Modern harmonic analysis: singular integrals, maximal functions and Littlewood-Paley theory

Mode n Ha monic Analysis: Singula In eg als, Maximal Func ions and Li lewood-Paley heo y Be na Ramis Vich Supe ised by James R. W igh (UoE) Cosupe ised by Albe Mas Blesa (UPC) Final Bachelo ’s Thesis Bachelo ’s deg ee in Ma hema ics Bachelo ’s deg ee in Enginee ing Physics May, 2023 Als Ma is, en especial a en Ma c. iii i Abs ac The ools de eloped in he 1950s by Calde ón and Zygmund, which led o he bi h o mode n ha monic analysis, a e s udied. Some o hese concep s a e he Ha dy-Li lewood maximal unc ion oge he wi h i s Lpes ima es, in e - pola ion heo ems and, o emos , he Calde ón-Zygmund decomposi ion. These echniques allow us o show ha some singula in eg als a e well de ined and bounded on Lpspaces. Al hough Euclidean space is he o iginal se ing whe e hese ideas we e de eloped, a main aim o he p ojec is o unde s and how hese es ima es gene alise o o he measu e me ic spaces and o ec o - alued singula in eg als. Despi e being powe ul, classical Calde ón-Zygmund heo y has i s limi a- ions. Fo example, he sphe ical maximal ope a o in oduced by S ein in he 1970’s alls ou side he scope o he o iginal heo y. Howe e , one can u ilise Li lewood-Paley heo y ia squa e unc ion es ima es o p o e op imal es i- ma es o he sphe ical maximal ope a o . Ne e heless, endpoin bounds o simila singula in eg al and maximal ope a o s emain as open p oblems. Resum S’es udien les eines desen olupades en la dècada de 1950 pe Calde ón i Zygmund, les quals a en po a al naixemen de l’anàlisi ha mònica mode na. Alguns d’aques s concep es són la unció maximal de Ha dy-Li lewood jun a- men amb les se es p opie a s d’aco ació en espais Lp, eo emes d’in e polació i, sob e o , la descomposició de Calde ón i Zygmund. Aques es ècniques ens pe me en demos a que algunes in eg als singula s es an ben de inides i i ades en els espais Lp. To i que l’espai euclidià os el con ex o iginal on o es aques- es idees es a en desen olupa , un objec iu p incipal del p ojec e és en end e com aques es p opie a s es gene ali zen a al es espais mè ics de mesu a i a in eg als singula s de alo s ec o ials. To i se po en , la eo ia clàssica de Calde ón-Zygmund é les se es limi a- cions. Pe exemple, l’ope ado maximal es è ic in oduï pe S ein en la dècada de 1970 cau o a de l’abas de la eo ia o iginal. To i així, un po u ili za la eo ia de Li lewood-Paley ia desigual a s de squa e unc ions pe demos- a desigual a s òp imes pe l’ope ado maximal es è ic. Pe con a, hi ha i es en els ex ems del ang de alo s dels exponen s ppe a ope ado s in eg als singula s i maximals que omanen com a p oblemes obe s. Resumen Se es udian las he amien as desa olladas en la década de 1950 po Calde ón y Zygmund, las cuales lle a on al nacimien o del análisis a mónico mode no. Algunos de es os concep os son la unción maximal de Ha dy-Li lewood jun o con sus p opiedades de aco ación en espacios Lp, eo emas de in e polación y, sob e odo, la descomposición de Calde ón y Zygmund. Es as écnicas nos pe mi en demos a que algunas in eg ales singula es es án bien de inidas y aco adas en los espacios Lp. Aunque el espacio euclídeo sea el con ex o o iginal donde odas es as ideas ue on desa olladas, un obje i o p incipal del p oyec o es en ende cómo es as p opiedades se gene alizan a o os espacios mé icos de medida y a in eg ales singula es con alo es ec o iales. A pesa de se po en e, la eo ía clásica de Calde ón-Zygmund iene sus limi aciones. Po ejemplo, el ope ado maximal es é ico in oducido po S ein en la década de 1970 cae ue a del alcance de la eo ía o iginal. Sin emba go, uno puede u iliza la eo ía de Li lewood-Paley ia desigualdades de squa e unc ions pa a demos a desigualdades óp imas pa a el ope ado maximal es- é ico. Aun así, hay co as en los ex emos del ango de alo es de los exponen es ppa a ope ado es in eg ales singula es y maximales simila es que pe manecen como p oblemas abie os. Keywo ds: Singula in eg als, Calde ón−Zygmund, maximal unc ion, Li lewood−Paley Pa aules clau: In eg als singula s, Calde ón−Zygmund, unció maximal, Li lewood−Paley Palab as cla e: In eg ales singula es, Calde ón−Zygmund, unción maxi- mal, Li lewood−Paley AMS: 43-01 i Acknowledgemen s Fi s o all, I hank my home cen e CFIS o he aluable educa ion I e- cei ed, as well as o keeping ack o my p og ess. On op o ha , I hank CFIS and Fundació P i ada Mi -Puig o making my mobili y expe ience possible. I ha e been eeling like he whole p og amme was ailo ed o me. In he o he end, I hank he Ma hema ical Analysis g oup o he Uni e si y o Edinbu gh o welcoming me and aking me in o accoun o he weekly e en s hey o ganise: semina s, wo king g oups and pub and dinne ou ings. Bo h esea che s and PhD s uden s augh me a lo and made i easy o build iendship. In pa icula , I hank my supe iso Jim W igh o his ou s anding labou and o he pile o hou s he spen wi h me. I ake his en husiasm, kindness and modes y as an inspi ing example. In pa allel, I hank my cosupe iso Albe Mas o his a en i e acking om UPC in Ba celona and o ecommending me Jim as a men o . On he one hand, I app ecia e ha ing sha ed a la wi h Gio gio, who la ou ed my ou ine wi h humou , and his pe sonal ad ice. On he o he hand, I am g a e ul ha ou landlo d Ch is made his lawless home ou s o he academic yea . Pe acaba , ull ag ai a ma ma e, mon pa e, la me a ge mana i els meus pad ins el seu supo al lla g de o s aques s anys i la esiliència que suposa es a lluny els uns dels al es la majo ia del emps. G acis! Con en s 1 In oduc ion and p elimina ies 1 1.1 In oduc ion o he opic ...................... 