On limiting embeddings of Besov spaces

Transcription

1 STUDIA MATHEMATICA 171 1) 25) On limiing embeddings of Besov spaces by V. I.Kolyada Karlsad) and A. K.Lerner Rama Gan) Absrac. We invesigae he classical embedding B s B s n1/p 1/q) q,θ. The sharp asympoic behavior as s 1 of he operaor norm of his embedding is fond. In pariclar, or resl yields a refinemen of he Borgain, Brezis and Mironesc heorem concerning an analogos problem for he Sobolev-ype embedding. We also give a differen, elemenary proof of he laer heorem. 1. Inrodcion. For < s < 1 and 1 p, θ < he Besov space B s Rn ) consiss of all fncions f L p R n ) sch ha f b s s ω p f, )) θ d ) 1/θ <, where ω p f, ) is he L p -modls of coniniy of f. Se also B s p B s p,p. One of he basic ineqaliies of embedding heory, which goes back o he classical Hardy Lilewood heorem on Lipschiz classes, says ha if γ n1/p 1/q) < s, hen see [11, 6.3]) 1) f b s γ q,θ c f b s 1 p < q <, 1 θ < ). Noe ha a simple argmen shows ha f b s as s 1, whenever f. The aim of his paper is o sdy he asympoic behavior of he opimal consan c in 1) as s 1. Or main resl is he following. Theorem 1.1. Le < s < 1 and p < q <. Le also γ n1/p 1/q). Assme ha γ < s and 1 θ <. If eiher p > 1, n 1 or p 1, n 2, hen for any f B s Rn ) we have 2) f b s γ q,θ A 1 s)1/θ s γ) 1/θ f b s, where θ = maxp, θ) and he consan A does no depend on s and f. 2 Mahemaics Sbjec Classificaion: 46E3, 46E35. Key words and phrases: Besov space, embedding heorem, rearrangemen esimae, modls of coniniy. [1]

2 2 V. I. Kolyada and A. K. Lerner This heorem fails for p = n = 1. We will also show ha he exponen 1/θ is sharp in a sense. We shold menion ha his work was inspired by recen resls of Borgain, Brezis, and Mironesc [1, 2] and Maz ya and Shaposhnikova [1]. Le W 1 p R n ) be he Sobolev space of all f L p R n ) for which every firs-order weak derivaive exiss and belongs o L p R n ). Firs, i was observed in [1] ha here exiss a limiing relaion beween Sobolev and Besov norms, ha is, for any f W 1 p R n ), 3) lim s 1 1 s) f p b s p f p L p. The main resl of [2] concerns he well known Sobolev-ype embedding Bp s L q, q = np/), proved in he lae sixies independenly by several ahors for he references, see [3, 18], [7, Sec. 1], [8, p. 56]). In [2], he sharp asympoics of he bes consan as s 1 in a relaed ineqaliy was fond, namely, if 1/2 s < 1 and 1 p < n/s, hen for any f BpR s n ), 4) f p L q c n 1 s ) p 1 f p b s p q = np ), where he consan c n depends only on n. In view of 3), he classical Sobolev ineqaliy f L np/n p) c f p, 1 p < n, can be considered as a limiing case of 4). Similarly, Theorem 1.1 allows s o find a relaion beween 1) and he following resl: if eiher p > 1, n 1 or p 1, n 2, and γ n1/p 1/q) < 1, hen for any f W 1 p R n ), 5) f b 1 γ c f q,p p p < q < ). For p > 1 his esimae was proved by Il in [3, Theorem 18.12]. In he case p 1, n 2 he proof was given in [6]. By 3), Theorem 1.1 shows ha ineqaliy 5) is a limiing case of 2). We now rern o ineqaliy 4). Noe ha he proof given in [2] is qie complicaed. Aferwards, Maz ya and Shaposhnikova [1] gave a simpler proof of his resl. Also hey sdied he behavior of he opimal consan as s. More precisely, hey proved ha he consan in 4) can be replaced by c p,n s1 s)) 1 p. An imporan poin of he mehod in [1] is he fac ha he spherically symmeric rearrangemen is bonded in Besov spaces. Observe ha he proof of his fac de o Wik [15]) is raher difficl. However, i was shown in [5, 14] ha sharp heorems on embedding ino Orlicz classes can be direcly obained from esimaes of rearrangemens via modli of coniniy. The proof of hese esimaes is mch simpler han ha of Wik s heorem menioned above.

