to $\frac{(N+2)^2}{4}$ , and the optimal constant in the Hydrogen Uncertainty Principle (HyUP) improves from $\frac{(N-1)^2}{4}$ to $\frac{(N+1)^2}{4}$ . As a consequence of our results we answer to the open question of Maz’ya (Integral Equations Operator Theory 2018) in the case $N=2$ regarding the HUP for divergence free vector fields.

Cazacu, C., Flynn, J., & Lam, N. (2022).
Journal of Functional Analysis
arXiv:2012.12667
Extendability and the $\bar\partial$ operator on the Hartogs triangle.

In this paper it is shown that the Hartogs triangle $T$ in $\mathbb{C}^2$ is a uniform domain. This implies that the Hartogs triangle is a Sobolev extension domain. Furthermore, the weak and strong maximal extensions of the Cauchy-Riemann operator agree on the Hartogs triangle. These results have numerous applications. Among other things, they are used to study the Dolbeault cohomology groups with Sobolev coefficients on the complement of $T$ .

Burchard, A., Flynn, J., Lu, G., & Shaw, M. C. (2022).
Mathematische Zeitschrift
arXiv:2106.09867
Short proofs of refined sharp Caffarelli-Kohn-Nirenberg inequalities.

This note relies mainly on a refined version of the main results of the paper by F. Catrina and D. Costa (J. Differential Equations 2009). We provide very short and self-contained proofs. Our results are sharp and minimizers are obtained in suitable functional spaces. As main tools we use the so-called \textit{expand of squares} method to establish sharp weighted L2-Caffarelli-Kohn-Nirenberg (CKN) inequalities and density arguments.

Cazacu, C., Flynn, J., & Lam, N. (2021).
Journal of Differential Equations
arXiv:2104.10372
Sharp Hardy identities and inequalities on Carnot groups.

In this paper we establish general weighted Hardy identities for several subelliptic settings including Hardy identities on the Heisenberg group, Carnot groups with respect to a homogeneous gauge and Carnot–Carathéodory metric, general nilpotent groups, and certain families of Hörmander vector fields. We also introduce new weighted uncertainty principles in these settings. This is done by continuing the program initiated by [N. Lam, G. Lu and L. Zhang, Factorizations and Hardy’s-type identities and inequalities on upper half spaces, Calc. Var. Partial Differential Equations 58 2019, 6, Paper No. 183; N. Lam, G. Lu and L. Zhang, Geometric Hardy’s inequalities with general distance functions, J. Funct. Anal. 279 2020, 8, Article ID 108673] of using the Bessel pairs introduced by [N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Math. Surveys Monogr. 187, American Mathematical Society, Providence, 2013] to obtain Hardy identities. Using these identities, we are able to improve significantly existing Hardy inequalities in the literature in the aforementioned subelliptic settings. In particular, we establish the Hardy identities and inequalities in the spirit of [H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 1997, 443–469] and [H. Brezis and M. Marcus, Hardy’s inequalities revisited. Dedicated to Ennio De Giorgi, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 1–2, 217–237] in these settings.

Flynn, J., Lam, N., & Lu, G. (2021).
Advanced Nonlinear Studies
Sharp L2-Caffarelli–Kohn–Nirenberg inequalities for Grushin vector fields.

In this paper sharp L2-Caffarelli–Kohn–Nirenberg (CKN) inequalities on Euclidean space equipped with Grushin vector fields are established. By use of a Kelvin-like transform, and Grushin cylindrical coordinates, it is shown here that these inequalities are sharp for all possible parameter values for these inequalities.

Flynn, J. (2020).
Nonlinear Analysis
Sharp Caffarelli–Kohn–Nirenberg-type inequalities on Carnot groups.

The main purpose of this paper is to establish several general Caffarelli–Kohn–Nirenberg (CKN) inequalities on Carnot groups $G$ (also known as stratified groups). These CKN inequalities are sharp for certain parameter values. In case $G$ is an Iwasawa group, it is shown here that the L2-CKN inequalities are sharp for all parameter values except one exceptional case. To show this, generalized Kelvin transforms $K_\sigma$ are introduced and shown to be isometries for certain weighted Sobolev spaces. An interesting transformation formula for the sub-Laplacian with respect to $K_\sigma$ is also derived. Lastly, these techniques are shown to be valid for establishing CKN-type inequalities with monomial and horizontal norm weights.

Flynn, J. (2020).
Advanced Nonlinear Studies

Preprints

Flynn, J. & Reznikov, J. (2023). General Conformally Induced Mean Curvature Flow, arXiv:2309.14679.
Do, A., Flynn, J., Lam, N., & Lu, G. (2023): $L^{p}$ -Caffarelli-Kohn-Nirenberg inequalities and their stabilities, arXiv:2310.07083.
Flynn, J., & Vétois, J. (2023): Liouville-type result for the CR Yamabe equation in the Heisenberg group, arXiv:2310.14048
Flynn, J., Lam, N., & Lu, G. (2023): $L^p$ –Hardy identities and inequalities with respect to the distance and mean distance to the boundary, arXiv:2310.18758
Flynn, J., Lu, G., & Yang, Q. (2023): Conformally Covariant Boundary Operators and Sharp Higher Order Sobolev Trace Inequalities on Poincaré-Einstein Manifolds, arXiv:2311.10070
Flynn, J., Lu, G., & Yang, Q. (2023): Conformally Covariant Boundary Operators and Sharp Higher Order CR Sobolev Trace Inequalities on the Siegel Domain and Complex Ball, arXiv:2311.09956