1 1.2 Essen ial ini ial ools and esul s .................. 2 1.2.1 The Fou ie ans o m ................... 2 1.2.2 Weak Lebesgue spaces ................... 2 1.2.3 Con olu ion in Lebesgue spaces .............. 3 1.2.4 App oxima ions o he iden i y .............. 4 1.2.5 Schwa z unc ions and empe ed dis ibu ions ...... 4 1.2.6 Ma cinkiewicz in e pola ion heo em ........... 6 1.2.7 The Ha dy-Li lewood maximal unc ion ......... 7 1.2.8 Laye cake ep esen a ion ................. 10 1.3 Mo i a ion o s udying singula in eg als ............. 11 1.3.1 Di ichle ’s p oblem o he Laplace equa ion on he uppe hal plane .......................... 11 1.3.2 The Hilbe ans o m ................... 14 1.3.3 Connec ion o he maximal unc ion wi h he ha monic ex ension on he uppe hal plane ............. 18 1.3.4 Non angen ial con e gence ................. 20 1.4 The Ha dy-Li lewood maximal unc ion on measu e me ic spaces 21 2 Calde ón-Zygmund heo y 23 2.1 Calde ón-Zygmund decomposi ion ................. 23 2.2 Bounding singula in eg al ope a o s ............... 28 2.2.1 Fi s s eps in he Euclidean space ............. 28 2.2.2 Singula ke nels on measu e me ic spaces ........ 32 3 Vec o - alued ex ensions 39 3.1 In eg a ion o ec o - alued unc ions ............... 39 3.2 Li lewood-Paley heo y ...................... 49 3.2.1 Squa e unc ions ...................... 53 4 Applica ions and examples 59 4.1 Hö mande mul iplie s ....................... 59 4.1.1 Ellip ic di e en ial ope a o s ................ 62 4.2 Ma cinkiewicz mul iplie s ..................... 65 5 Beyond he pa adigm 69 ii iii CONTENTS 5.1 The sphe ical maximal ope a o .................. 69 CHAPTER 1 In oduc ion and p elimina ies 1.1. In oduc ion o he opic By “singula in eg al ope a o s” we mean, in he i s ins ance, con olu ion op- e a o s in Rn he ke nel unc ion o which p esen s a singula i y, say, a he o i- gin. Singula in eg als show up in a numbe o p oblems o analy ic na u e. Fo ins ance, hey gene a e solu ions o some pa ial di e en ial equa ions, ei he ellip ic o hype bolic; hey a ise in complex analysis; hey unde pin appa en ly un ela ed se ings in geome ic measu e heo y, e c. Fo decades, analys s el uncom o able when u ilising such ools because he e was no knowledge ega ding hei boundedness p ope ies. We e hey handling con inuous ope a o s on Lpspaces o no ? Ha monic analysis is he na u al amewo k o s udying singula in eg al ope a o s. In he middle and end 20 h cen u y, he ield expe ienced a bu s . B illian ma hema icians con ibu ed o he expansion o he heo y conce ning singula in eg als. Calde ón, Zygmund, Li lewood, Paley, Ha dy, Bou gain and S ein a e jus some o he mos in luen ial d i ing o ces in he ield. Due o i s ubiqui y, in he li e a u e, heo y o singula in eg als is o en jus pa ially explained, because i se es o s ep o wa d a s ages wi hin p oblems o di e en na u es. The e o e, his documen in ends o collec mos o he heo y o singula in eg als, ga he ing al oge he all o i s pieces, pu ing hem in o con ex and depic ing hei mos emblema ic applica ions. This is, ins ead o ega ding i an auxilia y ool, we cen e hem in he spo ligh . 1 8In oduc ion and p elimina ies and w i e M (x) = sup >0 (b ∗ )(x).(1.5) In a simila way, we de ine a sibling o (1.3). De ini ion 1.19. Le ∈L1 loc(Rn)be a locally in eg able unc ion. The un- cen ed Ha dy-Li lewood maximal unc ion o is de ined as Munc (x) := sup B∋x 1 |B|ZB | (y)|dy, (1.6) whe e his ime he sup emum is ake o e all balls B ha con ain x. Rema k 1.20. Clea ly, M ≤Munc . Fu he mo e, he e e se inequali y holds up o a mul iplica i e cons an . Gi en x∈Rnand a ball B∋xo adius δ, conside a second ball, cen ed a xwi h adius 2δ, hus B⊆B2δ(x). Then, 1 |B|ZB | (y)|dy ≤2n |B2δ(x)|ZB2δ(x) | (y)|dy ≤2nM (x). Now aking he sup emum o e any ball Bcon aining x, we ge Munc ≤2nM. No e ha his has been possible hanks o he ollowing p ope y o balls in Rn:|B2 (x)|= 2n|B (x)|,∀x∈Rn, > 0. We a e ba ing i in mind, since we will need a simila condi ion a he ime o swapping he se ing o a gene ic measu e me ic space. Rema k 1.21. The Ha dy-Li lewood maximal unc ion is measu able, so i makes sense o compu e i s Lpno ms: I is easy o check ha he a e ages o e balls A(x, ) := 1 |B (x)|RB (x)| (y)|dy a e con inuous unc ions o x, hence mea- su able unc ions o x. Likewise, A(x, )a e con inuous unc ions o , he e o e aking he sup emum o he a e ages o e > 0and o e ∈Q>0yields he same esul . Tha he sup emum o a coun able collec ion o measu able unc- ions is a measu able unc ion is a basic ac om measu e heo y. One o he easons why he Ha dy-Li lewood maximal unc ion u ns ou so use ul in he heo y o singula in eg als is he ollowing heo em. I shows he beha iou o such ope a o on Lp(Rn)spaces. Consequen ly, i one p o es a bound o a gi en ope a o Tby he Ha dy-Li lewood maximal unc ion, hen boundedness o Tis concluded. Theo em 1.22 (Maximal heo em in Rn).Le be a measu able complex- alued unc ion on Rn. Then: 1.2 Essen ial ini ial ools and esul s 9 (a) I ∈Lp(Rn) o 1≤p≤ ∞,M (x)is ini e a.e. x∈Rn. (b) Fo e e y λ > 0and ∈L1(Rn), λ|{x∈Rn:M (x)> λ}| ≤ A∥ ∥1,(1.7) whe e Aonly