TRACES AND SOBOLEV EXTENSION DOMAINS

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 8, Pages S (06) Artcle electroncally ublshed on February 8, 2006 TRACES AND SOBOLEV EXTENSION DOMAINS PETTERI HARJULEHTO (Communcated by Davd Press) Abstract. Assume that Ω R n s a bounded doman and ts boundary Ω s m-regular, n 1 m<n. We show that f there exsts a bounded trace oerator T : W 1, (Ω) B n m 1 α ( Ω), 1 << and α =,and(1 λ)- Hölder contnuous functons are dense n W 1, (Ω), 0 λ<n m, then the doman Ω s a W 1, -extenson doman. 1. Introducton We say that a doman Ω R n s a W 1, -extenson doman f there exsts a bounded lnear extenson oerator E : W 1, (Ω) W 1, (R n ), so that Eu Ω = u for every u W 1, (Ω). From a hlosohcal ont of vew, when we extend a functon t does not matter how rregular the functon s nsde the doman as long as t s smooth enough near the boundary. Thus t s natural to try to characterze the extenson roerty by traces. A. Jonsson showed that a trace of a Sobolev functon u W 1, (R n )nanmregular F R n,0<m<n, belongs to a Besov sace B 1 n m (F ), [4, Theorem 3,. 4]. A bounded set F R n s m-regular f C 1 r m H m (B n (x, r) F ) Cr m for every x F and every r, 0<r dam(f ). There also exsts a bounded extenson oerator B 1 n m (F ) W 1, (R n ), [4, Theorem 3,. 4]. However, the roblem s that the extenson of u W 1, (Ω) should have weak dervatves across the boundary. Let 1 <<. Assume that Ω R n s a bounded doman and ts boundary Ω sm-regular, n 1 m < n. Assume that (1 λ)-hölder contnuous functons are dense n W 1, (Ω), 0 λ<n m. Assume that there exsts a bounded lnear trace oerator T : W 1, (Ω) B 1 n m ( Ω), so that Tu(x) = lm r 0 B n (x,r) Ω u(z)dhn (z) H m -almost everywhere n Ω. Then we show that the doman Ω s a W 1, -extenson doman. Receved by the edtors October 26, 2000 and, n revsed form, March 10, Mathematcs Subject Classfcaton. Prmary 46E35. Key words and hrases. Sobolev sace, Besov sace, trace oerator, extenson oerator c 2006 Amercan Mathematcal Socety Reverts to ublc doman 28 years from ublcaton Lcense or coyrght restrctons may aly to redstrbuton; see htt://

2 2374 PETTERI HARJULEHTO Jonsson s theorem mles the followng result. Assume that Ω R n s a bounded W 1, -extenson doman and ts boundary Ω sm-regular, n 1 m<n. Then there exsts a bounded trace oerator T : W 1, (Ω) B 1 n m ( Ω), so that Tu(x) = lm r 0 B n (x,r) Ω u(z)dhn (z) H m -almost everywhere n Ω. We obtan a characterzaton for Sobolev extenson domans. Assume that Ω R n s a bounded doman and ts boundary Ω sm-regular, n 1 m<n.then the followng two condtons are equvalent for every, n 1 << : (1) The doman Ω s a W 1, -extenson doman. (2) There exsts a bounded lnear trace oerator T : W 1, (Ω) B 1 n m ( Ω), so that Tu(x) = lm r 0 B n (x,r) Ω u(z)dhn (z) H m -almost everywhere n Ω and Lschtz contnuous functons are dense n W 1, (Ω). 