It has been known that sharp Sobolev embeddings into weak Lebesgue spaces are non-compact but the question of whether the measure of non-compactness of such an embedding equals to its operator norm constituted a well-known open problem. The existing theory suggested an argument that would possibly solve the problem should the target norms be disjointly superadditive, but the question of disjoint superadditivity of spaces $L^{p,\infty}$ has been open, too. In this paper, we solve both these problems. We first show that weak Lebesgue spaces are never disjointly superadditive, so the suggested technique is ruled out. But then we show that, perhaps somewhat surprisingly, the measure of non-compactness of a sharp Sobolev embedding coincides with the embedding norm nevertheless, at least as long as $p<\infty$. Finally, we show that if the target space is $L^{\infty}$ (which formally is also a weak Lebesgue space with $p=\infty$), then the things are essentially different. To give a comprehensive answer including this case, too, we develop a new method based on a rather unexpected combinatorial argument and prove thereby a general principle, whose special case implies that the measure of non-compactness, in this case, is strictly less than its norm. We develop a technique that enables us to evaluate this measure of non-compactness exactly.
We study totally bounded subsets in weighted variable exponent amalgam and Sobolev spaces. Moreover, this paper includes several detailed generalized results of some compactness criterions in these spaces.
The structure of non-compactness of optimal Sobolev embeddings of $m$-th order into the class of Lebesgue spaces and into that of all rearrangement-invariant function spaces is quantitatively studied. Sharp two-sided estimates of Bernstein numbers of such embeddings are obtained. It is shown that, whereas the optimal Sobolev embedding within the class of Lebesgue spaces is finitely strictly singular, the optimal Sobolev embedding in the class of all rearrangement-invariant function spaces is not even strictly singular.
Defect of compactness, relative to an embedding of two Banach spaces E and F, is a difference between a weakly convergent sequence in E and its weak limit taken up to a remainder that vanishes in the norm of F. For Sobolev embeddings in particular, defect of compactness is expressed as a profile decomposition - a sum of terms, called elementary concentrations, with asymptotically disjoint supports. We discuss a profile decomposition for the Sobolev space of a Riemannian manifold with bounded geometry, which is a sum of elementary concentrations associated with concentration profiles defined on…
Defect of compactness, relative to an embedding of two Banach spaces E and F, is a difference between a weakly convergent sequence in E and its weak limit taken up to a remainder that vanishes in the norm of F. For Sobolev embeddings in particular, defect of compactness is expressed as a profile decomposition - a sum of terms, called elementary concentrations, with asymptotically disjoint supports. We discuss a profile decomposition for the Sobolev space of a Riemannian manifold with bounded geometry, which is a sum of elementary concentrations associated with concentration profiles defined on…
We investigate the relationship between the compactness of embeddings of Sobolev spaces built upon rearrangement-invariant spaces into rearrangement-invariant spaces endowed with $d$-Ahlfors measures under certain restriction on the speed of its decay on balls. We show that the gateway to compactness of such embeddings, while formally describable by means of optimal embeddings and almost-compact embeddings, is quite elusive. It is known that such a Sobolev embedding is not compact when its target space has the optimal fundamental function. We show that, quite surprisingly, such a target space …
In this note, we give the affirmative answer of the question in [18], which is a compactness result of the non-radial Sobolev spaces. As an application, we show the existence of an extremal function of the critical Hardy inequality under spherical average zero. Next, we give an improvement of the compactness results of the radial Sobolev spaces in [8]. In Appendix, we give an alternative proof of Hardy type inequalities under spherical average zero.
Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces
在H^s空间中得到了一个改进的涉及Morrey范数的Sobolev不等式.这种改进产生了一个直接的证明的存在性的优化和紧凑的对称性的优化序列通常的Sobolev嵌入。更一般地说，它允许使用[K]中开发的位错空间的抽象方法导出[P. Gerard，ESAIM 1998]中获得的H^s中的轮廓分解的替代的、更透明的证明。Tintarev & K. H. Fieseler，Imperial College Press 2007]。我们还直接分析了Sobolev嵌入紧性的局部缺陷。Lions，Rev. Mat. Iberoamericana 1985]。作为模型应用，我们研究了一类次临界问题的渐近极限，得到了当s为整数（[O. Rey，Mandarin pta math.1989; Z.- C. Han，Ann. Inst. H.庞加莱肛门。Non Lineaire 1991]，[K. S. Chou & D. Geng，微分积分方程2000]）。
Cocompactness is a property of embeddings between two Banach spaces, similar to but weaker than compactness, defined relative to some non-compact group of bijective isometries. In presence of a cocompact embedding, bounded sequences (in the domain space) have subsequences that can be represented as a sum of a well-structured "bubble decomposition" (or defect of compactness) plus a remainder vanishing in the target space. This note is an exposition of different proofs of cocompactness for Sobolev-type embeddings, which employ methods of classical PDE, potential theory, and harmonic analysis.
