Fix the indices $\alpha$ and $\beta$, $1<\alpha<\beta<\infty$, and suppose $\varrho$ is an Orlicz gauge or Lorentz gamma norm on the real-valued functions on a set $X$ which are measurable with respect to a~$\sigma$-finite measure $\mu$ on it. Set

$$M(\gamma,X):=\{f\colon X\to\mathbb R \text{ with } \sup_{\lambda>0}\lambda \mu(\{x\in X: |f(x)|>\lambda\})^{\frac1{\gamma}}<\infty\},$$

$\gamma=\alpha,\beta$. In this paper we obtain, as a special case, simple criteria to guarantee that a linear operator $T$ satisfies $T\colon L_{\varrho}(X)\to L_{\varrho}(X)$, whenever $T\colon M(\alpha,X)\to M(\alpha, X)$ and $T\colon M(\beta,X)\to M(\beta, X)$.

We prove the connectedness and compute the diameter of the oriented graph of multi-graftings associated to exotic $\mathbb{CP}^1$-structures on a compact surface~$S$ with a given holonomy representation of Fuchsian type.

A recent theorem of S. Alesker, S. Artstein-Avidan and V. Milman characterises the Fourier transform on $ {\mathbb R}^{n} $ as essentially the only transform on the space of tempered distributions which interchanges convolutions and pointwise products. In this note we study the image of the Schwartz space on the Heisenberg group under the Fourier transform and obtain a similar characterisation for the Fourier transform on the Heisenberg group

We study the iteration of the family of maps given by $3$-step linear fractional recurrences. This family was studied earlier from the point of view of finding periodicities. In this paper we finish that study by determining all possible periods within this family. The novelty of our approach is that we apply the methods of complex dynamical systems. This leads to two classes of interesting pseudo automorphisms of infinite order. One of the classes consists of completely integrable maps. The other class consists of maps of positive entropy which have an invariant family of $K3$ surfaces.

We study the uniqueness sets, the weak interpolation sets, and convergence of the Lagrange interpolation series in radial weighted Fock spaces.

Aim of this paper is to show that weak solutions of the following fractional Laplacian equation $$ \begin{cases} (-\Delta)^s u=f &\text{in }\Omega\\ u=g &\text{in }\mathbb R^n\setminus\Omega \end{cases} $$ are also continuous solutions (up to the boundary) of this problem in the viscosity sense.

Here $s\in(0,1)$ is a fixed parameter, $\Omega$ is a bounded, open subset of $\mathbb R^n$ ($n\geqslant1$) with $C^2$-boundary, and $(-\Delta)^s$ is the fractional Laplacian operator, that may be defined as $$(-\Delta)^su(x):=c(n,s)\int\limits_{\mathbb R^n}\frac{2u(x)-u(x+y)-u(x-y)}{|y|^{n+2s}}\,dy,$$ for a suitable positive normalizing constant $c(n,s)$, depending only on $n$ and $s$.

In order to get our regularity result we first prove a maximum principle and then, using it, an interior and boundary regularity result for weak solutions of the problem.

As a consequence of our regularity result, along the paper we also deduce that the first eigenfunction of $(-\Delta)^s$ is strictly positive in $\Omega$.

In this note we will show that the dilation result obtained for fractional skew monoid rings, in the case of a cancellative left Ore monoid $S$ acting on a unital ring $A$ by corner isomorphisms, holds in full generality. We apply this result to the context of semigroup $C^*$-crossed products.

In this paper, we analyze the geometric structure of a Euclidean submanifold whose osculating spaces form a nonconstant family of proper subspaces of the same dimension. We prove that if the rate of change of the osculating spaces is small, then the submanifold must be a (submanifold of a) ruled submanifold of a very special type. We also give a sharp estimate of the dimension of the rulings.

We introduce a weaker variant of the concept of linear local connectivity, sufficient to guarantee the extendability of a conformal map $f\colon\mathbb D\to\Omega$ to the entire plane as a homeomorphism of locally exponentially integrable distortion. Additionally, we show that a conformal map as above cannot necessarily be extended in this manner if we assume that $\Omega$ is the image of $\mathbb D$ under a self-homeomorphism of the plane that has locally exponentially integrable distortion.

If a compact set $K \subset \mathbb{R}^{2}$ contains a positive-dimensional family of line-segments in every direction, then $K$ has positive measure.

We prove that the notions of compactness and weak compactness for a Hankel operator on $\mathit{BMOA}$ are identical.

Let $R$ be an associative unital ring with the unit group $U(R)$. Let $\mathcal{S}$ denote one of the following sets: the set of elements of $R$, of left ideals of $R$, of principal left ideals of $R$, or of ideals of $R$. Then the group $U(R)\times U(R)$ acts on the set $\mathcal{S}$ by left and right multiplication. In this note we are going to discuss some properties of rings $R$ with a finite number of orbits under the action of $U(R)\times U(R)$ on $\mathcal{S}$.

We give a characterization of hyperbolic Kato surfaces in terms of the existence of an automorphic Green function on a cyclic covering. This is achieved by analysing a naturally defined Levi-flat foliation, and by perturbing certain Levi-flat leaves to strictly pseudoconvex hypersurfaces.