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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.

In this article the authors study complex interpolation of Sobolev-Morrey spaces and their generalizations, Lizorkin-Triebel-Morrey spaces. Both scales are considered on bounded domains. Under certain conditions on the parameters the outcome belongs to the scale of the so-called diamond spaces., 44 pages, 1 figure

We prove a complex formulation of the real interpolation method, showing that the real and complex interpolation methods are not inherently real or complex. Using this complex formulation, we prove Stein interpolation for the real interpolation method. We apply this theorem to interpolate weighted $L^p$-spaces and the sectoriality of closed operators with the real interpolation method., 15 pages, slightly improved presentation

We consider the complex interpolation method in the context of protoquantum spaces. We apply it to construct an ℒp-protoquantization of an arbitrary normed space for 1 < p < ∞.

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 study interpolation spaces between global Morrey spaces and between local Morrey spaces. We prove that for a wide class of couples of these spaces the upper complex Calderón spaces are not described by the [math] -method of interpolation. A by-product of our results is that couples of Morrey spaces belonging to this class are not Calderón–Mityagin couples. A Banach couple [math] is said to have the universal [math] -property if all relative interpolation spaces from any Banach couple to [math] are relatively [math] -monotone. A couple of local Morrey spaces is proved to have the universal [math] -property once it is a Calderón–Mityagin couple.

Fil: Drelichman, Irene. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Ciudad Universitaria. Instituto de Investigaciones Matematicas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matematicas "Luis A. Santalo"; Argentina

We study complex interpolation of variable Triebel–Lizorkin spaces, especially we present the complex interpolation of F p ( ⋅ ) , q α and F p ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) spaces. Also, some limiting cases are given.

The Butterfly lemma we present can be considered a reiteration theorem for differentials generated from a complex interpolation process for families of K\"othe spaces. The lemma will be used to clarify the effect of different configurations in the resulting differential (because although interpolation is an orientation-free process, the obtention of differentials is not) and to round off a few aspects of Kalton's interpolation theorem., Comment: This paper is to appear in Nonlinear Analisis - TMA

A standard solution technique for linear operator equations of first kind is the Landweber scheme which is an iterative method that uses the negative gradient of the current residual as search direction, which is also called the Landweber direction. Though this method proves to be stable with respect to noisy data, it is known to be numerically slow for problems in Hilbert spaces and this behavior shows to be even worse in some Banach space settings. This is why the idea came up to use several search directions instead of the Landweber direction only which has led to the development of Sequential Subspace Optimization (SESOP) methods. This idea is related to the famous Conjugate Gradient (CG) techniques that are known to be amongst the most effective methods to solve linear equations in Hilbert spaces. Since CG methods decisively make use of the inner product structure, they have been inherently restricted to Hilbert spaces so far. SESOP methods in Banach spaces do not share the conjugacy property with CG methods. In this article we use the concept of generalized orthogonality in Banach spaces and apply metric projections to orthogonalize the current Landweber direction with respect to the search space of the last iteration. This leads to an accelerated SESOP method which is confirmed by various numerical experiments. Moreover, in Hilbert spaces our method coincides with the Conjugate Gradient Normal Error (CGNE) or Craig’s method applied to the normal equation. We prove weak convergence to the exact solution. Furthermore we perform a couple of numerical tests on a linear problem involving a random matrix and on the problem of 2D computerized tomography where we use different l p -spaces. In all experiments the orthogonalization of the search space shows superior convergence properties compared to standard SESOP. This especially holds for p close to 1. Letting p → 2 the more we recover the conjugacy property for the search directions and the more the convergence behaves independently of the size of the search space.

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 compute the K-functional related to some couple of spaces as small or classical Lebesgue space or Lorentz-Marcinkiewicz spaces completing the results of [11] . This computation allows to determine the interpolation space in the sense of Peetre for such couple. It happens that the result is always a GΓ-space, since this last space covers many spaces. The motivations of such study are various, among them we wish to obtain a regularity estimate for the so called very weak solution of a linear equation in a domain Ω with data in the space of the integrable function with respect to the distance function to the boundary of Ω.

