Marcinkiewicz multipliers are $L^p$ bounded for $1<p<\infty$ on the Heisenberg group ${ℍ}^n\simeq {ℂ}^n\times ℝ$, as shown by D. Müller, F. Ricci, and E. M. Stein. This is surprising in that these multipliers are invariant under a two-parameter group of dilations on ${ℂ}^n\times ℝ$, while there is no two-parameter group of automorphic dilations on ${ℍ}^n$. This lack of automorphic dilations underlies the failure of such multipliers to be in general bounded on the classical Hardy space $H^1$ on the Heisenberg group, and also precludes a pure product Hardy space theory.

We address this deficiency by developing a theory of flag Hardy spaces $H_{flag}^p$ on the Heisenberg group, $0<p\le 1$, that is in a sense “intermediate” between the classical Hardy spaces $H^p$ and the product Hardy spaces $H_{product}^p$ on ${ℂ}^n\times ℝ$ developed by A. Chang and R. Fefferman. We show that flag singular integral operators, which include the aforementioned Marcinkiewicz multipliers, are bounded on $H_{flag}^p$, as well as from $H_{flag}^p$ to $L^p$, for $0<p\le 1$. We also characterize the dual spaces of $H_{flag}^1$ and $H_{flag}^p$, and establish a Calderón–Zygmund decomposition that yields standard interpolation theorems for the flag Hardy spaces $H_{flag}^p$. In particular, this recovers some $L^p$ results of Müller, Ricci, and Stein (but not their sharp versions) by interpolating between those for $H_{flag}^p$ and $L^2$.

Characterization results for equality cases and for rigidity of equality cases in Steiner’s perimeter inequality are presented. (By rigidity, we mean the situation when all equality cases are vertical translations of the Steiner symmetral under consideration.) We achieve this through the introduction of a suitable measure-theoretic notion of connectedness and a fine analysis of barycenter functions for sets of finite perimeter having segments as orthogonal sections with respect to a hyperplane.

We state the Bohr–Sommerfeld conditions around a singular value of hyperbolic type of the principal symbol of a selfadjoint semiclassical Toeplitz operator on a compact connected Riemann surface. These conditions allow the description of the spectrum of the operator in a fixed-size neighborhood of the singularity. We provide numerical computations for three examples, each associated with a different topology.

We prove optimal high-frequency resolvent estimates for self-adjoint operators of the form

$$G=-\Delta +ib\left(x\right)\cdot \nabla +i\nabla \cdot b\left(x\right)+V\left(x\right)$$

on $L^2\left({ℝ}^n\right)$, $n\ge 3$, where $b\left(x\right)$ and $V\left(x\right)$ are large magnetic and electric potentials, respectively.

A classical pseudodifferential operator $P$ on ${ℝ}^n$ satisfies the $\mu$-transmission condition relative to a smooth open subset $\Omega$ when the symbol terms have a certain twisted parity on the normal to $\partial \Omega$. As shown recently by the author, this condition assures solvability of Dirichlet-type boundary problems for $P$ in full scales of Sobolev spaces with a singularity $d^{\mu -k}$, $d\left(x\right)=dist\left(x,\partial \Omega \right)$. Examples include fractional Laplacians ${\left(-\Delta \right)}^a$ and complex powers of strongly elliptic PDE.

We now introduce new boundary conditions, of Neumann type, or, more generally, nonlocal type. It is also shown how problems with data on ${ℝ}^n\setminus \Omega$ reduce to problems supported on $\overline{\Omega }$, and how the so-called “large” solutions arise. Moreover, the results are extended to general function spaces $F_{p,q}^s$ and $B_{p,q}^s$, including Hölder–Zygmund spaces $B_{\infty ,\infty }^s$. This leads to optimal Hölder estimates, e.g., for Dirichlet solutions of ${\left(-\Delta \right)}^{a}u=f\in L_{\infty }\left(\Omega \right)$, $u\in d^a{C}^a\left(\overline{\Omega }\right)$ when $0<a<1$, $a\ne \frac{1}{2}$.

In this article, we establish the unconditional uniqueness of solutions to an infinite radial Chern–Simons–Schrödinger (IRCSS) hierarchy in two spatial dimensions. The IRCSS hierarchy is a system of infinitely many coupled PDEs that describes the limiting Chern–Simons–Schrödinger dynamics of infinitely many interacting anyons. The anyons are two-dimensional objects that interact through a self-generated field. Due to the interactions with the self-generated field, the IRCSS hierarchy is a system of nonlinear PDEs, which distinguishes it from the linear infinite hierarchies studied previously. Factorized solutions of the IRCSS hierarchy are determined by solutions of the Chern–Simons–Schrödinger system. Our result therefore implies the unconditional uniqueness of solutions to the radial Chern–Simons–Schrödinger system as well.