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We study complex interpolation of Herz‐type Triebel–Lizorkin spaces by using the Calderon product method. Additionally we present complex interpolation between Herz‐type Triebel–Lizorkin spaces and Triebel–Lizorkin spaces F∞,βs. Moreover, we apply these results to obtain the complex interpolation of Triebel–Lizorkin spaces equipped with power weights and between bmo (or hp) spaces and Herz spaces.

We consider a non-autonomous Cauchy problem involving linear operators associated with time-dependent forms $a(t;.,.):V\times V\to {\mathbb{C}}$ where $V$ and $H$ are Hilbert spaces such that $V$ is continuously embedded in $H$. We prove $H$-maximal regularity under a new regularity condition on the form $a$ with respect to time; namely Holder continuity with values in an interpolation space. This result is best suited to treat Robin boundary conditions. The maximal regularity allows one to use fixed point arguments to some non linear parabolic problems with Robin boundary conditions.

In this paper we introduce and study the fractional Hardy spaces of the half space and of the unit ball in the quaternionic setting. In particular, we discuss their properties of invariance and of factorization in terms of functions in the Hardy space of the half space in the first case, and in terms of a suitable reproducing kernel Hilbert space in the case of the unit ball.

Let X be a quasi-Banach space, Y be a γ-Banach space (0

In the paper, abstract interpolation of Hardy spaces on circular domains is studied. In particular, an extension of Peter Jones's result on the interpolation of Hardy spaces on the disc to multiply-connected domains is presented. Moreover, some applications in approximation theory are given.

The aim of this paper is to establish the isomorphic classification of Besov spaces over [0, 1]d. Using the identification of the Besov space Bp,qα([0,1]d) with the lq-infinite direct sum (⊕n=1∞lpn)q of finite-dimensional spaces lpn (which holds independently of the dimension d≥1 and of the smoothness degree of the space α>0) we show that Bp,qα([0,1]d), 0

A sufficient condition for higher-order Sobolev-type embeddings on bounded domains of Carnot–Caratheodory spaces is established for the class of rearrangement-invariant function spaces. The condition takes form of a one-dimensional inequality for suitable integral operators depending on the isoperimetric function relative to the Carnot–Caratheodory structure of the relevant sets. General results are then applied to particular Sobolev spaces built upon Lebesgue, Lorentz and Orlicz spaces on John domains in the Heisenberg group. In the case of the Heisenberg group, the condition is shown to be necessary as well.

Given non-negative measurable functions ϕ,ψ on Rn we study the high dimensional Hardy operator Hf(x)=ψ(x)∫B(0,|x|)f(y)ϕ(y)dy between Orlicz–Lorentz spaces ΛXG(w), where f is a measurable function of x∈Rn and B(0,t) is the ball of radius t in Rn. We give sufficient conditions of boundedness of H:Λu0G0(w0)→Λu1G1(w1) and H:Λu0G0(w0)→Λu1G1,∞(w1). We investigate also boundedness and compactness of H:Λu0p0(w0)→Λu1p1,q1(w1) between weighted and classical Lorentz spaces. The function spaces considered here do not need to be Banach spaces. Specifying the weights and the Orlicz functions we recover the existing results as well as we obtain new results in the new and old settings.

Precise descriptions of the spaces associated with weighted Sobolev spaces on the real line are given.

In this paper, we study the well-posedness of the degenerate differential equations with fractional derivative Dα(Mu)(t)=Au(t)+f(t),(0≤t≤2π) in Lebesgue–Bochner spaces Lp(T;X), periodic Besov spaces Bp,qs(T;X) and periodic Triebel–Lizorkin spaces Fp,qs(T;X), where A and M are closed linear operators in a complex Banach space X satisfying D(A)⊂D(M), α>0 and Dα is the fractional derivative in the sense of Weyl. Using known operator-valued Fourier multiplier results, we completely characterize the well-posedness of this problem in the above three function spaces by the R-bounedness (or the norm boundedness) of the M-resolvent of A.

A systematic treatment is given of several classes of parametric Marcinkiewicz integrals. The boundedness on Triebel–Lizorkin spaces will be presented for these operators with rough kernels in , which relates to the Grafakos–Stefanov class. Moreover, the boundedness on Besov spaces for above operators is also considered.

This paper deals with dimension-controllable (tractable) embeddings of Besov spaces on n-dimensional torus into small Lebesgue spaces. Our techniques rely on the approximation structure of Besov spaces, extrapolation properties of small Lebesgue spaces and interpolation.