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

The aim of this thesis is threefold. Firstly, we study the stochastic evolution equations (driven by an infinite dimensional cylindrical Wiener process) in a class of Banach spaces satisfying the so-called H-condition. In particular, we deal with the questions of the existence and uniqueness of solutions for such stochastic evolution equations. Moreover, we analyse the Markov property of the solution. Secondly, we apply the abstract results obtained in the first part to the so-called Heath-Jarrow-Morton-Musiela (HJMM) equation. In particular, we prove the existence and uniqueness of solutions to the HJMM equation in a large class of function spaces, such as the weighted Lebesgue and Sobolev spaces. Thirdly, we study the ergodic properties of the solution to the HJMM equation. In particular, we analyse the Markov property of the solution and we find a sufficient condition for the existence and uniqueness of an invariant measure for the Markov semigroup associated to the HJMM equation (when the coefficients are time independent) in the weighted Lebesgue spaces.

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 paper we obtain well-posedness results for Poisson problems with Dirichlet, Neumann, or mixed boundary conditions for the Brinkman system with measurable coefficients and data in $$L^p$$ -based Sobolev and Besov spaces in Lipschitz domains on a compact Riemannian manifold M of dimension $$m\ge 2$$ . For the mixed problem we refer to partially vanishing traces on Ahlfors regular sets. We exploit the continuity property of an operator related to the variational formulation of such a boundary value problem on complex interpolation scales of $$L^p$$ -based Sobolev spaces defined on M or on a Lipschitz domain of M, $$p\in (1,\infty )$$ , and the property that this operator is an isomorphism for $$p=2$$ . Then the stability of the quality of being isomorphism on complex interpolation scales leads to the extension of the well-posedness results of analyzed boundary value problems from $$p\!=\!2$$ to p in a neighborhood of 2. First, we focus on a variational approach that reduces boundary problems of transmission, Dirichlet and mixed type for the Brinkman system to equivalent mixed variational formulations with data in $$L^p$$ -based Sobolev and Besov spaces. For $$p=2$$ , such a mixed variational formulation is well-posed. The mixed variational formulation is further expressed in terms of a linear continuous operator on $$H^{1,q}\times L^q$$ -Sobolev spaces for any $$q\in (1,\infty )$$ , which is also invertible on the solution space corresponding to $$q=2$$ . Working on complex interpolation scales allows us to extend the invertibility of the operator for $$q=2$$ to a neighborhood of 2, and then to extend the well-posedness result to $$L^p$$ -based Sobolev spaces with p in a neighborhood of 2. Well-posedness results for the analyzed transmission problems allow us to define the layer potentials for the nonsmooth coefficient Brinkman system and to obtain their properties in $$L^p$$ -based Sobolev and Besov spaces. Then the solution of the Poisson problem of Dirichlet type is constructed explicitly in terms of such layer potentials. Finally, the Poisson problem of Neumann type is also analyzed and the corresponding well-posedness result in $$L^p$$ -based Sobolev and Besov spaces is also obtained. In addition, we determine the unique solution of the Neumann problem in the case $$p=2$$ , by using a layer potential approach. We extend the well-posedness results obtained in Kohr and Wendland (Boundary value problems for the Brinkman system with measurable coefficients in Lipschitz domains on compact Riemannian manifolds: a variational approach, 2018) for boundary problems for the nonsmooth coefficient Brinkman system in $$L^2$$ -based Sobolev spaces on Lipschitz domains in compact Riemannian manifolds to a more general setting of $$L^p$$ -based Sobolev and Besov spaces.

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.

Let Ω be a bounded open subset of R n with a mild regularity property, let m ∈ N and p ∈ ( 1 , ∞ ) , and let W m , p ( Ω ) be the usual Sobolev space of order m based on L p ( Ω ) ; the closure in W m , p ( Ω ) of the smooth functions with compact support is denoted by W 0 m , p ( Ω ) . A special case of the results given below is that u ∈ W 0 m , p ( Ω ) if and only if all distributional derivatives of u of order m belong to L p ( Ω ) and u / d m ∈ L 1 ( Ω ) , where d ( x ) = dist ( x , ∂ Ω ) . In fact what is proved is the analogous result when the Sobolev space is based on a member of a class of Banach function spaces that includes both L p ( Ω ) and L p ( ⋅ ) ( Ω ) , the Lebesgue space with variable exponent p ( ⋅ ) satisfying natural conditions.

