Aales Academiæ Scietiarum Feicæ Mathematica Volume 41, 2016, 103 117 POINTWISE AND GRAND MAXIMAL FUNCTION CHARACTERIZATIONS OF BESOV-TYPE AND TRIEBEL LIZORKIN-TYPE SPACES Tomás Soto Uiversity of Helsiki, Deartmet of Mathematics ad Statistics P.O. Box 68, FI-00014 Uiversity of Helsiki, Filad; tomas.soto@helsiki.fi Abstract. I this ote, we establish characterizatios for the homogeeous Besov-tye saces Ḃ,R s,τ ad Triebel Lizorki-tye saces F,R s,τ, itroduced by Yag ad Yua, through fractioal Hajłasz-tye gradiets for suitable values of the arameters, ad τ whe 0 < s < 1, ad through grad Littlewood Paley-tye maximal fuctios for all admissible values of the arameters. These characterizatios exted the characterizatios obtaied by Koskela, Yag ad Zhou for the stadard homogeeous Besov ad Triebel Lizorki saces. 1. Itroductio The mai urose of this ote is to establish oitwise characterizatios of the Besov-tye ad Triebel Lizorki-tye fuctio saces, itroduced by Yag ad Yua i [14] ad [16], through Hajłasz-tye gradiets. Characterizatios of this tye go back to Hajłasz s oitwise characterizatio of the classical Sobolev saces [6], ad they have foud may alicatios i both the Euclidea settig as well as i the settig of more geeral metric measure saces. The families of fuctio saces cosidered i this aer iclude the stadard Besov ad Triebel Lizorki saces as well as the fractioal Morrey Sobolev saces for smoothess idices s 0, 1 as secial cases. To begi with, we first recall the defiitios of the stadard homogeeous Triebel Lizorki ad Besov saces. For a dimesio N := {1,2,3, }, which shall be fixed throughout the aer, we deote by SR the class of Schwartz fuctios, i.e. the class of comlex-valued C R fuctios φ for which φ Sk,m := su 1+ x m γ φx γ k, x R is fiite for all k, m N 0 := N {0}; here γ = γ 1 + + γ ad γ = γ 1 x 1 γ x for all multi-idices γ = γ 1,,γ N 0. The semiorms Sk,m iduce a locally covex toology o SR. We deote by S R the class of temered distributios, i.e. the class of cotiuous comlex-valued liear fuctioals o SR, ad eui S R with the toology iduced by the maigs f f,φ, φ SR. For stadard facts about the Schwartz sace ad temered distributios, articularly their Fourier-trasforms ad covolutios, we refer to e.g. [5]. doi:10.5186/aasfm.2016.4101 2010 Mathematics Subject Classificatio: Primary 42B35; Secodary 46E35. Key words: Besov-tye sace, fuctio sace, grad maximal fuctio, Hajłasz gradiet, Triebel Lizorki-tye sace. The author was suorted by the Fiish CoE i Aalysis ad Dyamics Research.
104 Tomás Soto Followig [3], ϕ ad ψ shall throughout the aer be fixed elemets of SR satisfyig ad su ϕ, su ψ { ξ R : 2 1 ξ 2 }, ϕξ, ψξ c > 0 whe 3/5 ξ 5/3 ϕ2 j ξ ψ2 j ξ = 1 whe ξ 0. j Z For φ SR ad j Z, we write φ j for the Schwartz fuctio x 2 j φ2 j x. For s R, 0 < < ad 0 <, the homogeeous Triebel Lizorki sace F, s R is defied as the class of temered distributios f for which ˆ f F,R s = R [ j Z ] / 1/ 2 js ϕ j fz dz is fiite, with the obvious modificatio made whe =. For s R, 0 < ad 0 <, the homogeeous Besov sace Ḃs,R is defied as the class of temered distributios f for which f Ḃs, R = j Z 1/ 2 js ϕ j f L R is fiite, with agai the obvious modificatio made whe =. It is well kow that after uotietig out the temered distributios whose Fourier-trasforms are suorted