3 Limiing embeddings of Besov spaces 3 In his paper we show ha 4) as well as a more general resl of [1] can be immediaely derived from he rearrangemen esimaes obained in [5]. Moreover, we show ha a sronger ineqaliy wih he Lorenz L q,p -norm on he lef-hand side of 4) holds: for any < s < 1 and 1 p < n/s, 6) f p L q,p R n ) c p,n s1 s) ) p f p b s pr n ). I is easy o see, by he sandard relaion beween Lebesge and Lorenz norms, ha 6) implies 4). Noe ha 6) is conained implicily in [1]. Observe ha esimaes of Lorenz norms via Besov norms wiho he sharp asympoics of he bes consan) are also well known [4, 12] see Secion 2 below). We show ha o ge 6) for 1/2 < s < 1, i sffices o se hese esimaes wih some inermediae Besov norm and hen apply Theorem 1.1. Ths, his heorem is a more precise resl han 6) in he cases p 1 or n 1, 1/2 < s < 1. We also noe ha he papers [1, 2] deal wih a slighly differen definiion of he B s p-norm. However i is well known ha hese definiions are eqivalen wih corresponding consans no depending on s. We recall briefly he proof of his fac below. The paper is organized as follows. Secion 2 conains necessary informaion and some axiliary resls concerning Hardy-ype ineqaliies, modli of coniniy, and non-increasing rearrangemens. In Secion 3 we prove Theorem 1.1, and in Secion 4 we prove 6). 2. Axiliary resls 2.1. Hardy-ype ineqaliies. We will se he following lemma of Hardy [13, p. 196]. Lemma 2.1. Le α > and 1 p <. Then for any non-negaive measrable fncion ϕ on, ), p ) 1/p 7) ϕ)d) α 1 p ) d ϕ)) p α 1 d α 1/p and p ) 1/p 8) ϕ)d) α 1 p 1/p. d ϕ)) p d) α 1 α We say ha a measrable fncion f on, ) is qasi-decreasing wih consan c > if f 1 ) cf 2 ) whenever < 2 < 1 <. We will need a Hardy-ype ineqaliy for qasi-decreasing fncions in he case < p < 1. Thogh he following saemen is apparenly known, we shall prove i for he sake of compleeness.

4 4 V. I. Kolyada and A. K. Lerner Proposiion 2.2. Le f be a non-negaive, qasi-decreasing fncion on, ) wih consan c. Sppose also ha α >, β > 1 and < p < 1. Then 9) α 1 ) p f) β c1 + β)) 1 p p d d α 1 f) β+1 ) p d. α Proof. We can assme ha 1 + β)p > α, oherwise he inegral on he righ-hand side of 9) diverges. Also, one can sppose ha f is bonded and compacly sppored. Define F) =Ì f)β d. Noe ha F) f) β+1 /cβ + 1). Therefore, inegraing by pars yields α 1 F)) p d = p α+β f)f)) p 1 d α c1 + β))1 p p α 1 f) β+1 ) p d, α as reqired Modli of coniniy. For any f L p R n ), 1 p <, and h R n, se ) 1/p. I p h) = fx + h) fx) p dx R n The modls of coniniy of a fncion f is defined by ω p f, δ) = sp I p h) h δ < δ < ). Observe ha ω p f, δ) is a non-decreasing and sbaddiive fncion. In pariclar, his implies ha ω p f, n δ)/δ is qasi-decreasing on, ) wih consan 2. We wish o poin o ha he papers [2, 1] deal wih he seminorm fx) f w s p fy) p ) 1/p x y n+sp dxdy R R n in place of f b s p. I is well known ha hese seminorms are eqivalen. We oline he proof of his fac js in order o show ha he corresponding consans do no depend on s. Proposiion 2.3. For any f B s pr n ) and all s, 1), nv n ) 1/p 2 n 2 f b s p f w s p n + p)v n ) 1/p f b s p, where v n is he volme of he ni ball in R n. Proof. Define ω p f, δ) = δ nì h δ Ip ph)dh) 1/p. I is easy o see ha 1) v 1/p n 2 n 2 ω p f, δ) ω p f, δ) v 1/p n ω p f, δ) δ > ).