We study the second-order quasi-linear stochastic partial differential equations (SPDEs) defined on C 1 -domains. The coefficients are random functions depending on t , x and the unknown solutions. We prove the uniqueness and existence of solutions in appropriate Sobolev spaces, and in addition, we obtain L p and Holder estimates of both the solution and its gradient.

We give sufficient conditions for the best approximation of convex, bounded, closed and solid sets in Kothe–Bochner spaces and apply this result to sequence spaces.

This article presents necessary and sufficient optimality conditions for weakly efficient solution, Henig efficient solution, globally efficient solution and superefficient solution of vector equilibrium problem without constraints in terms of contingent derivatives in Banach spaces with stable functions. Using the steadiness and stability on a neighborhood of optimal point, necessary optimality conditions for efficient solutions are derived. Under suitable assumptions on generalized convexity, sufficient optimality conditions are established. Without assumptions on generalized convexity, a necessary and sufficient optimality condition for efficient solutions of unconstrained vector equilibrium problem is also given. Many examples to illustrate for the obtained results in the paper are derived as well.

We define a Hardy type space, by taking in the maximal characterization of Hardy spaces, the Wiener amalgam norms of the maximal functions, instead of the Lebesgue norms. The functions in this space can then behave differently locally and at infinity. We prove that this space contains the classical Hardy space and obtain an atomic decomposition.

We investigate order asymptotically isometric copies of c 0 , l ∞ and l 1 in Kothe spaces. Firstly we prove a number of general relations between them and we discuss what happens with such copies while passing to Kothe dual spaces, or vice versa – passing to preduals. Finally, we apply the results to characterize certain classes of Kothe spaces as Orlicz spaces, Calderon–Lozanovskiĭ spaces, Orlicz–Lorentz spaces and Musielak–Orlicz spaces with such copies.

We emphasize a bridge between two areas of function theory: hilbertian Muntz spaces and model spaces of the Hardy space of the right half plane. We give miscellaneous applications of this viewpoint to hilbertian Muntz spaces.

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.

We prove that the maximal operators supported by submanifolds are bounded and continuous on the Triebel–Lizorkin spaces F s p , q ( R d ) and Besov spaces B s p , q ( R d ) for 0 s 1 and 1 p , q ∞ . As a corollary, we also show that these operators are bounded and continuous on the fractional Sobolev spaces W s , p ( R d ) for 0 ≤ s 1 and 1 p ∞ . Our main results represent significant improvements as well as natural extensions of what was known previously.

In this paper, the concept of \(b_v(s)\)-metric space is introduced as a generalization of metric space, rectangular metric space, b-metric space, rectangular b-metric space and v-generalized metric space. We next give proofs of the Banach and Reich contraction principles in \(b_v(s)\)-metric spaces. Using a new result, we provide short proofs which are different from of the original ones in metric spaces. The results we obtain generalize many known results in fixed point theory. We also provide a solution to an open problem.

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.

We study the composition operators with closed range on the weighted Bloch spaces, the weighted Dirichlet spaces, the Bergman spaces and the Hardy space, the space BMOA. In particular, we study the...

The upper and lower complex interpolation of -spaces and -spaces are considered. When , and u are finite, we prove that . Otherwise, the space is a closed subspace of . We also establish interpolation results of certain closed subspaces of -spaces and -spaces, which are the closure of linear subspaces satisfying the lattice condition.

Let $\phi$ be a quasiconformal mapping, and let $T_\phi$ be the composition operator which maps $f$ to $f\circ\phi$. Since $\phi$ may not be bi-Lipschitz, the composition operator need not map Sobolev spaces to themselves. The study begins with the behavior of $T_\phi$ on $L^p$ and $W^{1,p}$ for $1, Comment: 19 pages, 5 figures

In this paper we study the Hardy–Lorentz spaces for Banach space valued martingales. The dual spaces are characterized and several martingale inequalities are established. The proofs mainly depend on the classical tool of atomic decompositions. As usual, these conclusions are closely related to the geometrical properties of the underlying Banach spaces.

In this paper, we introduce a three step iteration method and show that this method can be used to approximate fixed point of weak contraction mappings. Furthermore, we prove that this iteration method is equivalent to Mann iterative scheme and converges faster than Picard-S iterative scheme for the class of weak contraction mappings. We also present tables and three graphics to support this result. Finally, we prove a data dependence result for weak contraction mappings using this three step iterative scheme.

A sufficient condition is given to ensure the boundedness of Fourier Haar multiplier operators from L 1 ( [ 0 , 1 ] , X ) to L 1 ( [ 0 , 1 ] , Y ) , where X and Y are arbitrary Banach spaces and the Fourier Haar multiplier sequences come not from R , as in the classical case, but from the space of bounded operators from the Banach space X to the Banach space Y.

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 ∞ .

We propose an approach to the construction of solutions and quasisolutions of a boundary-value problem for the Lyapunov equation in a Banach space. If the necessary and sufficient conditions for the solvability of this boundary-value problem are satisfied, then the corresponding solutions of the problem are constructed by using the generalized inverse operator. As an example, we consider the problem in the space of bounded sequences with countably dimensional matrices.

In this paper we consider some Banach spaces of analytic functions on the unit disk generated by the cone of analytic functions with monotone decreasing Taylor coefficients. We get that some of these spaces are Banach lattices with respect to this cone. Different ordered spaces of linear bounded operators acting between the previous spaces are also investigated, with emphasis on the so-called regular multipliers and Hankel operators.

It is well known that there exist hypercyclic composition operators on the Hardy spaces \(H^p\) for \(0 1\), all weighted Bergman spaces \(A^p_\beta \) with weight \(z\mapsto (1-|z|^2)^\beta \) (\(0 -1\)), and a class of weighted Dirichlet spaces, mimic the behavior of the Hardy spaces.

Let be the open unit disk in the complex plane and the class of all analytic functions on . Let be an analytic self-map of and . In this paper, the boundedness and compactness for product-type oper...

For , the unit ball of , we consider Bergman–Orlicz spaces of holomorphic functions in , which are generalizations of classical Bergman spaces. We characterize the dual space of large Bergman–Orlicz space, and bounded Hankel operators between some Bergman–Orlicz spaces and , where and are either convex or concave growth functions.

We study mappings inducing bounded composition operators on weighted Sobolev spaces on Carnot groups. We give an analytical description of such mappings in terms of the integrability of weighted distortion function. Bibliography: 17 titles.

We give an explicit formula to factorize functions in the Lorentz spaces L ( p , q ) . Some applications to centralizers, twisted sums and interpolation theory are included.

Let [Formula: see text] be a divergence form inhomogeneous higher order elliptic operator with complex bounded measurable coefficients. In this paper, for all [Formula: see text] and [Formula: see text] satisfying a weak ellipticity condition, the authors introduce the local Hardy spaces [Formula: see text] associated with [Formula: see text], which coincide with Goldberg’s local Hardy spaces [Formula: see text] for all [Formula: see text] when [Formula: see text] (the Laplace operator). The authors also establish a real-variable theory of [Formula: see text], which includes their characterizations in terms of the local molecules, the square functions or the maximal functions, the complex interpolation and dual spaces. These real-variable characterizations on the local Hardy spaces are new even when [Formula: see text] (the divergence form homogeneous second-order elliptic operator). Moreover, the authors show that [Formula: see text] coincides with the Hardy space [Formula: see text] associated with the operator [Formula: see text] for all [Formula: see text], where [Formula: see text] is some positive constant depending on the ellipticity and the off-diagonal estimates of [Formula: see text]. As an application, the authors establish some mapping properties for the local Riesz transforms [Formula: see text] on [Formula: see text], where [Formula: see text] and [Formula: see text].