CHARACTERIZATION OF REARRANGEMENT INVARIANT SPACES WITH FIXED POINTS FOR THE HARDY LITTLEWOOD MAXIMAL OPERATOR

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1 Annales Academiæ Scieniarum Fennicæ Mahemaica Volumen 31, 2006, CHARACTERIZATION OF REARRANGEMENT INVARIANT SPACES WITH FIXED POINTS FOR THE HARDY LITTLEWOOD MAXIMAL OPERATOR Joaquim Marín and Javier Soria Universia Auònoma de Barcelona, Deparmen of Mahemaics ES Bellaerra Barcelona, Spain; Universiy of Barcelona, Deparmen of Applied Mahemaics and Analysis ES Barcelona, Spain; Absrac. We characerize he rearrangemen invarian spaces for which here exiss a nonconsan fixed poin, for he Hardy Lilewood maximal operaor he case for he spaces L p R n was firs considered in [7]. The main resul ha we prove is ha he space L n/n 2, R n L R n is minimal among hose having his propery. 1. Inroducion The cenered Hardy Lilewood maximal operaor M is defined on he Lebesgue space L 1 loc Rn by 1 M fx = sup fx y dy, r>0 B r B r where B r denoes he measure of he Euclidean ball B r cenered a he origin of R n. In his paper we sudy he exisence of non-consan fixed poins of he maximal operaor M i.e., M f = f in he framework of he rearrangemen invarian r.i. funcions spaces see Secion 2 below. We will use some of he esimaes proved in [7], where he case L p R n was sudied, and show ha hey can be sharpened o obain all he rearrangemen invarian norms wih his propery in paricular we exend Korry s resul o he end poin case p = n/n 2, where he weak-ype spaces have o considered. The main argumen behind his problem is he exisence of a minimal space L n/n 2, R n L R n conained in all he r.i. spaces wih he fixed poin propery Mahemaics Subjec Classificaion: Primary 42B25, 46E30. The research of he firs-named auhor was parially suppored by Grans MTM , 2001SGR00069 and by Programa Ramón y Cajal MCYT and ha of he second auhor was parially suppored by Grans MTM , 2001SGR00069.

2 40 Joaquim Marín and Javier Soria 2. Background on rearrangemen invarian spaces Since we work in he conex of rearrangemen invarian spaces i will be convenien o sar by reviewing some basic definiions abou hese spaces. A rearrangemen invarian space X = XR n r.i. space is a Banach funcion space on R n endowed wih a norm XR n such ha f XR n = g XR n whenever f = g. Here f sands for he non-increasing rearrangemen of f, i.e., he non-increasing, righ-coninuous funcion on [0, equimeasurable wih f. An r.i. space XR n has a represenaion as a funcion space on X 0, such ha f XR n = f X 0,. Any r.i. space is characerized by is fundamenal funcion φ X s = χ E XR n here E is any subse of R n wih E = s and he fundamenal indices where I is well known ha log M X s β X = inf s>1 log s M X s = sup and β X = sup s<1 φ X s φ X, s > 0. 0 β X β X 1. log M X s, log s We refer he reader o [2] for furher informaion abou r.i. spaces. 3. Main resul Before formulaing our main resul, i will be convenien o sar wih he following remarks see [7]: Remark 3.1. By Lebesgue s differeniaion heorem one easily obains ha fx M fx a.e. x R n ; hus f is a fixed poin of M, if and only if f is posiive and 1 fy dy fx a.e. x R n, Bx, r Bx,r or equivalenly f is a posiive super-harmonic funcion i.e. f 0, where is he Laplacian operaor.

3 Invarian spaces wih fixed poins 41 Remark 3.2. If f is a non-consan fixed poin of M, and ϕ 0 belongs o he Schwarz class S R n, wih R n ϕx dx = 1, hen he funcion f x = f ϕ x, wih ϕ x = n ϕx/ is also a non-consan fixed poin of M which belongs o C R n noice ha using he Lebesgue differeniaion heorem, here exiss some > 0 such ha f is non-consan, since f is non-consan. In paricular if XR n is an r.i. space and f XR n is a non-consan fixed poin of M, since S R n L 1 R n L R n we ge ha f XR n C R n is a non-consan fixed poin of M. Remark 3.3. Using he heory of weighed inequaliies for M see [5], if M f = f, in paricular f A 1 he Muckenhoup weigh class, and hence fx dx defines a doubling measure. Hence, f / L 1 R n. Also, using he previous remark we see ha if f L p R n is a fixed poin, hen f L q R n, for all p q. Definiion 3.4. Given an r.i. space XR n, we define D I2 XR n = { f L 0 R n : I 2 f XR n < }, where I 2 is he Riesz poenial, I 2 fx = x y 2 n fy dy. R n I is no hard o see ha he space D I2 XR n is eiher rivial or is he larges r.i. space which is mapped by I 2 ino XR n, and is also relaed wih he heory of he opimal Sobolev embeddings see [4] and he references quoed herein. Theorem 3.5. Le XR n be an r.i. space. The following saemens are equivalen: 1 There is a non-consan fixed poin f XR n of M. 2 n 3 and x 2 n χ {x: x >1} x XR n. 3 n 3 and χ [0,1] + 2/n 1 χ [1, X 0,. 4 n 3 and L n/n 2, R n L R n XR n. 5 n 3 and D I2 XR n {0}. Proof. 1 2 Since if n = 1 or n = 2, he only posiive super-harmonic funcions are he consan funcions see [8, Remark 1, p. 210], necessarily n 3. Moreover, i is proved in [7] ha, if f C R n is a non-consan fixed poin of M, hen fx c x 2 n χ {x: x >1} x. Since f XR n, hen x 2 n χ {x: x >1} x XR n. 2 3 Since if x 2 n χ {x: x >1} x XR n, hen F x = χ {x: x 1} x + x 2 n χ {x: x >1} x XR n.