5 Limiing embeddings of Besov spaces 5 Nex, by Fbini s heorem, f p wp s = n + sp) s ω p f, )) p d. Ths, applying 1), we obain he desired resl. The following proposiion is also based on simple and well known argmens. Proposiion 2.4. For any f W 1 p R n ), ω p f, δ) 11) lim = f 1 p 1 p < ). δ + δ Proof. For each h R n and almos every x R n, fx + h) fx) = fx + h) hd see [9, p. 135]). By Minkowski s ineqaliy we ge I p h) h f p, and hence 12) ω p f, δ) δ f p δ > ). On he oher hand, for any ε > here exiss an infiniely differeniable fncion f ε wih a compac sppor sch ha 13) f f ε ) p < ε. I is easy o see ha µ ε δ) = sp f ε + h) f ε ) f ε ) h p /δ as δ. h =δ Ths, here exiss δ ε > sch ha µ ε δ) < ε for all < δ < δ ε. Frher, by 12) and 13), ω p f ε, δ) ω p f, δ) + εδ. Therefore, for all < δ < δ ε we ge f p f ε p + ε µ ε δ) + ω p f ε, δ)/δ + ε ω p f, δ)/δ + 3ε. Togeher wih 12), his yields 11). The las proposiion easily implies ha he limiing relaion 3) sill holds for or choice of he definiion of he Besov space. Moreover, we ge a more general resl. Proposiion 2.5. For any f Wp 1 R n ), ) 1 1/θ 14) lim 1 s 1 s)1/θ f b s = f L p 1 p, θ < ). θ

6 6 V. I. Kolyada and A. K. Lerner Proof. Se A = f L δ p. Fix ε >. By 11), here exiss δ > sch ha ω p f, )/) θ A θ < ε for any, δ). Hence, 1 s) f θ b s Aθ /θ 1 s) 1 s)θ 1 ω p f, )/) θ A θ d + Aθ θ δ1 s)θ s) s ω p f, )) θ d δ ε θ δ1 s)θ + Aθ θ δ1 s)θ f θ p 1 s). sθδsθ Now we can choose σ > sch ha ε θ δ1 s)θ + Aθ θ δ1 s)θ f θ p 1 s) < 3ε sθδsθ for any s 1 σ, 1), which proves 14). Finally, he proof of or main resl is based on he following heorem [6]. Theorem 2.6. Le 1 p < q < and γ n1/p 1/q) < 1. If eiher p > 1, n 1 or p 1, n 2, hen for any δ >, δ 15) γ 1 ω q f, )) p d ) 1/p c q,p,nδ γ 1 γ ω p f, )) q d ) 1/q. δ 2.3. Rearrangemens. The non-increasing rearrangemen of a measrable fncion f on R n is defined by f ) = sp inf fx) < < ). E = x E Define also f ) = 1 f )d > ). Le < p, r <. A measrable fncion f belongs o he Lorenz space L p,r R n ) if f p,r 1/p f )) r d ) 1/r <. I is well known see, e.g., [13, p. 192]) ha ) p 1/ν ) r 1/r 16) f p,ν f p,r < p <, r < ν). ν p The following heorem was proved in [5]. I plays a key role in or proof of 6).

