ON SHARP HIGHER ORDER SOBOLEV EMBEDDINGS

Transcription

1 Communicaion in Conemporary Mahemaic Vol. 6, No. 3 (24) c World Scienific Publihing Company ON SHARP HIGHER ORDER SOBOLEV EMBEDDINGS MARIO MILMAN Deparmen of Mahemaic, Florida Alanic Univeriy, Boca Raon, FL 33431, USA exrapol@bellouh.ne EVGENIY PUSTYLNIK Deparmen of Mahemaic, Technion Irael Iniue of Technology, Haifa 32, Irael evg@echunix.echnion.ac.il Received 3 December 22 Revied 5 Augu 23 Le Ω be an open domain in R n, le k N, p n. Uing a naural exenion of k he L(p, q) pace and a new form of he Pólya Szegö ymmerizaion principle, we exend he harp verion of he Sobolev embedding heorem: W k,p (Ω) L( np n kp, p) o he limiing value p = n. Thi reul exend a recen reul in [3] for he cae k k = 1. More generally, if Y i a r.i. pace aifying ome mild condiion, i i hown ha W k,y (Ω) Y n(, k) = {f : k/n (f () f ()) Y }. Moreover Y n(, k) i no larger (and in many cae eenially maller) han any r.i. pace X(Ω) uch ha W k,y (Ω) X(Ω). Thi reul exend, complemen, implifie and harpen recen reul in [1]. Keyword: Embedding; Sobolev; rearrangemen. Mahemaic Subjec Claificaion 2: 46E3, 26D1 1. Inroducion Le W k,p (Ω) denoe he uual Sobolev pace of funcion f defined on an open domain Ω R n, uch ha f and all i diribuional derivaive D α f, α k, are zero a Ω, and, moreover, uch ha D α f L p (Ω), α = k. The claical Sobolev embedding heorem aer ha if Ω an open domain in R n hen, W k,p (Ω) L q (Ω), 1 q = 1 p k n, 1 < p < n k. (1.1) The norm of he embedding blow up a p n k. In fac, if p = n k, we formally have q = and (1.1) i fale. Thu, in he limiing cae, i i neceary o go ouide he 495

2 496 M. Milman & E. Puylnik L q cale o find he correc arge pace. Indeed, i wa hown a by Trudinger [25] for k = 1, and Sricharz [23], for k > 1 (cf. alo [9] for furher hiorical reference and background), ha, if Ω <, we have W k, n k (Ω) L Φ nk (Ω), (1.2) where Φ n (x) i a Young funcion uch ha Φ n k n (x) n ex for large x. k k I i alo known ha neiher (1.1) nor (1.2) are harp. The harp form of (1.1) i provided by he O Neil Sein verion of he Sobolev embedding heorem (cf. [19]), which require he ue of he L(p, q) pace: W k,p (Ω) L(q, p)(ω), 1 q = 1 p k n, 1 < p < n k. (1.3) Again (1.3) fail when p = n k. Moivaed in par by (1.3), Hanon [12], and Brézi Wainger [6] improved on (1.2) and obained in he limiing cae p = n k, W k, n k where, for q > 1, H q (Ω) i he Lorenz pace defined by { ( Ω H q (Ω) = f : f f () H q(ω) = 1 + ln Ω (Ω) H n (Ω), (1.4) k ) q d } 1/q <. The reul i opimal among all r.i. pace; hi wa proved in [12] for Riez poenial and in [9] for Sobolev pace hemelve. In paricular, in [9] i i hown ha, for any r.i. pace X(Ω), W k, n k (Ω) X(Ω) H n (Ω) X(Ω). (1.5) I i cuomary o rea (1.3) and (1.4) (or (1.2)) a eparae problem wih heir correponding eparae proof. b In hi paper we hall how, exending he mehod developed in [3] for he cae k = 1, a unified mehod o prove he Sobolev embedding heorem and he correponding harp borderline cae. In fac, for he borderline cae, we acually improve on he claical reul ince our arge pace are rearrangemen invarian e which are ricly conained in he opimal pace decribed above. The wo main ingredien of our mehod are: (a) he ue of a very naural exenion of he L(p, q) pace recenly propoed in [3], c and (b) he ue of a new verion of he Pólya Szegö ymmerizaion principle ha i valid for higher order derivaive. The new mehod can be eaily generalized o give harp reul in he conex of Sobolev pace baed on general rearrangemen invarian pace. k a We have learned ha (1.2) wa alo obained by Yudovich [26]. In connecion wih (1.4) we hould alo menion heir connecion wih he capaciary inequaliie of [18] (cf. alo [17]). b The exponenial eimae (1.2) hough are uually derived a a conequence of a quanified form of (1.3) by exrapolaion (cf. [23]). c In paricular, hi allow u o rea he cae p = n, q = in a unified form wih he oher cae. k

