Underlying Measure Space Research Articles

Automatic continuity of separating and biseparating isomorphisms

Let p p be a fixed number with 1 ≤ p > ∞ 1 \leq p > \infty . It is shown that every surjective and biseparating linear map between L p L^p -spaces is continuous when the underlying measure space is non-atomic. We also prove that a separating isomorphism on l p l^p is both continuous and biseparating. Furthermore, these (bi-)separating maps take the form of a weighted composition operator. Our proofs are direct, elementary and do not invoke deep results about Riesz spaces or Banach lattices.

New Brezis–Van Schaftingen–Yung–Sobolev type inequalities connected with maximal inequalities and one parameter families of operators

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 of our approach comes from the fact that the underlying measure space incorporates the parameter as a variable. We also show that our framework can be adapted to treat related characterizations of Sobolev spaces obtained earlier by Bourgain–Nguyen. 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 and different areas of Analysis, suggesting a vast potential for future research. In a different direction, and inspired by methods originally due to Gagliardo and Garsia, we obtain new maximal inequalities which combined with mixed norm inequalities are applied to obtain Brezis–Van Schaftingen–Yung type inequalities in the context of Calderón–Campanato spaces. In particular, Log versions of the Gagliardo–Brezis–Van Schaftingen–Yung spaces are introduced and compared with corresponding limiting versions of Calderón–Campanato spaces, resulting in a sharpening of recent inequalities due to Crippa–De Lellis and Brué–Nguyen.

WEIGHTED COMPOSITION OPERATORS BETWEEN LORENTZ SPACES

AbstractWe investigate the boundedness, compactness, invertibility and Fredholmness of weighted composition operators between Lorentz spaces. It is also shown that the classes of Fredholm and invertible weighted composition maps between Lorentz spaces coincide when the underlying measure space is nonatomic.

COMPACT WEIGHTED COMPOSITION OPERATORS BETWEEN -SPACES

We provide complete characterisations for the compactness of weighted composition operators between two distinct $L^{p}$-spaces, where $1\leq p\leq \infty$. As a corollary, when the underlying measure space is nonatomic, the only compact weighted composition map between $L^{p}$-spaces is the zero operator.

Frames, their relatives and reproducing kernel Hilbert spaces

This paper considers different facets of the interplay between reproducing kernel Hilbert spaces (RKHS) and stable analysis/synthesis processes: first, we analyze the structure of the reproducing kernel of a RKHS using frames and reproducing pairs. Second, we present a new approach to prove the result that finite redundancy of a continuous frame implies atomic structure of the underlying measure space. Our proof uses the RKHS structure of the range of the analysis operator. This in turn implies that all the attempts to extend the notion of Riesz basis to general measure spaces are fruitless since every such family can be identified with a discrete Riesz basis. Finally, we show how the range of the analysis operators of a reproducing pair can be equipped with a RKHS structure.

Analytic computable structure theory and $$L^p$$-spaces part 2

Suppose $$p \ge 1$$ is a computable real. We extend previous work of Clanin, Stull, and McNicholl by determining the degrees of categoricity of the separable $$L^p$$ spaces whose underlying measure spaces are atomic but not purely atomic. In addition, we ascertain the complexity of associated projection maps.

Two weight estimates with matrix measures for well localized operators

In this paper, we give necessary and sufficient conditions for weighted L 2 L^2 estimates with matrix-valued measures of well localized operators. Namely, we seek estimates of the form \[ ‖ T ( W f ) ‖ L 2 ( V ) ≤ C ‖ f ‖ L 2 ( W ) , \| T(\mathbf {W} f)\|_{L^2(\mathbf {V})} \le C\|f\|_{L^2(\mathbf {W})}, \] where T T is formally an integral operator with additional structure, W , V \mathbf {W}, \mathbf {V} are matrix measures, and the underlying measure space possesses a filtration. The characterization we obtain is of Sawyer type; in particular, we show that certain natural testing conditions obtained by studying the operator and its adjoint on indicator functions suffice to determine boundedness. Working in both the matrix-weighted setting and the setting of measure spaces with arbitrary filtrations requires novel modifications of a T1 proof strategy; a particular benefit of this level of generality is that we obtain polynomial estimates on the complexity of certain Haar shift operators.