4 42 Joaquim Marín and Javier Soria An easy compuaion shows ha F χ [0,1] + 2/n 1 χ [1,. 3 4 Since f L n/n 2, R n L R n if and only if sup f W <, where W = max1, 1 2/n, we have ha f f L n/n 2, R n L R n W 1 and since W 1 = χ [0,1] + 2/n 1 χ [1, X 0, we have ha wih c = W 1 X 0,. f XR n = f X 0, c f L n/n 2, R n L R n 4 5 Since see [9] and [1] I 2 f c 1 2/n 1 f s ds + 0 f ss 2/n 1 ds c 2 I 2 f 0 where f 0 x = f c n x n, c n = measure of he uni ball in R n. Observe ha f 0 = f. Rewriing he middle erm in he above inequaliies, using Fubini s heorem, we ge I 2 f d 1 n n 2 f ss 2/n 1 ds d 2 I 2 f 0, where f = 1 0 f s ds. Thus, f D I2 XR n if and only if 1 f ss 2/n 1 ds <. X 0, Since F = is a decreasing funcion, and F 0 x = F c n x n χ [0,1] ss2/n 1 ds = c χ [0,1] + 2/n 1 χ [1, χ {x: x 1} x + x 2 n χ {x: x >1} x L n/n 2, R n L R n we ge ha χ [0,1] D I 2 XR n.

5 Invarian spaces wih fixed poins 43 Anoher argumen o prove his par is he following: Since, if n 3 see [2, Theorem 4.18, p. 228] I 2 : L 1 R n L n/n 2, R n and I 2 : L n/2,1 R n L R n is bounded, we have ha I 2 : L 1 R n L n/2,1 R n L n/n 2, R n L R n XR n is bounded, and hence L 1 R n L n/2,1 R n D I2 XR n. 5 1 Since n 3, we can use he classical formula of poenial heory see [10, p. 126] h = I 2 h o conclude ha here is a posiive funcion f = I 2 χ [0,1] XRn. Then 0 f = I 2 χ [0,1] ϕ XR n C R n and f 0. We now consider paricular examples, like he Lorenz spaces: Corollary 3.6. Le 1 p <, and assume Λ p R n, w is a Banach space i.e., w B p if 1 < p < or p B 1, if p = 1, see [3]. Then, here exiss a non-consan funcion f Λ p R n, w such ha M f = f if and only if n 3 and w d <. 1 p1 2/n In paricular, his condiion always holds, for p > 1 and n large enough. Proof. The inegrabiliy condiion follows by using he previous heorem. Now, if w B p, hen here exiss an ε > 0 such ha w B p ε, and hence i suffices o ake n > 2/ε. Observe ha if w = 1 and p = 1, hen Λ 1 R n, w = L 1 R n, which does no have he fixed poin propery for any dimension n. Corollary 3.7. Le 1 p, q if p = 1 we only consider q = 1. Then, here exiss a non-consan funcion f L p,q R n such ha M f = f if and only if n 3 and { n/n 2 < p or p = n/n 2 and q =. Corollary 3.8. See [7] Le 1 p. There exiss a non-consan funcion f L p R n such ha M f = f if and only if n 3 and n/n 2 < p. I is ineresing o know when given an r.i. space XR n, he space D I2 XR n is no rivial, or equivalenly 2 D I2 XRn { } := f L 0 [0, : f ss 2/n 1 ds < X 0, is no rivial. This will be done in erms of he fundamenal indices of X. We sar by compuing he fundamenal funcion of D I2 XR n.