3 On Sharp Higher Order Sobolev Embedding 497 To beer explain he ingredien of he proof we ar by recalling he L(, q) pace recenly inroduced in [3] in connecion wih he cae k = 1 of (1.3). We oberve ha if we formally le p n k in (1.3), and diregard he blow up conan, one i naurally led o conider he pace L(, n k ). However, i i well known, and eay o ee, ha wih he uual definiion L(, q) = {}. On he oher hand, imiaing Benne DeVore Sharpley [4], d we modify he definiion of he L(p, q) pace a follow. Le 1 < p, < q, le Ω be a domain in R n, and le M(Ω) be he e of meaurable funcion on Ω. We le L(p, q)(ω) = {f M(Ω) : f L(p,q) < }, wih { f L(p,q) = [(f () f ()) 1/p ] q d } 1/q, 1 < p, < q <, we ue he uual modificaion when q =. Thee pace make ene (and are no rivial) for 1 < p, < q. Moreover, he new pace acually coincide wih he claical L(p, q) pace e whenever 1 < p <, < q. In fac for funcion in he claical L(p, q) pace we have f { [(f () f ()) 1/p ] q d } 1/q { [f () 1/p ] q d } 1/q, 1 < p <, < q. The econd idea underlying our mehod i a uiable exenion of he Pólya Szegö ymmerizaion principle for higher order derivaive. I i eay o ee ha he andard form of hi principle, comparing he fir derivaive of a funcion f(x) and i non increaing rearrangemen f (), canno be generalized, even o econd derivaive, becaue here are infiniely many mooh funcion f(x) uch ha df d i no differeniable, even in he ene of diribuion. For example, he funcion f(x) = 1 + in x, < x < 3π 2, ha rearrangemen f () = (1 + co 2 )χ (,π)() + (1 + in )χ [π, 3π 2 ) (), hu df d ha a jump a he poin = π (a more deailed dicuion devoed o hi opic can be found in [7]). Neverhele, we hall how ha a uiable modificaion of Pólya Szegö hold for higher order derivaive. To ae our reul, we recall an inequaliy for radial pherically decreaing rearrangemen from [3] and [2]. Suppoe ha f i a mooh funcion uch ha f and f vanih a infiniy and le f denoe i radial pherically decreaing ymmeric rearrangemen. Then, here exi a univeral conan γ n uch ha 1/n (f () f ()) γ n f (). (1.6) d In heir pionering paper, hee auhor defined he L(, ) pace. e Thi juifie uing he ame noaion, L(p, q), for he new pace. f In wha follow, we ue he ymbol o denoe equivalence modulo conan, and he ymbol o indicae maller or equal modulo conan.