2. Notaton and relmnares Throughout ths aer C wll denote a constant whch may change even n a sngle strng of an estmate. We let Ω be a doman of Eucldean n-sace R n, n 2. The norm of x R n s x = x n x2 n. Byanoenballcenteredatx and wth radus r>0wemeanthesetb n (x, r) ={y R n : y x <r}. Byanoen cube we mean the set Q(x, r) ={y R n :max j=1,..,n y j x j <r}, andwewrte kq for the cube wth the same center as Q anddlatedbyafactork>0. We let dam(a) = su{ x y : x, y A} denote the dameter of a set A. The boundary of a doman Ω s denoted by Ω. Let A be the Lebesgue n-measure of a set A. Let H m be the m-dmensonal Hausdorff measure. We wrte u v f there exsts a constant C 1sothatC 1 u v Cu. By u(x)dx we denote the ntegral A average of the functon u over the set A wth A > 0. We say that a bounded set F R n s m-regular, 0 <m n, fthereexstsa constant C>0sothat C 1 r m H m (B n (x, r) F ) Cr m for every x F and every r, 0<r dam(f ). A tycal examle of an m-regular set s the boundary of Koch s snowflake doman wth m = ln 4 ln 3. For more detals about m-regular sets we refer to [6, Chater II]. By C(A) we denote the set of contnuous functons n a set A. By C 0,λ (A), 0 <λ 1, we denote the set of λ-hölder contnuous functons u C 0,λ (A) fthere exsts a constant C>0sothat u(x) u(y) C x y λ for every x, y A. Ifλ = 1 we say that the functon u s Lschtz contnuous. By L (D), 1,wedenotetheclassofall-ntegrable functons n an oen set D R n. By L 1, (D), 1, we denote the class of all functons whose frst weak artal dervatves belong to L (D). The classcal Sobolev sace n an oen set D s denoted by W 1, (D) =L (D) L 1, (D), 1. We equ W 1, (D) wth the norm u W 1, (D) = u L (D) + u L (D), where u s the weak gradent. Lcense or coyrght restrctons may aly to redstrbuton; see htt://

3 TRACES AND SOBOLEV EXTENSION DOMAINS 2375 By M 1, (D), 1 <,wedenotethehaj lasz sace u M 1, (D)fu L (D), and there exsts a non-negatve g L (D) sothat u(x) u(y) x y (g(x) +g(y)) for almost every x, y D. For an arbtrary doman Ω t holds that M 1, (Ω) W 1, (Ω). If a doman Ω s a W 1, -extenson doman, then M 1, (Ω) = W 1, (Ω), [1, Theorem 1,. 405]. These saces were defned by P. Haj lasz, [1]. Let F R n be an m-regular set, 0 <m n. The Besov sace B 1 n m (F ), 1 <<, conssts of all functons u L (F )forwhch u(x) u(y) (2.1) x y <1 x y n+2m dhm (x)dh m (y) <. TheBesovsacesequedwththenorm u B 1 n m ( (F ) = u L (F ) + x y <1 u(x) u(y) x y n+2m dhm (x)dh m (y) More detals about classcal Besov saces and Besov saces on an m-regular set can be found for examle n [8, Sectons 8.3 and 8.4] and [6, Chater V]. We say that a doman Ω R n s a W 1, -extenson doman f there exsts a bounded lnear extenson oerator E : W 1, (Ω) W 1, (R n ), such that Eu Ω = u and a constant C>0, so that ) 1. for every u W 1, (Ω). Eu W 1, (R n ) C u W 1, (Ω) 3. Extenson oerator Frst we construct an extenson oerator. Let Ω R n be a bounded doman and let ts boundary Ω bem-regular, n 1 m<n. In ths chater we assume there exsts a bounded trace oerator T : W 1, (Ω) B 1 n m ( Ω), 1 <<, such that Tu(x) = lm u(z)dh n (z) r 0 B n (x,r) Ω H m -almost everywhere n Ω. Let W 0 be the Whtney decomoston of R n \ Ω, [10, Theorem 1,. 