BV has the bounded approximation property


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1 The Journal of Geometric Analysis volume 15 (2005), number 1, pp. 17 [version, April 14, 2005] BV has the boune approximation property G. Alberti, M. Csörnyei, A. Pe lczyński, D. Preiss Abstract: We prove that the space BV (R n ) of functions with boune variation on R n has the boune approximation property. Keywors: BV functions in several variables, boune approximation property. Mathematics Subject Classification (2000): 4635, 46B28 (46G10) Introuction Given an integer n 1, BV (R n ) enotes the Banach space of real functions with boune variation on R n, namely the functions u in L 1 (R n ) whose istributional graient Du is a boune (vectorvalue) Raon measure. As usual, u BV := u 1 + Du, where Du is the total variation of the measure Du. A Banach space F has the boune approximation property if for every ε > 0 an every compact set K F one can fin a finiterank operator T from F into itself with norm T C, where C is a finite constant which epens only on F, so that Ty y F ε for every y K (cf. [4], Definition 1.e.11). In this efinition, it is clearly equivalent to consier only sets K which are finite an containe in a prescribe ense subset of F. We prove the following: Theorem 1.  The space BV (R n ) has the boune approximation property. Remarks.  (i) The operators T given in our proof are projections. (ii) The result can be extene to the space BV (R n, R k ) of BV functions value in R k, as well as the corresponing Banach spaces on over the complex scalars. Moreover, it hols for the space BV (Ω, R k ) of BV functions on an open subset Ω of R n with the extension property, namely when there exists a boune extension operator from BV (Ω) to BV (R n ). This class inclues all boune open sets Ω with Lipschitz bounary. We o not know if the result hols for Ω an arbitrary open set. (iii) A careful analysis of the proof of Theorem 1 shows that for every separable subspace X of BV (R n ) there is a sequence (P k ) of commuting finite rank projections from BV (R n ) into itself with uniformly boune norms such
2 2 G. Alberti et al. BV has the boune approximation property 3 that P k f f in BV (R n ) for every f X. Moreover there is a separable subspace Y of BV (R n ) with a Schauer basis which contains X. Specific aspects of Banach space theory an approximation theory of these spaces have been recently stuie in [3], [7] (non linear approximation by Haar polynomial, bouneness of some averaging projections), an [6] (failure of lattice structure). Theorem 1 provies further information on Banach space properties of spaces of functions of boune variation. Acknowlegements.  The secon author acknowleges the support of her research by the Royal Society Wolfson Research Merit Awar, in particular its support of the first an thir authors research visits to UCL. The visits of secon author to the Mathematics Department in Pisa was supporte by GNAFA. The secon author was partly supporte by the NSF grant No. DMS The thir author was partially supporte by KBN grant No. 2P03A Proof of the result Let us fix some notation: n is the Lebesgue measure on R n an is the imensional Hausorff measure. Let µ be a positive measure on R n, D a Borel set, an f : R n R k a function whose restriction to D is µsummable; if 0 < µ(d) <, we enote by f µ,d := ( f )/µ(d) the average of f on D with respect to the measure D µ. We set f µ,d := 0 when µ(d) = 0. If µ is the Lebesgue measure we simply write f D. Since no confusion may arise, we enote by 1 A the characteristic function of a set A in R n (an not the average of 1 on A). Given a finite