We study Legendrian embeddings of a compact Legendrian submanifold $L$ sitting in a closed contact manifold $(M,\xi)$ whose contact structure is supported by a (contact) open book $\mathcal{OB}$ on $M$. We prove that if $\mathcal{OB}$ has Weinstein pages, then there exist a contact structure $\xi^{\prime}$ on $M$, isotopic to $\xi$ and supported by $\mathcal{OB}$, and a contactomorphism $f : (M, \xi) \to (M, \xi^{\prime})$ such that the image $f(L)$ of any such submanifold can be Legendrian isotoped so that it becomes disjoint from the closure of a page of $\mathcal{OB}$.

We continue the study in [2] in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $(\mathbb{R}^{+})^d$. Our goal is to establish a large deviation principle in this setting specifying the rate function in terms of $P$-pluripotential-theoretic notions. As an important preliminary step, we first give an existence proof for the solution of a Monge–Ampère equation in an appropriate finite energy class. This is achieved using a variational approach.

We obtain improved fractional Poincaré inequalities in John domains of a metric space $(X, d)$ endowed with a doubling measure $\mu$ under some mild regularity conditions on the measure $\mu$. We also give sufficient conditions on a bounded domain to support fractional Poincaré type inequalities in this setting.

We study weakly trapped submanifolds of codimension two in a standard static spacetime. In this setting, we apply some generalized maximum principles in order to investigate the geometry of these trapped submanifolds. For instance, we establish sufficient conditions to guarantee that such a spacelike submanifold must be a hypersurface of the Riemannian base of the ambient spacetime. As a consequence, we prove that there do not exist $n$-dimensional compact (without boundary) trapped submanifolds immersed in an $(n+2)$-dimensional standard static spacetime. Such a nonexistence result was originally obtained for stationary spacetimes by Mars and Senovilla [20]. Furthermore, we also investigate parabolic weakly trapped submanifolds immersed in a standard static spacetime.

We consider weighted ray-transforms $P_W$ (weighted Radon transforms along oriented straight lines) in $\mathbb{R}^d, d \geq 2$, with strictly positive weights $W$. We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions on $\mathbb{R}^d$. In addition, the constructed weight $W$ is rotation-invariant continuous and is infinitely smooth almost everywhere on $\mathbb{R}^d \times \mathbb{S}^{d-1}$. In particular, by this construction we give counterexamples to some well-known injectivity results for weighted ray transforms for the case when the regularity of $W$ is slightly relaxed. We also give examples of continous strictly positive $W$ such that $\mathrm{dim} \: \mathrm{ker} \: P_W \geq n$ in the space of infinitely smooth compactly supported functions on $\mathbb{R}^d$ for arbitrary $n \in \mathbb{N} \cup \lbrace \infty \rbrace$, where $W$ are infinitely smooth for $d=2$ and infinitely smooth almost everywhere for $d \geq 3$.

A generalization of a result of Wermer concerning the existence of polynomial hulls without analytic discs is presented. As a consequence it is shown that there exists a Cantor set $X$ in $\mathbb{C}^3$ whose polynomial hull is strictly larger than $X$ but contains no analytic discs.

We characterize the indecomposable injective objects in the category of finitely presented representations of an interval finite quiver.

We consider the discrete Laplacian $\Delta$ on the cubic lattice $\mathbb{Z}^d$, and deduce estimates on the group $e^{i t \Delta}$ and the resolvent $(\Delta-z)^{-1}$, weighted by $\ell^q (\mathbb{Z}^d)$-weights for suitable $q \geqslant 2$. We apply the obtained results to discrete Schrödinger operators in dimension $d \geqslant 3$ with potentials from $\ell^p (\mathbb{Z}^d)$ with suitable $p \geqslant1$.

The geometric descriptions of the (essential) spectra of Toeplitz operators with piecewise continuous symbols are among the most beautiful results about Toeplitz operators on Hardy spaces $H^p$ with $1 \lt p \lt \infty$. In the Hardy space $H^1$, the essential spectra of Toeplitz operators are known for continuous symbols and symbols in the Douglas algebra $C + H^{\infty}$. It is natural to ask whether the theory for piecewise continuous symbols can also be extended to $H^1$. We answer this question in the negative and show in particular that the Toeplitz operator is never bounded on $H^1$ if its symbol has a jump discontinuity.

We study differentiability properties of a potential of the type $K \star \mu$, where $\mu$ is a finite Radon measure in $\mathrm{R}^N$ and the kernel $K$ satisfies ${\lvert \nabla^j K(x) \rvert \leq C \lvert x \rvert}^{-(N-1+j)} , j={0, 1, 2}$. We introduce a notion of differentiability in the capacity sense, where capacity is classical capacity in the de la Vallée Poussin sense associated with the kernel ${\lvert x \rvert}^{-(N-1)}$. We require that the first order remainder at a point is small when measured by means of a normalized weak capacity “norm” in balls of small radii centered at the point. This implies weak $L^{N/(N-1)}$ differentiability and thus $L^p$ differentiability in the Calderón–Zygmund sense for $1 \leq p \lt N / (N-1)$. We show that $K \star \mu$ is a.e. differentiable in the capacity sense, thus strengthening a recent result by Ambrosio, Ponce and Rodiac. We also present an alternative proof of a quantitative theorem of the authors just mentioned, giving pointwise Lipschitz estimates for $K \star \mu$. As an application, we study level sets of newtonian potentials of finite Radon measures.

We give a partial converse to [8, Theorem 1.3] (as a resolution of [2, Problem 8.4] for the quasiconformal $Q$-composition) for $Q_{0 \lt \alpha \lt 2^{-1}} (\mathbb{R}^{n \geq 2})$, and yet demonstrate that if $f : \mathbb{R}^2 \to \mathbb{R}^2$ is a homeomorphism then the boundedness of $u \mapsto u \circ f$ on $Q_{2^{-1} \lt \alpha \lt 1} (\mathbb{R}^2) \subset BMO (\mathbb{R}^2)$ yields the quasiconformality of $f$.

In a recent series of articles, the authors have studied the transition behavior of partial Bergman kernels $\Pi_{k, [E_1, E_2]} (z,w)$ and the associated DOS (density of states) $\Pi_{k, [E_1, E_2]} (z)$ across the interface $\mathcal{C}$ between the allowed and forbidden regions. Partial Bergman kernels are Toeplitz Hamiltonians quantizing Morse functions $H : M \to \mathbb{R}$ on a Kähler manifold. The allowed region is $H^{-1} ([E_1, E_2])$ and the interface $\mathcal{C}$ is its boundary. In prior articles it was assumed that the endpoints $E_j$ were regular values of $H$. This article completes the series by giving parallel results when an endpoint is a critical value of $H$. In place of the Erf scaling asymptotics in a $k^{-\frac{1}{2}}$ tube around $\mathcal{C}$ for regular interfaces, one obtains $\delta$-asymptotics in $k^{-\frac{1}{4}}$ tubes around singular points of a critical interface. In $k^{-\frac{1}{2}}$ tubes, the transition law is given by the osculating metaplectic propagator.