4 498 M. Milman & E. Puylnik Thi lead o 1/n (f () f ()) Y f Y, f C (R n ), (1.7) for any r.i. pace Y aifying he (P ) condiion (cf. Definiion 2.1 below). We will how g (cf. Theorem 3.4 below) ha, under mild condiion on a r.i. norm Y (Ω), 1 k < n, k/n (f () f ()) Y (Ω) k f Y (Ω), f C (Ω). (1.8) We prove ha condiion of hi ype are opimal by mean of reformulaing a neceary condiion for Sobolev embedding derived in [1] (cf. Theorem 3.6 below). More preciely we how ha if X(Ω) i a r.i. pace uch ha and Ω <, hen f X(Ω) k f Y (Ω) f X(Ω) c(ω) k/n (f () f ()) Y. (1.9) We are now ready o give our proof of he following harp form of he Sobolev embedding heorem. h Theorem 1.1. Le Ω be an open domain in R n, le 1 < p n k, 1 q = 1 p k n, wih he convenion ha q = if 1 q = (i.e. q = when p = k n ). Then wih W k,p (Ω) L(q, p)(ω), (1.1) f L(q,p) c k f L p, f C (Ω). (1.11) Moreover, he reul i opimal: if Ω <, hen for any r.i. pace X(Ω), 1 < p n k, ( ) np W k,p (Ω) X(Ω) L n kp, p X(Ω). Proof. A a conequence of Example 4.1 below i follow ha (1.8) hold wih Y = L p, 1 < p n k. Therefore (1.1) and (1.11) follow from he following rivial compuaion wih indice, ( ) np L n kp, p = {f : (f () f ()) k/n L p }, 1 < p n k, wih f L( np n kp,p) = (f () f ()) k/n L p. g The reaon we are able o obain a higher order reul eem o be conneced wih he formulaion of (1.6) and (1.7), which involve he quaniy 1/n (f () f ()) = 1 1/n (f ()). In he lieraure, auhor (cf. in paricular Taleni [24] or Kolyada [15]) ofen formulae he fir order eimae in erm of 1 1/n (f ()), which lead o an eimae ha i more difficul o ierae. h We hould noe here he conribuion of Maly Pick [16], ee Sec. 4.1 below.

5 On Sharp Higher Order Sobolev Embedding 499 Tha hi condiion i opimal now follow direcly from he previou calculaion and (1.9). To exend Theorem 1.1 o more general r.i. pace, i i herefore naural o urn he lef hand ide of (1.8) ino a definiion. Given a r.i. pace Y (Ω), le Y n (, k)(ω) = {f : k/n (f () f ()) Y (Ω)}, f Yn(,k) = k/n (f () f ()) Y. (1.12) The preceding dicuion lead o he following generalizaion of Theorem 1.1. Theorem 1.2. Le Ω be an open domain in R n, uppoe ha k < n. Le Y (Ω) be a r.i. pace aifying he condiion Q(k 1) (cf. Definiion 2.2 below) and (2.1) below. Then and W k,y (Ω) Y n (, k), (1.13) f Yn(,k) k f Y (Ω), f C (Ω). Moreover, if Ω <, and X(Ω) i a r.i. pace hen and W k,y (Ω) X(Ω) Y n (, k) X(Ω), f X f Yn(,k). Proof. If Y i a r.i. pace aifying he aumpion of he heorem hen, by Lemma 2.3 and Theorem 3.4 below, (1.8) hold. Thi i all we need o obain (1.13). The fac ha he Y n (, k) pace are opimal follow once again from (1.9). The la heorem obain a paricularly imple form if he pace Y n (, k) ielf i a r.i. pace. For example, if Y aifie he Q(k)-condiion (ee Definiion 2.2 below), i follow by Lemma 2.6 ha f Yn(,k) k/n f () Y. (1.14) Corollary 1.3. Le Ω be an open domain in R n, le Y (Ω) be a r.i. pace aifying he Q(k)-condiion. Then Y n (, k) i a r.i. pace wih norm provided by (1.14) and moreover, k/n f () Y f Yn(,k) f W k,y (Ω).

6 5 M. Milman & E. Puylnik Y n (, k) i he opimal arge pace for he Sobolev embedding W k,y (Ω) X(Ω) among all r.i. pace. The quai-normed pace defined by he quai-norm k/n f () Y i he opimal arge pace among he cla of quai-normed r.i. pace. Remark 1.4. The opimal arge pace for embedding of generalized Sobolev pace are decribed in he recen paper [1]. The decripion obained in [1] i indirec and doe no imply he previou Corollary. A verion of our Corollary 1.3 (wih ronger aumpion on Y and wihou a udy of opimal condiion) wa claimed much earlier in [14]. Unforunaely, he proof indicaed here i baed on wo incorrec argumen (a higher order Pólya Szegö principle a a mere ieraion of fir order reul, and a heorem on inerpolaion of r.i. pace ha wa laer hown o be fale). The re of he paper i organized a follow: in Sec. 2 we eablih ome baic fac abou r.i. pace ha are ued in Sec. 3 o eablih he main ymmerizaion eimae o which we have referred in he coure of he proof of he main heorem. In he final Sec. 4 we collec a few remark and open problem. 2. Preliminarie Le Ω R n be a domain, and le Y = Y (Ω) be a rearrangemen invarian pace (r.i. pace). Le k N and define he Sobolev pace W k,y (Ω) = {f : D α f Y, D α f vanihe on Ω, α k}, where D α f i he lengh of he vecor whoe componen are all he derivaive of f of order α. W k,y (Ω) i provided wih he eminorm f W k,y (Ω) = D a (f) Y. α =k and Le (cf. [1]) f = ( f,..., f ), f = x 1 x n n i=1 2 f x 2 i { k k/2 f for even k, f = ( (k 1)/2 )f for odd k. We hall uually formulae condiion on r.i. pace Y in erm of he Hardy operaor P f() = 1 f()d ; Qf() = f() d.