Borel measure λ on R n, real or R k value, λ is the variation of λ, while λ a an λ s are the absolutely continuous an the singular part of λ with respect to the Lebesgue measure. Given a positive locally finite measure µ on R n, λ λ is the RaonNikoým erivative of λ with respect to µ; so is efine even if λ is not absolutely continuous with respect to µ, in which case it is the RaonNikoým erivative of the absolutely continuous part of λ with respect to µ. Note that λ λ a an λ λ s are boune linear operators from the space of measures (R n ) into itself, while λ λ is a boune linear operator from (R n ) into L 1 (µ). If u is a BV function, we write D a u an D s u for the absolutely continuous an the singular part of the measure Du. We enote by BV loc (R n ) the class of all functions on R n which belong to BV (A) for every boune open set A R n ; in this case Du is a locally finite Borel measure on R n. We will nee the following scale version of Poincaré inequality (cf. [2], Remark 3.50): for every open cube in R n an every u BV () there hols u u n C iam() Du (). (1) The letter C enotes any constant that might epen only on the imension of the space n; the actual value of C may change at every occurrence. Lemma 2.  Let be a locally finite family of pairwise isjoint open cubes in R n whose closures cover R n. If u is a function in BV loc (R n ) with average 0 on each cube then Du C Du (Ω) where Ω is the the union of all. Proof.  We must show that Du () C Du (Ω) for := R n \ Ω. Since the covering is locally finite, is the union of all bounaries with. Thus is an (n 1)rectifiable set, an the restriction of the measure Du to is given by Du = tr + (u) tr (u) n 1, where tr + (u) an tr (u) are the traces of u on the two sies of with respect to some choice of orientation for, an n 1 enotes the restriction of the measure n 1 to the set (see [2], Theorem 3.77). Hence Du () = tr + (u) tr n 1 (u) tr + (u) + tr n 1 (u) = tr n 1 Ω (u) C Du () = C Du (Ω). The inequality in the fourth line is obtaine as follows. For every open cube with size 1 in R n an every u BV () there hols tr Ω (u) L 1 ( ) C u BV () by the continuity of the trace operator, cf. [2], Theorem 3.87 (here L 1 ( ) stans, as usual, for L 1 ( n 1 )). If in aition u has average 0 on then u L 1 () C Du () by (1), an therefore tr Ω (u) L 1 ( ) C Du ().
3 4 G. Alberti et al. BV has the boune approximation property 5 Since the last inequality is scaling invariant, it hols for cubes of any size. Definition 3.  Let µ be a locally finite, positive measure on R n, an let be a countable family of pairwise isjoint, boune Borel sets. We enote by Ω the union of all D an set an, for every map f L 1 (µ, R k ), σ := sup { iam(d) : D } (2) Uf := D f µ,d 1 D. (3) Lemma 4.  (a) The operator U efine in (3) is a boune linear operator from L 1 (µ, R k ) into L 1 (µ, R k ) with norm U = 1. (b) Given a sequence of coverings i, efine Ω i, σ i, an U i accoringly. If σ i 0 as i +, then (U i f 1 Ωi f) 0 in L 1 (µ, R k ) for every f in L 1 (µ, R k ), that is f f µ,d 0. (4) D i Proof.  Statement (a) is immeiate. To prove statement (b), we remark that since the operators U i an the truncation operators f 1 Ωi f are uniformly boune, it suffices to show that (U i f 1 Ωi f) 0 for f in a suitable a ense subset of L 1 (µ, R k ). For instance, we can take f continuous an compactly supporte, an then (4) is an immeiate consequence of the uniform continuity of f. Definition 5.  