Let X be a rearrangement-invariant Banach function space on Q where Q is a cube in and let be the Sobolev space of real-valued weakly differentiable functions f satisfying . We establish a reduction theorem for an embedding of the Sobolev space into spaces of Campanato, Morrey and Holder type. As a result we obtain a new characterization of such embeddings in terms of boundedness of a certain one-dimensional integral operator on representation spaces.

We study compact embeddings of weighted Sobolev spaces into Lebesgue spaces on the unit ball in . The weight is of slowly varyingly disturbed polynomial growth with a singularity at the origin. It extends [21], [27] to a wider class of weights. Special attention is paid to the influence of the growth rate of the weight on the quality of compactness, measured in terms of entropy and approximation numbers. In case of Hilbert spaces, the results are related to the distribution of eigenvalues of some degenerate elliptic operators.

Consider the fractional powers and of the Dirichlet and Neumann realizations of a second-order strongly elliptic differential operator A on a smooth bounded subset Ω of . Recalling the results on complex powers and complex interpolation of domains of elliptic boundary value problems by Seeley in the 1970's, we demonstrate how they imply regularity properties in full scales of -Sobolev spaces and Holder spaces, for the solutions of the associated equations. Extensions to nonsmooth situations for low values of s are derived by use of recent results on -calculus. We also include an overview of the various Dirichlet- and Neumann-type boundary problems associated with the fractional Laplacian.

We study functional calculus properties of C0-groups on real interpolation spaces using transference principles. We obtain interpolation versions of the classical transference principle for bounded groups and of a recent transference principle for unbounded groups. Then we show that each group generator on a Banach space has a bounded H1∞-calculus on real interpolation spaces. Additional results are derived from this.

We study the well-posedness of the second order degenerate differential equations with infinite delay: with periodic boundary conditions , where and M are closed linear operators in a Banach space satisfying , . Using operator-valued Fourier multiplier techniques, we give necessary and sufficient conditions for the well-posedness of this problem in Lebesgue-Bochner spaces , periodic Besov spaces and periodic Triebel-Lizorkin spaces .

We study Sobolev type spaces defined in terms of sharp maximal functions on Ahlfors regular subsets of R n and the relation between these spaces and traces of classical Sobolev spaces. This extends in a certain way the results of Shvartsman (20) to the case of lower dimensional subsets of the Euclidean space.

Let F be a Banach space. We establish necessary and sufficient conditions for the Dunford integration operator, from the space of F-valued Dunford integrable functions to the bidual of F, to belong to a given operator ideal. We also show how this fact can be used to characterize important classes of Banach spaces, such as Banach spaces with the Banach-Saks property, separable Banach spaces not containing c0, Banach spaces not containing c0 or l1 and Asplund spaces not containing c0.

An embedding inequality of Sobolev type is characterized in the paper with help of a duality principle and boundedness criteria for the Hardy–Steklov integral operator in weighted Lebesgue spaces.

We present a form of the Mean Value Theorem (MVT) for a continuous function f between metric spaces, connecting it with the possibility to choose the " 7→�(") relation of f in a homeomorphic way. We also compare our formulation of the MVT with the classic one when the metric spaces are open subsets of Banach spaces. As a consequence, we derive a version of the Mean Value Propriety for measure spaces that also possesses a compatible metric structure.

Let X = (X, d, μ)be a doubling metric measure space. For 0 < α < 1, 1 ≤p, q < ∞, we define semi-norms When q = ∞ the usual change from integral to supremum is made in the definition. The Besov space Bp, qα (X) is the set of those functions f in Llocp(X) for which the semi-norm ‖f ‖ is finite. We will show that if a doubling metric measure space (X, d, μ) supports a (1, p)-Poincare inequality, then the Besov space Bp, qα (X) coincides with the real interpolation space (Lp (X), KS1, p(X))α, q, where KS1, p(X) is the Sobolev space defined by Korevaar and Schoen [15]. This results in (sharp) imbedding theorems. We further show that our definition of a Besov space is equivalent with the definition given by Bourdon and Pajot [3], and establish a trace theorem (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We consider real interpolation methods defined by means of slowly varying functions and rearrangement invariant spaces, for which we present a collection of reiteration theorems for interpolation and extrapolation spaces. As an application we obtain interpolation formulas for Lorentz-Karamata type spaces, for Zygmund spaces , and for the grand and small Lebesgue spaces.