7 On Sharp Higher Order Sobolev Embedding 51 Recall ha a r.i. pace ha a repreenaion a a funcion pace on Y ˆ(, Ω ) uch ha f Y (Ω) = f Y ˆ(, Ω ). Moreover, ince he meaure pace will be alway clear from he conex i i convenien o drop he ha and ue he ame leer Y o indicae he differen verion of he pace Y ha we ue. We hall alo e our funcion equal o zero for x / Ω. Le (σ f)() = f( ),, >, and define he dilaion funcion d Y () by d Y () = (σ f)() Y = (σ f )() Y (ince (σ f) = (σ f )). Definiion 2.1. We hall ay ha Y aifie he (P )-condiion if P : Y (, ) Y (, ) define a bounded operaor (equivalenly Y aifie he (P )-condiion if and only if he upper Boyd index of Y i le han 1). In paricular, if Y aifie he (P )-condiion hen f Y i an equivalen norm on Y, f Y f Y. (2.1) Likewie we hall ay ha Y aifie he (Q)-condiion if Q : Y (, ) Y (, ) define a bounded operaor (equivalenly Y aifie he (Q)-condiion if and only if he lower Boyd index of Y i greaer han ). In wha follow we alo need o conider weighed norm inequaliie for Q wih power weigh. Thi lead o condiion on our pace. Definiion 2.2. Le k >, and le Y be a r.i. pace. We hall ay ha Y aifie he Q(k)-condiion if ( ) 1 d Q(k, Y ) = k/n d Y <. (2.2) 1 Noe ha if Y aifie he Q(k)-condiion for ome k hen i aifie a Q(b)- condiion for every b < k, including he Q() = Q-condiion. The following known reul (cf. [22, 17]) will be ueful in wha follow. Lemma 2.3. Le Ω be an open domain in R n, and le Y (Ω) be a r.i. pace aifying he (P ) and (Q)-condiion. Then f Y (Ω) D a (f) Y (Ω), f C (Ω), (2.3) α =2 hold wih conan of equivalence independen of f. Proof. We only need o remark ha he argumen in [22, 17] for L p exend o r.i. pace aifying he (P ) and (Q) condiion by a well known reul due o Boyd (cf. [5]).

8 52 M. Milman & E. Puylnik Example 2.4. In paricular if Ω i an open domain in R n, 1 < p <, hen f Lp (Ω) D a (f) Lp (Ω), f C (Ω). α =2 Lemma 2.5. If Y i a r.i. pace aifying he Q(k)-condiion, hen Q i bounded on he pace Y provided wih weigh k/n. More preciely, Proof. Le α = k n. We have k/n Qf() Y Q(k, Y ) k/n f() Y. (2.4) α Qf() = = 1 α d f() = Applying Minkowki inequaliy we obain α Qf() Y 1 1 f()() α α 1 d. 1 α d f() f()() α Y α 1 d d Y ( 1 ) α 1 d α f() Y. Lemma 2.6. Le Y be a r.i. pace aifying he Q(k)-condiion for ome k >. Then, for all meaurable funcion f wih f ( ) =, k/n (f () f ()) Y k/n f () Y Q(k, Y ) k/n (f () f ()) Y. (2.5) Proof. The fir inequaliy i rivial. To prove he econd inequaliy noe ha d d f () = f () f (). Therefore, by he fundamenal heorem of calculu, we have f () = (f () f ()) d = Q(f f )(). (2.6) The deired reul follow from Lemma 2.5. I i ueful o remark here ha, for he operaor P, he correponding weighed norm inequaliie for power weigh are auomaically rue. Lemma 2.7. Le Y be a r.i. pace. Then, for any α > α P (f)() Y 1 α α f() Y. Proof. Compuing d Y () for Y = L 1, and Y = L, we find, by inerpolaion, ha for any r.i. pace Y d Y () max{1, }

