A NOTE ON EXTREME CASES OF SOBOLEV EMBEDDINGS. Anisotropic spaces, Embeddings, Sobolev spaces

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1 A NOTE ON EXTREME CASES OF SOBOLEV EMBEDDINGS F.J. PÉREZ Absrac. We sudy he spaces of funcions on R n for which he generalized parial derivaives D r k k f exis and belong o differen Lorenz spaces L p k,s k. For his kind of funcions we prove a sharp version of he exreme case of he Sobolev embedding heorem using L(, s) spaces. Anisoropic spaces, Embeddings, Sobolev spaces. Inroducion In his paper we consider funcions f on R n wih generalized parial derivaives D r k k f r k f x r (r k k N). k Our main objecive is o obain an exreme case of a Sobolev ype inequaliy for hese funcions. More precisely, we wan o generalize he embedding W r n/r(r n ) L(, n/r)(r n ) (r, n N; r n) (Milman-Pusylnik [6], Basero-Milman-Ruiz [2] for r = ) o he case where he parial derivaives D r k k f of differen orders belong o differen Lorenz spaces L p k,s k. In order o inroduce he problem we recall some basic facs and review he lieraure. Le n, r N, p <. The Sobolev space Wp r (R n ) is he class of funcions f L p (R n ) wih all he generalized derivaives of order r belonging o L p (R n ). The classical Sobolev embedding heorem says ha if p < n/r hen Wp r (R n ) L q (R n ) q = np n rp. This heorem is well known and has been exensively considered in he lieraure. In his paper we deal wih he exreme case p = n/r (or Thanks o V.I. Kolyada for his useful ideas and suggesions. Research suppored in par by gran BMF C3-3 of he DGI, Spain. Mahemaics Subjec Classificaion (2): Primary 46E35, 46E3.

2 2 F.J. PÉREZ equivalenly, q = ). If = p = n/r i is known ha (see [4, ], [9]) W n (R n ) L (R n ). However, i is easy o see ha for < p = n/r, he funcions in W r n/r (Rn ) need no o be bounded. Many auhors have sudied which kind of embedding holds in his case. Hansson [6], and independenly and by differen mehods Brézis and Wainger [5], proved ha if Ω is an open domain in R n (n > ) wih Ω <, W n(ω) H n (Ω), () where W n(ω) is he closure of C (Ω) in Wn and [ ( ) n ] /n Ω f (s) ds H n (Ω) = {f : f Hn(Ω) = < }. + log Ω s s Moreover, Hansson [6] showed ha H n (Ω) is he opimal arge space in he class of rearrangemen invarian spaces. However, his resul can be improved in he following sense. Kolyada [, Lemma 5.](see also [9, p.7]) proved he inequaliy f () f (2) c /n ( f ) () >. (2) Basero,Milman and Ruiz (see [2, Remark (2.3)] showed ha f () f () c /n ( f ) () >. (3) Inequaliies (2) and (3) are equivalen (see Remark ). In [5,, 2] spaces relaed o inequaliy (3) were inroduced and sudied. I follows immediaely from (3) ha he Sobolev space w n, (R n ) = {f : f weak-l n (R n )} is conained in he Benne-De Vore-Sharpley space weak-l (R n ) = {f : f weak-l (R n ) = sup{f () f ()} < }. > Tha is (cf. []), w n, (R n ) weak-l (R n ). In [2], for q >, he (non linear) spaces L(, q)(r n ) are defined as he se of funcions f on R n such ha f L(,q) = [f () f ()] q d ) /q <. weak-l is no a linear space and. weak-l is no a norm.

3 A NOTE ON EXTREME CASES OF SOBOLEV EMBEDDINGS 3 The following sric inclusions hold for < p < q <, L = L(, ) L(, p) L(, q) weak-l. I follows from (3) ha (see [2]) W n(r n ) L(, n)(r n ). (4) Equivalen saemens had been proved wih differen mehods in [2, equaion (3.22)] and in [5]. In [2] i is shown how (4) improves (). From he recen resuls in [6], he embedding for derivaives of higher order W r n/r(r n ) L(, n/r)(r n ) (5) is derived 2. Now we can specify our objecive: find an embedding of ype (5) for funcions wih parial derivaives of differen orders. Exisence of mixed derivaives is no assumed. Le s explain more specifically which is he form of he embedding we are looking for. We consider he space of funcions f such ha he generalized parial derivaives D r k k f (k =,..., n) belong o differen spaces L p k. The corresponding classes of funcions naurally appear in he embedding heory as well as in applicaions. The mos exended heory of hese classes is conained in he monograph [4]. Furhermore, in his paper we allow he derivaives o belong o differen Lorenz spaces L p k,s k(r n ) (where p k, s k < and s k =, if p k = ). The use of Lorenz ype limiaions on he derivaives can be crucial in he esimaes of Fourier ransforms [, 3, 8], condiions for differeniabiliy [2], and embedding heorems [2]. Then our main problem is o find an embedding of ype (5) for funcions wih he derivaives D r k k f Lp k,s k(r n ) (k =,..., n). The answer is given a he following inequaliy, proved in Theorem below f L(,s)(R n ) c k= D r k k f p k,s k p = n/r, where r, p and s are suiable averages of he r k s, p k s and s k s o be defined laer, ha are frequenly used in his conex. Noe ha he mehods from [2, 5, 2] canno be used in our case since hey work for r = = r n = only. Moreover, he reasoning in [6] is no applicable because our r k s can be differen, and so, he exisence of mixed derivaives is no assumed. Thus, no inducion over 2 Noe ha if f W r n/r (Rn ), hen f W r n/r (Rn ), and, by he well known embedding of Sobolev spaces ino Lorenz spaces, f L n,n/r (R n ). From his and (3), he embedding (5) follows also.