at the origi, i.e. the olyomials, F, s R ad Ḃs, R become uasi- Baach saces, ideedet of the choice of ϕ i the sese that two admissible choices iduce euivalet uasiorms; see for istace [13]. The followig Triebel Lizorki-tye ad Besov-tye saces were itroduced by Yag ad Yua i [14] ad [16]. For s R, 0 < <, 0 < ad 0 τ <, the homogeeous Triebel Lizorki-tye sace F,R s,τ is defied as the class of temered distributios f for which ˆ [ f F, s,τ R = su 1 ] / 1/ 2 js ϕ x R, l Z Bx,2 l τ j fz dz Bx,2 l is fiite, with the obvious modificatio whe =. For s R, 0 <, 0 < ad 0 τ <, the homogeeous Besov-tye sace Ḃs,τ,R is defied as the class of temered distributios f for which f Ḃs,τ,R = su 1 1/ 2 js ϕ x R, l Z Bx,2 l τ j f L Bx,2 l is fiite, with agai the obvious modificatio whe =. Agai, these saces become uasiormed saces after uotietig out the olyomials. Actually, i the defiitios of [14] ad [16], the suremum is take over dyadic cubes istead of balls with dyadic radii, but it is uite easy to see that the above defiitios yield euivalet uasiorms. It is also kow that these saces are ideedet of the choice ofϕ[16, Corollary 3.1] ad that they are uasi-baach saces [11, Proositio 2.2] ad the j l j l
Characterizatios of Besov-tye ad Triebel Lizorki-tye saces 105 refereces therei. Here are a coule examles of how these saces coicide 1 with other well-kow fuctio saces: F, s,0r = F, s R ad Ḃs,0, R = Ḃs, R for all admissible arameters. F, s,1/ R = F, s R for all admissible arameters; i articular F 0,1/,2 R = BMOR. 1 Ḃs, u 1 u, R = N,u, s R for 0 < u < ad s R, where N,u, s R stads for the homogeeous Besov-Morrey sace, i.e. the Besov-tye sace based o the Morrey sace MuR istead of L R, itroduced i [9] ad [10]. F s,1 u 1 u, R = E,u, s R for 0 < u <, 0 < ad s R, where E,u, s R stads for the homogeeous Triebel Lizorki-Morrey sace, i.e. the Triebel Lizorki-tye sace based o the Morrey sace Mu R istead of L R. For 0 < α < mi1,, the sace F α,1 2 α 2 2,2 R coicides with the sace Q α R itroduced i [2]. For 1 < τ < ad all admissible values of, ad s, 1 F, s,τ R = F s+τ 1, R ad Ḃ s,τ, R = Ḃs+τ 1, R. Recall that for s > 0, F s, R ad Ḃs, R both coicide with the homogeeous Hölder Zygmud sace of order s; see e.g. [5, Theorem 6.3.6] ad [3, Sectio 5]. We refer to e.g. [11] ad [17] for the defiitios of the saces F,R s, BMOR, E,u,R s, N,u,R s ad Q α R which we shall ot eed i the seuel ad to [17, Proositio 1 ad Theorem 2] as well as the refereces therei for details about the above coicideces. We also refer to [11] for a detailed discussio of the history of the saces i uestio. Isired by [8], we ow defie fuctio saces aalogous tof,r s,τ adḃs,τ,r through Hajłasz-tye gradiets. If u : R C is a Lebesgue-measurable fuctio ad 0 < s <, D s u stads for the class of all fractioal s-hajłasz gradiets of u, i.e. the class of seueces g = g k k Z of measurable fuctios g k : R [0, ] for which there exists a set E R of measure zero such that, for all k Z, ux uy x y s g k x+g k y wheever x, y R \E ad 2 k 1 x y < 2 k. For 0 < <, 0 <, 0 < s < ad 0 τ <, we defie Ṁs,τ, R as the class of measurable fuctios u such that u Ṁs,τ,R := if su g D s u x R, l Z 1 Bx,2 l τ ˆ Bx,2 l / g k z dz k l 1/ 1 Here ad i the seuel, X = Y for fuctio sacesx ady meas that they embed cotiuously ito each other.