We study the Hardy-Littlewood maximal operator in the Musielak-Orlicz-Sobolev space $W^{1,φ}(\mathbb{R}^n)$. Under some natural assumptions on $φ$ we show that the maximal function is bounded and continuous in $W^{1,φ}(\mathbb{R}^n)$.
In this work we study some topological aspects of function spaces arising in Stieltjes differential calculus. Chief among them are compactness results related to the Ascoli-Arzelà and Kolmogorov-Riesz theorems, as well as their applications to Stieltjes-Sobolev spaces and decomposable functions.
We study the mapping property of the commutator of bilinear Hardy-Littlewood maximal operator in homogeneous Triebel-Lizorkin space. We also show that the commutator of bilinear Hardy-Littlewood maximal operator is a compact operator acting on product of Lebesgue spaces. The results are new even in the linear case.
Existence of extremal functions in higher-order affine Sobolev inequalities
Universite Claude Bernard Lyon 1, CNRS, Centrale Lyon, INSA Lyon, Université Jean Monnet, ICJ UMR5208（里昂第一大学、法国国家科学研究中心、里昂中央理工学院、里昂国立应用科学学院、让·莫内大学、ICJ联合研究实验室）
Given two measurable functions $V(r)\geq 0$ and $K(r)> 0$, $r>0$, we define the weighted spaces \[ H_V^1 = \{u \in D^{1,2}(\mathbb{R}^N): \int_{\mathbb{R}^N}V(|x|)u^{2}dx < \infty \}, \quad L_K^q = L^q(\mathbb{R}^N,K(|x|)dx) \] and study the compact embeddings of the radial subspace of $H_V^1$ into $L_K^{q_1}+L_K^{q_2}$, and thus into $L_K^q$ ($=L_K^q+L_K^q$) as a particular case. Both super- and sub-quadratic exponents $q_1$, $q_2$ and $q$ are considered. Our results do not require any compatibility between how the potentials $V$ and $K$ behave at the origin and at infinity, and essentially r…
It is common that a Sobolev space defined on $\mathbb{R}^m$ has a non-compact embedding into an $L^p$-space, but it has subspaces for which this embedding becomes compact. There are three well known cases of such subspaces, the Rellich compactness, for a subspace of functions on a bounded domain (or an unbounded domain, sufficiently thin at infinity), the Strauss compactness, for a subspace of radially symmetric functions in $\mathbb{R}^m$, and the weighted Sobolev spaces. Known generalizations of Strauss compactness include subspaces of functions with block-radial symmetry, subspaces of funct…
Defect of compactness for non-compact imbeddings of Banach spaces can be expressed in the form of a profile decomposition. This paper extends the profile decomposition for Sobolev spaces proved by Solimini (AIHP 1995) to the non-reflexive case p=1. Since existence of concentration profiles relies on weak-star compactness, the corresponding result is set in a larger, conjugate, space of functions of bounded variation. We prove existence of minimizers for related inequalities and generalizations for to spaces of bounded variation on Lie groups.
This paper concerns the following question: given a subset $E$ of $\mathbb{R}^n$ with empty interior and an integrability parameter $1<p<\infty$, what is the maximal regularity $s\in\mathbb{R}$ for which there exists a non-zero distribution in the Bessel potential Sobolev space $H^{s,p}(\mathbb{R}^n)$ that is supported in $E$? For sets of zero Lebesgue measure we apply well-known results on set capacities from potential theory to characterise the maximal regularity in terms of the Hausdorff dimension of $E$, sharpening previous results. Furthermore, we provide a full classification of all poss…
The optimal Orlicz target space is exhibited for embeddings of fractional-order Orlicz-Sobolev spaces in $\mathbb R^n$. An improved embedding with an Orlicz-Lorentz target space, which is optimal in the broader class of all rearrangement-invariant spaces, is also established. Both spaces of order $s\in (0,1)$, and higher-order spaces are considered. Related Hardy type inequalities are proposed as well. An extension theorem is proved, that enables us to derive embeddings for spaces defined in Lipschitz domains. Necessary and sufficient conditions for the compactness of fractional Orlicz-Sobolev…
The continouity and compactness of embedding operators in in Sobolev-Lions type spaces are derived. By applying this result separability properties of degenerate anisotropic differential operator equations, well-posedeness and Strichartz type estimates for solution of corresponding parabolic problem are established
Using concentration-compactness arguments we prove a variant of the Brezis-Lieb-Lemma under weaker assumptions on the nonlinearity than known before. An intermediate result on the uniform continuity of superposition operators in Sobolev space is of independent interest.