9 On Sharp Higher Order Sobolev Embedding 53 (cf. [5] for he deail). Since P i a poiive operaor we may uppoe ha f. Now, Thu, a deired. α P f() = 1 α P f() Y f() α d = 1 1 d Y ( 1 1 f()() α α d. ) α d α f() Y α 1 d α f() Y, 3. Symmerizaion Inequaliie for Higher Order Sobolev Space The main ool for our analyi i he following reul from [3]. Lemma 3.1. Suppoe ha Y i a r.i. pace aifying a (P )-condiion. Then for all mooh funcion f uch ha f ( ) =, 1/n (f () f ()) Y f Y. (3.1) Proof. The proof we give i he ame a he one given in [3] for Y = L p. However, i i imporan for our developmen o provide he complee deail here. Recall ha from Lemma 1 in [3] we have he poinwie inequaliy 1/n (f () f ()) f (), (3.2) where f (x) = f (c n x n ), denoe he radial pherically ymmerically decreaing rearrangemen of f, c n = meaure of he uni ball. Recall alo ha he Pólya Szegö ymmerizaion principle hold for r.i. pace (cf. [11, 13], a more recen reference i [8]): f Y f Y. (3.3) Applying he Y norm o (3.2), and uing ucceively he (P )-condiion and (3.3), we obain he deired reul. Our main reul in hi ecion i he higher order verion of Lemma 3.1. The fir ep of he inducion proce ha lead o higher order eimae i provided by he nex reul. Theorem 3.2. Le Ω be a domain in R n, and le Y (Ω) be a r.i. pace aifying he (P ) and Q(1) condiion. Then for all mooh funcion f uch ha f ( ) = f ( ) =, 2/n (f () f ()) Y f Y. (3.4)

10 54 M. Milman & E. Puylnik Proof. We hall prove below he elemenary eimae f D 2 f. (3.5) Applying Lemma 3.1 o f and combining wih (3.5), we obain 1/n ( f () f ()) Y f Y D 2 f Y. Therefore combining wih (2.3) we ge 1/n ( f () f ()) Y f Y. Combining he previou inequaliy wih Lemma 2.6 we obain 1/n f () Y f Y. By he generalized Pólya Szegö ymmerizaion principle (3.3) (applied o he r.i. norm defined by g B = 1/n g () Y ) we ee ha 1/n f () Y 1/n f () Y f Y. (3.6) Combining he la inequaliy wih he poinwie inequaliy (3.2) we find 2/n (f () f ()) Y 1/n f () Y. Inering he la eimae in (3.6) give u he deired eimae 2/n (f () f ()) Y f Y. I remain o prove (3.5). Uing ubindexe o indicae parial derivaive, we have g j = 1 g g = 1 g n g i g ij i=1 ( n n ) 2 g i g ij Therefore, by Cauchy Schwarz applied o inner um, we have j=1 i=1 1/2.

