TYPE AND COTYPE OF SOME BANACH SPACES

Type and cotype are computed for Banach spaces generated by some positive sublinear operators and Banach function spaces. Applications of the results yield that under certain assumptions Clarkson's inequalities hold in these spaces.


INTRODUCTION.
Given a Banach space X, we let for any n !1, p s 2 q < oo and s s < o, Ko'.)(X) and Kt,.s)(X be x C X, where {r" }'.denotes the sequence of Rademacher functions defined by for every choice of i}i.l r,(t) sign sin 2"tr for 0 s 1.If the left (resp.the right) inequality in (1,1) holds, X is of cotype (q, s) (resp.type (p,s)).If s 1, we say that X is of cotype q (resp.type p) (see [6]).
The notions of type and cotype have appeared in various problems involving the analysis of vector valued functions or random variables.One of the great advantages of the classification of Banach spaces in terms of type and cotype is the existence of a rather satisfactory geometric characterization of these notions.For example Maurey and Pisier [8] showed that a Banach space X is of type p for some p > (resp.cotype q for some q < oo) iffX does not contain l"s (resp.g's) uniformly.
the smallest constants for which The well-known examples of Banach spaces for which the above inequalities hold are L,-spaces (see [2]), p-Schatten ideals of compact operators on Hilbert spaces (see [9]), provided 1 < p < In [10] Milman showed, using interpolation techniques that if C: R" is a domain with minimally smooth boundary, then the inequality (1.2) applies to Sobolev spaces W() for < p 2. Further Cohos 3], using the above observation, proved that the inequalities (1.2) and (1.3) hold in W() for every domain C R" and < p < o.In the same way Cobos and Edmunds in [4] showed that some Besov spaces and Triebel-Sobolev spaces verify Clarkson's inequalities.
In this paper we compute the type and cotype for spaces of large class of Banach spaces generated by some positive sublinear operators and Banach function spaces.This class includes for example: interpolation spaces determined by the real method of interpolation, Besov spaces, Triebel-Sobolev spaces (see ], [11], [ 12])H'-spaces, an approximation space, L'(Ix,X)-spaces and the other (see for example [5]).
We also show that under some conditions Clarkson's inequalities hold in these spaces.2. PRELIMINARIES.
Let (,gt) be a complete o-finite measure space.If X is a Banach space, we denote by L (X) -L(f, ix,X)the F-space [i.e., complete and metrizable topological vector space of all equivalence classes of all Ix-Bochner measurable X-valued functions on .IfX R, then we write L LO(,l.t).
A Banach space E C L is called a Banach function space if Ix[ Y[ -a.e. on , x _ L and y IE E imply thatx tEE and [lll Recall that a Banach function space E is called p-convex (resp.p-concave), p < oo if there exists a constant M so that for all x,...,x,, E, we have The smallest possible value of M is denoted by M')(E) (resp.M0,E)).
In what follows let X be an F-space and let S be a positive sublinear operator defined on .X' taking values in L* -L*(, Ix); that is for every x, y tEX and any scalar .t he following hold: (i) Sx 0, (ii) S(),x) )q Sx, (iii) S(x + y) Sx + Sy.
For a given Banach function space E CL* and injective operator S:X.-L*, we define De(S)-{x X :Sx E).
If E -(L,, "11,), we write in short D, instead of De(S), where Ilxll, -(f= Il'd) ' for Throughout the paper, we assume that De(S) is a Banaeh space with the norm defined by Ilxll o-Sxll .
We say that a pair (E,S) is admissible provided that for any A with I.t(A) < =0, we have xSx,O in E for every sequence {x, } C De(S) such that x, 0 in X.Here Xa is a characteristic function ofA.
Let (T,v) and (, I.t) be measure spaces.In the sequel for any x x. tE.X" and f ,f.@L(T,v), we write f (R)xk(t)-A(t)xk for ttET.
(iii) lf a measure space (T,v) is finite, then for all x x.X and fl f.Lp, THEOREM 3.1.Assume that (T,v) is a finite measure space.Let E be a Banach function space and let f (i) IfE isp-convex, then for allxl,...,x X-De(S), we have (ii) If E is p-concave, then for all xl xn X, we have PROOF.First of all, suppose that , k 1, ...,n are step Nnctions.We can cerinly assume at cx,, withA C r measurable, paiise disjoint and r A. en for C -Me(E), we have i-I i-1 (by p convexity) Now assume that in the squence {f}k-, f with k 2 ,n are step functions.Take any sequence {gi}-in L'(T,v) f step functions such that gi "v-a.e, on T, and gi --*f in L'(T,v).From this, we easily get that and aj S gj (R) x + 2 (R) x, dv Since a Cb by previously proved inequality, we obtain desired inequality.Thus, by iterating the proof of (i) is complete.The proof of (ii) is similar.
Let us define on a F-spaceX a family of semi-norm "1 by Ix[ Sx(co) for every co .Nowt he main theorem of the paper is immediate.
THEOREM 3.2.Assume that < p < do and s < oo.Let E be a Banach function space and let K=D(S).
(i) If E is p-convex, < p 2, and s-concave, and for all xi ,x, GX k-1 then X is of type (p, s with Kte's)(X) C,(p s C M" )(E )Mts)(E).
PROPOSITION 3.1.Let (Le,S) be an admissible pair, ,:p <oo.Assume that D e CX with con- tinuous inclusion and that D e is a non-closed subspace in 3C.Then D e is not type r (resp.cotype r).for any r > p (resp..for any r < p).PROOF.The above assumptions imply that for any e > 0, D e contains (1 + e)-isomorphic copy of e (see [7]).Since type and cotype is inherited by subspaces, then the proof is finished.
In the theory of type and cotype the type and cotype indices of a Banach space B which are defined as follows p(B) sup{p: B is of type p}, q(B)= inf{q: B is of cotype p} are important (see [8] for details).

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: