On Functions of Integrable Mean Oscillation

Transcription

1 On Funcions of negrable Mean Oscillaion Oscar BLASCO * and M. Amparo PÉREZ Deparameno de Análisis Maemáico Universidad de Valencia 46 Burjasso Valencia Spain oblasco@uv.es Recibido: de diciembre de 24 Acepado: 2 de abril de 25 ABSTRACT Given f L (T we denoe by w mo(f he modulus of mean oscillaion given by w mo(f( = sup f(e iθ m dθ (f < where is an arc of T, sands for he normalized lengh of, and m (f = f(eiθ dθ. Similarly we denoe by w ho(f he modulus of harmonic oscillaion given by w ho (f( = sup f(e iθ P (f(z P z(e iθ dθ z < T where P z(e iθ and P (f sand for he Poisson kernel and he Poisson inegral of f respecively. is shown ha, for each < p <, here exiss C p > such ha [w mo(f(] p d [w ho (f(] p d Cp [w mo(f(] p d. Key words: mean oscillaion, BMO, modulus of coninuiy. 2 Mahemaics Subjec Classificaion: 46B25. * Parially suppored by Proyeco BMF22-4 and MTN E Rev. Ma. Complu. 25, 8; Núm. 2, SSN: 39-38

2 O. Blasco/M. A. Pérez On funcions of inegrable mean oscillaion. nroducion. As usual we denoe by BMO he space of funcions f L (T such ha f = sup f(e iθ m (f dθ T <, where is an arc of he circle T, sands for he normalized lengh of and m (f = f(eiθ dθ. We wrie f BMO = ˆf( + f. f f L (T and <, we define he modulus of mean oscillaion of f a he poin as w mo (f( = sup f(e iθ m (f dθ <. Clearly, for < s <, one has sup < s f(e iθ m (f dθ 2 f. Hence, for < s <, one has { w mo (f( w mo (f(s max w mo (f(, 2 f }. ( n paricular, f BMO if and only if w mo (f( < for some (or for all <. is known ha one can consider oher equivalen moduli o define BMO. For insance, for < q <, w mo,q (f( = ( sup f(e iθ m (f q dθ /q. < is also well known, by he John-Nirenberg lemma (see [7, 8], ha here exis C, C 2 > such ha for all λ > and any arc wih, { θ T : f(e iθ m > λ } From here one ges ha, for all >, One can also consider ( w mo(f( = sup 2 C e C 2 λ wmo(f(. w mo (f( w mo,q (f(. (2 f(e iθ f(e iϕ dθ dϕ Revisa Maemáica Compluense 25, 8; Núm. 2,

3 O. Blasco/M. A. Pérez On funcions of inegrable mean oscillaion or Clearly one ges ( ( w mo (f( = sup inf f(e iθ c dθ c and w mo (f( w mo(f( 2w mo (f( (3 w mo (f( w mo (f( 2 w mo (f(. A funcion f is said o have vanishing mean oscillaion, in shor f VMO, if lim f(e iθ m (f dθ =. This is a closed subspace of BMO, which can be characerized in many ways (see [7, 8, 5]. Theorem.. Le f BMO. The following saemens are equivalen: (i f VMO. (ii lim + T f f BMO =, where T f(e iθ = f(e i(θ. (iii lim r P r f f BMO =, where P r (e iθ = R( +re iθ re iθ. (iv f belongs o he closure of C(T in BMO. (v lim + w mo (f( =. A generalizaion of BMO is he space BMO(ρ, consising of funcions f L (T such ha w mo (f( = O(ρ( for a fixed funcion ρ wih cerain properies. The space BMO(ρ has been considered by various auhors (see [, 5, 7]. Our aim will be o analyze spaces where he funcion ρ is no explicily given, bu we do know is behavior a he origin in erms of cerain inegrabiliy condiions. Given < p <, we shall denoe by MO p (T he space of inegrable funcions such ha [w mo(f(] p d <. Due o (2, he spaces MO p q of funcions such ha [w mo,q(f(] p d < are all he same for < q <. These spaces were considered in [2] (see page 74, under a differen noaion. Also some spaces MO α s,r, which are closely relaed o he ones considered in his paper, were inroduced in [3]. We use he noaions ω (f( = sup f(e iθ f(e iϕ θ ϕ 467 Revisa Maemáica Compluense 25, 8; Núm. 2,