11 g 1 g = n i,j=1 = D 2 g. On Sharp Higher Order Sobolev Embedding 55 n n n gi 2 gij 2 i=1 j=1 i=1 g 2 ij Remark 3.3. The previou heorem could be conidered a a form of he Pólya Szegö principle for he Laplacian. A higher order verion of Theorem 3.2 can now be obained by ieraion. The main problem here i o check ha all requiremen o implemen he ieraion are fulfilled. Theorem 3.4. Le Ω be an open domain in R n, le k N, k < n, and le Y (Ω) be a r.i. pace aifying he (P ) and Q(k 1)-condiion. Then for all mooh funcion f uch ha m f ( ) =, m =, 1,..., k 1, he following inequaliy hold 1/2 k/n (f () f ()) Y k f Y. (3.7) Proof. Lemma 3.1 and Theorem 3.2 prove he cae k = 1, and k = 2. For he general cae we now proceed by finie inducion. Inducion urn ou o be he ame for odd or even number k. Le m be an arbirary number uch ha 2 < m k, m k (mod 2). Suppoe ha (3.7) i rue for k = m 2. Suppoe ha g i any funcion uch ha k g Y <, and le f = g. Applying (3.7) for k = m 2, we ge 1/2 (m 2)/n ( g () g ()) Y m 2 g Y. Combining wih Lemma 2.6 and he definiion of m give (m 2)/n ( g) () Y m g Y. (3.8) Conider now he r.i. pace B defined by he norm h B = (m 2)/n h () Y. By compuaion i i readily een ha d B () = d Y () (m 2)/n. Therefore B aifie he Q(1)-condiion. Moreover, by Lemma 2.3, (2.3) hold in B norm. Conequenly we may now apply Theorem 3.2 o derive 2/n (g () g ()) B g B = (m 2)/n ( g) Y. (3.9) On he oher hand, uing he propery uv Y 2 u v Y, which i valid for any r.i. pace Y, we derive h B (m 2)/n h () Y 1 2 (m 2)/n h() Y.

12 56 M. Milman & E. Puylnik Thu 2/n (g () g ()) B 1 2 m/n (g () g ()) Y. Combining he la eimae wih (3.8) and (3.9) yield m/n (g () g ()) Y m g Y, for any admiible m. I remain o pu m = k. Example 3.5. Le Ω be an open domain in R n, and le 1 < p < aifie all he hypohee of Theorem 3.4. n k 1. Then Lp (Ω) Proof. We give he deail for he convenience of he reader. By Hardy inequaliy we ee ha L p (Ω) aifie he (P )-condiion. Moreover ince d Y () = 1/p we ee ha for 1 < p < n k 1, Q(k, Y ) = 1 (k 1)/n 1/p d < proving ha L p alo aifie he Q(k 1) condiion. In paricular noe ha we may ake p = n k. We end hi ecion wih a reul ha how ha our condiion are be poible. For hi purpoe we recall a reul of [1] (ee p. 322, formulae (3.11), and p. 324, (3.16)) which ae ha: if X(Ω) i a r.i. pace, k < n, hen f X(Ω) k f Y (Ω), for all f C k (Ω), implie ha Ω f() k n d f Y, for all f. (3.1) X A a conequence we prove Theorem 3.6. Suppoe ha k < n, Ω <, and X(Ω) i a r.i. pace uch ha f X(Ω) k f Y (Ω), for all f C (Ω). Then, Y n (, k)(ω) X(Ω), (3.11) and moreover here exi a conan c = c( Ω, X) uch ha f X(, Ω ) c f Yn(,k).

13 On Sharp Higher Order Sobolev Embedding 57 Proof. By he fundamenal heorem of Calculu we have f () = = Ω (f () f ()) d (f () f ()) d + Ω f () d Therefore, = Ω (f () f ()) d + 1 Ω Ω f ()d. f X(, Ω ) f X(, Ω ) Ω (f () f ()) d + 1 Ω X(, Ω ) Ω f ()d χ Ω X(, Ω ). (3.12) I remain o eimae he fir erm in he previou inequaliy. Replacing f by k/n (f () f ()) in (3.1) we find Ω (f () f ()) d k/n (f () f ()) Y. X Therefore, inering he la eimae in (3.12), we find and (3.11) follow. f X(, Ω ) f Yn(,k) + 1 Ω Ω f ()d χ Ω X(, Ω ), 4. Final Remark 4.1. Equivalen condiion In [16] Maly Pick inroduced he pace { MP n (Ω) = f : f (/2) f () L n ((, Ω ), d }, and howed wih ad-hoc mehod ha for Ω <, W 1,n (Ω) MP n (Ω) H n (Ω) (4.1) (ee alo [15, Lemma 5.1]). In [3] i wa proved ha MP n (Ω) = L(, n)(ω). (4.2) For pk = n he fir half of Theorem 1.1 give ( W k,p (Ω) L, n ) (Ω). k

