quasi-Banach Function Spaces Research Articles

Maximal Estimates of Schrödinger Equations on Rearrangement Invariant Sobolev Spaces

We establish the maximal estimates for the solutions of some initial value problems on rearrangement-invariant quasi-Banach function spaces. Our result covers the cases for which the initial value problem is given by the Schrödinger equation.

Generalized Lebesgue Points for Hajłasz Functions

Let X be a quasi-Banach function space over a doubling metric measure space P. Denote by αX the generalized upper Boyd index of X. We show that if αX<∞ and X has absolutely continuous quasinorm, then quasievery point is a generalized Lebesgue point of a quasicontinuous Hajłasz function u∈M˙s,X. Moreover, if αX<(Q+s)/Q, then quasievery point is a Lebesgue point of u. As an application we obtain Lebesgue type theorems for Lorentz–Hajłasz, Orlicz–Hajłasz, and variable exponent Hajłasz functions.

Symmetrization, factorization and arithmetic of quasi-Banach function spaces

We investigate relations between symmetrizations of quasi-Banach function spaces and constructions such as Calderón–Lozanovskiĭ spaces, pointwise product spaces and pointwise multipliers. We show that under reasonable assumptions the symmetrization commutates with these operations. We determine also the spaces of pointwise multipliers between Lorentz spaces and Cesàro spaces. The methods that we develop may be regarded as an arithmetic of quasi-Banach function spaces and proofs of Theorem 3 , Theorem 4 , Theorem 6 give a kind of tutorial for these methods. Finally, the above results will be used in proofs of some factorization results.

Translation invariant maps on function spaces over locally compact groups

We prove that under adequate geometric requirements, translation invariant mappings between vector-valued quasi-Banach function spaces on a locally compact group G have a bounded extension between Köthe–Bochner spaces L r ( G , E ) . The class of mappings for which our results apply includes polynomials and multilinear operators. We develop an abstract approach based on some new tools as abstract convolution and matching among Banach function lattices, and also on some classical techniques as Maurey–Rosenthal factorization of operators. As a by-product we show when Haar measures which appear in certain factorization theorems for nonlinear mappings are in fact Pietsch measures. We also give applications to operators between Köthe–Bochner spaces.

FOURIER-TYPE TRANSFORMS ON REARRANGEMENT-INVARIANT QUASI-BANACH FUNCTION SPACES

AbstractWe establish the mapping properties of Fourier-type transforms on rearrangement-invariant quasi-Banach function spaces. In particular, we have the mapping properties of the Laplace transform, the Hankel transforms, the Kontorovich-Lebedev transform and some oscillatory integral operators. We achieve these mapping properties by using an interpolation functor that can explicitly generate a given rearrangement-invariant quasi-Banach function space via Lebesgue spaces.

Kadec\u2013Klee properties of some quasi-Banach function spaces

We study Kadec–Klee properties with respect to global (local) convergence in measure in quasi-Banach function spaces. First, we prove some general results which can be of independent interest. Next, we investigate these properties in symmetrizations E^{left( *right) }. Finally, we apply general results to study these properties in Marcinkiewicz and Lorentz spaces.

Hardy spaces for ball quasi-Banach function spaces

This article unifies the theory for Hardy spaces built on Banach lattices on R-n satisfying certain weak conditions on indicator functions of balls. The authors introduce a new family of function spaces, named the ball quasi-Banach function spaces, to define Hardy type spaces. The ones in this article extend classical Hardy spaces and include various known function spaces, for example, Hardy-Lorentz spaces, Hardy-Herz spaces, Hardy-Orlicz spaces, Hardy-Morrey spaces, Musielak-Orlicz-Hardy spaces, variable Hardy spaces and variable Hardy-Morrey spaces. Among them, Hardy-Herz spaces are shown to naturally arise in the context of any function spaces above. The example of Hardy-Morrey spaces shows that the absolute continuity of the quasi-norm is not necessary, which is used to guarantee the density of the set of functions having compact supports in Hardy spaces for ball quasi-Banach function spaces, but the decomposition result on these Hardy-type spaces never requires this absolute continuity of the quasi-norm. Moreover, via assuming that the powered Hardy-Littlewood maximal operator satisfies certain Fefferman-Stein vector-valued maximal inequality as well as it is bounded on the associate space, the atomic characterizations of Hardy type spaces are obtained. Although the results are based on the rather abstract theory of function spaces, they improve and extend the results for Orlicz spaces and Musielak-Orlicz spaces. Moreover, local Hardy type spaces and Hardy type spaces associated with operators in this setting are also studied.

Compactness in quasi-Banach function spaces and applications to compact embeddings of Besov-type spaces

There are two main aims of the paper. The first is to extend the criterion for the precompactness of sets in Banach function spaces to the setting of quasi-Banach function spaces. The second is to extend the criterion for the precompactness of sets in the Lebesgue spaces Lp(ℝn), 1 ⩽ p < ∞, to the so-called power quasi-Banach function spaces. These criteria are applied to establish compact embeddings of abstract Besov spaces into quasi-Banach function spaces. The results are illustrated on embeddings of Besov spaces , into Lorentz-type spaces.

Fourier integrals and Sobolev embedding on rearrangement invariant quasi-Banach function spaces

We extend the mapping properties for the fractional integral operators, the convolution operators, the Fourier integral operators and the oscillatory integral operators to rearrangementinvariant quasi-Banach function spaces. We also generalize the Fourier restriction theorem and the Sobolev embedding theorem to rearrangement-invariant quasi-Banach function spaces. We obtain the above results by introducing two families of rearrangement-invariant quasi-Banach function spaces. Furthermore, these two families of rearrangement-invariant quasi-Banach function spaces also give us some embedding and interpolation results of Triebel–Lizorkin type spaces and Hardy type spaces built on rearrangement-invariant quasi-Banach function spaces.