4 4 F.J. PÉREZ he order of he derivaives is possible. Insead, our approach is based on embeddings of Besov spaces and he ransiiviy of embeddings, ogeher wih resuls from [4]. 2. Some definiions Le S (R n ) be he class of all measurable, almos everywhere finie funcions f on R n, such ha for each y >, λ f (y) {x R n : f(x) > y} <. The non-increasing rearrangemen of f S (R n ) is a non-increasing funcion f on R + (, + ) ha is equimeasurable wih f. The rearrangemen f can be defined by he equaliy f () = sup inf f(x), < <. x E E = The following relaion holds [3, Ch.2] f(x) dx = sup E = In wha follows we se E f () = f (u)du. f (u)du. Assume ha < q, p <. A funcion f S (R n ) belongs o he Lorenz space L q,p (R n ) if ( f q,p /q f () ) ) /p p d <. We have he inequaliy [3, p.27] f q,s c f q,p ( < p < s < ), so ha L q,p L q,s for p < s. In paricular, for < p q L q,p L q,q L q. Le f be a measurable funcion on R n. Le j {,..., n}. We define he difference j (h)f(x) f(x + he j ) f(x), h R, where e j is he uni coordinae vecor. If r >, inducively, r j(h)f(x) j (h)[ r j (h)f](x). Le q <. The funcion ω j (f; δ) q = sup j (h)f q δ >, <h<δ

5 A NOTE ON EXTREME CASES OF SOBOLEV EMBEDDINGS 5 is called he modulus of coninuiy of f wih respec o he variable x j in he meric L q. For p < we denoe L p L p (R +, du/u); se also L L (R + )(see [7]). 3. Auxiliary lemmas Lemma. Le α >, θ. Le ψ() be a funcion on R +, nonnegaive, non-decreasing such ha α ψ() L θ. Then, for any δ > here exiss a funcion ϕ on R + coninuously differeniable such ha: i) ψ() ϕ(), ii) ϕ() α δ decreases and ϕ() α+δ increases, iii) α ϕ() L θ c α ψ() L θ where c is a consan ha only depends on δ and α. The proof follows he scheme of Lemma 2. of [4], so we don include i here. Le < α j <, θ j for j =,..., n. Denoe ( ) ( ) α = n ; θ = n. (6) α j α α j θ j j= Lemma 2. Le n N, < α j < and θ j for j =,..., n. Se α and θ as in (6). Se also j= < δ 2 min j n {α j}. For j =,..., n, le ϕ j be posiive and coninuously differeniable funcions on R +, saisfying ϕ j () α j L θ j. Suppose in addiion ha ϕ j () αj+δ increases and ϕ j () αj δ decreases. Then here exis posiive funcions δ,..., δ n on R + such ha n δ j () = ( > ); j= and for σ() n j= ϕ j(δ j ()) i holds ha /θ n [ αθ n σ() d) θ c α j ] α nα ϕ j () L θ j j, j= where c is a consan ha only depends on δ, r j and n. Proof. Le < a < b be wo posiive consans. A posiive funcion g on R + is said o be of power ype (a, b) if g() a and g() b. I is easy o see ha if g is of power ype (a, b), hen is inverse g exiss on R +, and i is of power ype (/b, /a).