14 58 M. Milman & E. Puylnik Therefore we could have ued he econd half of (4.1) combined wih (4.2) and (1.5) o prove ha he L(, n k ) pace are opimal. Likewie, ince i i alo hown independenly in [3] ha for Ω <, q > 1, L(, q)(ω) H q (Ω), we have ill anoher mehod o prove ha hee pace are opimal. In [16] he proof of he fac ha MP q (Ω) H q (Ω) (and herefore by (4.2) ha L(, q)(ω) H q (Ω)) i indirec. The auhor how ha L(, q)(ω) i no a linear pace. We prefer o give here a imple direc proof. To hi end we now exhibi f H q (Ω)\L(, q)(ω). Le f() = Σ i=1 χ (,c i)(), wih c i = e 2i. Then a compuaion how ha Therefore, for any q (1, ), 1 f () f () = 1 c i χ (ci, )(). i=1 (f () f ()) q d = 1 q ln e 1 i=1 1 i=1 c i c q i χ (c i,1)() d q q ci d q+1 (1 c q i ) =. i=1 On he oher hand, by Hardy inequaliy, ( 1 ( f ) q )1/q () d f Hq ( ci i=1 = 1 q 1 ( ln 1 ) ) q 1/q d 2 i(q 1)/q <. i=1 More generally, he mehod of proof of (4.2) given in [3] can be eaily exended o he pace Y n (, k) inroduced in hi paper o ha we can readily how ha ( ) f Yn(,k) (f k/n f ()). 2 The inereing paper by Y. Sagher and P. Shvarman [21] (ee alo he reference herein) conain an exenive comparion of he expreion f () f () and f ( 2 ) f (), and oher analogou expreion obained uing Peere K and E-funcional. Y

15 On Sharp Higher Order Sobolev Embedding Some queion In hi ecion we indicae ome queion and open problem raied by our reaearch for hi paper. I i inereing o oberve ha he condiion of our embedding heorem allow u o decribe he r.i. e conaining W k,p even for p bigger han he criical value n k,p k. A i well known in hi cae, W L. Our rearrangemen invarian e are conained in L, which i a omewha urpriing propery, ince for finie meaure pace L i he malle rearrangemen invarian pace. I could be of inere o udy hi phenomena in deail. We now give an example of a reul in hi direcion Example 4.1. Le n k < p < n k 1, f W k,p (Ω). Suppoe ha f ( 2 ) f () i a.e. equivalen o a monoone funcion on (, 1). Then f ( 2 lim ) f () =. k/n 1/p The reaon we were able o improve, in ome cae, well known opimal embedding heorem, i he fac ha he pace Y n (, k) are only r.i. e. I i herefore of inere o ak if he Y n (, k) pace are he opimal arge for he Sobolev embedding heorem among all r.i. e. In paricular, we conjecure ha for pk = n we have ( r.i. hull(w k,n (Ω)) = L, n ) (Ω). k Here he rearrangemen invarian hull of W k,p (Ω) = {f M(Ω) : g W k,p (Ω).. g = f }. In hi repec we refer he reader o [4] where i i hown ha for a cube Q, r.i. hull BMO(Q)) = L(, )(Q). I could alo be of inere o conider he modified rearrangemen invarian hull, a defined by Neruov [2] modified r.i. hull(w k,n (Ω)) = {f : g W k,p (Ω).. f g }. A cloely conneced problem i o give neceary and ufficien condiion on Y for he pace Y n (, k) o be a r.i. pace. I follow from our reul ha a ufficien condiion i for Y o aify he Q(k)-condiion. We conjecure ha he condiion i neceary a well. Thi conjecure i obviouly rue for L(p, q) pace and can be readily verified for oher pace udied in [9]. Finally we would like o ugge ha he Y n (, k) pace inroduced in hi paper hould alo be of inere in inerpolaion heory (cf. [21].) Acknowledgmen We hank he referee for hi/her criical reading of he paper and helpful uggeion. We are hankful o A. Cianchi for ueful informaion. Thi paper i in final form and no verion of i will be ubmied for publicaion elewhere.

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17 On Sharp Higher Order Sobolev Embedding 511 [25] N. S. Trudinger, On imbedding ino Orlicz pace and ome applicaion, J. Mah. Mech. 17 (1967) [26] V. Yudovich, On cerain eimae conneced wih inegral operaor and oluion of ellipic equaion, Dokl. Akad. Nauk SSSR 138(4) (1961) (Englih ranlaion: Sovie Mah. 2(3) (1961) )