167]. Let W be the famly of those cubes Q W 0,forwhch dam(q ) < mn{a, dam(ω)} for every. Here 0 <a 1saconstantwhchwe wll fx n the roof of Lemma 3.4. We wll wrte that W = Q W Q.Let(φ )bea artton of unty corresondng to the collecton W 0 :0 φ 1, φ C0 ( 9 8 Q ), φ Cdam(Q ) 1 and φ (x) = 1 for every x R n \ Ω; see [10,. 170]. For each cube Q W we ck x Ω sothatdst(q, Ω) = dst(q,x ). Let (3.1) a = Tu(y)dH m (y), B n (x,r ) Ω where Tu B 1 n m ( Ω) and r =dam(q ). If Q / W we set a =0. Lcense or coyrght restrctons may aly to redstrbuton; see htt://

4 2376 PETTERI HARJULEHTO We defne an extenson oerator E : W 1, (Ω) W 1, (R n ) by settng { Eu(x) = u(x), for x Ω; =1 a φ (x), for x R n \ Ω. Note that the extenson oerator s the same for every. It s also easy to see that Eu C (R n \ Ω) and Eu s zero n the comlement of Q W 9 8 Q Ω. The extenson oerator s equvalent to countng the sum only over those φ wth Q W. For techncal reasons we have chosen a slghtly dfferent extenson oerator than Jonsson [4, Theorem 3,. 4]. For every cube we frst ck the nearest ont x from the boundary and then set the value of a ; see (3.1). Jonsson just sets a = B n (z,cr ) Ω Tu(y)dH m (y), where z s the center of the cube Q W 0, r =dam(q )andc>0saconstant Lemma. Let 1 <<. Assume that Ω R n s a bounded doman and ts boundary Ω s m-regular, n 1 m<n. If there exsts a bounded trace oerator T : W 1, (Ω) L ( Ω), then there exsts a constant C, 0 <C<, such that for every u W 1, (Ω). Eu L (R n ) C u W 1, (Ω) Proof. Let x 9 8 Q 0 for some Q 0 W. Hence we obtan Eu(x) = =1 a φ (x) C su a, where the suremum s taken over all those cubes Q Wfor whch 9 8 Q 9 8 Q 0. Snce Ω sm-regular and r =dam(q ), we obtan (3.3) su ( a su B n (x,r ) Ω C dam(q 0 ) m ( Tu dh m) 1 B n (x 0,Cr 0 ) Ω Tu dh m) 1. Here C>0saconstant so that B n (x,r ) B n (x 0,Cr 0 ) for every, 9 8 Q 9 8 Q 0. It s ossble to choose ths constant, because for every two touchng cubes the dstance of corresondng onts n boundary s comarable to the dameters of cubes. Note that W s a collecton of Whtney cubes wth dam(q ) < mn{a, dam(ω)} for every =1, 2,... Let N k be a number of cubes from the k th generaton n the Whtney decomoston; then r j 2 k. We wrte Q k, Lcense or coyrght restrctons may aly to redstrbuton; see htt://

5 TRACES AND SOBOLEV EXTENSION DOMAINS 2377 =1,...,N k, to denote the Whtney cubes from the k th generaton. There exsts an nteger k 0 such that Q k W f and only f k k 0. Ths mles N k R n \ Ω Eu dh n k=k 0 =1 9 8 Qk Eu dh n N k C2 k( m) k=k 0 =1 N k C2 k(n m) k=k 0 =1 C k=k 0 2 k(n m) C Tu L ( Ω). Ω B n (x,cr ) Ω B n (x,cr ) Ω Tu dh m Tu dh m Q k Tu dh m The second-to-last nequalty follows from the fact that no ont belongs to more than C(n) balls B n (x,cr ), =1,...,N k. To see that, let x Ω. Assume that the ont x belongs to the balls B n (x,cr ), =1, 2, 3,... Snce dst(q,x) C2 k and Q =2 kn for every =1, 2, 3,..., and nterors of Whtney ( cubes ) are n C2 dsjont, we obtan that the ont x cannot belong to more than k 2 = C n k balls B n (x,cr ). There exsts a bounded trace oerator, and hence we obtan Ths comletes the roof of Lemma 3.2. Eu L (R n ) C u W 1, (Ω) Lemma. Let 1 <<. Assume that Ω R n s a bounded doman and ts boundary Ω s m-regular, n 1 m<n. If there exsts a bounded trace oerator T : W 1, (Ω) B 1 n m ( Ω), then there exsts a constant C, 0 <C<, such that Eu L (R n \ Ω) C u W 1, (Ω) for every u W 1, (Ω). Proof. If Q and Q 0 are two Whtney cubes from W, wth 9 8 Q 9 8 Q 0, then dam(q 0 ) dam(q ) and furthermore x x 0 Cdam(Q 0 ), where dst(q 0, Ω) =dst(q 0,x 0 ), x 0 Ω, and dst(q, Ω) = dst(q,x ), x Ω. We obtan for these cubes Q and Q 0 that a a 0 = ( C B n (x,r ) Ω B n (x,r ) Ω B n (x 0,r 0 ) Ω ( r 2m 0 B n (x 0,Cr 0 ) Ω Tu(z)dH m (z) B n (x 0,r 0 ) Ω B n (x 0,r 0 ) Ω Tu(w)dH m (w) Tu(z) Tu(w) dh m (z)dh m (w) Tu(z) Tu(w) dh m (z)dh m (w) ) 1 ) 1. Lcense or coyrght restrctons may aly to redstrbuton; see htt://

6 2378 PETTERI HARJULEHTO The constant C does not deend on the cube Q 0. Assume that Q j W s a Whtney cube from the k th generaton; then r j 2 k. Let x Q j. Ths mles that Eu(x) = a φ (x) = (a a j ) φ (x) =1 snce φ (x) = 1 for every x R n \ Ω. We obtan Eu dh n C su a a j C dam(q j ) Q j Q j Q Q j C2 2km 2 k 2 kn Tu(z) Tu(w) dh m (w)dh m (z) B n (x j,cr j ) Ω C2 k(n 2m) B n (x j,cr j ) Ω B n (x j,r j ) Ω B n (x j,r j ) Ω =1 We wrte W = Q W Q. Ths mles N k Eu dh m Eu dh n W Q j C N k k=k 0 j=1 B n (x j,cr j ) Ω k=k 0 j=1 2 k(n 2m) B n (x j,r j ) Ω Tu(z) Tu(w) dh m (w)dh m (z). Tu(z) Tu(w) dh m (w)dh m (z). Every ont x Ω belongs only to a unformly bounded number of balls B n (x j,r j ) from the k th generaton n the Whtney decomoston. Ths yelds Eu dh n C W k=k 0 2 k(n 2m) Tu(z) Tu(w) dh m (w)dh m (z) z w <C2 k z w <C2 k Tu(z) Tu(w) = C dh m (w)dh m (z). 2 k( n+2m) k=k 0 Note that W s a collecton of Whtney cubes Q W 0 wth dam(q) mn{a, dam(ω)}. We choose that a s small enough to guarantee that C2 k 0 < 1. By [5, Lemma 5.3,. 166] we obtan (3.5) Eu dh n Tu(z) Tu(w) C z w n+2m dhm (w)dh m (z). W z w <1 Lcense or coyrght restrctons may aly to redstrbuton; see htt://

7 Let Q k 0 1,...,Qk 0 N TRACES AND SOBOLEV EXTENSION DOMAINS 2379 be the largest generaton of the Whtney cubes belongng to W. ) \ Note that dam(q k 0 ) deends on mn{a, dam(ω)}. WewrteA = W. Assume that x A and x 9 8 Qk 0. Ths yelds N Eu(x) = a φ (x) =1 C su a j j C su a j. j C dam(q k 0 ) ( N =1 9 8 Qk 0 Here the suremum s taken over all cubes 9 8 Qk 0 j 9 8 Qk 0. As n (3.3) we obtan ( su a C Tu dh m) 1 ; here r 0 =dam(q k 0 ). Ths mles Eu(x) dh n A B n (x,cr 0 ) Ω N =1 C Tu dh m. B n (x,cr 0 ) Ω Snce no ont n Ω belongs to more than C(n) balls B n (x,cr 0 )weobtan (3.6) Eu(x) dh n (x) C Tu L ( Ω). A In R n \ ( Ω W A) the functon Eu s zero. Snce there exsts a bounded trace oerator we obtan by (3.5) and (3.6) that Eu L (R n \ Ω) C u W 1, (Ω). Ths comletes the roof of Lemma 3.4. The extenson of u W 1, (Ω) s a C -functon n R n \ Ω. We are left to rove that the weak gradent of Eu exsts across the boundary. Lemma 3.7 follows easly from the constructon of the extenson oerator Lemma. Let 1 << and 0 <λ 1. Assume that Ω R n s a bounded doman and there exsts a trace oerator T to the boundary Ω so that Tu = u Ω for every u W 1, (Ω) C( Ω). Ifu W 1, (Ω) C 0,λ (Ω), theneu C 0,λ (R n ). The followng result s due to Haj lasz and Marto, [2, Lemma 11,. 