Fix ū in BV (R n ) an set µ := Dū. Given a locally finite family of mutually isjoint open cubes whose closures cover R n, we efine σ as in (2), an for every u BV (R n ) we set T 1 u(x) := u 1 (x) T 2 u(x) := [ Du ] (x x n ) 1 (x) T 3 u(x) := [ Ds u Dū ] (ū(x) ū ) 1 (x) µ, (here x is the center of, i.e., the average of the ientity map x x over, an the ot stans for the usual scalar prouct in R n ). Finally we set Tu := T 1 u + T 2 u + T 3 u. (5) Lemma 6.  (a) The operator T efine in (5) is a boune linear operator from BV (R n ) into itself with norm T C(1 + σ). If in aition D a ū = 0, then T is a projection. (b) Given a sequence of families i of cubes which satisfy the assumptions of Definition 5, we efine σ i an T i accoringly, an if σ i 0 as i +, then T i u u in L 1 (R n ) for every u BV (R n ). If in aition D s u is of the form D s u = f Dū with f a scalar function, then T i u u in BV (R n ). Proof.  We first prove (a). The following estimates are immeiate: T 1 u 1 u 1 (6) T 2 u 1 σ D a u (7) T 3 u 1 Cσ D s u (8) (for the secon estimate we use that x x iam() σ an for the thir that ū ū C σ Dū () by estimate (1)). stimates (6,7, 8) imply Tu 1 u 1 + Cσ Du. (9) In orer to estimate D(Tu) we notice that Tu an u have the same average on every cube, an therefore we can use Lemma 2 to estimate the total variation of D(Tu u); enoting by Ω the union of the open cubes we obtain D(Tu) Du + D(Tu u) Du + C D(Tu u) (Ω) (1 + C) Du + C D(Tu) (Ω) (1 + C) Du + C D(T 2 u) () + D(T 3 u) (). (10) Since T 2 u is affine on an its graient is the average of Du over, we have n [ D(T 2 u) () = Du ] n n () Du n n Du (). The graient of T 3 u on Ω agrees with Dū times the average of the scalar prouct of Dsu an Dū with respect to the measure µ = Dū, an since = 1 µalmost everywhere, Dū [ D(T 3 u) () = Ds u Dū ] µ, µ() D su Du ().
4 6 G. Alberti et al. BV has the boune approximation property 7 Hence (10) becomes D(Tu) C Du. (11) stimates (9) an (11) imply the first part of statement (a). Concerning the secon part, assume that D a ū = 0. Notice that T 1, T 2, T 3 are mutually commuting projections, thus T is a projection, an its range is the irect sum of the ranges of T 1, T 2, T 3, namely the space of all BV functions whose restriction to each agrees with a linear combination of ū an an affine function. Let us prove statement (b). Inequality (1) yiels Ti 1 u u 1 = u u n Cσ i Du () Cσ i Du, i i an recalling estimates (7), (8) we get T i u u 1 T 1 i u u 1 + T 2 i u 1 + T 3 i u 1 Cσ i Du, which tens to 0 as σ i 0. To prove the secon part of statement (b), we enote by Ω i the union of all cubes i, an estimate D(T i u u) using Lemma 2 again an taking into account that D(T 1 i u)() = 0 for every i: D(T i u u) C D(T i u u) (Ω i ) = C i D(T i u) Du () C i D(T 2 i u) D a u () + D(T 3 i u) D s u (). (12) Setting g := Du n then = Dau, we have D(T 2 n i u) = g n on each i, an D(Ti 2 u) D a u () = g g n. (13) Since D s u = fdū, a simple computation yiels D(Ti 3u) = f µ,dū on each i, an then D(Ti 3 u) D s u () = f µ, f Dū. (14) Finally (12,13,14) yiel D(T i u u) C i [ g g ] n + f f µ,. By (4) both sums at the righthan sie of this inequality vanish as σ i 0, an the proof of statement (b) is complete. Lemma 7.  Let u j be a sequence of functions from BV (R n ). Then there exists ū BV (R n ) such that D a ū = 0, an for every j, D s u j can be written in the form D s u j = f j Dū for some scalar function f j L 1 ( D s ū ). Proof.  