Fractional difference sequence spaces have been studied in the literature recently. In this work, some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators on some difference sequence spaces of fractional orders are established. Some classes of compact operators on those spaces are characterized. The results of this work are more general and comprehensive then many other studies in literature.
Among the sets of sequences studied, difference sets of sequences are probably the most common type of sets. This paper considers some $\ell_{p}$ type fractional difference sequence spaces via Euler gamma function. Although we characterize compactness conditions on those spaces using the main tools of Hausdorff measure of noncompactness, we can only obtain sufficient conditions when the final space is $\ell _{\infty }$. However, we use some recent results to exactly characterize the classes of compact matrix operators when the final space is the set of bounded sequences.
The compactness of the commutators of bilinear fractional integral operators and point-wise multiplication, acting on products of Lebesgue spaces, is characterized in terms of appropriate mean oscillation properties of their symbols. The compactness of the commutators when acting on product of weighted Lebesgue spaces is also studied.
In this paper, we study a family of general fractional Sobolev spaces $\MsqpOm$ when $\Om=\Rn$ or $\Om$ is a bounded domain, having a compact, Lipschitz boundary $\Bdy$, in $\Rn$ for $n\geq2$. Among other results, some compact embedding results of $\MVsqpRn\hookrightarrow\LqRn$ and $\MVsqpRn\hookrightarrow\LlRn$ for suitable potential functions $V(x)$ are described.
We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by finite homogeneous Sobolev norm together with finite $L^2$ norm of the Riesz potentials. As a byproduct we prove also existence of maximizers for the interpolation inequalities in Sobolev spaces for radially symmetric fractional super and sub harmonic functions.
Let $I_α$ be the bilinear fractional integral operator, $B_α$ be a more singular family of bilinear fractional integral operators and $\vec{b}=(b,b)$. Bényi et al. in \cite{B1} showed that if $b\in {\rm CMO}$, the {\rm BMO}-closure of $C^{\infty}_{c}(\mathbb{R}^n)$, the commutator $[b,B_α]_{i}(i=1,2)$ is a separately compact operator. In this paper, it is proved that $b\in {\rm CMO}$ is necessary for $[b,B_α]_{i}(i=1,2)$ is a compact operator. Also, the authors characterize the compactness of the {\bf iterated} commutator $[Π\vec{b},I_α]$ of bilinear fractional integral operator. More precisel…
We introduce the new space $BV^α(\mathbb{R}^n)$ of functions with bounded fractional variation in $\mathbb{R}^n$ of order $α\in (0, 1)$ via a new distributional approach exploiting suitable notions of fractional gradient and fractional divergence already existing in the literature. In analogy with the classical $BV$ theory, we give a new notion of set $E$ of (locally) finite fractional Caccioppoli $α$-perimeter and we define its fractional reduced boundary $\mathscr{F}^α E$. We are able to show that $W^{α,1}(\mathbb{R}^n)\subset BV^α(\mathbb{R}^n)$ continuously and, similarly, that sets with (…
We prove a Hardy inequality on convex sets, for fractional Sobolev-Slobodecki\uı spaces of order $(s,p)$. The proof is based on the fact that in a convex set the distance from the boundary is a superharmonic function, in a suitable sense. The result holds for every $1<p<\infty$ and $0<s<1$, with a constant which is stable as $s$ goes to $1$.
In this paper we study mixed norm boundedness for fractional integrals related to Laplace--Beltrami operators on compact Riemannian symmetric spaces of rank one. The key point is the analysis of weighted inequalities for fractional integral operators associated to trigonometric Jacobi polynomials expansions. In particular, we find a novel sharp estimate for the Jacobi fractional integral kernel with explicit dependence on the type parameters.
We realize norms of most composition operators acting on the Hardy space with linear fractional symbol as roots of hypergeometric functions. This realization leads to simple necessary and sufficient conditions on the symbol to exhibit extremal non-compactness, establishes equivalence of cohyponormality and cosubnormality of composition operators with linear fractional symbol, and yields a complete classification of those linear fractional that induce composition operators whose norms are determined by the action of the adjoint on the normalized reproducing kernels in the Hardy space.
Let a:[0,1] -> R be a Lebesgue-almost everywhere positive function. We consider the Riemann-Liouville operator R^a of variable order a(.) as an operator from L_p[0,1] to L_q[0,1]. Our first aim is to study its continuity properties. For example, we show that R^a is always continuous in L_p[0,1] if p>1. Surprisingly, this becomes false for p=1. In order R^a to be continuous in L_1[0,1], the function a(.) has to satisfy some additional assumptions. In the second, central part of this paper we investigate compactness properties of R^a. We characterize functions a(.) for which R^a is a compact ope…
In this work we investigate the well-posedness for difussion equations associated to subelliptic pseudo-differential operators on compact Lie groups. The diffusion by strongly elliptic operators is considered as a special case and in particular the fractional diffusion with respect to the Laplacian. The general case is studied within the Hörmander classes associated to a sub-Riemannian structure on the group (encoded by a Hörmander system of vector fields). Applications to diffusion equations for fractional sub-Laplacians, fractional powers of more general subelliptic operators, and the corres…
We introduce and study fractional variable exponents Sobolev trace spaces on any open set in the Euclidean space equipped with the Lebesgue measure. We show that every equivalence class of Sobolev functions has a quasicontinuous representatives. We use the relative capacity to characterize completely the zero trace fractional variable exponents Sobolev spaces. We also give a relative capacity criterium for removable sets.
The Aubin-Lions lemma and its variants play crucial roles for the existence of weak solutions of nonlinear evolutionary PDEs. In this paper, we aim to develop some compactness criteria that are analogies of the Aubin--Lions lemma for the existence of weak solutions to time fractional PDEs. We first define the weak Caputo derivatives of order $γ\in (0,1)$ for functions valued in general Banach spaces, consistent with the traditional definition if the space is $\mathbb{R}^d$ and functions are absolutely continuous. Based on a Volterra type integral form, we establish some time regularity estimat…
We provide a new quantitative version of Helly's theorem: there exists an absolute constant $α>1$ with the following property: if $\{P_i: i\in I\}$ is a finite family of convex bodies in ${\mathbb R}^n$ with ${\rm int}\left (\bigcap_{i\in I}P_i\right )\neq\emptyset $, then there exist $z\in {\mathbb R}^n$, $s\leq αn$ and $i_1,\ldots i_s\in I$ such that \begin{equation*} z+P_{i_1}\cap\cdots\cap P_{i_s}\subseteq cn^{3/2}\left(z+\bigcap_{i\in I}P_i\right), \end{equation*} where $c>0$ is an absolute constant. This directly gives a version of the "quantitative" diameter theorem of Bárány, Katchalsk…
The bounded variation seminorm and the Sobolev seminorm on compact manifolds are represented as a limit of fractional Sobolev seminorms. This establishes a characterization of functions of bounded variation and of Sobolev functions on compact manifolds. As an application the special case of sets of finite perimeter is considered.
We obtain an improved Sobolev inequality in H^s spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of optimizing sequences for the usual Sobolev embedding. More generally, it allows to derive an alternative, more transparent proof of the profile decomposition in H^s obtained in [P. Gerard, ESAIM 1998] using the abstract approach of dislocation spaces developed in [K. Tintarev & K. H. Fieseler, Imperial College Press 2007]. We also analyze directly the local defect of compactness of the Sobolev embedding in term…