4 O. Blasco/M. A. Pérez On funcions of inegrable mean oscillaion and ( ω q (f( = sup f(e i(θ+u f(e iθ q dθ /q u T for < q <. Now, for < s < and < p, q, he Besov space Bq,p(T s consiss of funcions in L q (T such ha s ω q (f L p ((,, d. Of course he cases Bs q, where < q correspond o Lipschiz or Hölder classes, o be denoed Lip s (T insead of B, (T. s We denoe by X p he space consising of funcions L (T such ha ω (f L p ((,, d. From (3 we easily obain, for any >, w mo (f( Cω (f(. Hence X p MO p (T for any < p <. On he oher hand, if, J are arcs on T such ha J hen m J (f m (f J w mo(f( J. (4 Now, given wih, using he Lebesgue differeniaion heorem, one ges f(e iθ = lim n m n f, for a.a. θ, where n is a decreasing sequence of arcs conaining θ such ha n = 2 n+. Hence using (4, we have ha for any f MO (T Therefore we obain, for any >, f(e iθ m lim n m n f m f C C ω (f( 2C m k f m k f k= w mo (f(2 k k= w mo (f(s ds s. w mo (f(s ds s. (5 This implies ha MO (T Lip φ, where Lip φ sands for he space of coninuous funcions such ha f(e iθ f(e iϕ Cφ( θ ϕ, Revisa Maemáica Compluense 25, 8; Núm. 2,

5 O. Blasco/M. A. Pérez On funcions of inegrable mean oscillaion for φ( = sup{ w mo(f(s ds s : w mo(f( d }. BMO-ype characerizaions of hese spaces have been exensively considered in he lieraure. The reader is referred o [2, 3, 9] for he case p = and o [4, 5] for he cases < s and < p <. We shall consider a descripion of MO p (T where he averages over arcs are replaced by averages wih respec he Poisson kernel. We denoe by BMOH he space of funcions f L (T such ha f = sup f(e iθ P (f(z P z (e iθ dθ z <, T where denoes he open uni disc, P (f(z = T f(eiθ P z (e iθ dθ and P z(e iθ = R( +ze iθ. We wrie f ze iθ BMOH = P (f( + f. is no difficul o prove (see [7,8] ha f BMO if and only if f BMOH wih equivalen norms. n his siuaion we define he modulus of harmonic oscillaion of f a he poin as w ho (f( = sup z < T f(e iθ P (f(z P z (e iθ dθ. Hence, f BMO (respec. BMOH if and only if w mo (f( < (respec. w ho (f( <. For < p <, we denoe by HO p (T he space of f L (T such ha [w ho(f(] p d <. Of course one can also use oher moduli o define his space. For insance, for < q <, or w ho,q (f( = w ho (f( = ( sup f(e iθ P (f(z q P z (e iθ dθ /q, z < T sup z < inf f(e iθ c P z (e iθ dθ c T. The main objecive of he paper is o show ha, for < p <, we have MO p (T = HO p (T and wih equivalen norms. The paper is divided ino wo secions. The firs one is devoed o inroducing MO p (T and proving some of is properies and he second one o inroducing HO p (T and o showing ha HO p (T coincides wih MO p (T. 2. negrable mean oscillaion. We will see firs ha he modulus of mean oscillaion is coninuous. We shall use he following lemma. 469 Revisa Maemáica Compluense 25, 8; Núm. 2,

6 O. Blasco/M. A. Pérez On funcions of inegrable mean oscillaion Lemma 2.. Le f L (T. f { n } is a sequence of arcs such ha lim n n = for some arc wih > hen lim n Proof. Le us firs esimae f(e iθ m n (f dθ n n = f(e iθ m (f dθ. f(e iθ m n (f dθ n n n ( f(e iθ m (f dθ f(e iθ m (f dθ n + m n (f m (f f(e iθ m (f dθ n f(e iθ m (f dθ + m n (f m (f + 2 f ( n f(e iθ m (f dθ. Noice ha ν(a = A f(eiθ dθ and µ (A = A f(eiθ m (f dθ are a complex and a posiive measure respecively wih inegrable densiies. Therefore he resul follows passing o he limi as n goes o. Proposiion 2.2. Le f BMO. Then w mo (f is increasing and coninuous in (, ]. Proof. Monooniciy has already been proved in our formula (. Le < and le us prove ha i is lef coninuous a. Given ε > we find T such ha < and w mo (f( f(e iθ m (f dθ + ε 2. Le ( n be a sequence such ha n for all n N and converges o. f =, we can find n such ha lim n n =. Hence w mo (f( w mo (f( n f(e iθ m (f dθ n n f(e iθ m n (f dθ + ε 2. Now use Lemma 2. o ge lim n w mo (f( w mo (f( n =. f < here exiss n such ha n for n n. Hence w mo (f( w mo (f( n < ε 2 for n n. Revisa Maemáica Compluense 25, 8; Núm. 2,

