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Let $$\mathcal {E}$$ be a symmetrically $$\Delta $$ -normed ideal in B(H). For $$1\le p

For complex interpolation nodes, we study the problem of interpolation by series of exponential functions whose exponents form a set, which is condensed at infinity in a certain direction. We obtain a criterion for all sets of nodes from a special class. For arbitrary sets of nodes, we obtain a necessary condition for the solvability of a more general problem of interpolation by functions that can be represented as Radon integrals of an exponential function over a set of exponents. The paper also contains well-known results on interpolation, which, in particular, allow studying the multipoint holomorphic Vallee Poussin problem for convolution operators.

Classical results about peaking from complex interpolation theory are extended to polynomials on a closed disk, and on the complement of its interior. New results are obtained concerning interpolation by univalent polynomials on a Jordan domain whose boundary satisfies certain smoothness conditions.

We study the stability of the differential process of Rochberg and Weiss associated with an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of Köthe function spaces, we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of Köthe spaces) by showing that there is global (bounded) stability for families of up to three Köthe spaces distributed in arcs on the unit circle while there is no (bounded) stability for families of four or more Köthe spaces. In the context of arbitrary pairs of Banach spaces, we present some local stability results and some global isometric stability results.

Let $${\mathcal {M}}$$ be a diffuse von Neumann algebra with a faithful normal semi-finite trace $$\tau $$ , and let E be a symmetric quasi-Banach space. Then for any Orlicz function $$\varphi $$ , we can define the noncommutative Calderon–Lozanovskiĭ spaces $$E_\varphi ({\mathcal {M}})$$ . These spaces share many properties with their classical counterparts. In particular, new multiplication operator spaces and complex interpolation spaces of such spaces are given under a wide range of conditions. Moreover, letting $${\mathcal {A}}$$ be a maximal subdiagonal algebra of $${\mathcal {M}}$$ , we introduce the noncommutative Calderon–Lozanovskiĭ–Hardy spaces $$H^\varphi ({\mathcal {A}})$$ and transfer the recent results of the noncommutative Hardy spaces to the noncommutative Calderon–Lozanovskiĭ–Hardy spaces.

Let $$E_1,\;E_2$$ be symmetric quasi Banach function spaces on $$(0,\alpha )\;(0

In this article, we discuss the complex interpolation of various closed subspaces of Morrey spaces.We have been considering some closed subspaces of Morrey spacesin our earlier works.The main property that we need is the lattice property but in connectionwith the diamond spaces defined by Yuan et al. (2015),it seems to be natural to consider the convolution property as well.Our result will extend the resultsby Hakim and Sawano (2017) and Hakim et al. (2017).

In this paper we study a class of first-order Ventcel’ boundary value problems for second-order, elliptic Waldenfels integro-differential operators. More precisely, by using real analysis techniques such as Strichartz norms and the complex interpolation method we prove existence and uniqueness theorems in the framework of Sobolev and Besov spaces of $$L^{p}$$ type which extend earlier theorems due to Bony–Courrege–Priouret and Runst–Youssfi to the general degenerate case. Our proof is based on various maximum principles for second-order, elliptic Waldenfels operators with discontinuous coefficients in the framework of $$L^{p}$$ Sobolev spaces.

We determine the associate space of the logarithmic interpolation space (X0, X1)1,q,A where X0 and X1 are Banach function spaces over a σ-finite measure space (Ω, µ). Particularizing the results for the case of the couple (L1, L∞) over a non-atomic measure space, we recover results of Opic and Pick on associate spaces of generalized Lorentz-Zygmund spaces L(∞,q;A). We also establish the corresponding results for sequence spaces.

We proved a complex interpolation theorem of noncommutative Hardy spaces associated with semi-finite von Neumann algebras and extend the Riesz type factorization to the semi-finite case.

We prove that block spaces defined on ℝ n {\mathbb{R}^{n}} with an arbitrary Radon measure, which are known to be the preduals of Morrey spaces, are closed under the first and the second complex interpolation method. The proof of our main theorem uses the duality theorem in the complex interpolation method, the complex interpolation of certain closed subspaces of Morrey spaces, a characterization of the preduals of block spaces, and some formulas related to the Calderón product.

