Several results in the weighted spaces L^p(w) are known to be true if and only if the weight w belongs to the Muckenhoupt class A_{p} with the same exponent p. However, in several other cases, a weaker condition like w ∈ A_{∞} is sufficient for all p. In particular, a theory of weighted Besov and Triebel–Lizorkin spaces was developed with A_{∞} weights already in [H.-Q. Bui, Weighted Besov and Triebel spaces: interpolation by the real method, Hiroshima Math. J. 12 (1982), 581–605]. Following the recent activity with matrix A_{p} weights, including in the theory of function spaces, we are interested in matrix versions of such results. A version of the A_{∞} condition for matrix weights was already introduced in [A. Volberg, Matrix A_{p} weights via S-functions, J. Amer. Math. Soc. 10 (1997), 445–466], but seems to have been relatively little studied since then. To be precise, for matrix weights, A_{∞} splits into a family of weight classes called A_{p,∞}, where the exponent p of the space of interest L^p reappears; but they all reduce to the classical A_{∞} in the scalar case, and provide a genuine extension of the respective A_{p} class for matrices. In a recent preprint [F. Bu, T. Hytönen, D. Yang and W. Yuan, New characterizations and properties of matrix A_{∞} weights, Preprint (2023), arXiv: 2311.05974], we have pursued a systematic study of the matrix A_{p,∞} classes, obtaining several new characterisations and properties. In a follow-up paper by the same authors [F. Bu, T. Hytönen, D. Yang and W. Yuan, Besov–Triebel–Lizorkin-type spaces with matrix A_{∞} weights, Manuscript (2024), available on request], we apply these results to the theory of function spaces with such weights, extending various earlier results under the more restrictive A_{p} condition. In particular, we obtain the equivalence of two versions of weighted Triebel–Lizorkin sequence spaces,
f^{s}_{p,q}(W) = f^{s}_{p,q}({AQ}), (1)
where the space on the left is weighted in a pointwise way, and the one on the right via the sequence of so-called reducing operators associated with the weight on each dyadic cube. For matrix weights W ∈ A_{p,∞}, we obtain (1) for all p ∈ (0,∞), q ∈ (0,∞], extending Volberg’s [A. Volberg, Matrix A_{p} weights via S-functions, J. Amer. Math. Soc. 10 (1997), 445–466] result with q = 2 ≤ p < ∞. The talk will report on these topics.
The local smoothing conjecture for the Euclidean wave equation is a major open problem in harmonic analysis. In this talk I will discuss some recent work connecting local smoothing estimates and invariant spaces for Fourier integral operators, on manifolds in particular. The resulting estimates improve or complement those in the local smoothing conjecture, and they are essentially sharp on each compact manifold. This talk is based on joint work with Naijia Liu, Liang Song and Lixin Yan (Sun Yat-Sen University).
Sampling problems appear whenever a continuous, infinite-dimensional object x, has to be reconstructed from discrete, possibly finite, samples. This requires some assumptions on x. These assumptions can be either expressed by linear conditions, as it happens for Shannon-type results for band-limited functions, or by nonlinear conditions, as with sparsity in compressed sensing or with generative neural networks.
In this talk, I will discuss the case when x is not sampled directly: we sample F(x), where F is the so-called forward map, and wish to reconstruct x. This is the framework of inverse problems, in which an unknown quantity x has to be reconstructed from physical, indirect measurements F(x). Sampling appears in this context since, in practice, it is impossible to directly measure the infinite-dimensional quantity F(x), and only samples are available. I will discuss some abstract results, as well as some examples, including deconvolution and the inverse Radon transform.
In this talk we introduce variation seminorms of order ρ and consider the variation operator associated to the Ornstein-Uhlenbeck semigroup, taken with respect to t. We prove that this seminorm defines an operator of weak type (1,1) for the relevant Gaussian measure when ρ > 2.
This is based on a joint work with Valentina Casarino and Peter Sjögren.
Consider R^d × R^m with the group structure of a 2-step Carnot group and natural parabolic dilations. The maximal operator originally introduced by Nevo and Thangavelu in the setting of the Heisenberg groups is generated by (noncommutative) convolution associated with measures on spheres in R^d. We review some previous work about L^p boundedness and then talk about joint work with Jaehyeon Ryu in which the nondegeneracy condition in the known results on Métivier groups is dropped. The new results have the sharp L^p boundedness range for all two step Carnot groups with d ≥ 3. We also mention recent results regarding stability of the results under small perturbations.
Uniform resolvent estimates play a fundamental role in the study of spectral and scattering theory for Schrödinger equations. In particular, they are closely connected to global-in-time dispersive estimates, such as Strichartz estimates. In contrast with the Euclidean setting, a peculiar fact of the Schrödinger evolution equation associated to the sublaplacian on the Heisenberg group is that it fails to be dispersive, as shown by Bahouri, Gérard, and Xu. In fact, Strichartz or L^p-L^q estimates cannot hold in general. In this talk we will discuss uniform resolvent estimates on the Heisenberg group and their application to obtain certain smoothing effects for the Schrödinger equations.
Joint work with Luca Fanelli, Haruya Mizutani, and Nico Michele Schiavone.
Spherical t-designs are finite point sets on the unit sphere that enable exact integration of all polynomials of degree at most t using an equal-weight quadrature formula. The concept extends to spherical t-design curves, which are continuous curves on the sphere achieving the same integration property through normalized path integrals. However, explicit examples of t-design curves in the literature are rare, even for small t.
Here we construct new t-design curves for small t. We then introduce composite designs, which combine finite point sets and continuous curves to enable exact polynomial integration. Examples are derived using the vertices and edges of dual pairs of convex polytopes. We even construct a composite 19-design.
