Related papers: Shift-invariant subspaces of Sobolev type
We consider the family known as modified or generalized surface quasi-geostrophic equations (mSQG) consisting of the classical inviscid surface quasi-geostrophic (SQG) equation together with a family of regularized active scalars given by…
In a doubling metric measure space $(X,\rho,\mu)$ supporting a Poincar\'e inequality, we give a new characterisation of first-order Sobolev spaces by mean oscillations, and extend previous characterisations of constant functions in terms of…
We give a new characterization of Sobolev-Slobodeckij spaces W^{1+s,p} for n/p<1+s, where n is the dimension of the domain. To achieve this we introduce a family of curvature energies inspired by the classical concept of integral Menger…
We consider Kolmogorov-Fokker-Planck operators of the form $$ \mathcal{L}u=\sum_{i,j=1}^{q}a_{ij}(x,t)u_{x_{i}x_{j}}+\sum_{k,j=1}^{N} b_{jk}x_{k}u_{x_{j}}-\partial_{t}u, $$ with $\left( x,t\right) \in\mathbb{R}^{N+1},N\geq q\geq1$. We…
In this article, we introduce inhomogeneous Sobolev spaces that naturally generalise the standard Sobolev-Slobodeckij spaces. The inhomogeneity of these spaces is governed by a set function $\mu$, referred to as an environment. In the case…
In this article, we show the existence of large sets $\operatorname{LS}_2[3](2,k,v)$ for infinitely many values of $k$ and $v$. The exact condition is $v \geq 8$ and $0 \leq k \leq v$ such that for the remainders $\bar{v}$ and $\bar{k}$ of…
The admittable Sobolev regularity is quantified for a function, $w$, which has a zero in the $d$--dimensional torus and whose reciprocal $u=1/w$ is a $(p,q)$--multiplier. Several aspects of this problem are addressed, including zero--sets…
The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which…
We define distributions on an abstract measure space endowed with a sequence of partitions, and introduce analogues of Besov spaces with negative smoothness in this setting. In particular, we describe these spaces of distributions using…
The study of subshifts on groups different from $\mathbb{Z}$, such as $\mathbb{Z}^d$, $d\geq 2$, has been a subject of intense research in recent years. These investigations have unveiled aremarkable connection between dynamics and…
This paper introduces first order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally nonlinear class of generalised weakly differentiable functions and share key functional…
In this paper we show that the Fourier transform induces an isomorphism between the coorbit spaces defined by Feichtinger and Gr\"ochenig of the mixed, weighted Lebesgue spaces $L_{v}^{p,q}$ with respect to the quasi-regular representation…
In this paper, we study the traces and the extensions for weighted Sobolev spaces on upper half spaces when the weights reach to the borderline cases. We first give a full characterization of the existence of trace spaces for these weighted…
Let (G, V) be a prehomogeneous vector space, let O be any G(F_q)-invariant subset of V(F_q), and let f be the characteristic function of O. In this paper we develop a method for explicitly and efficiently evaluating the Fourier transform of…
In this article, we characterize the Beurling and Model subspaces of the Hardy-Hilbert space $H^2(\mathbb{D})$ invariant under the composition operator $C_{\phi_a}f=f\circ\phi_a$, where $\phi_a(z) = az + 1 - a$ for $a \in (0,1)$ is an…
One way to construct a maximal set of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space is by means of finite phase-space methods. MUBs obtained in this way are covariant with respect to some subgroup of the group of…
The purpose of this note is to prove the existence of a remarkable structure in an iterated sumset derived from a set $P$ in a Cartesian square $\mathbb{F}_p^n\times\mathbb{F}_p^n$. More precisely, we perform horizontal and vertical sums…
We find optimal conditions on $m$-linear Fourier multipliers to give rise to bounded operators from a product of Hardy spaces $H^{p_j}$, $0<p_j\le 1$, to Lebesgue spaces $L^p$. The conditions we obtain are necessary and sufficient for…
In this paper, we study different types of weighted Besov and Triebel-Lizorkin spaces with variable smoothness. The function spaces can be defined by means of the Littlewood-Paley theory in the field of Fourier analysis, while there are…
A characterization of the generalized Lipschitz and Besov spaces in terms of decay of Fourier transforms is given. In particular, necessary and sufficient conditions of Titchmarsh type are obtained. The method is based on two-sided estimate…