This note is a companion to the article On the mutually non isomorphic spaces published in this journal, in which P. Cembranos and J. Mendoza showed that is a collection of mutually non isomorphic Banach spaces [5]. We now complete the picture by allowing the non-locally convex relatives to be part of their natural family and see that, in fact, no two members of the extended class are isomorphic. Our approach is novel in the sense that we reach the isomorphism obstructions from the perspective of bases techniques and the different convexities of the spaces, both methods being intrinsic to quasi-Banach spaces.

A Sobolev type embedding for Triebel-Lizorkin-Morrey-Lorentz spaces is established in this paper. As an application of this result, the boundedness of the fractional integral operator on some generalizations of Hardy spaces such as Hardy-Morrey spaces and Hardy-Lorentz spaces are obtained.

We discuss the boundedness and compactness of some integral-type operators acting from spaces to mixed-norm spaces on the unit ball of .

Hudzik, Kaminska and Mastylo obtained some geometric properties of Calderon–Lozanovskiĭ function spaces which are defined on a nonatomic σ-measure space in Houston. J. Math. 22 (1996), but left the case of atomic measure unsolved. We studied the relevant problems for the sequence spaces and obtained the following main results: For the Calderon–Lozanovskiĭ sequence spaces , is order continuous if and only if and e is order continuous. Let Φ be strictly convex on , then the convex characteristic whenever e is not order continuous or ; if e is uniformly monotone and , then . For the Orlicz-Lorentz sequence space , if or , or ω is not regular; if Φ is strictly convex on , and ω is regular.

We introduce Lorentz spaces and with variable exponents. We prove several basic properties of these spaces including embeddings and the identity . We also show that these spaces arise through real interpolation between and . Furthermore, we answer in a negative way the question posed in [12] whether the Marcinkiewicz interpolation theorem holds in the frame of Lebesgue spaces with variable integrability.

We consider the uniform algebra of continuous and bounded functions that are analytic on the interior of the closed unit ball of a complex Banach function module X. We focus on norming subsets of , i.e., boundaries, for such algebra. In particular, if X is a dual complex Banach space whose centralizer is infinite-dimensional, then the intersection of all closed boundaries is empty. This also holds in case that X is an -sum of infinitely many Banach spaces and further, the torus is a boundary.

The Brezis-Gallouet-Wainger inequality describes a subtle embedding property into . The relation between the Brezis-Gallouet-Wainger inequality and the real interpolation functor together with the sharpness of the results is discussed in the present paper. As our first main results shows, it turns out that there are two intermediate terms between and the logarithmic boundedness, which is supposed to be the right-hand side of the Brezis-Gallouet-Wainger inequality. As the second result, the first result is extended to inequalities which reflect the meaning of the second index of Besov spaces and the interpolation theorem.

In this article, we show that the Valdivia–Vogt structure table—containing the sequence space representations of the most used spaces of smooth functions appearing in the theory of distributions—can be interpreted as a commutative diagram, i.e., there is an isomorphism between the space of infinitely differentiable functions and the space , where s is the space of rapidly decreasing sequences, such that its restriction to the other function spaces in the structure table yields an isomorphism between these spaces of smooth functions and their sequence space representation. This result answers the corresponding question of Prof. Dietmar Vogt formulated on the conference “Functional Analysis: Applications to Complex Analysis and Partial Differential Equations” held in Bȩdlewo in May 2012.

We apply the Calderon interpolation methods to Orlicz and weakly Orlicz function spaces with respect to a Banach-space-valued measure defined on a σ-algebra. The results we obtain generalize those in the case of Banach lattices of p-integrable and weakly p-integrable functions with respect to such a vector measure.

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 bounded geometry and, moreover, can be generalized to Triebel-Lizorkin spaces on manifolds, improving [11].

In this paper, we compare the recent approach of Hans Triebel to introduce smoothness spaces related to Morrey-Campanato spaces with Besov type and Triebel-Lizorkin type spaces. These two scales have been introduced some years ago and represent a further variant to measure smoothness by using Morrey spaces.

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order H-m (H-n), m is an element of N-n, under the heat kernel transform on H-n, using direct sum and direct integral of Bergmann spaces and certain unitary representations of H-n which can be realized on the Hilbert space of Hilbert-Schmidt operators on L-2 (R-n). We also show that the image of Sobolev space of negative order H-s (H-n), s(> 0) is an element of R is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on H-n under the heat kernel transform. (C) 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

We study traces in 2-microlocal Besov spaces with variable integrability on hyperplanes using atomic decompositions. Our results cover spaces of variable smoothness and also weighted spaces studied extensively in recent years.