7 O. Blasco/M. A. Pérez On funcions of inegrable mean oscillaion To see ha i is righ coninuous a, we shall argue as follows: Le ( n be a sequence such ha n for all n N and converges o. We shall find a subsequence ( nk such ha lim k w mo (f( nk = w mo (f(. Given ε > we find n T such ha < n n and w mo (f( n f(e iθ m n (f dθ n n + ε. Le F = {n N : n > }. f F is finie hen n for n n and w mo (f( n w mo (f( < ε for n n. Wihou loss of generaliy we assume n > for all n N. Call = n= k=n k. is easy o see ha is an arc and ha =. Take a subsequence n k such ha ( nk converges o. We have w mo (f( nk w mo (f( w mo (f( nk f(e iθ m (f dθ f(e iθ m nk (f dθ nk nk f(e iθ m (f dθ + ε. The proof is complee invoking Lemma 2.. Remark 2.3. Le f BMO and ake a(f = lim + w mo (f(. Hence f VMO if and only if a(f =. For each < p <, we define he quasi-norm (norm for p on MO p (T by ( f MO p = f L (T + [w mo (f(] p d /p. Alhough he nex resul is probably known, we include a proof for he sake of compleeness. Theorem 2.4. Le < p <. Then (MO p (T,. MO p is a complee space. Proof. Le {f n } be a Cauchy sequence in MO p (T. n paricular, here exiss f BMO such ha {f n } converges o f. Le, <. Using ha f n f in L (T we ge ha m (f n m (f and ha here exiss a subsequence (n k, such ha f nk f a.e. 47 Revisa Maemáica Compluense 25, 8; Núm. 2,

8 O. Blasco/M. A. Pérez On funcions of inegrable mean oscillaion Now f n (e iθ f(e iθ m (f n f dθ = f n (e iθ lim f nk (e iθ lim m (f n f nk dθ k k = lim f n (e iθ f nk (e iθ m (f n f nk dθ k Therefore Hence [ lim inf k lim inf k f n (e iθ f nk (e iθ m (f n f nk dθ w mo (f n f nk (. w mo (f n f( lim inf w mo (f n f nk (. k w mo (f n f( d ] p lim inf lim inf k k [w mo (f n f nk (] p d [w mo (f n f nk (] p d Finally, using ha f n is a Cauchy sequence we ge lim n f n f MO p = and ha f MO p. Proposiion 2.5. Le < p q < and s >. (i MO p (T MO q (T. (ii Lip s (T p> MOp (T MO (T C(T. (iii p> MOp (T VMO. Proof. (i is a consequence of he following fac: ( [w mo (f(] p d /p ( /p [w mo (f(2 k ] p. (ii Noe ha f Lip s if and only if w mo (f( C s. This gives he firs inclusion. The fac ha MO (T C(T follows by (5. (iii Observe ha, for any p > and >, one has w mo (f p ( log k= w mo (f p (u du u f MO p. Hence lim + w mo (f( = for f p> MO p (T. Revisa Maemáica Compluense 25, 8; Núm. 2,

9 O. Blasco/M. A. Pérez On funcions of inegrable mean oscillaion Le us poin ou some properies of BMO ha are shared by hese spaces. Proposiion 2.6. f f MO p (T hen f MO p (T. Proof. Le (, and T wih. Then f(e iθ m ( f dθ f(e iθ m (f dθ + m ( f m (f 2 f(e iθ m (f dθ 2 f(e iθ m (f dθ This shows ha w mo ( f ( 2 w mo (f( and he proof is complee. Recall ha T denoes he ranslaion operaor, ha is T f(e iθ = f(e i(θ. We have he following resul. Theorem 2.7. Le < p < and f MO p (T. Then lim s + T sf f MO p =. Proof. Due o (iii in Proposiion 2.5 f VMO. Now Theorem. gives ha lim s + T s f f BMO =. Noe ha w mo (T s f f( T s f f BMO for all <. On he oher hand w mo (T s f f( = sup (T s f f(e iθ m (T s f f dθ sup + sup T s f(e iθ m (T s f dθ = 2 w mo (f( f(e iθ m (f dθ The Lebesgue dominaed convergence heorem gives lim s + T s f f MO p =. 3. negrable harmonic oscillaion. Throughou his secion, given z \ {}, we denoe by z he open arc in T wih midpoin z z and lengh z = z. Given an arc T and λ we shall wrie λ for he arc wih he same midpoin and lengh λ. Le us collec several known facs o be used laer on. 473 Revisa Maemáica Compluense 25, 8; Núm. 2,

