2000 Mathematics Subject Classification: Primary: 46B03, 46B26. Secondary: 46E15, 54C35.We study the existence of pointwise Kadec renormings for Banach spaces of the form C(K). We show in particular that such a renorming exists when K is any product of compact linearly ordered spaces, extending the result for a single factor due to Haydon, Jayne, Namioka and Rogers. We show that if C(K1) has a pointwise Kadec renorming and K2 belongs to the class of spaces obtained by closing the class of compact...

We study the Kergin operator on the space $H_{Nb}\left(E\right)$ of nuclearly entire functions of bounded type on a Banach space E. We show that the Kergin operator is a projector with interpolating properties and that it preserves homogeneous solutions to homogeneous differential operators. Further, we show that the Kergin operator is uniquely determined by these properties. We give error estimates for approximating a function by its Kergin polynomial and show in this way that for any given bounded sequence of interpolation...

We study the class of all rearrangement-invariant ( = r.i.) function spaces E on [0,1] such that there exists 0 < q < 1 for which $\parallel {\sum }_{k=1}^n{\xi }_k{\parallel }_E\le C{n}^q$, where ${{\xi }_k}_{k\ge 1}\subset E$ is an arbitrary sequence of independent identically distributed symmetric random variables on [0,1] and C > 0 does not depend on n. We completely characterize all Lorentz spaces having this property and complement classical results of Rodin and Semenov for Orlicz spaces $exp\left(L_p\right)$, p ≥ 1. We further apply our results to the study of Banach-Saks index sets in...

In this paper we describe the structure of surjective isometries of the spaces of all absolutely continuous, singular, or discrete probability distribution functions on R equipped with the Kolmogorov-Smirnov metric. We also study the structure of affine automorphisms of the space of all distribution functions.

The isomorphic classification problem for the Köthe models of some $C^{\infty }$ function spaces is considered. By making use of some interpolative neighborhoods which are related to the linear topological invariant $D_{\phi }$ and other invariants related to the “quantity” characteristics of the space, a necessary condition for the isomorphism of two such spaces is proved. As applications, it is shown that some pairs of spaces which have the same interpolation property $D_{\phi }$ are not isomorphic.

Dick proved that all dyadic order 2 digital nets satisfy optimal upper bounds on the $L_p$-discrepancy. We prove this for arbitrary prime base b with an alternative technique using Haar bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds on the discrepancy function in Besov spaces with dominating mixed smoothness for a certain parameter range, and enlarge that range for order 2 digital nets. The discrepancy function in Triebel-Lizorkin and Sobolev spaces with dominating mixed...

The paper investigates the nonlinear function spaces introduced by Giaquinta, Modica and Souček. It is shown that a function from ${Cart}^p(\Omega ,{𝐑}^m)$ is approximated by ${𝒞}^1$ functions strongly in ${𝒜}^q(\Omega ,{𝐑}^m)$ whenever $q<p$. An example is shown of a function which is in ${cart}^p(\Omega ,{𝐑}^2)$ but not in ${cart}^p(\Omega ,{𝐑}^2)$.

Starting from a general formulation of the characterization by dyadic crowns of Sobolev spaces, the authors give a result of $L^p$ continuity for pseudodifferential operators whose symbol a(x,ξ) is non smooth with respect to x and whose derivatives with respect to ξ have a decay of order ρ with $0<\rho \le 1$. The algebra property for some classes of weighted Sobolev spaces is proved and an application to multi - quasi - elliptic semilinear equations is given.

We extend some results by Gol${}^{\prime }$dshtein, Kuz${}^{\prime }$minov, and Shvedov about the $L_p$-cohomology of warped cylinders to $L_{p,q}$-cohomology for $p\ne q$. As an application, we establish some sufficient conditions for the nontriviality of the $L_{p,q}$-torsion of a surface of revolution.