Consider the heat equation on (real) hyperbolic space H^n with initial data f. It is well-known that under mild conditions on f, the solution converges pointwise a.e. to f as time goes to zero. For rougher initial data, we characterize the weights v on H^n for which the solution converges pointwise a.e. to the initial data when the latter is in L^p(v), 1 ≤ p < ∞. As a tool, we also establish vector-valued weak type (1, 1) and L^p estimates (1 < p < ∞) for the local Hardy–Littlewood maximal function on H^n. Our results hold on arbitrary rank symmetric spaces and for alternative versions of the Laplacian (shifted, distinguished), as well as for the fractional heat equation and the Caffarelli-Silvestre extension.
I will survey the major results from pointwise ergodic theory over the last 90 years.
Let Ω ⊂ C be a simply connected domain such that ∂Ω is a rectifiable Jordan curve. We consider Hardy Smirnov spaces of pseudo holomorphic functions on Ω, namely solutions of ∂w = αw where α ∈ L^r(Ω), 2 ≤ r ≤ ∞. Functions in these spaces have boundary values in a suitable sense. We then consider the M. Riesz problem, namely finding a pseudo holomorphic function in F^p_α(Ω) with real part having a prescribed trace in L^p(∂Ω), 1 < p < ∞. When Ω is Lipschitz, this theory is applied to the Dirichlet problem with boundary data in L^p(∂Ω) for the equation div(σ∇u) = 0, where σ is a positive Lipschitz function in Ω. This is joint work with L. Baratchart and E. Pozzi.
A regular subriemannian manifold M carries a geometric hypoelliptic operator, the intrinsic sublaplacian. Due to a degeneracy of its symbol, geometric and analytic effects can be observed in the study of this operator, which have no counterpart in Riemannian geometry. During the last decades inverse spectral problems in subriemannian geometry have been studied by various authors. Typical approaches are based on the analysis of the induced subriemannian heat or wave equation. In this talk we survey some results in subriemannian geometry. In particular, we address the spectral theory of the sublaplacian in the case of certain compact nilmanifolds and for so-called H-type foliations. This talk is based on joint work with K. Furutani, C. Iwasaki, A. Laaroussi, I. Markina and S. Vega-Molino.
In the early 2000s, Auscher, Axelsson (Rosén) and McIntosh introduced a first-order approach to solve boundary value problems in an upper half-space for elliptic operators in divergence form with transversally independent coefficients. The idea is much alike the correspondence of harmonic functions in the plane with the Cauchy—Riemann equations in complex analysis. However, the Cauchy—Riemann system is perturbed in a rather intricate way by the coefficients of the elliptic operators and L^2-results already for very simple operators go beyond the solution of the famous Kato square root problem.
Later, Amenta, Auscher, Mourgoglou and Stahlhut have derived an impressive list of L^p-results in an interval of exponents p around 2 that is related to the L^p-bisectoriality of a specific perturbed Dirac operator at the boundary. Let p_+ be the upper endpoint of this interval. It had not been clear whether p_+ is an artefact of the first-order method or whether it has a precise meaning in terms of the underlying elliptic equation.
In my talk, I will give a gentle overview on the first-order approach, highlighting its connection and common roots with the Kato problem. I will then present a very recent result from a joint project with Pascal Auscher and Tim Böhnlein that answers the question on p_+: it simply is the `best’ exponent for weak reverse Hölder estimates of the gradient of weak solutions to the elliptic PDE.
(slides)
Originally motivated by physics, ergodic theorems have found applications in many areas of mathematics such as dynamical systems, number theory, stochastics, harmonic and functional analysis etc. We present some generalisations of the classical ergodic theorems and discuss recent developments in this area.
In the last few years, in collaboration with B. Franchi (University of Bologna) and P. Pansu (University of Orsay), we have proved Poincaré and Sobolev inequalities in Heisenberg groups, for Rumin differential forms. These inequalities can be seen as the analytical version of the well-known topological problem of whether a given closed form is exact. More precisely, one can ask whether a primitive can be upgraded to one that satisfies certain estimates; hence, geometric applications follow. If we look at the proofs of our inequalities, we see that at the very beginning we need estimates which, in turn, descend from the existence of a fundamental solution of a suitable Hodge-type Laplacian defined by M. Rumin in the Heisenberg groups. This is a hypoelliptic homogeneous operator. The estimates
that we obtain are sharp. In particular, for L^p, p > 1, the global estimates are a direct consequence of singular integral estimates related to the Folland-Stein theory of homogeneous kernels. When p = 1, the singular integral estimates are replaced by div-curl type inequalities, which go back to Bourgain-Brezis and Lanzani-Stein in Euclidean spaces, and
to Chanillo-Van Schaftingen and Baldi-Franchi-Pansu in Heisenberg groups. In the first part of the talk I would like to discuss the validity of these types of estimates in the Heisenberg groups. In the second part, I will discuss how this approach can be extended to general Carnot groups. The geometry of arbitrary Carnot groups is radically different from that of Heisenberg groups and one of the difficulties that one encounters is related to the fact that the structure of Rumin differential forms is much more complex. Although the same argument used in the Heisenberg case can still be applied in general Carnot groups, relying on the construction of a zero-order Laplacian, it involves singular integral operators that lack good homogeneity properties. While only non-sharp estimates are generally expected, sharp Sobolev inequalities have been obtained for top-degree forms, and in the specific case of the Cartan group (the free Carnot group of step 3 with 2 generators), for differential forms of any degree. In fact, in the Cartan group it is still possible to introduce homogeneous hypoelliptic Hodge-Laplacians for Rumin forms. These results also include recent joint work with F. Tripaldi (University of Leeds) and A. Rosa (INFN, Florence).