Result.Math. 39 (2001) 91-98 0378-6218/01/020091-8 $ 1.50+0.20/0 © Birkhiiuser Verlag, Basel, 2001 Results in Mathematics

THE (NON)TENSOR STRUCTURE OF BV-SPACES

MATTHIAS SCHMITT

ABSTRACT. We prove that the tensor product of the spaces BVX".", BVx, is not dense in the space BVX, x· .. xX" equipped with the usual BV-norm. After that we give an approximation result for functions of BVx by tensor product functions, similar to a well-known approximation theorem for functions of BVx by smooth functions. Finally we show that BVx is not, what we will call a symmetrized tensor product of the spaces BVx. (k = 1, ... , s), if we carry out the given construction with any reasonable crossnorm.

1. INTRODUCTION

The aim of the present paper is to investigate whether the space of functions of bounded variation on a product set possesses a kind of a tensor structure, in the sense that it can be obtained by the closures of the tensor product of the spaces of functions of bounded variation on the factor sets with respect to suitable norms. By some heuristical considerations, it can be seen rather easily, that there is no analogy to the L1 situation; i.e.: If the BV-spaces have a tensor structure, then it is not the most simple imaginable one.

In section 2 we give a definition for functions of bounded variation and introduce the concepts of tensor products and reasonable crossnorms. In section 3 we give a rigorous proof of the fact, that in general BV:, the tensor product of BVx" ... ,BVx. is not dense in BVx with respect to the BV -norm, where X is the cartesian product of X!, ... , Xs. Then we prove that each function of BVx can be approximated by a sequence of functions from BV: in L1-norm, in such a way, that the sequence of the variations converges to the variation of the limit-function. At last we make use of tensor product norms to obtain spaces, which we call symmetrized tensor products. For reasonable crossnorms, the symmetrized tensor products are spaces larger than the closure of BV: with respect to the BV­norm. At least for the case of the greatest reasonable crossnorm, the symmetrized tensor product is a subspace of the space BV x. But for all reasonable crossnorms, there are functions of BV x not being contained in the symmetrized tensor product.

The idea that led to this research originates from the theory of dynamical systems or to be more precise from the study of expanding product systems. A piecewise C 1 self-map 7 of X ~ ~ is called expanding, if inf{IID7(x)11 : x E X, IIxll ;;:; I} > 1. If v is a finite (signed) Borel measure on X absolutely continuous with respect to Lebesgue measure, then v 0 7-1 is also absolutely continuous; i.e. 7 induces a linear operator on LL the so-called Perron-Frobenius-Operator (PFO). Making use of a variation inequality for the PFO, Lasota and Yorke proved, in their pioneering work [10] concerning the I-dimensional case, the existence of an absolutely continuous invariant measure for an expanding, piecewise C2-transformation on an interval. In a C2-setting for X,

2000. Mathematics Subject Classification. 26A45, 54BlO. Key words and phrases. Functions of bounded variation, tensor products, reasonable crossnorms.

92 Schmitt

G6ra and Boyarsky [6] succeeded in generalizing these results for higher dimensions, particularly in generalizing the variation inequality. In [8] and [9] Keller was the first, who remarked the applicability of the famous spectral theorem of Ionescu-Thlcea and Marinescu [7] to PFOs, satisfying such a Lasota-Yorke-inequality, and so in particular operating on the space BVx. More recent results in this line of research, using bounded variation techniques, are the works of Tsujii [14] and Buzzi [1] about expanding, piecewise analytic maps in dimension 2 and the work of Buzzi and Keller [2]. In a piecewise affine setting, they identified the isolated eigenvalues of the PFO with the reciprocals of the poles of the dynamical zeta function. For finitely many systems (Xl, 7"d, ... , (X" 7".) the corresponding product system (X, T) is given by X := Xl x··· xX. and (TX)k = 7"kXk. We are interested in a relation between the spectral properties of the PFOs PTk acting on BVXk (k = 1, ... , s) on the one side, and the spectral properties of the PFO PT = ®~=l PTk acting on BVx on the other side. The results of this note show that additional work must be done in order to relate the spectrum of PT to those of PTl , ••• , PTs •

Acknowledgements. I wish to thank G. Keller who called my attention to this interesting question for many useful advices.