On the algebra of WCE operators

In this paper, we consider the algebra of WCE operators on $L^p$-spaces, and we investigate some al\-ge\-braic properties of it. For instance, we show that the set of normal WCE operators is a unital finite Von Neumann algebra, and we obtain the spectral measure of a normal WCE operator on $L^2(\mathcal {F})$. Then, we specify the form of projections in the Von Neumann algebra of normal WCE operators, and we obtain that, if the underlying measure space is purely atomic, then all projections are minimal. In the non-atomic case, there is no minimal projection. Also, we give a non-commutative operator algebra on which the spectral map is subadditive and submultiplicative. As a consequence, we obtain that the set of quasinilpotents is an ideal, and we get a relation between quasinilpotents and commutators. Moreover, we give some sufficient conditions for an algebra of WCE operators to be triangularizable, and consequently, that its quotient space over its quasinilpotents is commutative.

An exact Fatou lemma for Gelfand integrals: a characterization of the Fatou property

We establish an exact version of Fatou’s lemma for Gelfand integrals of functions and multifunctions in dual Banach spaces without any order structure, and under the saturation property on the underlying measure space. The necessity and sufficiency of saturation for the Fatou property is demonstrated. Our result has a direct application to the equilibrium existence result for saturated economies without convexity assumptions.

Higher-order Sobolev embeddings and isoperimetric inequalities

Optimal higher-order Sobolev type embeddings are shown to follow via isoperimetric inequalities. This establishes a higher-order analogue of a well-known link between first-order Sobolev embeddings and isoperimetric inequalities. Sobolev type inequalities of any order, involving arbitrary rearrangement-invariant norms, on open sets in Rn, possibly endowed with a measure density, are reduced to much simpler one-dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets. As a consequence, the optimal target space in the relevant Sobolev embeddings can be determined both in standard and in non-standard classes of function spaces and underlying measure spaces. In particular, our results are applied to any-order Sobolev embeddings in regular (John) domains of the Euclidean space, in Maz'ya classes of (possibly irregular) Euclidean domains described in terms of their isoperimetric function, and in families of product probability spaces, of which the Gauss space is a classical instance.

The Bang-Bang, Purification and Convexity Principles in Infinite Dimensions

The authors apply their recent work on the Lyapunov theorem in locally convex Hausdorff spaces to the bang-bang principle for control systems in infinite dimensions. They show that the bang-bang principle holds for every integrably bounded, measurable, weakly compact convex-valued multifunction if and only if the underlying measure space is saturated. They also demonstrate the equivalence of the bang-bang principle to what is termed the purification and convexity principles. Applications to variational problems with integral constraints are indicated.

Semigroups of partial isometries

We study self-adjoint semigroups of partial isometries on a Hilbert space. These semigroups coincide precisely with faithful representations of abstract inverse semigroups. Groups of unitary operators are specialized examples of self-adjoint semigroups of partial isometries. We obtain a general structure result showing that every self-adjoint semigroup of partial isometries consists of “generalized weighted composition” operators on a space of square-integrable Hilbert-space valued functions. If the semigroup is finitely generated then the underlying measure space is purely atomic, so that the semigroup is represented as “zero-unitary” matrices. The same is true if the semigroup contains a compact operator, in which case it is not even required that the semigroup be self-adjoint.

