Related papers: On Lipschitz spaces in the Dunkl setting -- semigr…

We define notions of semi-saturatedness and orthogonality for a Fell bundle over a quasi-lattice ordered group. We show that a compactly aligned product system of Hilbert bimodules can be naturally extended to a semi-saturated and…

Generalizing a result of Roe \cite{Roe} Strichartz proved in \cite{Str} that if a doubly-infinite sequence $\{f_k\}$ of functions on $\R^n$ satisfies $f_{k+1}=\Delta f_k$ and $|f_{k}(x)|\leq M$ for all $k=0,\pm 1,\pm 2,...$ and $x\in \R^n$,…

The initial part of this paper is devoted to the notion of pseudo-seminorm on a vector space $E$. We prove that the topology of every topological vector space is defined by a family of pseudo-seminorms (and so, as it is known, it is…

We extend the classical theorem of Uchiyama about constructive Fefferman-Stein decompositions of ${\rm BMO}$ functions by systems of singular integrals to the rational Dunkl setting. On $\mathbb{R}^N$ equipped with a root system $R$ and a…

We introduce and systematically investigate a scale of tent spaces that characterizes homogeneous Triebel-Lizorkin spaces $\mathrm{\dot F}^{\beta}_{p,q}$. These spaces generalize the classical spaces of Coifman, Meyer, and Stein, and are…

In this paper we introduce Lipschitz spaces with respect to the Gaussian measure, and study the boundedness of the fractional integral and fractional derivative operators on them.The methods are general enough to provide alternative proofs…

A closed subset of $\mathbb{R}^q$, definable in some given o-minimal structure, is Lipschitz normally embedded in $\mathbb{R}^q$ if and only if its one-point compactification is Lipschitz normally embedded in the unit sphere ${\bf S}^q$($ =…

Let $n\ge 2$ and $s\in (n-2,n)$. Assume that $\Omega\subset \mathbb{R}^n$ is a one-sided bounded non-tangentially accessible domain with $s$-Ahlfors regular boundary and $\sigma$ is the surface measure on the boundary of $\Omega$, denoted…

The algebra of polynomials in operators that represent generalized coordinate and momentum and depend on the Planck constant is defined. The Planck constant is treated as the parameter taking values between zero and some nonvanishing $h_0$.…

For a metric space $(K,d)$ the Banach space $\Lip(K)$ consists of all scalar-valued bounded Lipschitz functions on $K$ with the norm $\|f\|_{L}=\max(\|f\|_{\infty},L(f))$, where $L(f)$ is the Lipschitz constant of $f$. The closed subspace…

We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative,…

Let $P \subset \mathbb{R}^{d}$ be a closed convex cone. Assume that $P$ is pointed, i.e. the intersection $P \cap -P=\{0\}$ and $P$ is spanning, i.e. $P-P=\mathbb{R}^{d}$. Denote the interior of $P$ by $\Omega$. Let $E$ be a product system…

We develop a potential-theoretic and functional framework for the fractional--logarithmic Laplacian $(-\Delta)^{s+\ln}$ and its inhomogeneous counterpart $(\lambda I-\Delta)^{s+\ln}$ with $\lambda>1$. Their inverses yield logarithmic…

Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some `one particle space' $\K$ are formed by certain symmetrization procedures performed on the full Fock space. We investigate alternative ways of…

We characterise slice-regularity of functions over a real alternative *-algebra using operators that arise in Dunkl operator theory. We present a unifying perspective on hypercomplex analysis by defining a family of function spaces in the…

We study the classical spaces $L_{p}$ and $\ell_{p}$ for the whole range $0<p<\infty$ from a metric viewpoint and give a complete Lipschitz embeddability roadmap between any two of those spaces when equipped with both their ad-hoc distances…

The definition of a length operator in quantum cosmology is usually influenced by a~quantum theory for gravity considered. The semiclassical limit at the Planck age must meet the requirements implied in present observations. The features of…

We develop potential theory for $m$-subharmonic functions with respect to a Hermitian metric on a Hermitian manifold. First, we show that the complex Hessian operator is well-defined for bounded functions in this class. This allows to…

We discuss a framework for quantizing a Poisson manifold via the quantization of its symplectic groupoid, that combines the tools of geometric quantization with the results of Renault's theory of groupoid C*-algebras. This setting allows…

We introduce a definition of the fractional Laplacian $(-\Delta)^{s(\cdot)}$ with spatially variable order $s:\Omega\to [0,1]$ and study the solvability of the associated Poisson problem on a bounded domain $\Omega$. The initial motivation…