7 Limiing embeddings of Besov spaces 7 τ Theorem 2.7. Sppose ha f L p R n ), 1 p <. Then for any τ, δ >, 17) f ) f τ)) p d 2ωpf, p τ 1/n ) and 18) p/n f ) f )) p d d c p,nωpf, p δ)δ p. δ n Using 17) and Hölder s ineqaliy, we immediaely ge he esimae 19) f ) f ) 2 1/p 1/p ω p f, 1/n ). Noe ha in he one-dimensional case his was firs proved by Ul yanov [14]. We will also se he following resl menioned in he Inrodcion. Theorem 2.8. Le < s < 1 and 1 p < n/s. Sppose also ha 1 θ <. Then for any f B s Rn ) we have 2) f q,θ c f b s q = np ), where c = 2 1/p n 1/θ q. This heorem was proved in [4, 12]. In order o ge he explici vale of he consan c we will give an independen proof. Namely, applying 19), we obain f ) = [f ) f )] d 21/p 1 1/p ω p f, 1/n )d. From his and 8) we ge f θ q,θ 2θ/p θ/q 1 1 1/p ω p f, 1/n )d) θ d 2 1/p q ) θ θ1/q 1/p) ωpf, θ 1/n ) d = 2 1/p q ) θ n s ω p f, ) ) θ d, which yields 2). 3. Main resl Proof of Theorem 1.1. Se ω p f, ) = ω), ω q f, ) = η), and I f θ b s γ = s γ) ω q f, )) θ d q,θ.

8 8 V. I. Kolyada and A. K. Lerner Firs we sppose ha θ p. Using Fbini s heorem, we have ) I = p1 s) γ 1 1/p η)) θ α d p1 s) d, where α = 1 θ/p)1+p1 s)). Nex, applying in he case θ < p) Hölder s ineqaliy, we ge γ 1 1/p η)) θ α d γ 1 η)) p d ) θ/p d ) 1 θ/p 1+p1 s) = [p1 s)] θ/p 1 θ p)1 s) γ 1 η)) p d ) θ/p. I follows ha I p θ/p 1 s) θ/p θ1 s) γ 1 η)) p d ) θ/p 21) d. Applying 15), we obain θ1 s) γ 1 η)) p d ) θ/p d cθ θs γ) γ ω)) q d ) θ/q d, where c is a consan depending only on p, q, and n. Now we se he fac ha he fncion ω)/ is qasi-decreasing wih consan 2. Therefore, sing 9) wih β = q1 γ) 1, we ge θs γ) γ ω)) q d ) θ/q d 2q q1 γ)) 1 θ/q s ω)) θ d s γ)q = 2q q1 γ)) 1 θ/q f θ b s γ)q s. From his and from 21) we have 1 s)θ/p I A f θ b s γ s A depends only on p, q, θ and n). Assme now ha θ > p. Firs, by Fbini s heorem, I = θ1 s) γ 1 η)) θ d ) θ1 s) d. Frher, since η is non-decreasing, γ 1 η) 1 γ)p γ 1 η)) p d ) 1/p > ).

9 Limiing embeddings of Besov spaces 9 Applying his ineqaliy, we ge γ 1 η)) θ d = γ 1 η)) θ p γ 1 η)) p d 2 θ p γ 1 η)) p d ) θ/p. Therefore, from his and 15), I c1 s) θ1 s) γ 1 η)) p d ) θ/p d c1 s) θγ s) γ ω)) q d ) θ/q d. I remains o se 9) in he case θ < q or 7) if θ q, o obain I c 1 s s γ f θ b s, where c depends only on q, p, θ, and n. Remark 3.1. I is easy o show ha he asserion of Theorem 1.1 fails in he case p = n = 1. Indeed, assme ha 2) holds for p = n = 1. Le f = χ,1). A simple compaion shows ha ω p χ,1), ) = 2 1/p min 1/p, 1). Hence, χ,1) b s 1,θ = 2 1 1/θ θs1 s)) and ) χ,1) b s γ = 2 1/q 1 γ 1/θ. q,θ θ1 s)s γ) Le, for example, q = 2 and 1/2 < s < 1. Then we ge c 1 s for some fixed c >, which conradics 2) as s 1. Remark 3.2. The exponen 1/θ in 2) is sharp in a sense. Firs assme ha θ < p. Le α = 2 s n/p. We may sppose ha α. Se fx) = 1 α 1 x α )χ B x), where B is he ni ball in R n. Then a calclaion shows ha f p p c p,n, where he consan c p,n does no depend on α. Frher, fx) = x α 1 χ B x) and f p = c p,n 1 s) 1/p. Ths, ω p f, ) c p,n 1 s) 1/p for < 1 and ω p f, ) 2 f p c p,n for 1. Hence, 22) f b s c p,n 1 s) 1/p 1/θ.

