In this paper we survey some results on the Dirichlet problem

\[\begin{cases} L u =f&\text{in }\Omega \\ u=g &\text{in }\mathbb{R}^n\backslash\Omega \end{cases} \]

for nonlocal operators of the form

\[ Lu(x)=\operatorname{PV}\int_{\mathbb{R}^n}\bigl\{u(x)-u(x+y)\bigr\}K(y)\,dy. \]

We start from the very basics, proving existence of solutions, maximum principles, and constructing some useful barriers. Then, we focus on the regularity properties of solutions, both in the interior and on the boundary of the domain.

In order to include some natural operators~$L$ in the regularity theory, we do not assume any regularity on the kernels. This leads to some interesting features that are purely nonlocal, in the sense that they have no analogue for local equations.

We hope that this survey will be useful for both novel and more experienced researchers in the field.

In this paper, we study solutions to a nonlocal $1$-Laplacian equation given by

$$ -\int_{\Omega_J} J(x-y)\frac{u_\psi(y)-u(x)}{|u_\psi(y)-u(x)|}\,dy=0\quad\text{for $x\in\Omega$}, $$

with $u(x)=\psi(x)$ for $x\in \Omega_J\setminus\overline\Omega$. We introduce two notions of solution and prove that the weaker of the two concepts is equivalent to a nonlocal median value property, where the median is determined by a measure related to $J$. We also show that solutions in the stronger sense are nonlocal analogues of local least gradient functions, in the sense that they minimize a nonlocal functional. In addition, we prove that solutions in the stronger sense converge to least gradient solutions when the kernel $J$ is appropriately rescaled.

New constructions of braces on finite nilpotent groups are given and hence this leads to new solutions of the Yang--Baxter equation. In particular, it follows that if a group $G$ of odd order is nilpotent of class three, then it is a homomorphic image of the multiplicative group of a finite left brace (i.e.\ an involutive Yang--Baxter group) which also is a nilpotent group of class three. We give necessary and sufficient conditions for an arbitrary group $H$ to be the multiplicative group of a left brace such that $[H,H] \subseteq \operatorname{Soc} (H)$ and $H/[H,H]$ is a standard abelian brace, where $\operatorname{Soc} (H)$ denotes the socle of the brace $H$.

We prove that a locally complete metric space endowed with a doubling measure satisfies an $\infty$-Poincaré inequality if and only if given a null set, every two points can be joined by a quasiconvex curve which "almost avoids" that set. As an application, we characterize doubling measures on ${\mathbb R}$ satisfying an $\infty$-Poincaré inequality. For Ahlfors $Q$-regular spaces, we obtain a characterization of $p$-Poincaré inequality for $p>Q$ in terms of the $p$-modulus of quasiconvex curves connecting pairs of points in the space. A related characterization is given for the case $Q-1<p\leq Q$.

There are fourteen fine gradings on the exceptional Lie algebra $\mathfrak e_6$ over an algebraically closed field of zero characteristic. We provide their descriptions and a proof that any fine grading is equivalent to one of them.

In this paper we prove mixed norm estimates for Riesz transforms on the group $SU(2)$. From these results vector valued inequalities for sequences of Riesz transforms associated to Jacobi differential operators of different types are deduced.

We show that the product BMO space can be characterized by iterated commutators of a large class of Calderón--Zygmund operators. This result follows from a new proof of boundedness of iterated commutators in terms of the BMO norm of their symbol functions, using Hytönen's representation theorem of Calderón--Zygmund operators as averages of dyadic shifts. The proof introduces some new paraproducts which have BMO estimates.

In this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which the Galois correspondence is bijective is larger than the class of almost classically Galois extensions but not equal to the whole class. We show as well that the image of the Galois correspondence does not determine the Hopf Galois structure.

In this paper we introduce new techniques in order to deepen into the structure of a Leavitt path algebra with the aim of giving a description of the center. Extreme cycles appear for the first time; they concentrate the purely infinite part of a Leavitt path algebra and, jointly with the line points and vertices in cycles without exits, are the key ingredients in order to determine the center of a Leavitt path algebra. Our work will rely on our previous approach to the center of a prime Leavitt path algebra, "Centers of path algebras, Cohn and Leavitt path algebras," Bull. Malays. Math. Sci. Soc.. We will go further into the structure itself of the Leavitt path algebra. For example, the ideal $I(P_{ec} \cup P_{c} \cup P_l)$ generated by vertices in extreme cycles ($P_{ec}$), by vertices in cycles without exits ($P_c$), and by line points ($P_l$) will be a dense ideal in some cases, for instance in the finite one or, more generally, if every vertex connects to $P_l \cup P_c\cup P_{ec}$. Hence its structure will contain much of the information about the Leavitt path algebra. In the row-finite case, we will need to add a new hereditary set: the set of vertices whose tree has infinite bifurcations ($P_{b^\infty}$).

We call $G$ a $\operatorname{Hall}_{\mathcal X}$-group if there exists a normal nilpotent subgroup $N$ of $G$ for which $G/N'$ is an ${\mathcal X}$-group. We call $G$ a ${\mathcal T}_0$-group provided $G/\Phi(G)$ is a ${\mathcal T}$-group, that is, one in which normality is a transitive relation. We present several new local classes of groups which locally define $\operatorname{Hall}_{\mathcal X}$-groups and ${\mathcal T}_0$-groups where ${\mathcal X}\in\{ {\mathcal T},\mathcal {PT},\mathcal {PST}\}$; the classes $\mathcal {PT}$ and $\mathcal {PST}$ denote, respectively, the classes of groups in which permutability and S-permutability are transitive relations.