depends on he dimension n. (c) I ∈Lp(Rn),1< p ≤ ∞, hen M ∈Lp(Rn)and ∥M ∥p≤Ap∥ ∥p,(1.8) whe e Aponly depends on he dimension nand he exponen p. Es ima es o his Chebyshe kind as in (b) a e called weak- ype (1,1) es i- ma es. We shall con en ou sel es wi h he maximal unc ion being o weak- ype (1,1) and no ype (1,1). D as ically, i ∈L1(Rn)is no iden ically 0a.e., he co esponding maximal unc ion is ne e in L1(Rn). Rema k 1.23. I ∈L1(Rn)is no iden ically 0a.e., hen M /∈L1(Rn). P oo . S a by assuming ha ∈L1(Rn)has compac suppo . Le Bc(0) be a ball o adius c ha ully con ains he suppo o ,supp( )⊆Bc(0). Fo each x /∈Bc(0), we ha e M (x)≥1 |B2|x|(x)|ZB2|x|(x) | (y)|dy =1 |B2|x|(x)|ZBc(0) | (y)|dy =1 2n|B1(0)|ZBc(0) | (y)|dy 1 |x|n≡C( ) |x|n because Bc(0) ⊆B2|x|(x). This shows M (x)is no in eg able. Fo he gene al case when ∈L1(Rn)does no ha e compac suppo , es ic o a compac se Ksuch ha |Kis no iden ically 0a.e. in K. Hence, | | ≥ | |K|which implies M( )≥M( |K), so ha he p e ious a gumen applies. Since Theo em 1.22 is key in he esul s we will be using o bound singula in eg als, le us p o ide he p oo , aken om [10], Chap e 1, Sec ion 1.3. The p oo elies on a co e ing lemma. Lemma 1.24 (Vi ali- ype co e ing lemma in Rn).Le Ebe a measu able se in Rnco e ed by he union o a amily o balls o ini e diame e {Bk}k∈K. Then, om his amily we can subs ac a coun able sequence o disjoin balls {B′ k}k∈N⊆ {Bk}k∈Ksuch ha X k∈N |B′ k| ≥ C|E|. He e, C > 0is a cons an ha can be aken o be C= 5−n. 10 In oduc ion and p elimina ies P oo . (O Theo em 1.22) (a) ollows om (b) and (c) because unc ions in Lp(Rn)a e ini e a.e. Since he Ha dy-Li lewood maximal unc ion is i ially bounded on L∞(Rn), i we show ha i is a weak- ype (1,1) ope a o ( ha is, s a emen (b)) hen by Ma cinkiewicz in e pola ion heo em (Theo em 1.16), (c) ollows. So, le λ > 0, ∈L1(Rn)and conside E:= {x∈Rn:M (x)> λ}. Fo each x∈E, he e exis s a ball cen e ed a x,Bxsuch ha RBx| (y)|dy > λ|Bx|. We ha e ha he amily {Bx}x∈Eco e s E, hus by Lemma 1.24 he e exis s a coun able subse o disjoin balls {B′ k}k∈N⊆ {Bk}k∈Ksuch ha X k∈N |B′ k| ≥ C|E|, eaching λ|E| ≤ C−1λX k∈N |B′ k| ≤ C−1∥ ∥1. The la e heo em yields he celeb a ed Lebesgue di e en ia ion heo em. Co olla y 1.25 (Lebesgue di e en ia ion heo em in Rn).Whene e ∈L1 loc(Rn), lim →0 1 |B (x)|ZB (x) (y)dy = (x)a.e. x∈Rn.(1.9) The shocking aspec o he Lebesgue di e en ia ion heo em is ha we do no e en equi e egula i y in a locally in eg able unc ion o i s local a e ages a ound a poin o end o he alue on he poin , excep o a se o measu e ze o. The emaining p oo s o Lemma 1.24 and Co olla y 1.25 a e a ailable in [10], Chap e 1, Sec ion 1.3. 1.2.8. Laye cake ep esen a ion Nex , a lemma ha helps igu e ou some compu a ions. Lemma 1.26 (Laye cake ep esen a ion).Fo any 1≤p < ∞and ∈Lp(Rn), ZRn | (x)|pdx =pZ∞ 0 αp−1|{x∈Rn:| (x)|> α}| dα. P oo . S a wi h he igh hand side and p= 1. Z∞ 0 |{x∈Rn:| (x)|> α}|dα =Z∞ 0ZRn 1{| (x)|>α}(x)dx dα 1.3 Mo i a ion o s udying singula in eg als 11 Since he in eg and is posi i e, we a e en i led o apply Fubini-Tonelli heo em. Z∞ 0ZRn 1{| (x)|>α}(x)dx dα =ZRnZ∞ 0 1{| (x)|>α}(x)dα dx =ZRn | (x)|dx This way, we p o e he case p= 1. The emaining cases o punlock by conside ing a gene al ∈Lp(Rn)and g(x) = | (x)|p, wi h g∈L1(Rn)and applying he o mula o he case p= 1 o g. 1.3. Mo i a ion o s udying singula in eg als Singula in eg als a ise in a my iad o p oblems and se ings bo h in ma he- ma ics and physics. The e o e, ma hema ics needed a igo ous heo y o such ope a o s. In his sec ion, we p esen a a he simple ye in e es ing p oblem he solu ion o which demands knowledge on ou cen al opic. 1.3.1. Di ichle ’s p oblem o he Laplace equa ion on he uppe hal plane Le R+={(x, y)∈R2:y > 0} ⊂ R2be he uppe hal plane. Conside he ollowing Di ichle p oblem:      ∆u(x, y)=0,(x, y)∈R+ lim y→0u(x, y) = (x) (1.10) He e, is a eal- alued bounda y unc ion. Le i s ambien unc ion space be L2(R) o now. In conco dance, le he limi o he bounda y condi ion be unde s ood in he L2(R)sense. Conside u(x, y) = ZR ˆ ( )e2πix e−2π| |yd . (1.11) This is an absolu ely con e gen in eg al by Cauchy-Schwa z, since by Planche el heo em ˆ ∈L2(R), and e−2π|·|y∈L2(R). We a e also allowed o di e en ia e u(x, y)unde he in eg al sign wi h espec o ei he a iable hanks o he apid decay o he eal exponen ial unc ion. Indeed, ∆u(x, y) = ZR ˆ ( )(2πi )2e2πix e−2π| |yd +ZR ˆ ( )(−2π| |)2e2πix e−2π| |yd = 0 12 In oduc ion and p elimina ies hus uis a ha monic unc ion. Fo he bounda y alue, using Planche el heo em once again, we ha e ∥u(·, y)− (·)∥2=ˆ (·)e−2π|·|y−ˆ (·)2→0when y→0 by he domina ed con e gence heo em. This sol es p oblem (1.10) in he se - ing o L2(R). No ice ha no only does he solu ion (1.11) end o he ini ial da um in he L2sense bu also i is egula C2(R+). A s anda d a gumen o uniqueness o classical solu ion eads as ollows. Conside a con o mal map om he uni complex disk D o he uppe hal complex plane H(iden i ied wi h R+) like F:D→H F(z) = i1−z 1 + z. In pa icula , Fmaps he uni ci cle S1⊂C o he eal line {(x, y)∈R2: y= 0}, excep o he poin z=−1 ha “is mapped o in ini y”. Complex a iable unc ions heo y g an s ha i uis ha monic and Fis analy ic, hen u◦Fis ha monic. Bu now, u◦Fis a ha monic C2 unc ion on he open uni disk. The e o e, by he maximum p inciple o ha monic unc ions on a bounded egula domain, i wo solu ions o he Laplace equa ion ag ee on he bounda y, hen hey a e he same solu ion. Since u◦Fsol es uniquely Di ichle ’s p oblem on he uni disk, and Fis in e ible F−1(z) = i−z i+z, he solu ion on R+,u, is he unique solu ion ha anishes a in ini y. All in all, his means ha (1.11) is he classical solu ion o (1.10) ha ends o 0a in ini y. In iew o solu ion (1.11): De ini ion 1.27. Fo x∈Rnand y > 0, call Py(x) = ZRn e2πix e−2π| |yd (1.12) he Poisson ke nel in Rn. One may w i e he solu ion o (1.10) as he con olu ion o he bounda y alue unc ion wi h he Poisson ke nel in R: u(x, y) = (Py∗ )(x). Fu he mo e, a compu a ion (see Chap e 3, Sec ion 2.1, P oposi ion 5 in [10]) shows ha he Poisson ke nel (in Rn) can be ew i en as ollows. 