10 O. Blasco/M. A. Pérez On funcions of inegrable mean oscillaion Lemma 3.. There exis consans < C, C, C 2, C 3 < such ha (i z e iθ z C ( z, e iθ z, and z. (ii C z P z(e iθ C 2 z, eiθ z, and z. (iii 4 k z P z (e iθ C 3 4 k, eiθ z 2 k z \ 2 k z, k {, 2,..., N + }, where N = [log 2 z ] and z. Proof. All he saemens follow from he esimaes z e iθ z e iθ z + ( z z and e iθ z e iθ z + ( z. z For < p < we define ( f HO p = f L (T + [w ho (f(] p d /p o ge a quasi-norm in he space HO p (T. Proposiion 3.2. f f L (T and < hen w mo (f( c w ho (f(. Proof. Le T be an arc such ha. Consider z for which = z. From z = z we have z <. Using (ii in Lemma 3. we have f(e iθ m (f dθ f(e iθ P (f(z dθ z z + m (f P (f(z 2 f(e iθ P (f(z dθ z z ( π C Cw ho (f( f(e iθ P (f(z P z (θ dθ π Now aking he supremum over all arcs we ge w mo (f( C w ho (f(. Theorem 3.3. Le < p <. Then HO p (T = MO p (T wih equivalen quasinorms. Revisa Maemáica Compluense 25, 8; Núm. 2,

11 O. Blasco/M. A. Pérez On funcions of inegrable mean oscillaion Proof. HO p (T MO p (T follows from Proposiion 3.2. Assume now ha f MO p (T. Le us show ha f HO p (T and f HO p C f MO p. Le (, ]. For any z wih z = we consider he arc = z, which gives z = z =. Take N and k (k =,,..., N so ha k = 2 k z where =, N T, and N = T. Using (iii in Lemma 3. we have f(e iθ P (f(z P z (e iθ dθ T 2 f(e iθ m (f P z (e iθ dθ T C( f(e iθ m (f P z (e iθ dθ N + f(e iθ m (f P z (e iθ dθ k= k \ k ( C f(e iθ m (f dθ N + 4 k f(e iθ m (f dθ k= k \ k ( N C w mo (f( + 2 k f(e iθ m (f dθ. k k On he oher hand, by (4, f(e iθ m (f dθ k k k k k= ( k dθ f(eiθ m k (f + m j (f m j (f j= f(e iθ m k (f dθ k k k + m j (f m j (f j= f(e iθ m k (f dθ k k k w mo (f( k + 2w mo (f( j j= ( + 2kw mo (f( k. + k j= j j w mo(f( j 475 Revisa Maemáica Compluense 25, 8; Núm. 2,

12 O. Blasco/M. A. Pérez On funcions of inegrable mean oscillaion Combining boh esimaes one ges f(e iθ P (f(z P z (e iθ dθ ( N C w mo (f( + T k= + 2k 2 k w mo (f( k. Taking he supremum over {z : z = } and using k = 2 k we obain sup f(e iθ P (f(z P z (e iθ dθ ( z = T C w mo (f( + where N = N = [log 2 ] +. This implies ha For < p < we have ( w ho (f( C w mo (f( + N k= N k= + 2k 2 k w mo (f(2 k. + 2k 2 k w mo (f(2 k ( N [w ho (f(] p C p [w mo ((] p ( + 2k p + 2 pk [w mo (f(2 k ] p. k= For p we apply Hölder s inequaliy o obain N [w ho (f(] p C p ([w mo ((] p ( + 2k p + 2 k [w mo (f(2 k ] p. Now inegraing, and aking ino accoun ha k N = [log 2 ] + is equivalen o < 2 k, we ge [w ho (f(] p d C p k= [w mo ((] p d + C p C p f p MO p + C p C p f p MO p + C p C f p MO p. k= k= ( + 2k p 2 k min{p,} ( + 2k p 2 k min{p,} N + k= 2 k Puing ogeher all he esimaes we have he resul. ( + 2k p 2 [w mo(f(2 k ] p d k min{p,} [w mo (f(2 k ] p d [w mo (f(] p d Revisa Maemáica Compluense 25, 8; Núm. 2,

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