237]: If a comact set K R n satsfes H n λ (K) =0,where0 λ<1, then for every, 1, W 1, (R n \ K) C 0,1 λ (R n ) W 1, (R n ). Snce Ω sm-regular, H n λ ( Ω) = 0 for every λ, 0 λ<n m, the general case follows by Lemmata 3.2, 3.4 and 3.7 when we assume that W 1, (Ω) C 0,1 λ (Ω) s dense n W 1, (Ω) for some λ, 0 λ<n m Theorem. Let 1 <<. Let Ω R n be a bounded doman so that ts boundary Ω s m-regular, n 1 m<n. Assume that W 1, (Ω) C 0,1 λ (Ω) s dense n W 1, (Ω) for some λ, 0 λ<n m. If there exsts a bounded lnear trace oerator T : W 1, (Ω) B 1 n m ( Ω), Lcense or coyrght restrctons may aly to redstrbuton; see htt://

8 2380 PETTERI HARJULEHTO so that Tu = u Ω for every u W 1, (Ω) C( Ω), then the doman Ω s a W 1, - extenson doman: there exsts a bounded lnear extenson oerator so that Eu Ω = u for every u W 1, (Ω). E : W 1, (Ω) W 1, (R n ), 3.9. Remark. If Ω R n s a bounded W 1, -extenson doman, >n 1, then H n ( Ω) = 0, [7, Theorem 6.5,. 28] and [3, Corollary 4.9,. 184]. Thus t s natural to assume that m s strctly less than n. Lschtz contnuous functons are dense n W 1, (Ω) f M 1, (Ω) = W 1, (Ω), [1, Theorem 5,. 408]. In artcular M 1, (Ω) = W 1, (Ω) f Ω s a W 1, -extenson doman, [1, Theorem 1,. 405]. The reverse does not hold; there exsts a bounded doman Ω R n wth M 1, (Ω) = W 1, (Ω), but Ω s not a W 1, -extenson doman, [9]. It s an oen roblem whether the condton about the densty of Hölder contnuous functons s necessary n Theorem 3.8. By the roof of Lemmata 3.2 and 3.4 we obtan the followng corollary Corollary. Let 1 <<. Assume that Ω R n s a bounded doman and ts boundary Ω s m-regular, n 1 m<n. If there exsts a bounded trace oerator T : W 1, (Ω) B 1 n m ( Ω), then there exsts a bounded lnear extenson oerator E : B 1 n m ( Ω) W 1, (Ω). 4. Characterzaton for Sobolev extenson domans Usng Theorem 3.8 we gve a characterzaton for Sobolev extenson domans. We need the followng trace theorem, whch s due to Jonsson Theorem ([4, Theorem 3,. 4], [6, Theorem 1,. 182]). Let 1 <<. Assume that Ω R n s a bounded doman and ts boundary Ω s m-regular, n 1 m<n. Then the oerator T : W 1, (R n ) B 1 n m ( Ω), Tu(x) = lm r 0 B n (x,r) u(z)dhn (z) H m -almost everywhere n Ω, s bounded and lnear. Assume that Ω R n s a bounded doman and ts boundary Ω sm-regular, n 1 m<n. We have to show that f Ω s a W 1, -extenson doman, then lm u(z)dh n (z) = lm r 0 r 0 Eu(z)dH n (z), B n (x,r) Ω B n (x,r) for every x Ω \ F,wthH m (F )=0. The roof s smlar to the roof n [11, Theorem 1,. 121] as soon as we know that Ω s n-regular. The case m = n 1 has been roved n [6, Prooston 2,. 206]. A bounded W 1, -extenson doman, n 1 <<, sn-regular by [7, Theorem 6.5,. 28] and [3, Corollary 4.9,. 184]. Lcense or coyrght restrctons may aly to redstrbuton; see htt://

