Related papers: A clustering theorem in fractional Sobolev spaces

This paper is devoted to the study of large time bounds for the Sobolev norms of the solutions of the following fractional cubic Schr{\"o}dinger equation on the torus :$$i \partial\_t u = |D|^\alpha u+|u|^2 u, \quad u(0, \cdot)=u\_0,$$where…

Let $\Omega$ be a smooth, bounded domain of $\mathbb{R}^{N}$, $\omega$ be a positive, $L^{1}$-normalized function, and $0<s<1<p.$ We study the asymptotic behavior, as $p\rightarrow\infty,$ of the pair $\left( \sqrt[p]{\Lambda_{p}%…

In the present paper, we establish sharp Sobolev estimates for solutions of fully nonlinear parabolic equations, under minimal, asymptotic, assumptions on the governing operator. In particular, we prove that solutions are in…

We consider parabolic equations with operators $\mathcal{L}=\partial_{t}+a^{ij}D_{ij}+b^{i}D_{i}-c$ with $a$ being almost in VMO, $b$ in a Morrey class containing $ L_{d+2}$ and $c $ in a Morrey class containing $L_{(d+2)/2}$. We prove the…

The $p$-set, which is in a simple analytic form, is well distributed in unit cubes. The well-known Weil's exponential sum theorem presents an upper bound of the exponential sum over the $p$-set. Based on the result, one shows that the…

We extend the well-known result that any $f \in W^{1,n}(\Omega,\mathbb{R}^n)$, $\Omega \subset \mathbb{R}^n$ with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces $W^{s,\frac{n}{s}}(\Omega)$…

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…

We prove the $W^{1,2}_{p}$-solvability of second order parabolic equations in nondivergence form in the whole space for $p\in (1,\infty)$. The leading coefficients are assumed to be measurable in one spatial direction and have vanishing…

We describe a new, short proof of some facts relating the gap lengths of the spectrum of a potential of Hill's equation to its regularity. For example, a real potential is in a weighted Gevrey-Sobolev space if and only if its gap lengths…

The paper provides an extension, to fractional order Sobolev spaces, of the classical result of Murat and Brezis which states that the positive cone of elements in $H^{-1}(\Omega)$ compactly embeds in $W^{-1,q}(\Omega)$, for every $q < 2$…

Let $Y(\mathcal{X})$ be a ball quasi-Banach function space on the space of homogeneous type $(\mathcal{X},\rho,\mu)$ satisfying some mild additional assumptions, $q\in(0,\infty)$, and $\dot{W}^{s,q}_Y(\mathcal{X})$ with $s\in(0,1)$ be the…

Given a compact manifold $N^n \subset \mathbb{R}^\nu$, $s \ge 1$ and $1 \le p < \infty$, we prove that the class of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s, p}(Q^m; N^n)$ when…

We describe some sufficient conditions, under which smooth and compactly supported functions are or are not dense in the fractional Sobolev space $W^{s,p}(\Omega)$ for an open, bounded set $\Omega\subset\mathbb{R}^{d}$. The density property…

We consider the generalized two-dimensional Zakharov-Kuznetsov equation $u_t+\partial_x \Delta u+\partial_x(u^{k+1})=0$, where $k\geq3$ is an integer number. For $k\geq8$ we prove local well-posedness in the $L^2$-based Sobolev spaces…

When a function belonging to a fractional-order Sobolev space is supported in a proper subset of the Lipschitz domain on which the Sobolev space is defined, how is its Sobolev norm as a function on the smaller set compared to its norm on…

On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space $\mathcal{D}^{1,p}_0$ into $L^q$ and the summability properties of the distance function. We prove that in the…

This paper is devoted to the study of a generalization of Sobolev spaces for small $L^{p}$ exponents, i.e. $0<p<1$. We consider spaces defined as abstract completions of certain classes of smooth functions with respect to weighted…

For $1<p<\infty$ we give a characterization of the Sobolev space $\dot W^{1,p}(\mathbb R^d)$ in terms of the oscillations of a function on balls of varying centers and radii. Our work is motivated both by the study of trace ideal properties…

For a large class (heuristically most) of irreducible binary cubic forms $F(x,y) \in \mathbb Z[x,y]$, Bennett and Dahmen proved that the generalized superelliptic equation $F(x,y)=z^l$ has at most finitely many solutions in $x,y \in \mathbb…

In this paper, we prove the following inequality \begin{equation*} \|\big(\int_{\mathbb{R}^n}\frac{|f(\cdot+y)-f(\cdot)|^q}{|y|^{n+sq}}dy\big)^{\frac{1}{q}}\|_{L^{p,\infty}(\mathbb{R}^n)}\lesssim\|f\|_{\dot{L}^p_s(\mathbb{R}^n)},…