In this paper we develop a theory of Besov and Triebel–Lizorkin spaces on general noncompact connected Lie groups endowed with a sub-Riemannian structure. Such spaces are defined by means of hypoelliptic sub-Laplacians with drift, and endowed with a measure whose density with respect to a right Haar measure is a continuous positive character of the group. We prove several equivalent characterizations of their norms, we establish comparison results also involving Sobolev spaces of recent introduction, and investigate their complex interpolation and algebra properties.

The complex method of interpolation, going back to Calderon and Coifman et al., on the one hand, and the Alexander–Wermer–Slodkowski theorem on polynomial hulls with convex fibers, on the other hand, are generalized to a method of interpolation of real (finite-dimensional) Banach spaces and of convex functions. The underlying duality in this method is given by the Legendre transform. Our results can also be interpreted as new properties of solutions of the homogeneous complex Monge–Ampere equation.

We develop a special multilinear complex interpolation theorem that allows us to prove an optimal version of the bilinear Hormander multiplier theorem concerning symbols that lie in the Sobolev space $$L^r_s({\mathbb {R}}^{2n})$$ , $$2\le r2n$$ , uniformly over all annuli. More precisely, given such a symbol with smoothness index s, we find the largest open set of indices $$(1/p_1,1/p_2 )$$ for which we have boundedness for the associated bilinear multiplier operator from $$L^{p_1}({\mathbb {R}}^{ n})\times L^{p_2} ({\mathbb {R}}^{ n})$$ to $$ L^p({\mathbb {R}}^{ n})$$ when $$1/p=1/p_1+1/p_2$$ , $$1

Let $${\mathcal M}$$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $$\tau $$ . We show that the symmetrically $$\Delta $$ -normed operator space $$E({\mathcal M},\tau )$$ corresponding to an arbitrary symmetrically $$\Delta $$ -normed function space $$E(0,\infty )$$ is an interpolation space between $$L_0({\mathcal M},\tau )$$ and $${\mathcal M}$$ , which is in contrast with the classical result that there exist symmetric operator spaces $$E({\mathcal M},\tau )$$ which are not interpolation spaces between $$L_1({\mathcal M},\tau )$$ and $${\mathcal M}$$ . Besides, we show that the $${\mathcal K}$$ -functional of every $$X\in L_0({\mathcal M},\tau )+{\mathcal M}$$ coincides with the $${\mathcal K}$$ -functional of its generalized singular value function $$\mu (X)$$ . Several applications are given, e.g., it is shown that the pair $$(L_0({\mathcal M},\tau ),{\mathcal M})$$ is $${\mathcal K}$$ -monotone when $${\mathcal M}$$ is a non-atomic finite factor.

Characterization of the mapping properties such as boundedness, compactness, measure of non-compactness and estimates of the approximation numbers of Hardy-type integral operators in Banach function spaces are given.

Our understanding of the interplay between Poincare inequalities, Sobolev inequalities and the geometry of the underlying space has changed considerably in recent years. These changes have simultaneously provided new insights into the classical theory and allowed much of that theory to be extended to a wide variety of different settings. This paper reviews some of these new results and techniques and concludes with an example on the preservation of Sobolev spaces by the maximal function. [Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].

In this paper we present a new characterization of the Sobolev space $W^{1,p}$, $1\lt p\lt \infty$ which is a higher dimensional version of a result of Waterman [32]. We also provide a new and simplified proof of a recent result of Alabern, Mateu, and Verdera [2]. Finally, we generalize the results to the case of weighted Sobolev spaces with respect to a Muckenhoupt weight.

We define by interpolation a scale analogous to the Hardy $H^p$ scale for complete Pick spaces, and establish some of the basic properties of the resulting spaces, which we call $\mathcal{H}^p$. In particular, we obtain an $\mathcal{H}^p-\mathcal{H}^q$ duality and establish sharp pointwise estimates for functions in $\mathcal{H}^p$.

