We study band-dominated operators on (subspaces of) $L_p$-spaces over metric measure spaces of bounded geometry satisfying an additional property. We single out core assumptions to obtain, in an abstract setting, definitions of limit operators, characterizations of compactness and Fredholmness using limit operators; and thus also spectral consequences. In this way, we recover and unify the classical and recent results on limit operator techniques, but also gain new insights and are able to treat further applications.
For many known non-compact embeddings of two Banach spaces $E\hookrightarrow F$, every bounded sequence in $E$ has a subsequence that takes form of a \emph{profile decomposition} - a sum of clearly structured terms with asymptotically disjoint supports plus a remainder that vanishes in the norm of $F$. In this paper we construct a profile decomposition for arbitrary sequences in the Sobolev space $H^{1,2}(M)$ of a Riemannian manifold with bounded geometry, relative to the embedding of $H^{1,2}(M)$ into $L^{2^*}(M)$, generalizing the well-known profile decomposition of Struwe to the case of any…
We give a new characterization of the space of functions of bounded variation in terms of a pointwise inequality connected to the maximal function of a measure. The characterization is new even in Euclidean spaces and it holds also in general metric spaces.
In the setting of a metric space equipped with a doubling measure and supporting a Poincaré inequality, and based on results by Björn and Shanmugalingam (2007), we show that functions of bounded variation can be extended from any bounded uniform domain to the whole space. Closely related to extensions is the concept of boundary traces, which have previously been studied by Hakkarainen et al. (2014). On spaces that satisfy a suitable locality condition for sets of finite perimeter, we establish some basic results for the traces of functions of bounded variation. Our analysis of traces also prod…
In this paper, the fractional integral operator on non-homogeneous metric measure spaces is introduced, which contains the classic fractional integral operator, fractional integral with non-doubling measures and fractional integral with fractional kernel of order $α$ and regularity $ε$ introduced by García-Cuerva and Gatto as special cases. And the $(L^{p}(μ),L^{q}(μ))$-boundedness for fractional integral operators on non-homogeneous metric measure spaces is established. From this, the $(L^{p}(μ),L^{q}(μ))$-boundedness for commutators and multilinear commutators generated by fractional integra…
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…
In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below and positive injectivity radius. Denote by L the Laplace-Beltrami operator on M. We assume that the kernel associated to the heat semigroup generated by L satisfies a mild decay condition at infinity. We prove that if m is a bounded holomorphic function in a suitable strip of the complex plane, and satisfies Mihlin-Hormander type conditions of appropriate order at infinity, then the operator m(L) extends to an operator of weak type 1. This partially extends a celebrated resul…
We show that every metric space with bounded geometry uniformly embeds into an explicit reflexive Banach space (a direct sum of l^p spaces). In the case of discrete groups we show the analogue of a-T-menability. That is, we construct a metrically proper affine isometric action on this Banach space.
A separable Banach space X contains $\ell_1$ isomorphically if and only if X has a bounded wc_0^*-stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded wc_0^*-biorthogonal system.
We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best choice. We consider a more general definition subject to different local coordinates and give sufficient conditions on the corresponding coordinates resulting in equivalent norms. Our main application is the computation of traces on submanifolds with the help of Fermi coordinates. Our results also hold for corresponding spaces defined on vector bundles of bounde…
We define and study a noncommutative Fourier transform on every homogeneous complex bounded domain. We then give an application in noncommutative differential geometry by defining noncommutative Baumslag-Solitar tori.
We prove that if $τ$ is a large positive number, then the atomic Goldberg-type space $\mathfrak{h}^1(N)$ and the space $\mathfrak{h}_{\mathcal R_τ}^1(N)$ of all integrable functions on $N$ whose local Riesz transform $\mathcal R_τ$ is integrable are the same space on any complete noncompact Riemannian manifold $N$ with Ricci curvature bounded from below and positive injectivity radius. We also relate $\mathfrak{h}^1(N)$ to a space of harmonic functions on the slice $N\times (0,δ)$ for $δ>0$ small enough.
Let X be a separable metric space and let βbe the strict topology on the space of bounded continuous functions on X, which has the space of τ-additive Borel measures as a continuous dual space. We prove a Banach-Dieudonneé type result for the space of bounded continuous functions equipped with β. As a consequence, this space is hypercomplete and a Pták space. Additionally, the closed graph, inverse mapping and open mapping theorems holds for linear maps between space of this type.
It is known that the space of boundedly finite integer-valued measures on a complete separable metric space becomes itself a complete separable metric space when endowed with the weak-hash metric. It is also known that convergence under this topology can be characterised in a way that is similar to the weak convergence of totally finite measures. However, the original proofs of these two fundamental results assume that a certain term is monotonic, which is not the case as we give a counterexample. We manage to clarify these original proofs by addressing specifically the parts that rely on this…
We present simple conditions which ensure that a strongly elliptic operator $L$ generates an analytic semigroup on Hölder spaces on an arbitrary complete manifold of bounded geometry. This is done by establishing the equivalent property that $L$ is "sectorial", a condition that specifies the decay of the resolvent $(λI - L)^{-1}$ as $λ$ diverges from the Hölder spectrum of $L$. As one step, we prove existence of this resolvent if $λ$ is sufficiently large, and on this general class of manifolds, use a geometric microlocal version of the semiclassical pseudodifferential calculus. The properties…
Let $(\cx,\,d,\,μ)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors show that for the maximal Calderón-Zygmund operator associated with a singular integral whose kernel satisfies the standard size condition and the Hörmander condition, its $L^p(μ)$ boundedness with $p\in(1,\infty)$ is equivalent to its boundedness from $L^1(μ)$ into $L^{1,\infty}(μ)$. Moreover, applying this, together with a new Cotlar type inequality, the authors show that if the Calderón-Zygmund operator $T$ is bounded on $L^2(μ…
We establish a sharp estimate for a minimal number of binary digits (bits) needed to represent all bounded total generalized variation functions taking values in a general totally bounded metric space $(E,ρ)$ up to an accuracy of $\varepsilon>0$ with respect to the ${\bf L}^1$-distance. Such an estimate is explicitly computed in terms of doubling and packing dimensions of $(E,ρ)$. The obtained result is applied to provide an upper bound on the metric entropy for a set of entropy admissible weak solutions to scalar conservation laws in one-dimensional space with weakly genuinely nonlinear fluxe…
