Banach Function Lattices Research Articles

Weakly compact sets and weakly compact pointwise multipliers in Banach function lattices

AbstractWe prove that the class of Banach function lattices in which all relatively weakly compact sets are equi‐integrable sets (i.e. spaces satisfying the Dunford–Pettis criterion) coincides with the class of 1‐disjointly homogeneous Banach lattices. New examples of such spaces are provided. Furthermore, it is shown that Dunford–Pettis criterion is equivalent to de la Valleé Poussin criterion in all rearrangement invariant spaces on the interval. Finally, the results are applied to characterize weakly compact pointwise multipliers between Banach function lattices.

Stability of the inverses of interpolated operators with application to the Stokes system

We study the stability of isomorphisms between interpolation scales of Banach spaces, including scales generated by well-known interpolation methods. We develop a general framework for compatibility theorems, and our methods apply to general cases. As a by-product we prove that the interpolated isomorphisms satisfy uniqueness-of-inverses. We use the obtained results to prove the stability of lattice isomorphisms on interpolation scales of Banach function lattices and demonstrate their application to the Calderón product spaces as well as to the real method scales. We also apply our results to prove solvability of the Neumann problem for the Stokes system of linear hydrostatics on an arbitrary bounded Lipschitz domain with a connected boundary in mathbb {R}^n, nge 3, with data in some Lorentz spaces L^{p,q}(partial Omega , mathbb {R}^n) over the set partial Omega equipped with a boundary surface measure.

Compact Hankel Operators Between Distinct Hardy Spaces and Commutators

The paper is devoted to the study of compactness of Hankel operators acting between distinct Hardy spaces generated by Banach function lattices. We prove an analogue of Hartman’s theorem characterizing compact Hankel operators in terms of properties of their symbols. As a byproduct we give an estimation of the essential norm of such operators. Furthermore, compactness of commutators and semicommutators of Toeplitz operators for unbounded symbols is discussed.

Stability of Fredholm properties on interpolation Banach spaces

The main aim of this paper is to prove novel results on stability of the semi-Fredholm property of operators on interpolation spaces generated by interpolation functors. The methods are based on some general ideas we develop in the paper. This allows us to extend some previous work in literature to the abstract setting. We show an application to interpolation methods introduced by Cwikel–Kalton–Milman–Rochberg which includes, as special cases, the real and complex methods up to equivalence of norms and also some other well known methods of interpolation. A by-product of these results get the stability of isomorphisms on Calderón products of Banach function lattices. We also study the important characteristics in operator Banach space theory, the so-called modules of injection and surjection, and we prove interpolation estimates of these modules of operators on scales of the Calderón complex interpolation spaces.

Calderón–Zygmund singular operators in extrapolation spaces

We study the boundedness of the Hardy–Littlewood maximal operator in abstract extrapolation Banach function lattices and their Köthe dual spaces. The extrapolation spaces are generated by compatible families of Banach function lattices on quasi-metric measure spaces with doubling measure. These results combined with a variant of the integral Coifman–Fefferman inequality imply that every Calderón–Zygmund singular operator is bounded in considered extrapolation spaces. We apply these results to extrapolation spaces determined by compatible families of Calderón–Lozanovskii spaces, in particular to compatible families of Orlicz spaces that are interpolation of weighted Lp-spaces (1<p<∞) with Ap weights defined on spaces of homogeneous type.

Essential norm estimates for multilinear singular and fractional integrals

We derive lower two-weight estimates for the essential norm (measure of noncompactness) for multilinear Hilbert and Riesz transforms, and Riesz potential operators in Banach function lattices. As a corollary we have appropriate results in weighted Lebesgue spaces. From these statements we conclude that there is no \((m+1)\)-tuple of weights \((v,w_1, \dots, w_m)\) for which these operators are compact from \(L^{p_1}_{w_1} \times \dots \times L^{p_m}_{w_m}\) to \(L^q_v\).

Carleson measures on circular domains and canonical embeddings of Hardy spaces into function lattices

We study general variants of spaces of holomorphic functions on circular domains on the complex plane. We define Hardy-type spaces generated by Banach function lattices, for which we prove the Carleson theorem. We also analyze canonical embeddings of such spaces into appropriate function lattices. Finally, we study composition operators on Hardy-type spaces on circular domains and characterize order-boundedness of such maps.

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.

Pietsch--Maurey--Rosenthal factorization of summing multilinear operators

The main purpose of this paper is the study of a~new class of summing multilinear operators acting from the product of Banach lattices with some nontrivial lattice convexity. A mixed Pietsch--Maurey--Rosenthal type factorization theorem for these operators is proved under weaker convexity requirements than the ones that are needed in the Maurey--Rosenthal factorization through products of $L^q$-spaces. A by-product of our factorization is an extension of multilinear operators defined by a~$q$-concavity type property to a product of special Banach function lattices which inherit some lattice--geometric properties of the domain spaces, as order continuity and $p$-convexity. Factorization through Fremlin's tensor products is also analyzed. Applications are presented to study a special class of linear operators between Banach function lattices that can be characterized by a strong version of $q$-concavity. This class contains $q$-dominated operators, and so the obtained results provide anew factorization theorem for operators from this class.