2. PRELIMINARIES

By L~ we denote the space of all equivalence classes of real-valued functions defined on n that are integrable with respect to a given measure. If x = (x}, ... , Xd) is a vector with d coordinates we write Xl := (Xl' ... ' Xl-I, Xl+ I, ... , Xd). If f is a function of d variables we denote by ix l the function given by Xl f--t f(XI, xt}. (The argument of f must be interpreted correctly; i.e. the pair (Xl, Xl) is to be identified with x.)

2.1. Functions of bounded variation. The functions of bounded variation can be characterized as integrable functions whose partial derivatives in the sense of distributions are finite signed Borel measures. We use a characterization that reduces the variation of functions of several variables to the well-known variation of functions of one variable. Good references for multivariate functions of bounded variation are Chapter 5 in [4], Chapter 5 in [15] and Part I of [5].

Definition 2.1. Let I ~ lR be an interval. The variation of a function cp : 1-+ lR is

(1) var cp := sup{t Icp(~i) - cp(~i-I)1 : ~o ~ ... ~ ~r;[~O, ~r] ~ I; r E N}. .=1

The variation of an element f E L} is

(2) var f := inf var cpo <pEj

Remark 2.2. f E L} is of bounded variation if and only if it possesses a representative that can be written as a difference of two monotonous functions.

Definition 2.3. Let X ~ lRd be an a.e. open set. Then the variation of f E L5; is

d

var f := L / var ix l dx/. k=l

(3)

We call f of bounded variation if var f < 00 and denote the class of all such functions by BVx.

Theorem 2.4. For all sequences (fn)nEN ~ L5; converging to f E L5; in Ll-norm we have

(4) var f ~ lim infvar fn n-too

Schmitt 93

Remark 2.5. a) Endowed with the norm

(5) II/IIBV := var 1+ 11/111

BVx is a Banach space. b) With respect to the Ll-norm BVx is a dense subspace of L},;.

2.2. Tensor products of function spaces and product norms. First in this section we will introduce the tensor product of function spaces Ul ~ Lh1 , • •• , Us ~ Lhs where (Dk, Ak, J.1-k) is an underlying measure space (for k = 1, ... , s).

Definition 2.6. The tensor product of functions 'Pk : Dk -t lR (k = 1, ... ,s) is the map

(6) ~ {Dl x ... x Ds -t lR '61 'Pk : s k=l (Xl, ... ,Xs) >--+ I1k=l 'Pk(Xk)'

The tensor product of elements IkE Lhk (k = 1, ... , s) is that I E Lhl x ... xfl.s with

s } s { ® 'Pk : 'Pk Elk; k = 1, ... ,s ~ I =: ® fk. k=l k=l

(7)

Definition 2.7. The tensor product of the spaces Uk (k = 1, ... , s) is the linear space

® Uk := { t ® /k,i : Ik,i E Uk; k = 1, ... ,s; i = 1, ... ,n; n EN}. k=l i=l k=l

(8)

Definition and Remark 2.8. For linear functionals <I>k on the Uk,

(9) (~<I>k) (~~ Ik'i) := ~ 11 <I>k(!k,d

defines a linear functional ®~=l <I>k on ®~=l Uk·

From now on the spaces Uk should be equipped with norms Nk. The normed dual spaces are denoted by (Uk,Nk)' We discuss two canonical norms for the tensor product of normed linear spaces, namely the greatest and the least reasonable crossnorm. As references to this subject we give Chapter 1 of [11], Chapter VIII of [3] and the classical [12].

Definition 2.9. A norm N on ®~=l Uk is called a a) crossnorm if for all Ik E Uk

(10) N(~/k) = 11 Nk(/k).

b) reasonable product norm if for all <I>k E Uk

(11) N(~<I>k) = IT Nk(<I>k)'

Definition and Remark 2.10. Let f = 2.:7=1 ®~=l Ik,i E ®k=l Uk· Define r(f) by

(12) r(f):= inf{~g Nk(A,j): ~~jk,j = f} and )..(f) by

(13) )..(1):= sup { (®<I>k)(f): <I>k E Uk; Nk(<I>k) = l;k = 1, .. . ,s}. k=l

94 Schmitt

It is easy to check that for alII = 1, ... , s

(14) )..(f) = SUp { N{t (II Wk(fk,d )!t,i) : Wk E Uk; Nk(Wk) = 1; k f= I}-i=1 kil

Theorem 2.11. a) 'Y and)" are reasonable crossnorms. b) 'Y is the greatest and), is the least reasonable crossnorm; i. e. for any other reasonable crossnorm N we have

(15)

For completeness we give the proof of the following Lemma, because we couldn't find it in the literature.