1.3 Mo i a ion o s udying singula in eg als 13 P oposi ion 1.28. Py(x) = cn y (|x|2+y2)n+1 2 , cn=Γ(n+1 2) πn+1 2 (1.13) F om (1.13), i is s aigh o wa d o obse e ha he Poisson ke nel is a well-de ined in eg able unc ion Py∈L1(Rn) o all y > 0, meaning ha , o ∈Lp(Rn),1≤p≤ ∞, he Poisson in eg al u(x, y)=(Py∗ )(x)belongs o Lp(Rn)as a unc ion o x(as ema ked in he p elimina s, P oposi ion 1.5). Wha is mo e, Py∈Lp(Rn) o all y > 0and 1≤p≤ ∞. Besides his, i is no ha d o see ha (1.13) de ines an app oxima ion o he iden i y in he sense o De ini ion 1.6.1 Once a ha monic eal- alued unc ion on he plane u(x, y)is ob ained, a lici que y is inding he ha monic conjuga e unc ion (x, y), in he amewo k o complex a iable unc ions heo y. This ha monic conjuga e is gi en by he solu ion o he espec i e Cauchy-Riemann equa ions: ∂u ∂x =∂ ∂y ∂u ∂y =−∂ ∂x. (1.14) No ice ha u(x, y) = ZR Py(x− ) ( )d beha es nicely in o de o di e en ia e unde in eg al sign, bo h wi h espec o xand y, hanks o he apid decay o he Poisson ke nel (1.13), so in essence we a e in e es ed in checking (1.14) o he Poisson ke nel Pyand i s ha - monic conjuga e ke nel, say Qy, bo h hough o as unc ions o wo a iables P(x, y) := Py(x)and Q(x, y) := Qy(x) o (x, y)∈R+. F om he i s Cauchy- Riemann equa ion: ∂P(x, y) ∂x =−c1 2xy (x2+y2)2 Q(x, y) = Z∂P(x, y) ∂x (x, y)dy +g(x) = c1 x x2+y2+g(x) o a ce ain unc ion gdepending only on x. Imposing now he second Cauchy- Riemann equa ion (1.14), −∂Q(x, y) ∂x =−c1 x2+y2−2x2 (x2+y2)2+∂g(x) ∂x =c1 x2+y2−2y2 (x2+y2)2=⇒ g(x)≡0. 14 In oduc ion and p elimina ies So we a e le wi h Qy(x) = Q(x, y) = c1 x x2+y2=P(y, x) = Px(y),(1.15) which sol es he Cauchy-Riemann equa ions, hus (x, y) = (Qy∗ )(x)is he ha monic conjuga e o u(up o an addi i e cons an ha we se o 0 o a oid in eg abili y issues in wha ollows). No e ha is a bounded unc ion o x o e e y y > 0by Cauchy-Schwa z inequali y3. The unc ion uwas gene a ed by he bounda y unc ion . This is no he case o , ob ained as he ha monic conjuga e o u. This cons uc ion leads us o wonde ing abou he bounda y limi o (see Figu e 1.1). Fi s , ake he poin wise limi o (1.15). lim y→0Qy(x, y) = lim y→0c1 x x2+y2=1 πx i x= 0 (1.16) Fo mally, (1.16) is he ke nel o he Hilbe ans o m! We a e going o make p ecise he meaning o “H (x) = lim y→0 (x, y) = lim y→0(Qy∗ )(x)” and unco e i s p ope ies la e on. To be ai , he de ini ion o his ans o m equi es aking in o accoun he sub le y o he singula i y i p esen s a he o igin, no o men ion he lack o in eg abili y. He e is he i s ime we en- coun e a singula ke nel, which d i es us o wo ying abou i s de ini ion and he boundedness p ope ies o he con olu ion ope a o i de ines, bo h in he se ing o L2(R)and he es o he Lp(R)spaces. 1.3.2. The Hilbe ans o m The e a e se e al easons why he ope a o o he Hilbe ans o m dese es ha ing i s own name. A guably, i ca ied he i s singula ke nel ma hema i- cians wo ied abou . In e es ingly, i is a use ul ool in applied sciences such as spec oscopy in chemis y Due o he ac ha he unc ion :R→R, (x) = 1 xis no in eg able a ound he o igin, i makes no sense o in eg a e i s aigh away agains any Lp(R) unc ion. Fo ins ance, le g:R→R,g(x) = 1[−1,1](x)∈Lp(R) o any 1≤p≤ ∞, ye ZR (x)g(x)dx =Z1 −1 1 xdx 3Wha is mo e, i ∈Lp(R) o a ixed 1≤p < ∞and no icing ha Qy∈Lq(R) ∀1< q ≤ ∞ and y > 0, by applying he well known Young con olu ion inequali y one ge s ha Qy∗ ∈L (R)∀ > p. 1.3 Mo i a ion o s udying singula in eg als 15 H ∆u= 0 ∆ = 0 Figu e 1.1: Rise o he Hilbe ans o m in Di ichle ’s p oblem o Laplace’s equa ion. Fi s , le be de ined in he axis y= 0. Ob ain usuch ha ∆u= 0 and is i s bounda y alue. Then, ob ain he conjuga e ha monic unc ion o u( he one ha u ns u(x, y) + i (x, y)in o a holomo phic unc ion on he complex plane, se ing he addi i e cons an o 0). Finally, ob ain he Hilbe ans o m o ,H by compu ing he limi lim y→0 (x, y). is no compu able in he Lebesgue sense. Howe e , we s ill ha e hope o a de ini ion o he con olu ion o agains o he unc ions because o he ac ha is an odd unc ion. We would like o exploi his ea u e o achie e enough cancella ion o o e come he e ec o he singula i y. De ini ion 1.29. Le :R→Rbe a unc ion wi h a singula i y a ound he o igin. I s in eg al is compu ed in he p incipal alue sense as p. . ZR (x)dx := lim ϵ→0+Z|x|>ϵ (x)dx. This way, i ·1Bϵ(0)c∈L1(R)∀ϵ > 0, hen he limi o he abo e in eg al as ϵ→0+may also yield a eal numbe . P oposi ion 1.30. De ine he linea unc ional ac ing on φ∈S: p. . 1 x(φ) := p. . ZR φ(x) xdx This yields a empe ed dis ibu ion. 16 In oduc ion and p elimina ies P oo . In oduce a φ(0) e m aking ad an age o he oddness o 1 xand spli he domain o in eg a ion o a sepa a e ea men o he in eg abili y issues. p. . 