We study the interrelation between the limit $$L_p(\Omega )$$-Sobolev regularity $$\overline{s}_p$$ of (classes of) functions on bounded Lipschitz domains $$\Omega \subseteq \mathbb {R}^d$$, $$d\ge 2$$, and the limit regularity $$\overline{\alpha }_p$$ within the corresponding adaptivity scale of Besov spaces $$B^\alpha _{\tau ,\tau }(\Omega )$$, where $$1/\tau =\alpha /d+1/p$$ and $$\alpha >0$$ ($$p>1$$ fixed). The former determines the convergence rate of uniform numerical methods, whereas the latter corresponds to the convergence rate of best N-term approximation. We show how additional information on the Besov or Triebel–Lizorkin regularity may be used to deduce upper bounds for $$\overline{\alpha }_p$$ in terms of $$\overline{s}_p$$ simply by means of classical embeddings and the extension of complex interpolation to suitable classes of quasi-Banach spaces due to Kalton et al. (in: De Carli and Milman (ed) Interpolation theory and applications, American Mathematical Society, Providence, 2007). The results are applied to the Poisson equation, to the p-Poisson problem, and to the inhomogeneous stationary Stokes problem. In particular, we show that already established results on the Besov regularity for the Poisson equation are sharp.

We show that the Kottman constant $K(\cdot)$, together with its symmetric and finite variations, is continuous with respect to the Kadets metric, and they are log-convex, hence continuous, with respect to the interpolation parameter in a complex interpolation schema. Moreover, we show that $K(X)\cdot K(X^*)\geqslant 2$ for every infinite-dimensional Banach space $X$. We also consider the isomorphic Kottman constant (defined as the infimum of the Kottman constants taken over all renormings of the space) and solve the main problem left open in [CaGoPa17], namely that the isomorphic Kottman constant of a twisted-sum space is the maximum of the constants of the respective summands. Consequently, the Kalton--Peck space may be renormed to have Kottman's constant arbitrarily close to $\sqrt{2}$. For other classical parameters, such as the Whitley and the James constants, we prove the continuity with respect to the Kadets metric., 14 pp

We prove various extensions of the Coifman–Rubio de Francia–Semmes multiplier theorem to operator-valued multipliers on Banach function spaces. Our results involve a new boundedness condition on sets of operators which we call ℓ r ( ℓ s ) {\ell ^{r}(\ell ^{s})} -boundedness, which implies R \mathcal {R} -boundedness in many cases. The proofs are based on new Littlewood–Paley–Rubio de Francia-type estimates in Banach function spaces which were recently obtained by the authors.

In this work we study the following three space problem for operator spaces: if X is an operator space with base space isomorphic to a Hilbert space and X contains a completely isomorphic copy of the operator Hilbert space OH with respective quotient also completely isomorphic to OH, must X be completely isomorphic to OH? This problem leads us to the study of short exact sequences of operator spaces, more specifically those induced by complex interpolation, and their splitting. We show that the answer to the three space problem is negative, giving two different solutions.

The Orlicz spaces $$X^{\varPhi }$$ associated to a quasi-Banach function space X are defined by replacing the role of the space $$L^1$$ by X in the classical construction of Orlicz spaces. Given a vector measure m, we can apply this construction to the spaces $$L^1_w(m),$$ $$L^1(m)$$ and $$L^1(\Vert m\Vert )$$ of integrable functions (in the weak, strong and Choquet sense, respectively) in order to obtain the known Orlicz spaces $$L^{\varPhi }_w(m)$$ and $$L^{\varPhi }(m)$$ and the new ones $$L^{\varPhi }(\Vert m\Vert ).$$ Therefore, we are providing a framework where dealing with different kind of Orlicz spaces in a unified way. Some applications to complex interpolation are also given.

