The Gustavsson–Peetre method for several Banach spaces

We extend the Gustavsson–Peetre method to the context of N ‐tuples of Banach spaces. We give estimates for the norm of the interpolated operator. The method is applied to tuples of weighted L _p ‐spaces and to tuples of Orlicz spaces identifying the outcoming spaces in both cases. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interpolation of bilinear operators in Marcinkiewicz spaces

A theorem on interpolation of bilinear operators in symmetric Marcinkiewicz spaces is proved. It follows from the general bilinear results for the Peetre and Peetre-Gustavsson interpolation functors. Translated fromMatematicheskie Zametki, Vol. 60, No. 4, pp. 483–494, October, 1996.     

On the trace problem for Lizorkin–Triebel spaces with mixed norms

The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by x₁ = 0 and x_n = 0 are applied to functions belonging to quasi‐homogeneous, mixed norm Lizorkin–Triebel spaces ; Sobolev spaces are obtained from these as special cases. Spaces admitting traces in the distribution sense are characterised up to the borderline cases; these are also covered in case x₁ = 0. For x₁ the trace spaces are proved to be mixed‐norm Lizorkin–Triebel spaces with a specific sum exponent; for x_n they are similarly defined Besov spaces. The treatment includes continuous right‐inverses and higher order traces. The results rely on a sequence version of Nikol'skij's inequality, Marschall's inequality for pseudodifferential operators (and Fourier multiplier assertions), as well as dyadic ball criteria. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tractable embeddings of Besov spaces into small Lebesgue spaces

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.     

Representation,interpolation, and reiteration theorems for generalized approximation spaces

We show that the representation theorem for classical approximation spaces can be generalized to spaces A(X,l ^q (ℬ))={f∈X:{E _n (f)}∈l ^q (ℬ)} in which the weighted l ^q -space l ^q (ℬ) can be (more or less) arbitrary. We use this theorem to show that generalized approximation spaces can be viewed as real interpolation spaces (defined with K-functionals or main-part K-functionals) between couples of quasi-normed spaces which satisfy certain Jackson and Bernstein-type inequalities. Especially, interpolation between an approximation space and the underlying quasi-normed space leads again to an approximation space. Together with a general reiteration theorem, which we also prove in the present paper, we obtain formulas for interpolation of two generalized approximation spaces. Received: December 6, 2001; in final form: April 2, 2002?Published online: March 14, 2003     

Ёквивалентные нормировки пространствфункций, ассоциированных с обобшенным сдвигом ГегенбауЭра

Taylor-Delsart formula is elaborated in the paper for functions of the generalized Gegenbauer shift. This formula is utilized to construct a version of the Gegenbauer shift modulus of smoothness of order k which for k = 1 reduces to the modulus of smoothness of the first order. By means of this modulus and Peetre’s K-functional, an interpolation theorem is obtained. Equivalent normalizations are obtained for functional spaces associated with the generalized Gegenbauer shift.     

Further criteria for the indecomposability of finite pseudometric spaces

We continue the study of indecomposable finite (consisting of a finite number of points) pseudometric spaces (i.e., spaces whose only decomposition into a sum is the division of all distances in equal proportion). We prove that the indecomposability property is invariant under the following operation: connect two disjoint points by an additional simple chain, which is the inverted copy of the shortest path connecting these points. The indecomposability of the spaces presented by the graphsK _m,n (m ≥ 2,n ≥ 3) with edges of equal length is also proved. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 421–424, March, 1998.     

Description of K-spaces by means of J-spaces and the reverse problem in the limiting real interpolation

We establish conditions under which K-spaces in the limiting real interpolation involving slowly varying functions can be described by means of J-spaces and we also solve the reverse problem. To this end, we prove several versions of the fundamental lemma of the real interpolation theory. We apply our results to obtain density theorems for the corresponding limiting interpolation spaces.     

A characterization of the interpolation spaces ofH 1 andL ∞ on the line

The Calderón-Mitjagin theorem characterizes all interpolation spaces of the pair of Lebesgue spaces (L ¹,L ) as the rearrangement-invariant spaces. The results of this paper show that the interpolation spaces ofH ¹(R) andL (R) consist of elements whose nontangential maximal functions lie in rearrangement-invariant spaces.Communicated by Jaak Peetre.     

Interpolation und indexbedingungen auf rearrangement—invarianten funktionenräumen. I. Grundlagen und die K-methode

In this paper the K-interpolation method of J. Peetre is built up for rearrangement invariant norms ? on (0, ∞). The spaces (X₁, X₂)_θ,?;K (?∞ < θ < ∞), defined by the norm ∥f∥_θ,?;K = ?(t^?θK(t,f)), are shown to be intermediate spaces of the Banach spaces X₁ and X₂ if the condition α < θ ? 1 upon the upper index α of ? is assumed. For these spaces an interpolation theorem of M. Riesz-Thorin-type as well as theorems of reiteration and stability are valid, again under certain conditions upon the indices of ?. These index-conditions, which turn out to be of central importance in the interpolation theory on rearrangement invariant spaces, are shown to be equivalent to a generalized Hardy-Littlewood inequality, which is established in the first part of the paper.     