On some martingale inequalities for mean oscillations in weak spaces

Given a Banach function space \(X\) over a nonatomic probability space, we define, in a natural way, a quasi-Banach function space which is denoted by \(\mathrm {w}\text {-}{X}\). In this paper, we consider some inequalities in \(\mathrm {w}\text {-}{X}\) involving the mean oscillation and the maximal function of a martingale. The main results give necessary and sufficient conditions for the inequalities to be valid.

Pointwise tensor products of function spaces

By means of a mixture of linear and nonlinear techniques, the paper treats quasi-Banach function spaces as L∞-modules and studies the basic algebraical constructions and their interactions, placing the emphasis on tensor products. Sample result: if 0→Y→X→Z→0 is an exact sequence of L∞-modules and homomorphisms, where Y and Z are function spaces, and V is another function space, then the tensorized sequence 0→Y⊗V→X⊗V→Z⊗V→0 is exact too.

Approximation by Trigonometric Polynomials in Rearrangement Invariant Quasi Banach Function Spaces

We investigate the approximation properties of trigonometric polynomials in rearrangement invariant quasi Banach function spaces. Relations between moduli of smoothness of the derivatives of a functions and those of the functions itself are studied. We prove a theorem on the relationship between the modulus of smoothness and the best approximation in rearrangement invariant quasi Banach function spaces.

Approximation by polynomials in rearrangement invariant quasi Banach function spaces

In the present work we deal with the approximation properties of certain linear polynomial operators in rearrangement invariant quasi Ba- nach function spaces. We obtain some Jackson type direct theorem and sharp converse theorem of trigonometric approximation with respect to fractional positive order moduli of smoothness in these spaces.

The one-sided Ap conditions and local maximal operator

AbstractWe introduce the one-sided local maximal operator and study its connection to the one-sided Ap conditions. We get a new characterization of the boundedness of the one-sided maximal operator on a quasi-Banach function space. We obtain applications to weighted Lebesgue spaces and variable-exponent Lebesgue spaces.

Characterizations of BMO by $A_{p}$ Weights and $p$-Convexity

We show that the Lebesgue spaces for defining BMO can be replaced by $p$-convex rearrangement-invariant quasi-Banach function spaces associated with $A_{p}$-weighted measures.

Littlewood-Paley spaces

We introduce the Littlewood-Paley spaces in which the Lebesgue spaces, the Hardy spaces, the Orlicz spaces, the Lorentz-Karamata spaces, the r.-i. quasi-Banach function spaces and the Morrey spaces reside. The Littlewood-Paley spaces provide a unified framework for the study of some important function spaces arising in analysis.

A boundedness criterion for general maximal operators

We consider maximal operators MB with respect to a basis B. In the case when MB satisfies a reversed weak type inequality, we obtain a boundedness criterion for MB on an arbitrary quasiBanach function space X. Being applied to specific B and X this criterion yields new and short proofs of a number of well-known results. Our principal application is related to an open problem on the boundedness of the two-dimensional one-sided maximal function M + on L p.

A new characterization of the Muckenhoupt $A_p$ weights through an extension of the Lorentz-Shimogaki Theorem

Given any quasi-Banach function space X over R n , an index α x is defined that coincides with the upper Boyd index α x when the space X is rearrangement-invariant. This new index is defined by means of the local maximal operator m λ f. It is shown then that the Hardy-Littlewood maximal operator M is bounded on X if and only if α x 0 there is a constant c such that for any cube Q and any measurable subset E ⊂ Q, |E| |Q| log e (|Q| |E|) ≤ c (u(E) u(Q)) 1/p The case e = 0 is false corresponding to the class A p,1 . Other applications are given, in particular within the context of the variable L p spaces.

Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals

We present an extrapolation theory that allows us to obtain, from weighted L p inequalities on pairs of functions for p fixed and all A ∞ weights, estimates for the same pairs on very general rearrangement invariant quasi-Banach function spaces with A ∞ weights and also modular inequalities with A ∞ weights. Vector-valued inequalities are obtained automatically, without the need of a Banach-valued theory. This provides a method to prove very fine estimates for a variety of operators which include singular and fractional integrals and their commutators. In particular, we obtain weighted, and vector-valued, extensions of the classical theorems of Boyd and Lorentz–Shimogaki. The key is to develop appropriate versions of Rubio de Francia's algorithm.

Analytic functions with values in lattices and symmetric spaces of measurable operators

AbstractLet 0 < p,pi ≤ ∞, 0 < q,qi < ∞ (i = 1, 2) such thatLet E be a quasi-Banach lattice which fails to contain c0 and whose α-convexity constant is equal to 1 for some 0 < α < ∞. Then for every f∈H(E(q)) there exist g∈Hp, 0(E(q0)), h∈Hp1(E(q1)) such thatConsequently, E is q-concave for some finite q if and only if E is uniformly H1-convexifiable in the sense of [24]. Analogous results are also obtained for symmetric spaces of measurable operators. Another result proved in the paper says that if E is a symmetric quasi-Banach function space on (0, ∞) having the analytic Radon–Nikodym property then LE(M, τ) also possesses this property for any semifinite von Neumann algebra (M, τ).