This result is essentially containe in [1], but not explicitly state there. In the following proof, all numbere statements an efinitions are taken from [1]; we omit to recall the full statements. Let µ := j α j D s u j, where the α j are positive numbers chosen so to make the series converge. Let (µ, x) R n be the normal space to µ at every x R n in the sense of Definition 2.3, namely a Borel map which takes every x R n into a linear subspace (x) of R n, satisfies Du (x) (x) for µa.e. x an every u BV (R n ), an is µminimal with respect to inclusion. It follows immeiately that Dsuj (x) (µ, x) for µa.e. x an every j, an then (µ, x) contains nonzero vectors for µ a.e. x. Then we can choose a Borel map f L 1 (µ, R n ) so that f(x) (µ, x) an f(x) 0 for µa.e. x (cf. Proposition 2.11), an since µ is a singular measure, we can also assume that f(x) = 0 for n a.e. x. Set now µ := µ + n. By Proposition 2.6(iii), f(x) (µ, x) for µ a.e. x, an we can apply Theorem 2.12 to f an µ to obtain a BV function ū such that Dū (x) = f(x) for µ a.e. x. Since f(x) = 0 for n a.e. x, then Dū is singular, an since f(x) 0 for µa.e. x, then µ Dū. In particular D s u j Dū for every j. Finally, by Theorem 3.1 the space (µ, x) has imension at most 1 for every x because µ is a singular measure, an since both Duj Dū (x) an (x) belong (µ, x), then they must be parallel for µa.e. x. The following lemma is wellknown (see, for instance, [2, Corollary 3.89], [5, Section I.8, Theorem 2], [7,Proposition 5]): Lemma 8.  Let be any open cube in R n with egelength r 1, an let U be the associate truncation operator, that is, U u := 1 u. U is a linear projection on BV (R n ) with norm boune by some universal constant C. Proof of Theorem 1.  Let be given ε > 0 an a finite set K = {u j } of BV functions with compact support in R n. Take ū as in Lemma 7. Take a sequence of families i of cubes which satisfy the assumptions of Definition 5, so that σ i 1 an σ i 0 as i + (cf. (2)), an consier the corresponing operators T i. By Lemma 6 these operators are
5 8 G. Alberti et al. projections, their norms T i are boune by a universal constant, an there is i so large that T i u j u j BV < ε for every j. However, T i has not finite rank. To solve this problem, we choose an open cube which contains the support of all u j, take U as in Lemma 8, an set T := U T i. If we have chosen so that every cube in i which intersects is actually containe in, then T is a projection, too. References [1] G. Alberti, Rank one property for erivatives of functions with boune variation. Proc. Roy. Soc. inburgh Sect. A 123 (1993), [2] L. Ambrosio, N. Fusco, D. Pallara, Functions of boune variation an free iscontinuity problems. Oxfor Mathematical Monographs. Oxfor Science Publications, Oxfor, [3] A. Cohen, W. Dahmen, I. Daubechies, R. DeVore, Harmonic analysis of the space BV. Rev. Mat. Iberoamericana 19 (2003), [4] J. Linenstrauss, L. Tzafriri, Classical Banach Spaces I. rgebnisse er Mathematik un ihrer Grenzgebiete, 92. Springer, BerlinNew York, Reprinte in the series Classics in Mathematics. Springer, BerlinHeielberg, [5] V.G. Mazya, T. Shaposhnikova, Multipliers in spaces of ifferentiable functions. Leningra University Press, Leningra, 1986 (in Russian). [6] A. Pe lczyński, M. Wojciechowski, Spaces of functions with boune variation an Sobolev spaces without local unconitional structure. J. Reine Angew. Math. 558 (2003), [7] P. Wojtaszczyk, Projections an nonlinear approximation in the space BV (R ). Proc. Lonon Math. Soc. (3) 87 (2003), Giovanni Alberti: Department of Mathematics, University of Pisa, L.go Pontecorvo 5, Pisa, Italy ( m.unipi.it). Marianna Csörnyei: Department of Mathematics, University College Lonon, Gower Street, Lonon WC1 6BT, UK ( math.ucl.ac.uk). Aleksaner Pe lczyński: Institute of Mathematics, Polish Acaemy of Sciences, ul. Śniaeckich 8, Warszawa, Polan ( impan.gov.pl). Davi Preiss: Department of Mathematics, University College Lonon, Gower Street, Lonon WC1 6BT, UK ( math.ucl.ac.uk).
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