We present a multiplier theorem on anisotropic Hardy spaces. When m satisfies the anisotropic, pointwise Mihlin condition, we obtain boundedness of the multiplier operator Tm: → , for the range of p that depends on the eccentricities of the dilation A and the level of regularity of a multiplier symbol m. This extends the classical multiplier theorem of Taibleson andWeiss.

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.

In this paper an anisotropic grand Sobolev–Morrey spaces are introduced. With the help of integral representation we study differential and differential-difference properties of functions from these spaces. Keywords: Anisotropic grand Sobolev–Morrey spaces, Integral representation, Embedding theorem, Hölder spaces

Density of Lipschitz functions in Newtonian spaces based on quasi-Banach function lattices is discussed. Newtonian spaces are first-order Sobolevtype spaces on abstract metric measure spaces defined via (weak) upper gradients. Our main focus lies on metric spaces with a doubling measure that support a Poincare inequality. Absolute continuity of the function lattice quasi-norm is shown to be crucial for approximability by (locally) Lipschitz functions. The proof of the density result uses, among other facts, the fact that a suitable maximal operator is locally weakly bounded. In particular, various sufficient conditions for such boundedness on quasi-Banach function lattices (and rearrangement-invariant spaces, in particular) are established and applied.

We consider a Cauchy problem associated to a second-order evolution inclusion in non separable Banach spaces under Filippov type assumptions and we prove the existence of mild solutions.

We study conditions on Banach spaces close to \(L^1\) guaranteeing the existence of Random Unconditional Convergence and Divergence systems. Special attention is given to the Haar system and to Cesaro spaces.

We show that, under the continuum hypothesis, “to be universally separably injective” is not a 3-space property, as we wrongly claimed in the paper mentioned in the title.

It is known that the space of homogeneous type introduced by Coifman and Weiss (1971) provides a very natural setting for establishing a theory of Hardy spaces. This paper concentrates on how the geometrical conditions of the space of homogeneous type play a crucial role in building a theory of Hardy spaces via the Littlewood-Paley functions.

Recently more and more attention has been paid to subspaces of Morrey spaces. The description of interpolation results for many of these spaces is found. However, the ones for smoothness Morrey subspaces are missing. The aim of this paper is to describe the output by the first and the second complex interpolations of these smoothness Morrey subspaces.

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.

We introduce grand Hardy spaces defined on a probability space. Analogous to the classical theory, we prove Doob’s maximal inequality and obtain atomic characterization of grand Hardy martingale spaces. Finally, we investigate the John–Nirenberg theorem in the frame of grand Hardy spaces.

In this paper we study some generalization of invariant means on Banach spaces. We give some sufficient condition for the existence of the invariant mean and some examples where we have not it.

In this paper, we study the bounded approximation property for the weighted space$\mathcal{HV}$(U) of holomorphic mappings defined on a balanced open subsetUof a Banach spaceEand its predual$\mathcal{GV}$(U), where$\mathcal{V}$is a countable family of weights. After obtaining an$\mathcal{S}$-absolute decomposition for the space$\mathcal{GV}$(U), we show thatEhas the bounded approximation property if and only if$\mathcal{GV}$(U) has. In case$\mathcal{V}$consists of a single weightv, an analogous characterization for the metric approximation property for a Banach spaceEhas been obtained in terms of the metric approximation property for the space$\mathcal{G}_v$(U).

We study unboundedness of smoothness Morrey spaces on bounded domains Ω ⊂ Rn in terms of growth envelopes. It turns out that in this situation the growth envelope function is finite—in contrast to the results obtained by Haroske et al. (2016) for corresponding spaces defined on Rn. A similar effect was already observed by Haroske et al. (2017), where classical Morrey spaces Mu,p(Ω) were investigated. We deal with all cases where the concept is reasonable and also include the tricky limiting cases. Our results can be reformulated in terms of optimal embeddings into the scale of Lorentz spaces Lp,q(Ω).