Fractional integral operators between Banach function lattices

We study the generalized fractional integral transforms associated to a measure on a quasi-metric space. We give a characterization of those measures for which these operators are bounded between Lp-spaces defined on nonhomogeneous spaces. The key in the proof of one of the main theorems is the boundedness of the modified sublinear Hardy–Littlewood maximal operator in the classical Lebesgue space with general measure. We also provide necessary and sufficient conditions for some classes of integral operators to be bounded from Lorentz to Marcinkiewicz spaces.

Carleson measures and embeddings of abstract Hardy spaces into function lattices

We apply interpolation techniques to study behaviour of the canonical inclusion maps of quasi-Banach spaces of analytic functions on the open unit disk of the plane into (quasi)-Banach function lattices on the closed or open unit disk equipped with a Borel measure. These results are applied to abstract Hardy spaces generated by symmetric spaces. We investigate relationships between boundedness or compactness of the canonical inclusion maps and generalized variants of Carleson measures and show applications to composition operators on abstract Hardy spaces. We specialize our results to Hardy–Lorentz spaces.

The multisublinear maximal type operators in Banach function lattices

The main aim of this paper is to study a general multisublinear operators generated by quasi-concave functions between weighted Banach function lattices. These operators, in particular, generalize the Hardy–Littlewood and fractional maximal functions playing an important role in harmonic analysis. We prove that under some general geometrical assumptions on Banach function lattices two-weight weak type and also strong type estimates for these operators are true. To derive the main results of this paper we characterize the strong type estimate for a variant of multilinear averaging operators. As special cases we provide boundedness results for fractional maximal operators in concrete function spaces.

Banach lattice sums with hereditary Fatou properties

We present a characterization of Banach function lattices with the Fatou property generated by the interpolation sum applied to infinite families of Banach lattices with the Fatou property. We also discuss the Köthe duality between the sum and the intersection constructions for Banach spaces.

Asymptotically isometric and isometric copies of \(\ell_1\) in some Banach function lattices

We identify the class of Caldern-Lozanovskii spaces that do not contain an asymptotically isometric copy of \(\ell_1\), and consequently we obtain the corresponding characterizations in the classes of Orlicz-Lorentz and Orlicz spaces equipped with the Luxemburg norm. We also give a~complete description of order continuous Orlicz-Lorentz spaces which contain (order) isometric copies of \(\ell_1^{(n)}\) for each integer \(n \geq 2\).~As an application we provide necessary and sufficient conditions for order continuous Orlicz-Lorentz spaces to contain an (order) isometric copy of \(\ell_1\).~In particular we give criteria in Orlicz and Lorentz spaces for (order) isometric containment of \(\ell_1^{(n)}\) and \(\ell_1\).~The results are applied to obtain the~description of universal Orlicz-Lorentz spaces for all two-dimensional normed spaces.

Köthe dual of Banach lattices generated by vector measures

We study the Kothe dual spaces of Banach function lattices generated by abstract methods having roots in the theory of interpolation spaces. We apply these results to Banach spaces of integrable functions with respect to Banach space valued countably additive vector measures. As an application we derive a description of the Banach dual of a large class of these spaces, including Orlicz spaces of integrable functions with respect to vector measures.

Calderón–Lozanovskiı̆ construction on weighted Banach function lattices

We show that the Calderón–Lozanovskiı̆ construction ϕ(·) commute with arbitrary weighted Banach function lattices, that is, ϕ( E 0( w 0), E 1( w 1))= ϕ( E 0, E 1)( w) with w=1/ ϕ(1/ w 0,1/ w 1) if and only if ϕ is equivalent to a power function. From this result we can also get the Stein–Weiss theorem on interpolation of weighted L p spaces and its generalizations.

On strictly singular and strictly cosingular embeddings between Banach lattices of functions

Let E be a Banach function lattice such that L∞[0, 1] [rarrhk ] E [rarrhk ] L1[0, 1]. We characterize the strict singularity of the embedding L∞[0, 1] [rarrhk ] E and the strict cosingularity of E [rarrhk ] L1[0, 1] in terms of functionals defined by using characteristic functions.

On property (β) in Banach lattices, Calderón-Lozanowskii and Orlicz-Lorentz spaces

The geometry of Calderon-Lozanowskii spaces, which are strongly connected with the interpolation theory, was essentially developing during the last few years (see [4, 9, 10, 12, 13, 17]). On the other hand many authors investigated property (β) in Banach spaces (see [7, 19, 20, 21, 25, 26]). The first aim of this paper is to study property (β) in Banach function lattices. Namely a criterion for property (β) in Banach function lattice is presented. In particular we get that in Banach function lattice property (β) implies uniform monotonicity. Moreover, property (β) in generalized Calderon-Lozanowskii function spaces is studied. Finally, it is shown that in Orlicz-Lorentz function spaces property (β) and uniform convexity coincide.

Calderón Couples of Lorentz Spaces

We present some sufficient conditions under which pairs of Banach function lattices are Calderón couples with special attention to the case of classical Lorentz spaces.

On the geometry of some Calderón-Lozanovskiǐ interpolation spaces

Geometric properties of Calderón-Lozanovskiǐ spaces between Banach function lattices and L ∝ are considered. It is shown that under some general conditions these spaces posses certain monotonicity, rotundity and uniform nonsquareness properties. Some applications to renorming of Lorentz-Orlicz and Orlicz spaces are given.