1 x(φ)≤lim ϵ→0+Zϵ<|x|<1 |φ(x)−φ(0)| |x|dx +Z|x|≥1 |φ(x)| |x|dx ≤lim ϵ→0+Zϵ<|x|<1 |φ′(c)||x| |x|dx +Z|x|≥1 |xφ(x)| |x|2dx ≤ ∥ φ∥0,1Z|x|<1 dx +∥φ∥1,0Z|x|≥1 1 |x|2dx =C1∥φ∥0,1+C2∥φ∥1,0 We used he mean alue heo em (wi h some c∈[0, x]appea ing) and he semino ms om De ini ion 1.8. So he esul ollows om Theo em 1.10. The p incipal alue is a ool ha allows us o p ope ly de ine he Hilbe ans o m. De ini ion 1.31. Le φ∈S(Rn)be a Schwa z unc ion. The Hilbe ans- o m o φ,Hφ is de ined as he con olu ion o he p incipal alue dis ibu ion 1 πp. . 1 xagains he Schwa z unc ion φ(see De ini ion 1.12): Hφ(x) := 1 πp. . 1 x∗φ(x) = 1 πp. . ZR φ(x−y) ydy Na u ally, we would like o ake ad an age o he densi y o S(Rn)in Lp(Rn),1≤p < ∞ o ex end his de ini ion o any Lp(Rn) unc ion. This is one o he goals o Chap e 2. Al e na i ely, i is possible o de ine he Hilbe ans o m as a mul iplie ope a o . De ini ion 1.32. In he se ing o L2(Rn), whe e Planche el heo em o he Fou ie ans o m holds, an essen ially bounded unc ion m∈L∞(Rn)is called amul iplie , when we conside an associa ed ope a o Tm:L2(Rn)→L2(Rn) such ha d Tm (ξ) = m(ξ)ˆ (ξ),∀ ∈L2(Rn). The in ui ion one ge s om he de ini ion is ha he ole o a mul iplie unc ion is o modi y he equency spec um o a unc ion. The opic o mul i- plie s is once again a b oad wo ld on i s own. The e a e some examples linking his opic wi h he heo y o Chap e 2in Chap e 4. De ini ion 1.33. The Hilbe ans o m is he ope a o mapping H:L2(Rn)→ L2(Rn)de ined ia he mul iplie m(ξ) = −isgn(ξ). This is, d H (ξ) := −isgn(ξ)ˆ (ξ). 1.3 Mo i a ion o s udying singula in eg als 17 Again, i is desi able o somehow ex end his de ini ion o he es o Lp(Rn) spaces, elying, o example, on he ac ha L2(Rn)∩Lp(Rn)is dense in Lp(Rn) o 1≤p < ∞. No e he ollowing cohe ence: P oposi ion 1.34. De ini ion 1.31 and de ini ion 1.33 a e equi alen : 1 πp. . 1 xˆ = −isgn(·) in he sense o empe ed dis ibu ions. P oo . Le us i s show ha 1 πp. . 1 x= lim y→0Qy holds in he sense o empe ed dis ibu ions, whe e Qydeno es he conjuga e Poisson ke nel (unde s and he igh hand side as in he ollowing compu a ion). Take any φ∈S(Rn)and combine he limi on he igh -hand side wi h he one in he p incipal alue dis ibu ion: 1 πp. . 1 x(φ)−lim y→0Qy(φ) = lim y→0Z|x|>y 1 πx −1 π x x2+y2φ(x)dx + lim y→0Z|x|≤y−1 π x x2+y2φ(x)dx = lim y→0Z|x|>11 π 1 x(x2+ 1)φ(yx)dx + lim y→0Z|x|≤1−1 π x x2+ 1φ(yx)dx yields a e a scaling change o a iables. Realise ha 1(x) := 1 π 1 |x|(x2+ 1)1{|x|>1}(x) 2=1 π |x| x2+ 11{|x|≤1}(x) a e bo h in eg able unc ions. This, combined wi h he ac ha φis a Schwa z unc ion, allows us o in oke he domina ed con e gence heo em ha leads o 1 πp. . 1 x(φ)−lim y→0Qy(φ) =Z|x|>11 π 1 x(x2+ 1)φ(0)dx +Z|x|≤1−1 π x x2+ 1φ(0)dx = 0 because o oddness. 24 Calde ón-Zygmund heo y Theo em 2.1 (Calde ón-Zygmund Lemma in Rn).Le ∈L1(Rn)and λ > 0. (a) The e exis s a pa i ion Rn=F⊔Ω, such ha (b) | (x)| ≤ λa.e. x∈F, and (c) Ωcan be w i en as a coun able union o cubes Qkwi h disjoin in e io , Ω = Fk∈NQkmo eo e sa is ying λ≤1 |Qk|ZQk | (x)|dx ≤2nλ, ∀k∈N.(2.1) Essen ially, wha his heo em ells us is: o any wild unc ion jus subjec o being Lebesgue in eg able, he e exis s a decomposi ion o i s domain, Rn, in o wo disjoin se s such ha is essen ially bounded in one o hem, and al hough i may no be in he o he , he a e ages o o e some (almos ) disjoin cubes a e bounded. Al hough a gene alisa ion o his heo em is going o come up la e , we lea e he e he p oo , which consis s in an elegan s opping- ime a gumen wo h explaining. P oo . Mesh Rnin o cubes {Q0 k}k∈Nwi h disjoin in e io and o he same size, la ge enough so ha he a e ages o | |a e bounded abo e by he gi en λon all o he cubes in he mesh: 1 |Q0 k|ZQ0 k | (x)|dx < λ ∀k∈N. This is possible because is in eg able, 1 |Q0 k|ZQ0 k | (x)|dx ≤∥ ∥1 |Q0 k|, so choose he size o he cubes such ha |Q0 k|>∥ ∥1 λ. We a e going o un an algo i hm. Se Ω = ∅and he s ep s= 1. We spli each o he cubes {Q0 k}k∈Nin o 2ndyadic descenden cubes o he same size {Q1 k}k∈N. Case 1: Fo each descenden cube in s ep s( ha is, o each k∈Z), i 1 |Qs k|ZQs k | (x)|dx > λ, (2.2) hen Qs kis selec ed o ake pa in he se Ω, so ac ualise Ωnew = Ωold ∪Qs k. Fo such a cube Qs k, assume ha Qs−1 is i s di ec ances o cube. Then, by (2.2) and he ac ha Qs−1 ell in o Case 2, λ < 1 |Qs k|ZQs k | (x)|dx ≤2n |Qs−1 |ZQs−1 | (x)|dx ≤2nλ 2.1 Calde ón-Zygmund decomposi ion 25 which p o es (2.1) o Qs k. Case 2: Ins ead, i 1 |Qs k|ZQs k | (x)|dx ≤λ, hen we i e a e and u he di ide Qs kin o 2niden ical descenden cubes, and check in o which o he wo cases alls each o hem. Ac ualise snew =sold + 1 and le he algo i hm un ecu si ely. This way, we ob ain a pa i ion like in (a), plus (c) has been e i ied o all cubes Qs k ha we e selec ed o Case 1. Fac (b) yields om he Lebesgue di e en ia ion heo em (Co olla y 1.25)1because (x) = lim j→∞ 1 |Qj k|ZQj k | (x)|dx ≤λ∀k∈N since he e all o he in e ening cubes all in o Case 2. Le us now s a e he c ucial Calde ón-Zygmund decomposi ion o an in e- g able unc ion as a co olla y. Co olla y 2.2. Le ∈L1(Rn)and λ > 0. The e exis s a decomposi ion o as sum o wo unc ions, =g+bsuch ha gis an essen ially bounded unc ion, and such ha he suppo o bcan in i s u n be decomposed in o a union o cubes wi h disjoin in e io , in each o which bhas ze o a e age. Mo e p ecisely, he e exis s a decomposi ion =g+bsuch ha