司靈得 (Daniel Spector)

We review some recent results in compensated compactness, concerning primarily concentration effects of PDE-constrained sequences. We show that Müller's \( L \log L \) bound

\( \Phi(Du) \geq 0, \, Du \in L^q(\mathbb{R}^n) \implies \Phi(Du) \in L \log L_{\text{loc}} \)

for \( \Phi = \det \) and \( q = n \) holds for quasiconcave \( \Phi \) which are homogeneous of degree \( q > 1 \). This contrasts similar Hardy bounds which hold only for null Lagrangians.

Among all cylindrical beams of a given cross-sectional area, those with circular cross sections are the most resistant to twisting forces. The general dimensional analogue of this fact is the Saint-Venant inequality, which says that balls have the largest torsional rigidity among subsets of Euclidean space with a fixed volume. We discuss recent results showing that for a given set \( E \), the gap in the Saint-Venant inequality quantitatively controls the \( L^2 \) difference between solutions of the Poisson equation on \( E \) and on the nearest ball, for any Hölder continuous right-hand side. We additionally prove quantitative closeness of all eigenfunctions of the Dirichlet Laplacian.

This talk is based on joint work with Mark Allen and Dennis Kriventsov.

We present new developments in the theory of compact Calderón-Zygmund operators. In particular, we give a new formulation of the \( T1 \) theorem for compactness of CZ operators, which, compared to existing compactness criteria, more closely resembles David and Journé’s classical \( T1 \) theorem for boundedness and follows from a simpler argument. Our methods generalize to treat a class of "localized" operators on a Hilbert space — we apply this abstraction to characterize the compact pseudodifferential operators on \( L^2(\mathbb{R}^n) \). Additionally, we discuss the extension of compact CZ theory to weighted Lebesgue spaces via sparse domination methods.

This talk is based on joint works with Mishko Mitkovski, Paco Villarroya, Cody Waters, and Brett Wick.

The sharp fixed-time estimates for the solution of the Cauchy problem associated with the standard Euclidean Laplacian on Lebesgue and Hardy spaces were first studied independently by A. Miyachi and J.C. Peral in 1980. However, the sharp fixed-time estimate is still not available for many operators, especially on Hardy spaces for \(0 < p < 1\).

In this talk, we shall discuss fixed-time estimates for the solution of the wave equation associated with the twisted Laplacian. This talk is based on a joint work with K. Jotsaroop.

This presentation collects joint work with C. Mora-Corral, J. Cueto, H. Schönberger, P. Radu and M. Foss.

Interest in nonlocal gradients has increased in the last decades due to development of nonlocal modeling in a variety of fields, including mechanics and materials science. We start by defining nonlocal gradients in a general context, where their calculation depends on a general kernel. Our goal is to explore the structural properties of spaces associated with these gradients. From a functional analysis perspective, we seek kernels that make these spaces useful for studying variational problems and, consequently, applicable to physical models. Beyond the theoretical groundwork, we delve into the mathematical intricacies of these new functional spaces. These spaces are essential for understanding nonlocal phenomena and capturing behavior that local models might miss. Nonlocal gradients find practical applications in solid mechanics, particularly in finite elasticity. Additionally, we establish connections between nonlocal models derived from nonlinearly elastic models and the well-known Eringen’s nonlocal model of linear elasticity. Remarkably, these solid mechanics models can be seen as a special case of state-based peridynamics, a continuum theory designed to address material failure where classical elasticity theories fall short.

In this talk, we introduce the concept of \beta-dimensional BMO space \( BMO^{\beta}(\mathbb{R}^n) \) and the associated John-Nirenberg inequality.

We will discuss the mapping properties of Riesz potentials within \( BMO^{\beta} \) spaces, focusing specifically on the Morrey spaces and weak Lebesgue spaces \( L^{\infty,\lambda}(\mathbb{R}^n) \).

Additionally, we present that \( I_{\alpha}f \in BMO^{\beta+\alpha} \) is actually a necessary and sufficient condition for \( f \in BMO^{\beta} \) when \( f \) is a non-negative function.

Schatten class estimates of the commutator of Riesz transform in \( \mathbb{R}^n \) link to the quantised derivative of A. Connes. A general setting for quantised calculus is a spectral triple \( (\mathcal{A}, \mathcal{H}, D) \), which consists of a Hilbert space \( \mathcal{H} \), a pre-\( C^* \)-algebra \( \mathcal{A} \), represented faithfully on \( \mathcal{H} \) and a self-adjoint operator \( D \) acting on \( \mathcal{H} \) such that every \( a \in \mathcal{A} \) maps the domain of \( D \) into itself and the commutator \( [D, a] = Da - aD \) extends from the domain of \( D \) to a bounded linear endomorphism of \( \mathcal{H} \). Here, the quantised differential \( \partial a \) of \( a \in \mathcal{A} \) is defined to be the bounded operator \( i[\mathrm{sgn}(D), a] \), \( i^2 = -1 \).

We provide full characterisation of the Schatten properties of \( [M_b, T] \), the commutator of Calder\'{o}n--Zygmund singular integral \( T \) with symbol \( b \, (M_b f(x) := b(x) f(x)) \) on stratified Lie groups \( \mathbb{G} \). We show that, when \( p \) is larger than the homogeneous dimension \( Q \) of \( \mathbb{G} \), the Schatten \( \mathcal{L}_p \) norm of the commutator is equivalent to the Besov semi-norm \( B^{Q/p}_{p,p} \) of the function \( b \); but when \( p \leq Q \), the commutator belongs to \( \mathcal{L}_p \) if and only if \( b \) is a constant. For the endpoint case at the critical index \( p = Q \), we further show that the Schatten \( \mathcal{L}_{Q,\infty} \) norm of the commutator is equivalent to the Sobolev norm \( W^{1,Q} \) of \( b \). Our method at the endpoint case differs from existing methods of Fourier transforms or trace formula for Euclidean spaces or Heisenberg groups, respectively.

This talk is based on my recent work joint with Xiao Xiong and Fulin Yang.