In this paper, we study operator-theoretic properties (boundedness and compactness) of Hankel operators on the Fock-sobolev spaces $ \mathscr{F}^{p,m} $ in terms of $ \mathcal{BMO}_r^p $ and $ \mathcal{VMO}_r^p $ spaces, respectively, for a non-negative integers $ m $, $ 1 \leq p < \infty $ and $ r > 0 $. Along the way, we also study Berezin transform of Hankel operators on $ \mathscr{F}^{p,m} $. The results in this article are analogous to Zhu's characterization and Perälä's characterization of bounded and compact Hankel operators on the Bergman spaces of unit disc and the weighted Fock space…
Motivated by the recent characterization of Sobolev spaces due to Brezis-Van Schaftingen-Yung we prove new weak-type inequalities for one parameter families of operators connected with mixed norm inequalities. The novelty here comes from the fact that the underlying measure space incorporates the parameter as a variable. The connection to classical and fractional order Sobolev spaces is shown through the use of generalized Riesz potential spaces and the Caffarelli-Silvestre extension principle. Higher order inequalities are also considered. We indicate many examples and applications to PDE's a…
Critical fractional $p$-Hardy Sobolev equations: Global compactness and multiplicity of positive solutions
Indian Institute of Science Education and Research Pune（印度科学教育研究所浦那分校）；Gandhi Institute of Technology and Management (GITAM) University（甘地科技与管理大学）；Krea University（克雷亚大学）
In this paper, we characterize operator-theoretic properties (boundedness, compactness, and Schatten class membership) of Toeplitz operators with positive measure symbols on weighted Fock-Sobolev spaces of fractional order.
In this note we establish the boundedness properties of local maximal operators $M_G$ on the fractional Sobolev spaces $W^{s,p}(G)$ whenever $G$ is an open set in $\mathbb{R}^n$, $0<s<1$ and $1<p<\infty$. As an application, we characterize the fractional $(s,p)$-Hardy inequality on a bounded open set $G$ by a Maz'ya-type testing condition localized to Whitney cubes.
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.
We prove non-autonomous maximal $L^p$-regularity results on UMD spaces replacing the common Hölder assumption by a weaker fractional Sobolev regularity in time. This generalizes recent Hilbert space results by Dier and Zacher. In particular, on $L^q(Ω)$ we obtain maximal $L^p$-regularity for $p \ge 2$ and elliptic operators in divergence form with uniform $VMO$-modulus in space and $W^{α,p}$-regularity for $α> \frac{1}{2}$ in time.
For limiting non-compact Sobolev embeddings into continuous functions we study behavior of Approximation, Gelfand, Kolmogorov, Bernstein and Isomorphism s-numbers. In the one dimensional case the exact values of the above-mentioned strict s-numbers were obtained and in the higher dimensions sharp estimates for asymptotic behavior of strict s-numbers were established. As all known results for s-numbers of Sobolev type embeddings are studied mainly under the compactness assumption then our work is an extension of existing results and reveal an interesting behavior of s-numbers in the limiting ca…
We develop a functional framework suitable for the treatment of partial differential equations and variational problems on evolving families of Banach spaces. We propose a definition for the weak time derivative that does not rely on the availability of a Hilbertian structure and explore conditions under which spaces of weakly differentiable functions (with values in an evolving Banach space) relate to classical Sobolev--Bochner spaces. An Aubin--Lions compactness result is proved. We analyse concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we…
For a general open set, we characterize the compactness of the embedding $W^{1,p}_0\hookrightarrow L^q$ in terms of the summability of its torsion function. In particular, for $1\le q<p$ we obtain that the embedding is continuous if and only if it is compact. The proofs crucially exploit a {\it torsional Hardy inequality} that we investigate in details.
We obtain (essentially sharp) boundedness results for certain generalized local maximal operators between fractional weighted Sobolev spaces and their modifications. Concrete boundedness results between well known fractional Sobolev spaces are derived as consequences of our main result. We also apply our boundedness results by studying both generalized neighbourhood capacities and the Lebesgue differentiation of fractional weighted Sobolev functions.
In this paper, we solve the Dirichlet problem for Orlicz-Sobolev maps between singular metric spaces that extends the corresponding result of Guo et al. [arXiv 2021]. As an intermediate step, we develop a version of Rellich-Kondrachov compactness theorem for Orlicz-Sobolev mappings between metric spaces that extends a previous result of Guo and Wenger [Comm. Anal. Geom. 2020]. Another crucial ingredient is an Orlicz-Sobolev extension of the trace theory for metric valued Sobolev maps developed by Korevaar and Schoen [Comm. Anal. Geom. 1993].
In this paper we investigate some questions related to the continuity of maximal operators in $W^{1,1}$ and $BV$ spaces, complementing some well-known boundedness results. Letting $\widetilde M$ be the one-dimensional uncentered Hardy-Littlewood maximal operator, we prove that the map $f \mapsto \big(\widetilde Mf\big)'$ is continuous from $W^{1,1}(\mathbb{R})$ to $L^1(\mathbb{R})$. In the discrete setting, we prove that $\widetilde M: BV(\mathbb{Z}) \to BV(\mathbb{Z})$ is also continuous. For the one-dimensional fractional Hardy-Littlewood maximal operator, we prove by means of counterexample…