10 1 V. I. Kolyada and A. K. Lerner On he oher hand, if < < 1, /8 x /4, and h =, hen fx) fx + h) 2 5 α, and hs, ω q f, ) c α+n/q < < 1), where c is a posiive consan ha does no depend on s. From his and 22), f b s γ q,θ c1 s) 1/θ c 1 s) 1/p f b s c is posiive and does no depend on s). This shows ha he exponen 1/p of 1 s in 2) is sharp θ < p). If θ > p, i sffices o consider he fncion fx) = max1 x, ). Remark 3.3. If eiher p 1 or n 1, 1/2 < s < 1, and 1 p < n/s, hen Theorem 1.1 easily implies 6): f p q,p c 1 s p,n ) p f p b q = np ), s p which is a more precise resl han 4). Indeed, se θ = p, γ = s 1/2, and q = np/n γp). Then q = nq/n q/2). Applying Theorems 2.8 and 1.1, we have f p q,p = f p nq/n q/2),p c n,p ) p f bq,p 1/2 c p,n 1 s ) p f p b s p, as reqired. In he nex secion we will show a differen way of proving 6) for any p and n. 4. On he Borgain Brezis Mironesc ineqaliy. In Secion 2 we applied ineqaliy 19) o prove Theorem 2.8. We now se sharper rearrangemen esimaes Theorem 2.7) o prove he following. Theorem 4.1. Le < s < 1 and 1 p < n/s. Then for any f BpR s n ), 23) f p L q,p R n ) c s1 s) p,n ) p f p b s pr n ) q = np ). Observe ha, by 16), f p Lq f p n L,p, q and hence 23) immediaely implies 4) wih he consan c p,n s1 s)) 1 p. Proof of Theorem 4.1. Sppose firs ha 1/2 s < 1. Using he ideniy f ) = f τ) f τ)) dτ τ,

11 Limiing embeddings of Besov spaces 11 we obain f p q,p = f )) p d sp/n f )) p d sp/n ) np p f ) f )) p d sp/n ) np p = 1 s)p 1 s)p p/n f ) f )) p d d n ) np p 1 s)p f ξ) f )) p dξ d d 1 s c p,n ) p s ω p f, )) p d = c 1 s p,n ) p f p b s p 1 s)p n p/n we applied Hardy ineqaliy 8), Fbini s heorem, Hölder s ineqaliy and 18), respecively). Consider now he case < s < 1/2. Using he elemenary esimae f ) f τ/e) f τ)) dτ τ, we have f p np q,p np = np np e ) np p sp 2e n ) p ) p sp sp/n n ) p sp sp/n n ) p sp sp/n n = 2en p p p+1 s ) p f p b s p f /e) f )) p d sp/n f /e) f )) p d d f /e) f )) p d d f ) f )) p d d sp/n ω p pf, 1/n ) d here we sed Hardy ineqaliy 8), Fbini s heorem and 17)). Combining boh cases, we ge 23).