g(x)≤2nλ a.e. x ∈Rn,1 |Qk|ZQk b(x)dx = 0 ∀k∈N, 1 |Qk|ZQk |b(x)|dx ≤2nλ, supp(b) = G k∈N Qk, b ≤ . (2.3) gand ba e usually e e ed o as he “good” and he “bad” pa o . I is a wo hwhile ade o gain such boundedness p ope ies on gand b o he p ice o ha ing o deal wi h wo unc ions ins ead o only one. P oo . Fix any λ > 0and apply Calde ón-Zygmund Lemma, Theo em 2.1, o ge he decomposi ion Rn=F⊔Ω, as in he s a emen . De ine g(x) := ( (x), x ∈F 1 |Qk|RQk (x)dx, x ∈Qk,∀k∈N. 1Being me iculous, he Lebesgue di e en ia ion heo em was p esen ed in Chap e 1in he o m o a e ages o e balls, no cubes. Howe e , his heo em s ill holds i he amily o se s o e which one a e ages is so-called egula . In pa icula , he amily o all cubes in Rnis a egula amily. Such condi ion o being egula esembles he doubling condi ion. Fo de ails, see [10], Chap e 1, Sec ion 1.8. 26 Calde ón-Zygmund heo y Di ec ly no ice ha gis essen ially bounded by 2nλ. Consis en ly, le b(x) := (x)−g(x) = (0, x ∈F (x)−1 |Qk|RQk (x)dx, x ∈Qk,∀k∈N, which immedia ely implies (2.3) and he p oo is comple e. This is he igh app oach o ge he es ima es needed in he coming-up Theo em 2.9. In con as , he se ing o his chap e , unless o he wise speci ied, is a gene ic σ- ini e measu e space o e a me ic space equipped wi h a egula measu e ((X, d),Σ, µ)enjoying he doubling p ope y. Theo em 2.3 (Calde ón-Zygmund lemma, gene al se ing).Le ∈L1(X) and λ > 0. (a) The e exis s a pa i ion o he space X=F⊔Ω,Fbeing a closed se and Ωan open se , such ha (b) | (x)| ≤ λa.e. x∈F, and (c) Ωcan be w i en as a coun able disjoin union o smalle se s Ω = Fk∈NΩk mo eo e sa is ying 1 µ(Ωk)ZΩk | (x)|dµ(x)≤Cλ, ∀k∈N(2.4) o some eal cons an C > 0. In he same way as in he case o Rn: Co olla y 2.4. Le ∈L1(X)and λ > 0. The e exis s a decomposi ion o as sum o wo unc ions, =g+bsuch ha gis an essen ially bounded unc ion, and such ha he suppo o bcan in i s u n be decomposed in o a union disjoin se s {Qk}k∈N, in each o which bhas ze o a e age. Mo e p ecisely, he e exis s a decomposi ion =g+bsuch ha g(x)≤Cλ a.e. x ∈X, 1 µ(Qk)ZQk b(x)dµ(x)=0 ∀k∈N, 1 µ(Qk)ZQk |b(x)|dµ(x)≤Cλ, supp(b) = G k∈N Qk, b ≤ (2.5) o some C > 0. 2.1 Calde ón-Zygmund decomposi ion 27 An analogous a gumen as ha in Co olla y 2.2 p o es he new co olla y. No ice he sligh di e ences wi h espec o he o me Theo em 2.1 in he se ing o Rn. The downside o Theo em 2.3 is ha i lacks he lowe bound o he a e ages o e he Ωk. As an upside, we ge a opological cha ac e iza ion o he se s Fand Ω. These disag eemen s owe o he ac ha he p oo o Calde ón-Zygmund Lemma 2.1 canno be analogously gene alised o a gene ic measu e space, because pa i ioning he space Rnin o a pe ec ly i ing mesh o disjoin cubes (up o a se o measu e ze o) is a speci ici y o Rn. Acco dingly, Theo em 2.3 demands a o ally di e en p oo , hea ily elying on he Ha dy- Li lewood maximal unc ion. Fu he mo e, we a e going o be needing a con enien Vi ali- ype co e ing lemma o he p oo . Le us in oduce some no a ion: le B=B (x)be a ball o adius and cen e xin a me ic space. Deno e by B∗an enla ged dila ion o he ball B, sha ing he same cen e. Tha is, say c∗>1is he dila ing ac o , hen B∗=Bc∗ (x). Lemma 2.5 (Vi ali- ype co e ing lemma, gene al se ing).Le (X, d)be a me - ic space enjoying he he ollowing engul ing p ope y: he e exis s c1>1such ha o all x, y ∈Xand δ > 0 Bδ(x)∩Bδ(y)=∅=⇒Bδ(y)⊂Bc1δ(x). Le F⊆Xbe a nonemp y closed se . Then, he e exis s a sequence o balls (Bk)k∈Nand wo amilies o each dila ions, (B∗ k)k∈Nand (B∗∗ k)k∈N, such ha (a) (Bk)k∈Na e pai wise disjoin , (b) SkB∗ k=Fc, and (c) B∗∗ k∩F=∅,∀k. The in e es o his lemma is ha each amily o balls exhibi s di e en use ul p ope ies: (Bk)ka e pai wise disjoin , (B∗ k)kga he oge he o eco e Fcand he las dila ion is la ge enough so ha any ball in (B∗∗ k)kmee s he bo de o F. We cla i y ha each amily o dila ions sha es he same dila ion ac o . Check [12], Chap e 1, Sec ion 3.2., o he p oo o he lemma. Rema k 2.6. I is con enien o ex ac ano he sequence o se s om Lemma 2.5. Take he i s elemen in (B∗ k)k∈Nand de ine Q1:= B∗ 1. Nex , de ine Q2:= B∗ 2∖(Q1). By an induc i e p ocess, build Qk:= B∗ k∖ k−1 [ j=1 Qj!. The sequence (Qk)khas he p ope ies ha hei se s a e pai wise disjoin and SkQk=Fc. We paid he p ice ha Qka e no longe balls, bu o he less elemen a y se s. 28 Calde ón-Zygmund heo y The name Qko such new se s is inspi ed by hei ole in he p oo o Theo em 2.12, which mimics he one ca ied ou by he cubes in he p oo o he X=Rncase. P oo . (O Calde ón-Zygmund Lemma, Theo em 2.3)Le ∈L1(X)and ix λ > 0. Choose F:= {x∈X:M (x)≤λ}and so Ω := {x∈X:M (x)> λ}. Accoun ing ha a e aging o e balls A(x, ) := 1 µ(B (x)) RB (x)| (y)|dµ(y)is a con inuous unc ion o x, and ha he measu e µis assumed o be egula , i is easy o see ha he se Fso de ined is a closed se ; hence, Ωopen. Wo king i s wi h he se F, le us use he Lebesgue di e en ia ion heo em 1.40 (and Theo em 1.39). λ≥M (x) = sup >0 1 µ(B (x)) ZB (x) | (y)|dµ(y) ≥lim →0 1 µ(B (x)) ZB (x) | (y)|dµ(y) = | (x)|,a.e. x∈F, so (b) is shown. In o de o p o e (c), ake in o accoun Lemma 2.5 and Rema k 2.6. Fo each Bkin he sequence (Bk)k∈Ngi en by he lemma, choose a poin pk∈B∗∗ k∩F ( he lemma ensu es his se is nonemp y). By he de ini ion o F, λ≥M (pk)≥CuncMunc (pk)≥Cunc µ(B∗∗ k)ZB∗∗ k | (x)|dµ(x) ≥Cunc µ(B∗∗ k)ZQk | (x)|dµ(x)≥Cunc C∗∗ 1 µ(Qk)ZQk | (x)|dµ(x). The wo las inequali ies s em om he ac s ha Bk⊆Qk⊆B∗∗ kand he doubling p ope y: µ(Qk)≤µ(B∗∗ k)≤C∗∗µ(Bk)≤C∗∗µ(Qk). Since (Qk)k∈N pa i ion Ω,Ω = FkΩk≡FkQk, he p oo is comple e. No e ha his p oo un eils he p ecise iden i y o he se s Fand Ω, which a e de ined in e ms o he Ha dy-Li lewood maximal unc ion. In exac ly he same way as in Co olla y 2.2, he Calde ón-Zygmund decom- posi ion o an in eg able unc ion ∈L1(X)is deduced. 