6 6 F.J. PÉREZ Also, if g is of power ype (a, b ) and g 2 is of power ype (a 2, b 2 ), hen g g 2 is of power ype (a + a 2, b + b 2 ) and g g 2 is of power ype (a a 2, b b 2 ). Noe ha he funcions ϕ j are of power ype (α j δ, α j + δ). Se now Define for > n Φ(s) = s j= ϕ j (ϕ n (s)), s >. (7) δ n () = Φ (), δ j () = ϕ j (ϕ n (δ n ())) j =,..., n. (8) Of course, for j =,..., n, he funcions δ j are of power ype for some (a j, b j ). From his i follows ha Moreover, by (7) a j δ j() δ j () b j. (9) n δ j () = Φ(δ n ()) =. j= And by (8) ( i, j n) ϕ i (δ i ()) = ϕ j (δ j ()), () which implies ha σ() = nϕ j (δ j ()) (j =,..., n). Finally, using (), Hölder s inequaliy wih exponens nα jθ j, (9), and θα he change of variable δ j () = z we ge ( ( /θ n [ ] ) θα /θ αθ n σ() d) θ ϕj (δ j ()) nα j d = n δ j () α j n ( n [ ϕj (δ j ()) j= δ j () α j ] ) θj θ d j α nα j j= c n ( α j ϕ j () L θ j ) α/nα j. Lemma 3. Le n N, α,..., α n >, θ,..., θ n. Se α and θ as in (6). Then, for any q < and any f S (R n ), here exiss a non negaive funcion σ() on R + such ha and j= f () f (2) + /q σ() ( > ) () αθ/n σ() θ d ) /θ c α j ω j (f; ) q L θ j, (2) j=

7 A NOTE ON EXTREME CASES OF SOBOLEV EMBEDDINGS 7 where c is a consan ha doesn depend on f. Proof. Wihou loss of generaliy we can suppose ha he righ hand side of (2) is finie. As f S (R n ), hen he ω j (f; ) q are posiive funcions 3. Applying Lemma o he above menioned modulus wih δ = 2 min{α j} we conclude ha here exis coninuously differeniable funcions ϕ j () on R + such ha and < ω j (f; ) q ϕ j () ( > ), (3) ϕ j () α j δ, ϕ j () α j+δ, α j ϕ j () L θ j c α j ω j (f; ) q L θ j. (4) Now, noe ha he funcions ϕ j saisfy he condiions of Lemma 2. Hence, here exis posiive funcions δ,..., δ n on R + such ha n j= δ j() = and for σ() n j= ϕ j(δ j ()) he following inequaliy holds αθ/n σ() θ d ) /θ n c ( α j ϕ j () L θ j ) α nα j. (5) j= Las, using Lemma.3 of [2] we have f () f (2) + c /q ω j (f; δ j ()) q. From his and (3) we ge (). The esimae (2) is he consequence of (5), (4) and he inequaliy beween arihmeic and geomeric averages. j= 4. Embedding heorem Theorem. Le n 2, r j N, p j, s j < for j =,..., n and s j = if p j =. Se ( ) ( ) r = n, p = n ( ), s = n. r j= j r p j= j r j r s j= j r j Assume ha p = n/r. Then, for all f S (R n ) ha possess weak derivaives D r j j f Lp j,s j (R n ) (j =,..., n), i holds ha [f () f ()] s d ) /s c D r j j f p j,s j. 3 oherwise, since f S (R n ), we have ha f and he resul is obvious. j=

8 8 F.J. PÉREZ Proof. We fix q > max j n {p j r j }. Now we apply Theorem 3. of [4] wih he parameers q j ha are choosen in he said heorem aking he value of he q ha we have jus fixed. By his fac (i.e. q j = q, j =,..., n) and he assumpion p = n/r, i follows ha he parameers ρ j, κ j, α j and θ j appearing in ha heorem are ρ j = p j, j= κ j = p j q, α j = p jr j q, = κ j θ j s Thus we ge [h α j r j j (h)f q] θ dh ) /θj j c h k= + κ j s j. (6) D r k k f p k,s k. (7) Noe ha he lef hand side of (7) is a sum of Besov ype seminorms. Then, [7, Chap.4] as < α j <, α j ω j (f; ) q L θ j c [h α j r j j (h)f q] θ dh ) /θj j. (8) h By Lemma 3 we have [f () f (2)] θ d and ) /θ αθ/n σ() θ d ) /θ c θ/q σ() θ d ) /θ (9) α j ω j (f; ) q L θ j, (2) where (by (6), (6) and p = n/r 4 ). The value of α is ( ) ( ) q α = n = n = n α i p i r i q. i= So, he righ hand side of (9) and he lef hand side of (2) coincide. Moreover, from (6) and (6), we have ( ) θ = n ( [ κj = q + κ ] ) j = s. α θ j α j sα j s j α j Finally, j= f () f () = 2 4 which is he same as n j= p jr j =. j= j= i= f (u)du 2 /2 f (2u)du = (f (u) f (2u))du. (2)