106 Tomás Soto is fiite, ad for 0 <, 0 <, 0 < s < ad 0 τ <, we defie Ṅ, s,τr as the class of measurable fuctios u such that 1/ u Ṅs,τ,R := if su g D s u x R, l Z 1 Bx,2 l τ k l g k L Bx,2 l is fiite, with the obvious modificatios for = i both cases. Ṁ,R s,τ ad Ṅ,R s,τ become uasiormed saces after uotietig out the fuctios that are costat almost everywhere. For τ = 0, they coicide with the saces Ṁ, s R ad Ṅ, s R itroduced i [8]. Note that aalogous fuctio saces could well be defied o ay metric measure sace istead of just R ; see [8] ad [4] for the case τ = 0. Our mai result is the followig oitwise characterizatio of the elemets of F, s,τr ad Ḃs,τ, R for s 0,1 which geeralizes the result obtaied for τ = 0 i [8, Theorem 3.2]. Theorem 1.1. 1 s, we have Ṁs,τ ii For s 0,1, Ṅ s,τ,r = Ḃs,τ i For s 0,1,,,, ] ad τ [0, 1 + +s +s, R = F, s,τr with euivalet uasiorms., ], 0, ] ad τ [0, 1 + 1 s, we have +s,r with euivalet uasiorms. Note that if 0 < s < 1 ad τ 1/+1 s/, the s+τ 1/ 1, so i view of 1 we do ot exect that the rage of τ i the theorem could be imroved. The theorem follows from Theorems 1.2 ad 1.3 below, which are of ideedet iterest ad whose settig we shall exlai ext. Isired by [7] ad [8], we defie grad couterarts of F,R s,τ ad Ḃs,τ,R. For all N N 0 { 1}, ad m, l N 0, let { ˆ } A l N,m = φ SR : x γ φxdx = 0 whe γ N ad φ SN+l+1,m 1, R where the momet coditio is iterreted to be void whe N = 1. For s R, 0 < <, 0 <, 0 τ < ad N, m ad l as above, we defie the grad Triebel Lizorki-tye sace A l N,mF,R s,τ as the class of temered distributios f for which f A l N,m F, s,τ R 1 := su x R, l Z Bx,2 l τ ˆ Bx,2 l [ j l 1 ] 2 js su φ j fz dz φ A l N,m is fiite, with the obvious modificatio for =. For s R, 0 <, 0 <, 0 τ < ad N, m ad l as above, we defie the grad Besov-tye sace A l N,mḂs,τ, R as the class of temered distributios f for which 1 1 f A l N,m Ḃ,R s,τ := su 2 js su φ x R, l Z Bx,2 l τ j f φ A l L Bx,2 l N,m is fiite, where the suremum is take oitwise ad the obvious modificatio is made for =. The two uatities defied above are uasiorms whe N = 1, ad whe N N 0, they become uasiorms after uotietig out the olyomials with degree at most N. We obviously have A l N,mF, s,0r = A l N,mF, s R ad j l
Characterizatios of Besov-tye ad Triebel Lizorki-tye saces 107 A l N,mḂs,0,R = A l N,mḂs,R for all admissible arameters, where A l N,mF,R s ad A l N,mḂs, R are the grad Triebel Lizorki ad grad Besov saces defied i [7] ad [8]. The followig result exteds [8, Theorem 3.1]. Theorem 1.2. i Let0 < <,0 <,s R adj = /mi1,,. If the itegers N 1, m 0 ad l 0 satisfy 2 N +1 > maxs,j s ad m > maxj,+n +1, the F,R s,τ = A l N,mF,R s,τ for all τ [0, 1]. ii Let 0 <, 0 <, s R ad J = /mi1,. If the itegers N 1, m 0 ad l 0 are related by 2, the Ḃs,τ,R = A l N,mḂs,τ,R for all τ [0, 1]. iii The results of i ad ii actually hold for all τ [0, 1 + ǫ, where 2 ǫ = mi 2N +1 s, m J, 2N +1++s J > 0. The roof of the theorem is reseted i sectio 2. We also have the followig result which geeralizes the aalogous results for τ = 0 obtaied i [8, Theorem 3.2]. Theorem 1.3. Suose that s 0,1 ad m +1 or s = 1 ad m +2. i For,,, ], τ [0, 1 + 1 s ad l N +s +s 0, we have Ṁ,R s,τ = A l 0,mF,R s,τ with euivalet uasiorms. ii For, ], 0, ], τ [0, 1+ 1 s ad l N +s 0, we have Ṅs,τ,R = A l 0,mḂs,τ,R with euivalet uasiorms. The roof of this is reseted i sectio 3. Previously, the Besov-tye ad Triebel Lizorki tye saces have bee characterized through differece itegrals see e.g. [1] where the o-homogeeous versios of the saces are treated ad through Peetre-tye maximal fuctios as well as local meas i [15]. We refer to [11] for a variety of other characterizatios. We ed this sectio with some otatio covetios. We write A for the -dimesioal Lebesgue measure of a measurable set A, ad A f orf A for A 1 fxdx A wheever the latter uatity is well-defied. For two o-egative fuctios f ad g with the same domai, we write f g if f Cg for some ositive ad fiite costat C, usually ideedet of some aramters; f g meas that f g ad g f. For real umbers x ad y, we may write x y for mix,y. For a dyadic cube Q := 2 j k +[0,1], j Z, k Z, we let lq = 2 j ad x Q = 2 j k. For a ball B of R, we deote by λb, λ > 0, the ball cocetric with B with radius λ times the radius of B. We write PR for the vector sace of olyomials i R ad P N R for the vector sace of olyomials with degree at most N whe N N 0 ; whe N < 0, we let P N R := {0}. We shall freuetly abuse otatio by writig F,R s,τ for both the class of temered distributios f such that f Ḟs,τ, R is fiite as well as the fuctio sace {f S R /PR : f F, s,τ R < }, ad similarly for other saces defied i this sectio as well. Whe talkig about F, s,τr ad Ḃs,τ, R i the same setece with some idicated arameter rage, it is uderstood that the ossibility = is excluded i the case of F,R s,τ, ad similarly for the other airs of saces defied i this sectio.