2.2. Bounding singula in eg al ope a o s 2.2.1. Fi s s eps in he Euclidean space He e is whe e he Lpboundedness heo em o con olu ion- ype ope a o s will shine. We will ge o he desi ed heo em in he b oades se ing a e in o- 2.2 Bounding singula in eg al ope a o s 29 ducing he speci ic hypo heses and i s co esponding e sion in Rnwhich se es as inspi a ion. To s a wi h, gene al con olu ion- ype ope a o s in Rno e Lp(Rn) unc- ions looks like (T )(x) = ZRn K(x−y) (y)dy, o ∈Lp(Rn),(2.6) being K:Rn→Ca unc ion called he con olu ion ke nel. Assume o he momen he ideali y ha he ke nel is in eg able, K∈L1(Rn). Then, •i p= 1,T ∈L1(Rn)and mo eo e , ∥T ∥1=∥K∥1∥ ∥1by Fubini heo em. •i 1< p ≤ ∞,T ∈Lp(Rn)and mo eo e , ∥T ∥p≤ ∥ K∥1∥ ∥pby Minkowski in eg al inequali y. As easy as ha o an in eg able ke nel: Tis a bounded ope a o on Lp(Rn) o any 1≤p≤ ∞. Howe e , ou in e es elies on ke nels ha a e no in eg able due o a single singula i y, say in he o igin o he ke nel, K:Rn∖{0} → C. Always keep in mind he example o he Hilbe ans o m, (1.16). The in eg abili y issue o singula ke nels is al eady a hassle o he asso- cia ed ope a o o be de ined. One can encoun e many di e en app oaches digging in he li e a u e. The ypical s a egies o o e come such p oblems a e: (a) The ollowing is inspi ed by he concep o p incipal alue. Conside he unca ions o he ke nel a ound he singula i y Kϵ(x) := (K(x)i |x| ≥ ϵ 0i |x|< ϵ and so Tϵ (x) := ZRn Kϵ(x−y) (y)dy This way, i is usually a simple ma e o show ha Tϵis well de ined and bounded on some Lp(Rn)spaces o all ϵ > 0. The subsequen p ocedu e is de ining T:= limϵ→0Tϵin a sui able way and p o ing i inhe i s he boundedness p ope y om Tϵ.Usually, hese kind o app oaches in ol e se e al uni o m and Lp(Rn)-no m con e gence a gumen s. See [10], Chap e 2. (b) En e he wo ld o empe ed dis ibu ions. In his app oach, one would say ha ini ially T∈S∗(Rn)is a empe ed dis ibu ion. Howe e , one 30 Calde ón-Zygmund heo y imposes ha Tag ees wi h a measu able unc ion Kaway om he o igin, namely ⟨K, φ⟩=ZRn K(x)φ(x)dx o any φ∈S(Rn)wi h 0/∈supp(φ)( his way, he singula i y is dodged). I is implici ly assumed ha Kφ ∈L1(Rn). Then, one de ines he ope a o Tby means o he con olu ion o he empe ed dis ibu ion K∈S∗(Rn) agains he Schwa z unc ion φ∈S(Rn), as in De ini ion 1.12; he esul o such a con olu ion is a smoo h unc ion wi h a mos polynomial g ow h. Tφ(x) := K∗φ(x) Fo his s a egy, one would hope ully p o e he a p io i e sion o he desi ed es ima es o such a de ini ion o Tand hen use a densi y a gumen o he class o Schwa z unc ions o ob ain he es ima es in he se ing o Lp(Rn)spaces o unc ions. (c) We a e going o ollow an app oach aligned wi h he p e ious one, bu wo king wi h ano he class o dense unc ions: Lp(Rn)∩Lq(Rn). I is simila o he p e ious one in he sense ha i ies o a oid he singula i y by choosing con enien ly suppo ed unc ions. See [3], Chap e 5. Essen ially, wo hypo hesis on he ke nel a e equi ed o succeed in ou mission. The i s o hem is a oo hold on a pa icula Lp(Rn)space: a e co ec ly de ining he Tope a o , assume ha T:Lq(Rn)→Lq(Rn)bound- edly: ∥T ∥q≤A∥ ∥q. This se es as an ing edien o in oke Ma cinkiewicz in e pola ion heo em. The second hypo hesis is a echnical one. De ini ion 2.7. A con olu ion ke nel Kon Rnis said o sa is y he Hö man- de condi ion i sup |y|>0Z|x|≥2|y| |K(x−y)−K(x)|dx =B < ∞,(2.7) whe e B > 0is a ini e numbe . Since he in eg al is compu ed o e he egion {x∈Rn:|x|>2|y|}, he singula i y o he ke nel is a oided bo h o x−y,|x−y|≥|x|−|y| ≥ 2|y|−|y|= |y|>0and o x,|x| ≥ 2|y|>0. In some sense, we a e asking ha he global a ia ion o he ke nel is no so wild ha is no in eg able. Ne e heless, he Hö mande condi ion is usually seen as a weakened e sion o he s onge condi ion |∇K(x)| ≤ C |x|n+1 (2.8) 2.2 Bounding singula in eg al ope a o s 31 o K∈C1(Rn∖{0}). E en hough condi ion (2.8) is nea e han he Hö man- de condi ion, we a e s ill in e es ed in keeping he la e since some ke nels ul il he Hö mande condi ion, bu no condi ion (2.8) (in his ega d, we discuss he Hö mande mul iplie s in Chap e 4, Sec ion 4.1). P oposi ion 2.8. I a ke nel K∈C1(Rn∖{0}) ul ils (2.8), hen i sa is ies he Hö mande condi ion (2.7). P oo . Using he mul idimensional mean alue heo em o K, Z|x|≥2|y| |K(x−y)−K(x)|dx ≤Z|x|≥2|y| |∇K(c)||y|dx o some cin he segmen joining x−yand x. We may assume ha his segmen does no con ain he o igin, because he se o x o which he segmen con ains he o igin is o measu e ze o. Applying he g adien bound, Z|x|≥2|y| |∇K(c)||y|dx ≤CZ|x|≥2|y| |y| |c|n+1 dx. Now cis compa able in modulus o x, since i lies in he segmen joining xand x−yand bo h endpoin s a e compa able in modulus o x, (1 2|x|≤|x−y| ≤ 3 2|x|). This means |c| ≥ A|x| o a uni e sal cons an A. CZ|x|≥2|y| |y| |c|n+1 dx ≤AC Z|x|≥2|y| |y| |x|n+1 dx Wi h he change o a iables z=x |y|, AC Z|x|≥2|y| |y| |x|n+1 dx =AC Z|z|≥2 1 |z|n+1 dz which is now a ini e numbe independen o y, yielding he uni o m bound. We a e in he posi ion o s a ing a heo em o bound singula in eg als on Lp(Rn)spaces in he se ing o Rn. Theo em 2.9. Le Tbe a linea ope a o such ha he e exis s a measu able ke nel unc ion Ksuch ha T (x) = ZRn K(x−y) (y)dy con e ges absolu ely whene e ∈L2(Rn)and x /∈supp( ). Suppose he ol- lowing: (i) Tis bounded on L2(Rn):∥T ∥2≤A∥ ∥2. 