We discuss some questions on strong summability of Fourier series in the context of periodic generalized Morrey spaces. By using periodic Lizorkin–Triebel–Morrey spaces as well as periodic Nikol’skij–Besov–Morrey spaces we are able to derive some if and only if assertions in this field. In addition we derive some conclusions on the local regularity of functions in terms of generalized Holder–Zygmund spaces. Finally, we characterize the asymptotic behaviour of the approximation numbers with respect to the identity mapping from periodic Nikol’skij–Besov–Morrey spaces into the space of continuous functions.

In this paper, we obtain some properties of invertible operators; convex, balanced, absorbing sets; and $\mathcal {D}$ -boundedness in Serstnev spaces. We prove that some PN spaces (V,ν,τ,τ ∗), which are not Serstnev spaces, in which the triangle function τ ∗ is not Archimedean can be endowed with a structure of a topological vector space, and we give suitable example to illustrate this result. Also, we show that the topological spaces obtained in such a manner are normable under certain given conditions: some examples are given.

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

In this note, first, we show that the asymptotic subcone, the ultralimit, the completion of an asymptotically PT−1 space are still an asymptotically PT−1 space. Secondly, we consider two kinds of metric spaces, which have been considered by Ibragimov and Gromov, respectively. We show that they are asymptotically PT−1 spaces under particular conditions, which provide some concrete examples of asymptotically PT−1 spaces.

The purpose of this paper is to prove some fixed point results using JS-G-contraction on G-metric spaces, also to prove some fixed point results on Gb-complete metric space for a new contraction. Our results extend and improve some results in the literature. Moreover, some examples are presented to illustrate the validity of our results.

We are dealing with general boundary-value problem for linear integro-differential equations on a segment of the real axis. In the case under consideration the order of internal differential operators is higher than the order of exterior one. We prove that the problem is wellposed in the Hadamard sense in a new pair of weight-free Sobolev spaces.

We consider general linear approximation spaces $$X^b_q$$ based on a quasi-Banach space X, and we analyze the degree of compactness of the embedding $$X^b_q \hookrightarrow X$$ . Applications are given to periodic Besov spaces on the d-torus, including spaces of generalized and logarithmic smoothness. In particular, we obtain the exact asymptotic behavior of approximation and entropy numbers of embeddings of such Besov spaces in Lebesgue spaces and in Besov spaces of logarithmic smoothness.

We study the action of the Laplace transform $$\mathcal L$$ on rearrangement-invariant function spaces. We focus on the optimality of the range and the domain spaces.

We give an alternative proof of recent results by the authors on uniform boundedness of dyadic averaging operators in (quasi-)Banach spaces of Hardy–Sobolev and Triebel–Lizorkin type. This result served as the main tool to establish Schauder basis properties of suitable enumerations of the univariate Haar system in the mentioned spaces. The rather elementary proof here is based on characterizations of the respective spaces in terms of orthogonal compactly supported Daubechies wavelets.

The named space denoted by V p q k consists of L q functions on [ 0 , 1 ) d of bounded p -variation of order k ∈ N . It generalizes the classical spaces V p ( 0 , 1 ) ( = V p ∞ 1 ) and B V ( = [ 0 , 1 ) d ) ( V 1 q 1 where q ≔ d d − 1 ) and is closely related to several important smoothness spaces, e.g., to Sobolev spaces over L p , B V and B M O and to Besov spaces. The main approximation result concerns the space V p q k of smoothness s ≔ d 1 p − 1 q ∈ ( 0 , k ] . It asserts the following: Let f ∈ V p q k be of smoothness s ∈ ( 0 , k ] , 1 ≤ p q ∞ and N ∈ N . There exist a family Δ N of N dyadic subcubes of [ 0 , 1 ) d and a piecewise polynomial g N over Δ N of degree k − 1 such that ‖ f − g N ‖ q ⩽ C N − s ∕ d | f | V p q k . This implies similar results for the above mentioned smoothness spaces, in particular, solves the going back to the 1967 Birman–Solomyak paper (Birman and Solomyak, 1967) problem of approximation of functions from W p k ( [ 0 , 1 ) d ) in L q ( [ 0 , 1 ) d ) whenever k d = 1 p − 1 q and q ∞ .