6 44 Joaquim Marín and Javier Soria Lemma 3.9. Le X be an r.i. space on R n, n 3. Le Y be given by 2. Then φ Y s s n/2 P 1 2/n χ [0,s] X where P 1 2/n f = 2/n 1 0 fss 2/n ds. Proof. s n/2 P 1 2/n χ [0,s] s n/2 χ [0,s] + χ [0,s] rr2/n 1 dr. 1 2/n s χ [s, Theorem Le X be an r.i. space on R n, n 3. Le Y be given by 2. Then 1 If β X < 1 2/n, hen Y {0}. 2 If Y {0} hen β X 1 2/n. Thus Proof. 1 Le χ r = χ [0,r]. Then P 1 2/n χ r = 1 0 χ r ξ dξ ξ c 2 k1 n/2 χ n/2 2 r. k P 1 2/n χ r X c 2 k1 n/2 φ X 2 k r cφ X r 2 k1 n/2 M X 2 k. Le ε > 0 be such ha β X + ε < 1 2/n. Then by he definiion of β X i follows readily ha here is a consan c > 0 such ha and hence 2 k1 n/2 M X 2 k M X 2 k c2 k β X +ε, 2 k1 n/2 β X ε <, which implies ha χ r Y. 2 Since Y {0} if and only if P 1 2/n χ [0,1] X < and 3 supp 1 2/n χ [0,1] φ X P 1 2/n χ [0,1] X <,

7 Invarian spaces wih fixed poins 45 and easy compuaions show ha 3 implies ha 4 1 sup 1 hen, by 4 M X a = max sup 1/a = max sup max 1/a a 1 2/n Thus, if a < 1, using again 4 we ge which implies ha M X a a 1 2/n φ X = c <, 1 2/n φ X a φ X, sup <1/a φ X a φ X φ X a a 1 2/n, sup a 1 2/n φ X <1/a 1 2/n sup 1/a φ X, sup <1/a φ X a φ X 1 2/n sup 1/a φ X a1 2/n β X 1 2/n. φ X a φ X. Le us see ha he converse in he previous heorem is no rue. Proposiion There are rearrangemen invarian spaces X such ha 1 Y {0} and β X 1 2/n. 2 Y = {0} and β X < 1 2/n. Proof. Le ϕ = a χ [0,1] + b χ [1,, wih 0 a, b 1. Le { X = f L 0 [0, } : sup f ϕ <. Since ϕ is a quasi-concave funcion, we have ha and β X = mina, b, ϕ = φ X βx = maxa, b. On he oher hand, he space Y defined by 2 is no rivial if and only if b 1 2/n. Now, o prove 1 ake b 1 2/n and a 1 2/n. And o see 2 ake b > 1 2/n and a 1 2/n.

8 46 Joaquim Marín and Javier Soria and Remark If we consider { X 0 = f L 0 [0, : sup } f 1 2/n 1 + log + < { X 1 = f L 0 [0, : sup f 1 2/n } 1 + log + < hen β Xi = β Xi = 1 2/n, Y 0 = {0} and Y 1 {0}. Remark I was proved in [7] ha if we consider he srong maximal funcion i.e., he maximal operaor associaed o cenered inervals in R n, hen here were no fixed poins in any L p R n space, regardless of he dimension. The same argumen works o show ha L p R n canno be replaced by any differen r.i. space. Also, if we sudy his quesion for oher kind of ses, like, e.g., Buseman Feller differeniaion bases see [6], hen he only possible fixed poins are he consan funcions. This observaion applies o any non-cenered maximal operaor wih respec o balls, cubes, ec.. References [1] Benne, C., and K. Rudnick: On Lorenz Zygmund Spaces. - Disseraiones Mah. Rozprawy Ma. 175, [2] Benne, C., and R. Sharpley: Inerpolaion of Operaors. - Academic Press, Boson, [3] Carro, M., L. Pick, J. Soria, and V. Sepanov: On embeddings beween classical Lorenz spaces. - Mah. Inequal. Appl. 4, 2001, [4] Edmunds, D. E., R. Kerman, and L. Pick: Opimal Sobolev imbeddings involving rearrangemen-invarian quasinorms. - J. Func. Anal. 170, 2000, [5] García-Cuerva, J., and J. L. Rubio de Francia: Weighed Norm Inequaliies and Relaed Topics. - Norh-Holland Mah. Sud. 116, Norh-Holland, Amserdam, [6] de Guzmán, M.: Differeniaion of Inegrals in R n. - Lecure Noes in Mah. 481, Springer-Verlag, Berlin New York, [7] Korry, S.: Fixed poins of he Hardy Lilewood maximal operaor. - Collec. Mah. 52, 2001, [8] Lieb, E. H., and M. Loss: Analysis. - American Mahemaical Sociey, Providence, [9] Sawyer, E.: Boundedness of classical operaors on classical Lorenz spaces. - Sudia Mah. 96, 1990, [10] Sein, E.: Singular Inegrals and Differeniabiliy Properies of Funcions. - Princeon Universiy Press, New Jersey, Received 9 Sepember 2004