108 Tomás Soto 2. Proof of Theorem 1.2 The argumet below modifies the roof of [8, Theorem 3.1]; we have left out some details that carry over uchaged. Proof of Theorem 1.2. i First, if a temered distributio f belogs to A l N,mF, s,τr, the f F, s,τ R f A l N,mF, s,τ R < sice our fixedϕis a costat multile of a elemet of A l N,m. I this sese we have Al N,mF,R s,τ F,R s,τ. Coversely, if a temered distributio f belogs to F, s,τr, it is show i the roof of [16, Lemma 4.2] that γ ψ j ϕ j f L R 2 j γ +/ s τ f F s,τ, R, for all itegers j ad multi-idices γ, so the discussio i [3,. 153 155] alies: if γ > s+τ /, the j 1 γ ψ j ϕ j f L R <, so there exists olyomials P K K N i P L R with L = s+τ / ad a olyomial P f such that f +P f = lim ψ j ϕ j f +P K K j= K with covergece i S R ; here the olyomial P f is uiue modulo P L R i the sese that if ψ i, ϕ i, PK i K N ad Pf i are, for i {1,2}, two choices of admissible fuctios as above, the P 1 f P 2 f P L R. Furthermore, L < s+ ǫ N+1, ad 2 it is easily checked that for f 1, f 2 F, s,τr, f 1 +P f1 f 2 +P f2 is a elemet of P N R if ad oly if f 1 f 2 is a olyomial. The rule f f +P f =: f thus yields a well-defied, liear ad ijective maig from the fuctio sace F, s,τr ito S R /P N R. The la ow is to show that this maig actually takes F, s,τr cotiuously ito A l N,mF, s,τr. Sice m > +N +1, by the roof of [7, Theorem 1.2] we have φ j fz 3 a QR t R Q 1/2 χ Q z lq=2 j R for all φ A l N,m, z R ad j Z, with the imlied costat ideedet of these arameters. Here the outer sum is take over all dyadic cubes Q of R with side legth 2 j ad the ier sum over all dyadic cubes R of R ; t R deotes f, ψ R with ψ R x = 2 i/2 ψ i x x R for all dyadic cubes R with side legth 2 i ; ad 4 a QR = 2 i j /2+N+1 1+2 mii,j x Q x R m for all cubes Q ad R with side legths 2 j ad 2 i resectively. The cubes Q i 3 are airwise disjoit, so for ay dyadic cube P with side legth 2 l we have ˆ [ 1 fz ] / 1/ 2 js su φ P τ j dz P φ A j l 1/ 1 ˆ [ / 2 js a P τ QR t R Q /2 χ Q z] dz P j l lq=2 j R
Characterizatios of Besov-tye ad Triebel Lizorki-tye saces 109 ˆ 1 [ Q s/ /2 P τ P Q P R a QR t R, R Q f,r s,τ a QR t R χ Q z] / dz 1/ where f, s,τr stads for the sace of seueces b Q Q of comlex umbers, idexed by the dyadic cubes of R, such that ˆ [ b Q Q f,r s,τ = su 1 ] / 1/ Q s/ 1/2 b P P τ Q χ Q x dx, P Q P where the suremum is take over all dyadic cubes P of R, is fiite. From 4 it is easy to check that for ǫ > 0 as i the statemet of art iii, we have s [ J ǫ lq lq 2 J+ ǫ ] x Q x R lr 2 a QR 1+ mi, lr maxlq,lr lr lq for all dyadic cubes Q ad R, i.e. that the oerator b Q Q a QR b R is ǫ-almost diagoal o f, s,τr [16, Defiitio 4.1]. Now τ < 1 + ǫ, so accordig 2 to [16, Theorem 