Related papers: Nonnegative kernels and $1$-rectifiability in the …

In this paper we consider a large class of fully nonlinear integro-differential equations. The class of our nonlocal operators we consider is not spatial homogeneous and we put mild assumptions on its kernel near zero. We prove the H\"older…

We study the horizontally regular curves in the Heisenberg groups $H_n$. We show the fundamental theorem of curves in $H_n$ $(n\geq 2)$ and define the concept of the orders for horizontally regular curves. We also show that the curve…

Given an associative $\mathbb{C}$-algebra $A$, we call $A$ strongly rigid if for any pair of finite subgroups of its automorphism groups $G, H,$ such that $A^G\cong A^H$, then $G$ and $H$ must be isomorphic. In this paper we show that a…

We prove that the Heisenberg Riesz transform is $L_2$--unbounded on a family of intrinsic Lipschitz graphs in the first Heisenberg group $\mathbb{H}$. We construct this family by combining a method from \cite{NY2} with a stopping time…

The main aim of this article is to establish boundedness of singular integrals with non-smooth kernels on product spaces. Let $L_1$ and $L_2$ be non-negative self-adjoint operators on $L^2(\mathbb{R}^{n_1})$ and $L^2(\mathbb{R}^{n_2})$,…

The only known examples of noncompact Einstein homogeneous spaces are standard solvmanifolds (special solvable Lie groups endowed with a left invariant metric), and according to a long standing conjecture, they might be all. The…

For a strictly convex set $K\subset \mathbb{R}^2$ of class $C^2$ we consider its associated sub-Finsler $K$-perimeter $|\partial E|_K$ in $\mathbb{H}^1$ and the prescribed mean curvature functional $|\partial E|_K-\int_E f$ associated to a…

Let G be a reductive group over a commutative ring R. We say that G has isotropic rank >=n, if every normal semisimple reductive R-subgroup of G contains (G_m)^n. We prove that if G has isotropic rank >=1 and R is a regular domain…

The fundamental role of the Cauchy transform in harmonic and complex analysis has led to many different proofs of its $L^2$ boundedness. In particular, a famous proof of Melnikov-Verdera [18] relies upon an iconic symmetrization identity of…

We study singular integral operators induced by Calder\'on-Zygmund kernels in any step-$2$ Carnot group $\mathbb{G}$. We show that if such an operator satisfies some natural cancellation conditions then it is $L^2$ bounded on all intrinsic…

In this short technical note, we extend a recently published result [Liao2017] on the Perron root (or the spectral radius) of non-negative matrices to real-valued non-negative kernels on an arbitrary measurable space $(\mathrm{E},…

We introduce a class of non-commutative Heisenberg like infinite dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the…

We consider nonconstant periodic constrained minimizers of semilinear elliptic equations for integro-differential operators in $\mathbb{R}$. We prove that, after an appropriate translation, each of them is necessarily an even function which…

Suppose that a Lie algebra $L$ admits a finite Frobenius group of automorphisms $FH$ with cyclic kernel $F$ and complement $H$ such that the characteristic of the ground field does not divide $|H|$. It is proved that if the subalgebra…

We show, by means of a class of examples, that if $K_1$ and $K_2$ are two positive definite kernels on the unit disc such that the multiplication by the coordinate function on the corresponding reproducing kernel Hilbert space is subnormal,…

A $G$-kernel is a group homomorphism from a group $G$ to the outer automorphism group of a C$^*$-algebra. Inspired by recent work of Evington and Gir\'{o}n Pacheco in the stably finite case, we introduce a new invariant of a $G$-kernel…

The paper introduces a new characterisation of strictly positive definiteness for kernels on the 2-sphere without assuming the kernel to be radially (isotropic) or axially symmetric. The results use the series expansion of the kernel in…

Explicit generators are found for the group $G_2$ of automorphisms of the algebra $\mS_2$ of one-sided inverses of a polynomial algebra in two variables over a field of characteristic zero. Moreover, it is proved that $$ G_2\simeq…

Given $k:\mathbb{R}^n\setminus\{0\} \to \mathbb{R}^n$ a kernel of class $C^2$ and homogeneous of degree $1-n$, we prove existence and uniqueness of H\"older regular solutions for some non-linear transport equations with velocity fields…

In this paper, we consider fully nonlinear integro-differential equations with possibly nonsymmetric kernels. We are able to find different versions of Alexandroff-Backelman-Pucci estimate corresponding to the full class $\cS^{\fL_0}$ of…