Lemma 2.12. Let N be any crossnorm. If each fk E Uk (for k = 1, ... ,s) can be approximated in Nk-norm by elements of a linear subspace Vk, then with respect to N the space ®~=1 Vk is dense in the closure of ®~=1 Uk·

PROOF. It suffices to see that each f E ®~=1 Uk can be approximated in N-norm by elements

of ®~=1 Vk· For this we will show that each f = 2:~=1 ®~=1 A,i E ®~-:,\ Vk 0 ®~=l Uk can be approximated by elements of ®~=1 Vk 0 ®~=l+1 Uk for I = 1, ... ,s. W.l.o.g. let fk,i f= 0 a.e. for all i = 1, ... ,n and all k = 1, ... ,s. Let f > O. For i = 1, ... ,n there is 9i E Vi so that

f

NMl,i - 9i);::; TI N (f .) n kil k k,1

Choosing 9 := 2:~1 ®~-:,\ ik,i 09i 0 ®~=1+1 fk,i E ®~=1 Vk 0 ®~=l+1 Uk we get:

NU-g)=N(t0A,i0U/,i-gd0 0 A,i) ~tN/u/,i-gi)IINk(!k,i)~€ i=l k=l k=l+l i=l k,pl

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3. TENSOR PRODUCTS OF BV-SPACES

Let Xl ~ ]Rd1 , ••• ,Xs ~ ]Rd, bea.e. open sets and X := XIX" ,xXs ~]Rd whered:= d1+· ·+ds .

An analogous equation to (3) , adapted to this setting, holds for all f E L~:

s

(16) var f = L ! var fxk d:h k=l

For the tensor product of the spaces BVXk (k = 1, ... , s) we use for brevity the notation BV:. For its closure with respect to a norm N we write BV!lN.

3.1. The closure with respect to the BV-norm. A direct calculation leads to the following formula for the variation of a pure product function ®~=1 fk E ®~=1 Lt:

s

(17) varQ9fk = LvarflII Ilfklh k=1 1=1 kil

The inclusion BV: <;; BVx is an immediate consequence. Because of the completeness of (BVx, II IIBv) we even have

(18)

This is a strict inclusion, as the following simple example already shows.

Schmitt

Example 3.1. Let [:= [0,1] and A:= {(x,y) E [2: x ~ y}. For the characteristic function XA we have

var XA = J var(XA}x dx + J var(XA}y dy = 2 < 00,

and so XA E BVI 2 .

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But XA 1: BVI 18> BV BVI: We take h = L~l Ii 18> gi E BVI 18> BVI. It follows from Remark 2.2 that the set of discontinuities of h is contained in the union of at most count ably many parallels to the x- and to the y-axis. I.e. except for at most countably many y we have:

lim (XA - h)y(x} - lim (XA - h)y(x} = 1 x-ty+ x-ty-

Therefore we get:

var(XA - h}y ~ 1

for these allowed y. Thus

var(XA - h} ~ J var(XA - h)y dy ~ 1 > O.

<> As we have seen now a function I E BVx can not be approximated in general by functions from

BV: in the sense of the BV-norm. In the following we give a positive approximation result in a weaker sense.

Definition and Remark 3.2.

Tx := { t aiXA,nx : Ai is a d-dimensional interval; ai E lR; i = 1, ... , nj n EN} ,=1

forms a linear subspace of BVx. It is not difficult to see that

(19) Tx =Q9TXk ' k=l

The following theorem is a variant of Theorem 5.3.3 in [15] concerning the approxima tion of functions of bounded variation by smooth functions. This approximation theorem is used in order to develop a satisfying spectral theory for tensor product operators on BV-spaces. Under the additional assumption that the spectral projections corresponding to the isolated eigenvalues of the factor operators are Ll-bounded, this was done in [13]. A result without this restrictive assumption is work in progress.