4.1], the oerator described above is bouded o f, s,τr. Also, by [16, Theorem 3.1], the oerator u u, ψ Q is bouded from F s,τ Q, R to f,r s,τ. Combiig these results with the estimates above yields f A l N,m F, s,τ R R a QR t R Q R f,r s,τ Q t Q Q f s,τ,r f F, s,τ R, i.e. the maig described above is a cotiuous embeddig of F, s,τr ito A l N,mF, s,τr. ii We have A l N,mḂs,τ,R Ḃs,τ,R i the same sese as above. Coversely, if a temered distributio f belogs to Ḃs,τ, R, agai by the roof of [16, Lemma 4.2] ad [3,. 153 155] it suffices to show that f A l N,m Ḃ,R s,τ, where f is as i art i, is cotrolled by a costat times f Ḃs,τ,R. Usig 3, oe ca check that f A l N,m Ḃ,R s,τ a QR t R, ḃs,τ R Q,R where ḃs,τ, R stads for the sace of seueces b Q Q of comlex umbers, idexed by the dyadic cubes of R, such that b Q Q ḃs,τ,r = su 1 [ P P τ j=j P Q P, lq=2 j Q s/ 1/2+1/ b Q ] / 1/,
110 Tomás Soto where the suremum is take over all dyadic cubes P of R ad j P = log 2 lp, is fiite. Combiig this estimate with [16, Theorems 4.1 ad 3.1] as above the yields f A l N,m Ḃ,R s,τ f Ḃ,R s,τ. iii This is cotaied i the argumets above. 3. Proof of Theorem 1.3 We ow tur to the idetificatio of the Hajłasz-tye saces with the saces defied through grad maximal fuctios. We shall eed the followig Sobolev-tye embeddig, which is a secial case of [8, Lemma 2.3]. Lemma 3.1. Let s 0,1] ad 0 < ǫ < ǫ < s. The there exists a ositive costat C such that for all x R, k Z, measurable fuctios u ad g D s u, ˆ if c C ˆBx,2 uy c dy C2 kǫ 2 js ǫ g j y dy. k j k 2 Bx,2 k+1 To talk about the idetificatio of the saces of measurable fuctios defied through Hajłasz gradiets ad the saces of temered distributios defied through grad maximal fuctios, we eed the followig basic lemma. The techiues of the roof are similar to the oes emloyed i [7, Theorem 1.1] ad [8, Theorem 3.2]. Lemma 3.2. i Let u Ṁs,τ, R or u Ṅs,τ, R with s 0,1],, ], 0, ] ad τ 0. The u defies a temered distributio i +s the sese that for all fuctios φ SR, uφ is itegrable ad ˆ uφ Cu φ S0,N R for some iteger N deedig o, s, ad τ. ii Suose that f S R belogs to A l 0,mF,R s,τ or A l 0,mḂs,τ,R with s 0,,, ], 0, ], τ [0, ad l, m N +s 0. The f coicides with a locally itegrable fuctio i the sese that there exists a fuctio f L 1 loc R such that ˆ f,φ = fφ R for all φ SR with comact suort. Proof. i Let u deote either u Ṁs,τ,R or u Ṅ,R s,τ, whichever is fiite. Fix ǫ ad ǫ such that 0 < ǫ < ǫ < s ad <. For ay x R, Lemma 3.1 yields if c C ˆBx,1 uy c dy if so u is locally itegrable. 2 js ǫ g D s u j 2 if g D s u j 2 ˆ Bx,2 2 js ǫ g j L Bx,2 2 js ǫ u <, j 2 g j y dy