Wavelet decompositions of L 2-functionals

Based on distribution-theoretical definitions of L ₂ and Sobolev spaces given by Werner in [P. Werner (1970). A distribution-theoretical approach to certain Lebesgue and Sobolev spaces. J. Math. Anal. Appl., 29, 19–78.] real interpolation, Besov type spaces and approximation spaces with respect to multiresolution approximations are considered. The key for the investigation are generalized moduli of smoothness introduced by Haf in [H. Haf (1992). On the approximation of functionals in Sobolev spaces by singular integrals. Applicable Analysis, 45, 295–308.]. Those moduli of smoothness allow to connect the concept of L ₂-functionals with more recent developments in multiscale analysis, see e.g. [W. Dahmen (1995). Multiscale analysis, approximation, and interpolation spaces. In: C.K. Chui and L.L. Schumaker (Eds.), Approximation Theory VIII, Vol. 2: Wavelets and Multilevel Approximation, pp. 47–88.]. In particular, we derive wavelet characterizations for the Sobolev spaces introduced by Werner and establish stable wavelet decompositions of L ₂-functionals. Generalizations to more general spaces of functionals and applications are also mentioned.     

Monotonicity properties of interpolation spaces

For any interpolation pair (A ₀ A ₁), Peetre’sK-functional is defined by: $$K\left( {t,a;A_0 ,A_1 } \right) = \mathop {\inf }\limits_{a = a_0 + a_1 } \left( {\left\| {a_0 } \right\|_{A_0 } + t\left\| {a_1 } \right\|_{A_1 } } \right).$$ It is known that for several important interpolation pairs (A ₀,A ₁), all the interpolation spacesA of the pair can be characterised by the property ofK-monotonicity, that is, ifa∈A andK(t, b; A₀, A₁)≦K(t, a; A₀, A₁) for all positivet thenb∈A also. We give a necessary condition for an interpolation pair to have its interpolation spaces characterized byK-monotonicity. We describe a weaker form ofK-monotonicity which holds for all the interpolation spaces of any interpolation pair and show that in a certain sense it is the strongest form of monotonicity which holds in such generality. On the other hand there exist pairs whose interpolation spaces exhibit properties lying somewhere betweenK-monotonicity and weakK-monotonicity. Finally we give an alternative proof of a result of Gunnar Sparr, that all the interpolation spaces for (L _v ^p , L _w ^q ) areK-monotone.     

A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces

In this paper we work in separated locally convex spaces where we give equivalent statements for the formulae of the conjugate function of the sum of a convex lower‐semicontinuous function and the precomposition of another convex lower‐semicontinuous function which is also K ‐increasing with a K ‐convex K ‐epi‐closed function, where K is a nonempty closed convex cone. These statements prove to be the weakest constraint qualifications given so far under which the formulae for the subdifferential of the mentioned sum of functions are valid. Then we deliver constraint qualifications inspired from them that guarantee some conjugate duality assertions. Two interesting special cases taken from the literature conclude the paper. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On K-groups of operator algebras on HD n spaces and QD n spaces

We obtain the K-groups of the operator ideals contained in the class of Riesz operators. And based on the results, we calculate the K-groups of the operator algebras on HD nspaces and QDn spaces.     

Reiteration theorems for the K‐interpolation method in limiting cases

Sharp reiteration theorems for the K‐interpolation method in limiting cases are proved using two‐sided estimates of the K‐functional. As an application, sharp mapping properties of the Riesz potential are derived in a limiting case. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Infinite Dimensional Geometric Properties of Real Interpolation Spaces

Estimates for the moduli of noncompact convexity of l_p-sums and real interpolation spaces for finite families of spaces are given. It is proved that such an interpolation preserves nearly uniform convexity and property (β).     

Interpolation of Herz Spaces and Applications

C. Herz introduced in [Hr] some new spaces to study properties of functions. An Interesting account, with many applications, of some particular cases of the generalized Herz spaces is given in [BS]. In this paper we first identify the duals of the generalized Herz spaces. Then, we characterize their intermediate spaces when the complex method of interpolation for families of spaces Is used. Applications are given that show the bounded ness of many operators on the generalized Herz spaces.     

Intrinsic characterizations of Besov spaces on Lipschitz domains

The aim of this paper is to study the equivalence between quasi‐norms of Besov spaces on domains. We suppose that the domain Ω ? ?ⁿ is a bounded Lipschitz open subset in ?ⁿ. First, we define Besov spaces on Ω as the restrictions of the corresponding Besov spaces on ?ⁿ. Then, with the help of equivalent and intrinsic characterizations (the Peetre‐type characterization 3.10 and the characterization via local means 3.13) of these spaces, we get another equivalent and intrinsic quasi‐norm using, this time, generalized differences and moduli of smoothness. We extend the well‐known characterization of Besov spaces on ?ⁿ described in Theorem 2.4 to the case of Lipschitz domains.