Theorem 3.3. Let Q <;;; JRd be a bounded d-dimensional interval. Then for all I E BVQ there exists a sequence (fn}nEN <;;; TQ so that

(20) IIIn - 1111 ~ 0 and var In ~ var f.

PROOF. W.l.o.g. we let Q := [O,l]d. For n E N let I~n) be a partition of Q in cubes with edges of length 2-n . Then the sequence

In := E(fII~n)} = 2dn L j I(x} dx· XI

IEI~n)

of conditional expectations with respect to the I~n) satisfies (20). The proof of the first part of (20) follows from the martingale convergence theorem. We will show that var In ~ var I for all n E N, which together with Theorem 2.4 yields the second

96 Schmitt

part of our assertion. The case d = 1 follows directly from the representation of In given above.

Let d ~ 2. We fix iii E J E I~~l' 2n-1 (i+1)2-n i2-n

var(fn)iiI = L 12dn (1 r I(XI, Xl) dXI dXI -1 J. I(XI , xt} dXI dXI) 1 i=l J Ji2- n J (,-1)2 - n

2n-1 (i+l)2-n i2-n

;£ 2(d-1)n r L 12n ( r h (xt} dXI - { lx, (Xl) dXI) 1 dXI J J i=l Ji2- n Jr.-l)2- n

= 2(d-1)n 1 var E(fx,IIin») dXI ;£ 2(d-1)n 1 var h dit,

using the d = 1 result for the last inequality. An integration leads us to the end of the proof. 0

Corollary 3.4. If the Xk are bounded dk-dimensional intervals (for k = 1, ... , s), then for all I E BVx there exists a sequence (fll)nEN ~ BV!j such that (20) holds.

3.2. Symmetrized tensor products. Motivated by (17) we will investigate whether it is possible to obtain the whole space BVx from the space BV: by the following construction. We start with a prescription to endow the tensor product of s normed linear spaces (Ul ' N 1), •. . , (Us, N s) with a norm a. Later we will inspect the cases where a = I or a = A. For l = 1, ... ,s we close BV!j with respect to the norm al that one gets by thinking of the l-th factor BVx, to be equipped with the BV-norm, but the remaining s - 1 factors to be equipped with the corresponding L1-norms. The space that we are interested in is the intersection of these s spaces which we will denote by Bv:a. Endowed with the sum norm E7=1 al this intersection of Banach spaces becomes a Banach space on its own. We will restrict ourselves to reasonable crossnorms.

Definition and Remark 3.5. For all reasonable crossnorms a the space

(21) Tl := BVx, ® Tx, := TXl ® ... ® TXI _ 1 ® BVx, ® TXI +1 Q9 ..• ® Tx.

is dense in the space BV:a, where we wrote XI := Xl x . . . XI- l X XI+1 X . .. x Xs .

Theorem 3.6. For all f E BV:~' we have

(22) 1M) = ! var IXI dXI + Ilflh·

PROOF . The equation II (f) = J IIh lIav dXI follows from the proof of Theorem 1.15 in [11) applied to our special setting. 0

Remark 3.7. a) On the space BV!j the norm II is weaker than the BV-norm. It follows that BV®BV C BV®~' and so x - x

(23) BV!jBV ~ BV:~ .

b) We use (22) to define II for all f E L~. By standard arguments one can prove that II is a norm on BV} := {J E L~ : II(f) < oo} and that this space is a Banach space. That is why

BV:~' ~ BVi' and expression (22) gives us an extension of the product norm II to a larger space. It follows already from the calculation in Example 3.1 that this space actually is a strictly larger space in general. Analogous statements for the intersections are true:

(24)

c) In view of b) we have got the following question, which we couldn't answer yet. Is inclusion (23) a strict one or have we got equality?

Schmitt 97

Remark 3.8. a) An application of (14) to our special situation leads to

(25) AM) = sup{ III qJ; hk(Xk) )JXI dXl!LV : hk E L~kj Ilhkll oo = Ij k # I}

for all J E BVJ!, noticing L~ = (L1-)*. Because this expression is well-defined for all J E L1- we will use the notation A1U) also in this case. The inequality A1U) ;;:; flU) holds for all such f. b) The rather technical proof of the fact that Al defines a norm on the space B~~l := {J E L1- : AM) < oo} can be found in [13]. c) We do not know if the whole space BVJ!' is included in BV,y. And because it is not clear whether the BV~1 are Banach spaces at all, we even do not know if B~~A is included in L1-.