Characterizatios of Besov-tye ad Triebel Lizorki-tye saces 111 Now write N for the smallest iteger strictly greater tha max,+s+τ. For ay φ SR we the have ˆ ˆ ux ub0,1 φx dx uxφx dx u B0,1 φ S0,N + R k 1 φ S0,N u B0,1 + knˆ 2 k 1 φ S0,N u B0,1 + ˆ k 12 kn B0,2 k \B0,2 k 1 B0,2 k B0,2 k ux u B0,1 dx ux u B0,1 dx φ S0,N u B0,1 + k ˆ 2 kn ux u B0,2 i dx k 1 i=0 B0,2 i φ S0,N u B0,1 + ˆ 2 in ux u B0,2 i dx. i 0 B0,2 i As above, the itegral i the ith term of the latter sum ca be estimated by ˆ ˆ ux u B0,2 i dx if 2 iǫ 2 js ǫ g B0,2 i j y dy g D s u j i 2 B0,2 i+1 if 2 iǫ 2 js ǫ B0,2 i+1 1 gj L B0,2 i+1 g D s u j i 2 2 iǫ +τ 2 js ǫ u 2 is+τ u. Thus, ˆ R uxφx dx φ S0,N j i 2 u B0,1 + i 0 2 i+s+τ N u where the uatity iside the latter aretheses is fiite because+s+τ N < 0. ii Let f deote either f A l 0,m F, s,τ R or f A l 0,mḂs,τ,R, whichever is fiite. Fix a comactly suorted ϑ SR such ϑ = 1. It is well kow that that R f = ϑ f + ϑ j+1 ϑ j f with covergece i S R, so it suffices to show that ϑ j+1 ϑ j f L 1 B < j 0 j=0 wheever B R is a ball with radius 1. For 1, this is almost immediate: ϑ 1 ϑ is a costat multile of a elemet of A l 0,m, so ϑ j+1 ϑ j f L 1 B 2 js 2 js ϑ 1 ϑ j f L 2 js f <. B j 0 j 0 j 0 Suose ow that < < 1. If x +s R ad y Bx,2 j, j N 0, the φz := φz 2 j x y is for all φ A l 0,m a uiform costat multile of some,
112 Tomás Soto elemet of A l 0,m, ad we have φ j x = φ j y. Thus, ˆ su φ j fx su φ j fy dy Bx,2 j 2 j js 2 js su 1/ φ j f 2 j js jτ f. L Bx,2 j Usig this estimate i a arbitrary ball B of radius 1 we thus get ϑ 1 ϑ j f L 1 B ϑ 1 ϑ j f 1 L B ϑ 1 ϑ j f L B j 0 j 0 1 [ 2 j js jτ f 2 js f ] j 0 j 0 2 j 1 s f <, sice 1 s < +s1 s = 0. +s We are ow ready to give the roof of Theorem 1.3. The methods are based o the case = of the roof of [8, Theorem 3.2]. Proof of Theorem 1.3. We shall first rove i ad ii uder the assumtio that s 0,1 ad m +1. i We start by establishig the embeddig Ṁs,τ, R A l 0,mF, s,τr with the assumtio that <. To this directio, fix ǫ ad ǫ so that 0 < ǫ < ǫ < s ad < mi,. Let u Ṁs,τ, R, ad choose g D s u so that ˆ 1 / 1/ g Bx,2 l τ k y dy 2 u Ṁs,τ,R. su x R, l Z Bx,2 l k l Recall that by Lemma 3.2 above, u defies a temered distributio. Sice m +1, we have 5 su φ k uz 2 k 2 ˆ j1 ǫ 2 is ǫ g i y dy j k i j 2 Bz,2 j+1 for all k Z ad z R, where the imlied costat does ot deed o these two arameters; a similar estimate is established i [8,. 15 16], but 5 ca also be deduced i a maer similar to the roof of art i of Lemma 3.2. We ow cosider a ball B := Bx,2 l with l Z ad use the estimate 5 for z B ad k l. The terms of the sum with j l ca be estimated as i the roof of Lemma 3.2: 6 ˆ Bz,2 j+1 g i y dy 2 j gi L Bz,2 j+2 2 j jτ u Ṁs,τ,R, ad sice 1 s+ τ > 0, we get 2 ˆ j1 ǫ 2 is ǫ g i y dy j l i j 2 Bz,2 j+1 j l 2 j js+j jτ u Ṁs,τ,R 2 l1 s+ τ u Ṁs,τ,R.