12 12 V. I. Kolyada and A. K. Lerner Remark 4.2. The cases < s < 1/2 and 1/2 s < 1 correspond o small and big smoohness, respecively. Ineqaliy 17) is sharp only in he case of small smoohness, while 18) is sharp in boh cases see [5] for deails). Therefore, o prove Theorem 4.1 for small s, i sffices o apply 17), while he case of big s reqires a sronger ineqaliy 18). Remark 4.3. Observe ha in he case p = n = 1 he proof of 23) is exremely simple. Indeed, exacly as in [5, Theorem 1], one can prove ha 24) f ) 2ω 1 f, )/. Therefore, f 1/1 s),1 = s f ) d 1/s 2s d ω 1 f, ) 1+1/s = 2s f b s, 1 and 23) is proved. We also oline, for he sake of compleeness, he proof of 24). I sffices o assme ha f has a compac sppor. Then for any x R and all >, fx) fx) 1 fx + y)dy + 1 j+1) j+2) fx + y)dy fx + y)dy j= j j+1) 1 fx + y) fx) dy + 1 fy + ) fy) dy. R Hence, f ) 1 2 fy + ) f ) 1 dy + 1 fy + ) fy) dy 2ω 1 f, )/, and 24) is proved. Acknowledgmens. The firs named ahor was parially sppored by gran BFM 2-26-C4-3 of he DGI, Spain. We are graefl o he referee for his/her sefl remarks. R References [1] J. Borgain, H. Brezis and P. Mironesc, Anoher look a Sobolev spaces, in: Opimal Conrol and Parial Differenial Eqaions. In honor of Professor Alain Bensossan s 6h Birhday, J. L. Menaldi, E. Rofman and A. Slem eds.), IOS Press, Amserdam, 21,

13 Limiing embeddings of Besov spaces 13 [2] J. Borgain, H. Brezis and P. Mironesc, Limiing embedding heorems for W s,p when s 1 and applicaions, J. Analyse Mah ), [3] O. V. Besov, V. P. Il in and S. M. Nikol skiĭ, Inegral Represenaions of Fncions and Embedding Theorems, Vols. 1-2, Winson, Washingon, DC, Halsed Press, New York, [4] C. S. Herz, Lipschiz spaces and Bernsein s heorem on absolely convergen Forier ransforms, J. Mah. Mech ), [5] V. I. Kolyada, Esimaes of rearrangemens and embedding heorems, Ma. Sb ), 3 23 in Rssian); English ransl.: Mah. USSR-Sb ), [6], On relaions beween modli of coniniy in differen merics, Trdy Ma. Ins. Seklova ), in Rssian); English ransl.: Proc. Seklov Ins. Mah ), [7], Rearrangemen of fncions and embedding of anisoropic spaces of Sobolev ype, Eas J. Approx ), [8] L. D. Kdryavsev and S. M. Nikol skiĭ, Spaces of Differeniable Fncions of Several Variables and Embedding Theorems, Encyclopedia Mah. Sci. 26, Springer, Berlin, 1991, [9] E. H. Lieb and M. Loss, Analysis, Grad. Sd. in Mah. 14, Amer. Mah. Soc., [1] V. Maz ya and T. Shaposhnikova, On he Borgain, Brezis, and Mironesc heorem concerning limiing embeddings of fracional Sobolev spaces, J. Fnc. Anal ), [11] S. M. Nikol skiĭ, Approximaion of Fncions of Several Variables and Embedding Theorems, Springer, Berlin, [12] J. Peere, Espaces d inerpolaion e héorème de Soboleff, Ann. Ins. Forier Grenoble) ), no. 1, [13] E. M. Sein and G. Weiss, Inrodcion o Forier Analysis on Eclidean Spaces, Princeon Univ. Press, [14] P. L. Ul yanov, Embedding of cerain fncion classes H ω p, Izv. Akad. Nak SSSR Ser. Ma ), ; English ransl.: Mah. USSR-Izv ), [15] I. Wik, Symmeric rearrangemen of fncions and ses in R n, Repor no ), Dep. of Mah., Univ. of Umeå, Deparmen of Mahemaics Karlsad Universiy Karlsad, Sweden [email protected] Deparmen of Mahemaics Bar-Ilan Universiy 529 Rama Gan, Israel [email protected] Received Janary 5, 24 Revised version Ags 17, )