We return to the situation of Example 3.1. We even can show that XA ¢:. BVI ®al BVI where a is an arbitrary reasonable crossnorm.

Theorem 3.9. If a is a reasonable crossnorm, then BVI 2 is not contained in BVI ®a BVI.

PROOF. Since a ;;;; A, we need to consider only the case a = A. Let L~=l J; ® XAi E '0\ with Ai =]ai-l, ail (for i = 1, ... , n) where 0 ;;:; ao < ... < an :s; 1. M I t h ",n ",n h A ] ai_1+ai ] d A ]ai_1+ai ] oreover we e := L..i=l XAi,1 - L..i=l XAi,2 were i,l:= ai-I, --2- an i,2 := --2-' ai . Then h E L'I with Ilhlloo = 1. For a.e. x we have:

I ( ) ~ () () ~ ()ai-ai- l ~ )ai-ai- l h Y ~ J; X XAi Y dy = ~ J; x 2 - ~ J;(x 2 = 0

~l ~l ~l

Thereby we obtain:

Al(XA- tJ;®XAi) ;;;;var(! h(Y)XA(.,y)dy- I h(y)tJ;(')xAi(Y)dY)

;;;; til h(y) (XA (ai- 12+ ai, Y) - XA (ai-I, Y) ) dyl

+ til h(y) (XA(a;,y) _XA(ai-l/ai,y)) dyl

= ~I! h(Y)Xlai_l,ai-~+ail(Y)dYI + ~I! h(Y)XJ"i_~+ai,a;J(Y)dYI = t I XAi,l (y) dy + t I XAi.2(Y) dy = 1 > 0

;=1 i=1

REFERENCES

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[1] J. Buzzi, A.c.i.m. 's for arbitrary expanding piecewise lR-analytic mappings of the plane, Ergod. Th. & Dynam. Sys. 20 (2000), 697-708.

[2] J. Buzzi, G. Keller, Zeta functions and transfer operators for multidimensional piecewise affine and expanding maps, Ergod. Th. & Dynam. Sys. (to appear).

[3] J. Diestel, J. J. Uhl jr., Vector Measures, Mathematical surveys number 15, American Mathematical Society, Providence, Rhode Island (1977).

[4] L. e. Evans, E. W. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, eRe Press (1992).

[5] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhauser: Boston-Basel-Stuttgart, 1984. [6] P. Gora, A. Boyarsky, Absolutely continuous invariant measures for piecewise expanding C2 transformations in

lRN , Israel J. Math. 67 (1989), 272-286.

98 Schmitt

[7] C. Ionescu-Thlcea, G. Marinescu, Theorie ergodique pour des classes d'operations non completement continues, Ann. Math. 52 (2) (1950), 140-147.

[8] G. Keller, ErgodiciU et measures invariantes pour les transformations dilatantes par morceaux d'une region bornee du plan, C.R. Acad. Sci. Paris, Serie A 289 (1979), 625-627.

[9] G. Keller, Un theoreme de la limite centrale pour une classe de transformations monotones par morceaux, C.R. Acad. Sci. Paris, Serie A 291 (1980), 155-158.

[10] A. Lasota, J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 286 (1984), 481-488.

[11] W. A. Light, E. W. Cheney, Approximation Theory in Tensor Product Spaces, Lecture Notes in Mathematics 1169, Springer (1985) .

[12] R. Schatten, A Theory of Cross-Spaces, Princeton University Press (1950), Reprint: Kraus Reprint Corporation (1965).

[13] M. Schmitt, Tensorprodukte von BV -Riiumen und BV -Raum-Operatoren mit Tensorstruktur, Dipiomarbeit, Eriangen (1998).

[14] M. Tsujii, Absolutely continuous invariant measures for piecewise real-analytic expanding maps on the plane, Commun. Math. Phys. 208 (2000), 605-622.

[15] W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics 120, Springer (1989).

LEHRSTUHL FUR ANGEWANDTE MATHEMATIK II, UNIVERSITAT ERLANGEN-NURNBERG, MARTENSSTR. 3, 91058 ERLANGEN, GERMANY

E-mail address:schmittmillam.uni-er1angen.de

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