Characterizatios of Besov-tye ad Triebel Lizorki-tye saces 113 For the terms with j > l, we have Bz,2 j+1 2B for all z B, so that 7 ˆ Bz,2 j+1 g i y dy M g i χ 2B z, where M is the Hardy Littlewood maximal fuctio. Combiig these estimates, we have ˆ 1 [ ] 2 ks su φ B τ k uz dz B 8 B B τ k l + 1 ˆ B τ k l B 2 k1 s 2 l1 s+ τ u Ṁs,τ,R [ 2 k1 s 2 j1 ǫ 2 is ǫ M g i χ 2B k>l l<j k i j 2 Sice 1 s > 0, the first uatity ca be estimated easily: B 2 2 k1 s l1 s+ B τ τ u Ṁs,τ,R k l z u Ṁ s,τ,r. ] dz. For the secod uatity, we exchage the order of summatio i the itegrad as follows: [ 2 k1 s 2 ] j1 ǫ 2 is ǫ M g i χ 2B z k>l l<j k i j 2 [ 2 k1 s 2 is ǫ M k>l i l 1 [ 2 k1 s 2 i1 s M k>l l 1 i k 2 +2 ks ǫ i k 1 2 is ǫ M g i χ 2B z g i χ 2B z ] g i χ 2B z l<j mik,i+2 2 j1 ǫ Now, usig Hölder s ieuality whe > 1 ad the sub-additivity of t t whe < 1, the latter uatity ca be estimated from above by a costat times +s [ 2 k1 s 1 2 i1 s 1 M g i χ 2B z k>l i l 1 l 1 i k 2 +2 ks ǫ 1 M i k 1 g i χ 2B z. 2 is ǫ 1 M ] g i χ 2B z ]
114 Tomás Soto Sice > 1 ad > 1, the Fefferma-Stei vector valued maximal ieuality thus yields ˆ 1 [ 2 k1 s 2 ] j1 ǫ 2 is ǫ M g B τ i χ 2B z dz B k>l 1 ˆ M B τ B i l 1 ˆ 1 B τ 2B i l 1 l<j k i j 2 g i χ 2B z dz g i z dz u Ṁ s,τ,r. All i all, takig the suremum over all admissible B i 8 yields u A l 0,m F, s,τ R u Ṁ,R s,τ, i.e. Ṁ, s,τr A l 0,mF, s,τr. The case = ca be hadled i a similar but easier maer. Now suose that f A l 0,mF, s,τr. Accordig to Lemma 3.2, f coicides with a locally itegrable fuctio, so fixig a comactly suorted ϑ SR such that ϑ = 1, we have f = lim R j ϑ j f oitwise almost everywhere. I articular, if x ad y are distict Lebesgue oits of f ad k Z satisfies 2 k 1 x y < 2 k, we have fx fy ϑ k fx ϑ k fy + ϑ 1 ϑ j fx + ϑ 1 ϑ j fy, j k ad sice both ϑ 1 ϑ ad z ϑz ϑz 2 k [x y] are costat multiles of elemets of A l 0,m, we get where for all k Z ad x R. Thus, fx fy x y s h k x+h k y, h k x = 2 ks j k 1 u Ṁs,τ,R su x R, l Z Bx,2 l τ su φ j fx ˆ Bx,2 l / 1/ h k z dz, ad sice for ay l Z ad z R we have the oitwise estimates h k z = 2 ks 2 js su 2 js φ j fz k l k l j k we coclude that k l j l 2 ks 1 j k k l 2 js 1 su 2 js φ j fx 2 js su φ j fz, u Ṁs,τ i.e. A l 0,m F s,τ,r Ṁs,τ,R.,R u A l 0,m F, s,τ R,
Characterizatios of Besov-tye ad Triebel Lizorki-tye saces 115 ii That Ṅ,R s,τ A l 0,mḂs,τ,R for < < ca be established usig +s the same estimates as above; here all values 0, ] are ermitted because we oly eed the L -boudedess of M, where ǫ 0,s satisfies >, istead of the Fefferma Stei maximal ieuality. Let u be a fuctio i Ṅs,τ,R ad g D s u a fractioal s-gradiet that yields the uasiorm of u u to a costat multile of at most two. Lettig B := Bx,2 l for arbitrary x R ad l Z, arguig as i 5, 6 ad 7 we have 2 ks su φ k uz 2 k1 s 2 l1 s+ τ u Ṅs,τ,R +2 k1 s 2 j1 ǫ l<j k i j 2 2 k1 s 2 l1 s+ τ u Ṅs,τ,R +2 k1 s i l 1 for all k l ad z B, which further yields 2 ks su φ k f L B B τ 2 l k1 s u Ṅ s,τ,r +2 k1 s +2 ks ǫ 2 is ǫ M g i k 1 i 2 is ǫ M 2 is ǫ 2 mik,i+21 ǫ M χ 2B B τ 2 l k1 s u Ṅ,R s,τ +2 k1 s 1 2 i1 s 1 M +2 ks ǫ 1 l 1 i k 2 i k 1 2 is ǫ 1 M l 1 i k 2 g g i B τ 2 l k1 s u Ṅ s,τ,r +2 k1 s 1 +2 ks ǫ 1 i k 1 2 is ǫ 1 g i L 2B. i L B χ 2B χ 2B 2 i1 s M g l 1 i k 2 L B L B g i χ 2B z g i χ 2B z, i χ 2B 2 i1 s 1 g i L 2B L B Dividig out by B τ 2B τ, takig first the l / orm over k l ad the takig the suremum over all admissible balls B yields u A l 0,m Ḃ s,τ,r u Ṅ s,τ,r, i.e. Ṅ, s,τr A l 0,mḂs,τ, R. The case = ca be hadled i a similar but easier maer. O the other had, if f A l 0,mḂs,τ,R where < <, the f is a locally +s itegrable fuctio ad h k k Z is a costat times a elemet of D s f, where the
116 Tomás Soto fuctios h k are as i art i. For ay ball B := Bx,2 l, we have B τ h k L B B τ 2 ks 11 1 j k 2 js 11 1 2 js su φ j f, L φ A l B 0,m so takig the l orm over k l ad the the suremum over all admissible balls B yields f Ṅs,τ,R f A l 0,mḂs,τ,R, which meas that A l 0,mḂs,τ,R Ṅs,τ,R. The case = ca be hadled i a similar but easier maer. The case with s 0,1 ad m + 1 is thus rove. The case with s = 1 ad m +2 ca be rove usig exactly the same argumet, excet for the fact that 1 s is ot ositive. To remedy this, we use the assumtio that m +2 to relace 5 with the estimate su φ k uz 2 2k 2 ˆ j2 ǫ 2 is ǫ g i y dy. j k i j 2 Bz,2 j+1 The above roof ca thus be carried out by relacig 1 s with 2 s where ecessary. We omit the details. Ackowledgemets. The author would like to thak Eero Saksma ad Yua Zhou for readig the mauscrit ad makig several valuable remarks. The author also wishes to thak IPAM at UCLA for suortig his articiatio i the rogram Iteractios Betwee Aalysis ad Geometry i Srig 2013, durig which art of this ote was writte. Refereces [1] Drihem, D.: Characterizatios of Besov-tye ad Triebel Lizorki-tye saces by differeces. - J. Fuct. Saces Al. 2012, Art. ID 328908, 1 24. [2] Essé, M., S. Jaso, L. Peg, ad J. Xiao: Q saces of several real variables. - Idiaa Uiv. Math. J. 49:2, 2000, 575 615. [3] Frazier, M., ad B. Jawerth: A discrete trasform ad decomositios of distributio saces. - J. Fuct. Aal. 93:1, 1990, 34 170. [4] Gogatishvili, A., P. Koskela, ad Y. Zhou: Characterizatios of Besov ad Triebel Lizorki saces o metric measure saces. - Forum Math. to aear. [5] Grafakos, L.: Moder Fourier aalysis. Secod editio. - Grad. Texts i Math. 250, Sriger, New York, 2009. [6] Hajłasz, P.: Sobolev saces o a arbitrary metric sace. - Potetial Aal. 5:4, 1996, 403 415. [7] Koskela, P., D. Yag, ad Y. Zhou: A characterizatio of Hajłasz Sobolev ad Triebel Lizorki saces via grad Littlewood Paley fuctios. - J. Fuct. Aal. 258:8, 2010, 2637 2661. [8] Koskela, P., D. Yag, ad Y. Zhou: Poitwise characterizatios of Besov ad Triebel Lizorki saces ad uasicoformal maigs. - Adv. Math. 226:4, 2011, 3579 3621. [9] Kozoo, H., ad M. Yamazaki: Semiliear heat euatios ad the Navier Stokes euatio with distributios i ew fuctio saces as iitial data. - Comm. Partial Differetial Euatios 19:5-6, 1994, 959 1014. [10] Mazzucato, A.: Besov Morrey saces: fuctio sace theory ad alicatios to o-liear PDE. - Tras. Amer. Math. Soc. 355:4, 2003, 1297 1364. [11] Sawao, Y., D. Yag, ad W. Yua: New alicatios of Besov-tye ad Triebel Lizorkitye saces. - J. Math. Aal. Al. 363:1, 2010, 73 85.
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