9 A NOTE ON EXTREME CASES OF SOBOLEV EMBEDDINGS 9 And from his and Hardy s inequaliy [3, pg.24], [f () f ()] s d ) /s c [f () f (2)] s d ) /s. (22) Puing ogeher (22), (9), (2), (8), (7) we obain he resul. Remark. In his paragraph we show ha esimaes (3) and (2) are equivalen. I is easy o see ha f (/2) f () 2(f () f ()). (23) So, (3) implies (2). Noe ha (2) is easily proved oo. From (2), using (2) and he fac ha for any g S (R n ) g () increases in, he esimae (3) follows. Inequaliy (23) appears in [2, Theorem 4.]. Noe also ha inequaliies equivalen o (2) are used in [8, Lemma 5] and [2, Theorem 4.]. Remark 2. In he case r j = r, s j = p j = p ( j n), Theorem implies he embedding (5). References [] Basero, J., Milman, M., and Ruiz, F.: On he connecion beween weighed norm inequaliies, commuaors and real inerpolaion, Seminario García de Galdeano 8 (996). [2] Basero, J., Milman, M., and Ruiz, F.: A noe on L(, q) spaces and Sobolev embeddings, Indiana Univ. Mah. J. 52(5) (23), [3] Benne, C., and Sharpley, R.: Inerpolaion of Operaors, Academic Press, 988. [4] Besov, O.V., Il in, V.P., and Nikol skiĭ, S.M.: Inegral Represenaion of Funcions and Imbedding Theorems, vol. 2, Winson, Washingon D.C., Halsed, New York Torono London, 978. [5] Brézis, H. and Wainger, S.: A noe on limiing cases of Sobolev embeddings and convoluion inequaliies, Comm. Parial Diff. Eq. 5, No. 7 (98), [6] Hansson, K.: Imbedding heorems of Sobolev ype in poenial heory, Mah. Scand. 45 (979), [7] Herz, C.: Lipschiz spaces and Bernsein s heorem of absoluely convergen Fourier ransform, J. Mah. Mech. 8 No. 8 (968), [8] Kolyada, V.I.: On imbedding in classes φ(l), Izv. Akad. Nauk SSSR Ser. Ma. 39 No. 2 (975) , 472; English ransl, in Mah. USSR Izvesija 9 No. 2 (975) [9] Kolyada, V.I.: Esimaes of rearrangemens and imbedding heorems, Ma. Sb. (N.S.) 36(78) No. (988), 3 23; English ransl. in Mah. USSR-Sb. 64 No. (989) 2. [] Kolyada, V.I.: Rearrangemens of funcions and embedding heorems, Uspehi maem. nauk 44 No. 5 (989), 6 95; English ransl. in Russian Mah. Surveys 44 No. 5 (989), 73 8.

10 F.J. PÉREZ [] Kolyada, V.I.: Esimaes of Fourier ransforms in Sobolev spaces, Sudia Mah. 25 No. (997), [2] Kolyada, V.I.: Rearrangemens of funcions and embedding of anisoropic spaces of he Sobolev ype, Eas J. on Approximaions 4 No. 2 (998), 99. [3] Kolyada, V.I.: Embeddings of fracional Sobolev spaces and esimaes of Fourier ransforms, Ma. Sb. 92 No. 7 (2), 5 72; English ransl. in Sbornik: Mahemaics 92 No. 7 (2), 979. [4] Kolyada, V.I., and Pérez, F.J.: Esimaes of difference norms for funcions in anisoropic Sobolev spaces, Ma. Nachr. 267, (24). [5] Maly, J. and Pick, L.: An elemenary proof of sharp Sobolev embeddings, Proc. Amer. Mah. Soc. 3 (22), [6] Milman, M. and Pusylnik, E.: On sharp higher order Sobolev embeddings, Comm. Con. Mah. 6 No. 3 (24), [7] Nikol skiĭ, S.M.: Approximaion of Funcions of Several Variables and Imbedding Theorems, Springer Verlag, Berlin Heidelberg New York, 975. [8] Pelczyński, A., and Wojciechowski, M.: Molecular decomposiions and embedding heorems for vecor-valued Sobolev spaces wih gradien norm, Sudia Mah. 7 (993), 6. [9] Pelczyński, A., and Wojciechowski, M.: Sobolev Spaces, Handbook of he Geomery of Banach Spaces, Vol. 2, (W.B. Johnson and J. Lindensrauss Eds), Elsevier Science, 23, p [2] Sein, E.M.: The differeniabiliy of funcions in R n, Annals of Mahemaics 3 (98), [2] Tarar, L.: Imbedding heorems of Sobolev spaces ino Lorenz spaces, Bolleino U.M.I. 8 No. B (998), Francisco Javier Pérez Lázaro, Deparameno de Maemáicas e Informáica, Universidad Pública de Navarra, Campus de Arrosadía, 36 Pamplona, Spain., francisco.perez@unavarra.es