9 TRACES AND SOBOLEV EXTENSION DOMAINS 2381 By Theorem 4.1 we obtan the followng result Corollary. Let n 1 <<. Assume that Ω R n s a bounded W 1, - extenson doman and ts boundary Ω s m-regular, n 1 m < n. Then there exsts a bounded lnear trace oerator T : W 1, (Ω) B 1 n m ( Ω), so that Tu(x) = lm r 0 B n (x,r) Ω u(z)dhn (z) H m -almost everywhere n Ω. By Theorem 3.8, Corollary 4.2 and Remark 3.9 we obtan the followng characterzaton for Sobolev extenson doman Theorem. Let n 1 <<. Assume that Ω R n s a bounded doman and ts boundary Ω s m-regular, n 1 m < n. Then the followng two condtons are equvalent: (1) The doman Ω s a W 1, -extenson doman: there exsts a bounded lnear extenson oerator E : W 1, (Ω) W 1, (R n ), such that Eu Ω = u for every u W 1, (Ω). (2) There exsts a bounded lnear trace oerator T : W 1, (Ω) B 1 n m ( Ω), such that Tu(x) = lm r 0 B n (x,r) Ω u(z)dhn (z) H m -almost everywhere n Ω, and Lschtz contnuous functons are dense n W 1, (Ω) Remark. By Theorem 3.8 the set of Lschtz contnuous functons n Theorem 4.3(2) can be relaced by the set W 1, (Ω) C 0,1 λ (Ω), where 0 λ<n m. Acknowledgments I thank P. Haj lasz for suggestng that I study ths toc, my teacher R. Hurr- Syrjänen for her helful gudance and the referee for many crucal comments and correctons. References [1] P. Haj lasz: Sobolev saces on an arbtrary metrc sace, Potental Anal. 5(4) (1996), MR (97f:46050) [2] P. Haj lasz and O. Marto: Traces of Sobolev functons on fractal tye sets and characterzaton of extenson domans, J. Funct. Anal. 143(1) (1997), MR (98d:46034) [3] D. A. Herron and P. Koskela: Unform, Sobolev extenson and quasconformal crcle domans, J. Anal. Math. 57 (1991), MR (94m:46059) [4] A. Jonsson: The trace of otentals on general sets, Ark. Mat. 17(1) (1979), MR (80:46029) [5] A. Jonsson and H. Walln: A Whtney extenson theorem n L and Besov saces, Ann. Inst. Fourer, Grenoble 28(1) (1978), MR (81c:46024) [6] A. Jonsson and H. Walln: Functon saces on subsets of R n, Harwood Academc Publsher, London, 1984, Mathematcal Reorts, Volume 2, Part 1. MR (87f:46056) [7] P. Koskela: Caacty extenson domans, Ann. Acad. Sc. Fenn. Ser. A I Math. Dssertatones 73 (1990), MR (91e:46041) [8] A. Kufner, O. John and S. Fučík: Functon saces, Noordhoff Internatonal Publshng, Leyden, MR (58:2189) Lcense or coyrght restrctons may aly to redstrbuton; see htt://

10 2382 PETTERI HARJULEHTO [9] A. S. Romanov: On a generalzaton of Sobolev saces, Sberan Math. J. 39(4) (1998), MR (99k:46059) [10] E. M. Sten: Sngular ntegrals and dfferentablty roertes of functons, Prnceton Unversty Press, Prnceton, N.J., MR (44:7280) [11] H. Walln: The trace to the boundary of Sobolev saces on a snowflake, Manuscrta Math. 73(2) (1991), MR (92k:46053) Deartment of Mathematcs and Statstcs, P.O. Box 68 (Gustav Hällströmn katu 2B), FIN Unversty of Helsnk, Fnland E-mal address: etter.harjulehto@helsnk.f Lcense or coyrght restrctons may aly to redstrbuton; see htt://