Multipliers for continuous frames in Hilbert spaces

In this paper, we examine the general theory of continuous frame multipliers in Hilbert space. These operators are a generalization of the widely used notion of (discrete) frame multipliers. Well-known examples include anti-Wick operators, STFT multipliers or Calderón–Toeplitz operators. Due to the possible peculiarities of the underlying measure spaces, continuous frames do not behave quite as their discrete counterparts. Nonetheless, many results similar to the discrete case are proven for continuous frame multipliers as well, for instance compactness and Schatten-class properties. Furthermore, the concepts of controlled and weighted frames are transferred to the continuous setting.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Bisimulations for non-deterministic labelled Markov processes

We extend the theory of labelled Markov processes to include internal non-determinism, which is a fundamental concept for the further development of a process theory with abstraction on non-deterministic continuous probabilistic systems. We define non-deterministic labelled Markov processes (NLMP) and provide three definitions of bisimulations: a bisimulation following a traditional characterisation; a state-based bisimulation tailored to our ‘measurable’ non-determinism; and an event-based bisimulation. We show the relations between them, including the fact that the largest state bisimulation is also an event bisimulation. We also introduce a variation of the Hennessy–Milner logic that characterises event bisimulation and is sound with respect to the other bisimulations for an arbitrary NLMP. This logic, however, is infinitary as it contains a denumerable . We then introduce a finitary sublogic that characterises all bisimulations for an image finite NLMP whose underlying measure space is also analytic. Hence, in this setting, all the notions of bisimulation we consider turn out to be equal. Finally, we show that all these bisimulation notions are different in the general case. The counterexamples that separate them turn out to be non-probabilistic NLMPs.

On the Rademacher maximal function

This paper studies a new maximal operator introduced by Hytonen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The Lp-boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to - finite measure spaces with filtrations and the L p -boundedness is shown not to depend on the underlying measure space or the filtration. Martingale techniques are applied to prove that a weak type inequality is sucient for L p -boundedness and also to provide a characterization by concave functions.

The maximal theorem for weighted grand Lebesgue spaces

We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0; 1) R, and the maximal function is localized in (0; 1). Moreover, we prove that the inequality kMfkp);w ckfkp);w holds with some c inde- pendent of f i w belongs to the well known Muckenhoupt class Ap, and therefore i kMfkp;w ckfkp;w for some c independent of f.

The K-divisibility constant for couples of Banach lattices

It is shown that the Brudnyi–Krugljak K-divisibility constant for an arbitrary couple of Banach lattices on the same underlying measure space is bounded above by 4.

Finitely purely atomic measures: Coincidence and rigidity properties

This paper is a natural continuation of the paper [2] by the same author. We shall prove that several coincidence and rigidity phenomena which usually do not appear are possible only in case the underlying measure space is trivial (i.e. is a finite union of atoms). Examples: coincidence of twoL p spaces, reflexivity ofL 1, Radon—Nikodym property ofL ∞, coincidence of Dunford, Pettis or Bochner integrability, coincidence of theL p space and of the weakL p space.

Some Uniform Ergodic Inequalities in the Nonmeasurable Case

Uniform and nonmeasurable versions of some classical ergodic inequalities (all of them going back to the Hopf–Yosida–Kakutani maximal ergodic theorem) are estab- lished. Usually, uniformity involves nonmeasurable suprema and all the technical difficulties arising from this. In the present paper, a simplification is achieved by extending the given operator (a positive L1-contraction) to the class of all (i.e., not necessarily measurable) functions on the underlying measure space. This not only leads to technical improvements and clarifications of the proofs, but also to remarkable generalizations of known results. In particular, it turns out that the “operator” under consideration need not even be an extension of an L1-contraction, but has only to fulfill some mild conditions such as positivity, super-additivity, and a certain contractivity property involving upper integrals.

ON MEASURE SPACES WHERE EGOROFF'S THEOREM HOLDS

A measure space (X, S, µ) is called almost f inite if X is a union of a setof finite measure and finite many atoms of infinite measure. It is shown that Egoroff’sTheorem for sequences of measurable functions holds if and only if the underlyingmeasure space is almost finite. As a consequence we obtain several theorems on theinteraction between convergences almost everywhere, almost uniform and in measure,respectively, with no preliminary conditions on the measure space (X, S, , µ), thusextending results from [2],[4],[6],[10]. It is proved further that if (X, S, µ) is almostfinite (is not almost finite) then f : R ? R preserves almost uniform convergenceand convergence in measure, respectively, if and only if f is continuous (is uniformlycontinuous), thus augmenting a result of [3].