Several results concerning multipliers of symmetric Banach function spaces are presented firstly. Then the results on multipliers of Calderon-Lozanovskiǐ spaces are proved. We investigate assumptions on a Banach ideal space E and three Young functions φ1, φ2 and φ, generating the corresponding Calderon-Lozanovskiǐ spaces so that the space of multipliers of all measurable x such that x y ∈ Eφ for any can be identified with . Sufficient conditions generalize earlier results by Ando, O'Neil, Zabreǐko-Rutickiǐ, Maligranda-Persson and Maligranda-Nakai. There are also necessary conditions on functions for the embedding to be true, which already in the case when E = L1, that is, for Orlicz spaces give a solution of a problem raised in the book 26. Some properties of a generalized complementary operation on Young functions, defined by Ando, are investigated in order to show how to construct the function φ2 such that . There are also several examples of independent interest.

We consider local means with bounded smoothness for Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Based on those we derive characterizations of these spaces in terms of Daubechies, Meyer, Bernstein (spline) and more general r-regular (father) wavelets, finally in terms of (biorthogonal) wavelets which can serve as molecules and local means, respectively. Hereby both, local means and wavelet decompositions satisfy natural conditions concerning smoothness and cancellation (moment conditions). Moreover, the given representations by wavelets are unique and yield isomorphisms between the considered function spaces and appropriate sequence spaces of wavelet coefficients. These wavelet representations lead to wavelet bases if, and only if, the function spaces coincide with certain classical Besov-Triebel-Lizorkin spaces.

We prove an estimate for the trace operator in spaces whose weak derivatives are in Orlicz spaces without constraints upon the growth of the generating Young function. This extends the known trace theorems established in Orlicz-Sobolev spaces, where the complementary Young function had to fulfill the Δ2-condition.

Entropy numbers of operators acting between vector-valued sequence spaces are estimated using information about the coordinate mappings. To do this some new ideas of combinatorial type are used. The results are applied to give sharp two-sided estimates of the entropy numbers of some embeddings of Besov spaces. For instance, our main result allows us to give exact two-sided estimates of the entropy numbers of the natural embedding of in where Q = (0, 1)d; θ1, θ2, p1, p2 ∈ (0, ∞], when the condition 1/θ1 − 1/θ2 ≥ 1/p1 − 1/p2 > 0 is satisfied. This work enables us to construct an example showing that the behaviour under real interpolation of entropy numbers can be even worse than in the example of 7.

In this paper, the authors establish the grand Lusin-area characterization of the Hajlasz-Sobolev spaces and, more generally, the Triebel-Lizorkin spaces on both Euclidean spaces and RD-spaces, namely, metric spaces enjoying both doubling and reverse doubling properties.

Dedicated to Professor V. I. Burenkov on the occasion of his 70th birthday We characterize the traces of vector-valued Besov and Lizorkin-Triebel spaces. Therefrom we derive the corresponding assertions for the vector-valued Sobolev spaces . Here we do not assume the UMD property for the Banach space E.

In this paper, we study anisotropic Bessel potential and Besov spaces, where the anisotropy measures the extra amount of regularity in certain directions. Some basic properties of these spaces are established along with applications to elliptic boundary value problems.

The purpose of this paper is to introduce five martingale Orlicz-Hardy spaces and to establish the atomic decomposition theorem. As applications we show the relation among five martingale Orlicz-Hardy spaces and the duality, namely, the dual of martingale Orlicz-Hardy spaces are generalized martingale Campanato spaces. Further, we prove a John-Nirenberg type inequality for generalized martingale Campanato spaces when the stochastic basis is regular.

Recently, importance of the Besov space has been acknowledged by analysts studing such subsets with lower dimension than the whole space as fractals in the Euclidean space. On the other hand, by taking an extension K of local field K′, K′ is contained in K as a subset with lower dimension than the whole space K and the present author showed that it can be viewed as a d-set of K in terms of fractal analysis. In this article, Besov spaces on separable extension of the field of p-adic numbers and on its d-set will be consistently introduced. After that, a trace theorem showing a relationship between those Besov spaces will be presented and finally a penetrating stochastic process into K′ will be addressed as an application of the trace theorem.