32 Calde ón-Zygmund heo y (ii) The ke nel K e i ies he Hö mande condi ion (2.7)wi h cons an B. Then, (a) Tis bounded on Lp(Rn),1<p<∞, and ∥T ∥p≤Cn,p ∥ ∥p o ∈Lp(Rn)and Cn,p only depending on n,p,Aand B. (b) Tis weak- ype (1,1), i.e, o all λ > 0and ∈L1(Rn) λ|{x∈Rn:|T (x)|> λ}| ≤ Cn∥ ∥1 whe e Cnis a cons an only depending on he dimension n,Aand B. The p oo s o (some sligh a ian s o ) his heo em a e a ailable in [10], Chap e 2, Sec ion 2 and [3], Chap e 5, Sec ion 1. O en, condi ion (i) o Theo em 2.9 is encapsula ed in o he , pe haps mo e p ac ical, hypo heses, such as he Fou ie ans o m o he ke nel being uni o mly bounded, ˆ K≤A. We a e skipping he p oo o he heo em because a mo e gene al one is going o be discussed in de ail in due ime. Co olla y 2.10. The Hilbe ans o m is a bounded ope a o on Lp(R) o 1<p<∞. P oo . Since he mul iplie unc ion o he Hilbe ans o m mH(ξ) = −isgn(ξ) is a bounded unc ion, condi ion (i) in Theo em 2.9 is ul illed. Tha he g a- dien condi ion (2.8) is sa is ied by he ke nel 1 x, and hus so is he Hö mande condi ion o (ii), is s aigh o wa d and comple es he p oo . 2.2.2. Singula ke nels on measu e me ic spaces The i s signi ican issue ha we encoun e when a emp ing o gene alise wha was done in he p e ious sec ion is he ac ha in a gene al ((X, d),Σ, µ), sub ac ing poin s x−ymakes no sense because we lack a g oup s uc u e. Thus, we can no longe unde s and con olu ion as we did in Rn. Since we una oidably need wo a iables o inpu in o K(an in eg a ion a iable and a a iable o he esul ing unc ion T ), we a e going o ge a ound his obs acle by conside ing 2- a iable ke nels, K(x, y), ha a e assumed o blow up and be oublesome a ound x=y. Immedia ely a e wa ds, we need o e o mula e he Hö mande condi ion. As poin ed ou , we subs i u e K(x−y)by K(x, y). Wha do we swap K(x) o , hen? We should no gi e p e e ence o any pa icula poin in X; we do 2.2 Bounding singula in eg al ope a o s 33 no e en ha e an o igin now, so ins ead we a e going o in oduce K(x, y0) and include y0in he sup emum. This choice is going o lead o success in he co esponding p oo . Finally, we shall d op he 2 ac o in he in eg a ion domain and in oduce a ce ain cons an C > 1 o e sa ili y and gene ali y. All in all: De ini ion 2.11. A ke nel Kon he p oduc measu e space ((X, d),Σ, µ)× ((X, d),Σ, µ)is said o sa is y he Hö mande condi ion i sup y,y0∈XZd(x,y)≥Cd(y,y0) |K(x, y)−K(x, y0)|dµ(x) = B < ∞(2.9) o some cons an s B > 0and C > 1. Theo em 2.12. Le Tbe a linea ope a o such ha he e exis s a measu able ke nel unc ion Ksuch ha T (x) = ZX K(x, y) (y)dµ(y) con e ges absolu ely whene e ∈Lq(X)and x /∈supp( ). Suppose he ollow- ing: (i) Tis bounded on Lq(X) o some 1< q ≤ ∞:∥T ∥q≤A∥ ∥q. (ii) The ke nel measu able unc ion K e i ies he Hö mande condi ion (2.9) wi h cons an s Band C. Then, (a) Tis bounded on Lp(X)∩Lq(X),1< p < q, and ∥T ∥p≤Cp∥ ∥p o ∈Lp(X)∩Lq(X), and Cponly depending on p,q,A,Band C. (b) Tis weak- ype (1,1) in he sense ha o any λ > 0, λ µ{x∈X:|T (x)|> λ} ≤ C1∥ ∥1 o ∈L1(X)∩Lq(X)and some cons an C1depending on q,A,Band C. P oo . We aim a showing ha Tis weak- ype (1,1) on L1(X)∩Lq(X)so ha Ma cinkiewicz in e pola ion heo em applies. Wi h his pu pose, le λ > 0and ake ∈L1(X)∩Lq(X). Since ∈Lq(X),T is well de ined and belongs o Lq(X)by assump ion (i). Also, hanks o he ac ha ∈L1(X), we a e allowed o in oke he Calde ón-Zygmund decomposi ion, Co olla y 2.4, on a 40 Vec o - alued ex ensions Namely, he unc ion ∥F∥B:x→ ∥ F(x)∥Bis measu able1as a eal- alued unc ion. On he o he hand, by assuming (i), we a e enabling he o hcoming de - ini ion o in eg a ion o ec o - alued unc ions ia an ex ension om a dense class. No e ha we could ha e assumed, wi h less gene ali y, ha Bis sepa able s aigh away. We app ecia e ha ing an analogue o he Lebesgue spaces. De ini ion 3.2. Le (X, Σ, µ)be a measu e space and le Bbe a Banach space. Fo e e y 1≤p≤ ∞ le Lp B(X) := {F:X→Bmeasu able :ZX ∥F(x)∥p Bdµ(x)<∞} (3.1) (in he con en ional unde s anding ha he in eg al is subs i u ed by an essen ial sup emum when p=∞) be he se o (equi alence classes o ) ec o - alued measu able unc ions he p- h powe o he no m o which is in eg able (in he eal Lebesgue sense). Call hem ec o - alued Lebesgue spaces, o simply Lebesgue spaces i no con usion may occu . I u ns ou ha o all 1≤p≤ ∞ and Banach space B,Lp B(X)is also a Banach space, equipped wi h he expec able no m ∥F∥Lp B(X)=ZX ∥F(x)∥p Bdµ(x)1 p o ∥F∥L∞ B(X)= ess sup x∈X ∥F(x)∥B. Simila ly, one may de ine he weak-Lp B(X)spaces. De ini ion 3.3. Le (X, Σ, µ)be a measu e space and le Bbe a Banach space. Fo e e y 1≤p≤ ∞, le Lp,∞ B(X) := {F:X→Bmeasu able : sup λ>0 λpµ({x∈X:∥F(x)∥B> λ})<∞} (3.2) 1Cha ac e ise he no m by duali y: ∥F(x)∥B= sup∥b∗∥B∗≤1| ⟨b∗, F(x)⟩ | (we can ake B∗ o be he dual space o BFins ead o B, whene e x∈Xallows us). We in oke migh y heo ems om unc ional analysis. Fo ou Banach space B, we know ha he dual closed uni ball B∗ 1(0) is compac in he weak∗ opology. No only his, bu since Bis sepa able, B∗ 1(0) equipped wi h he weak∗ opology is me izable. Now, any compac me ic space is sepa able, hus B∗ 1(0) wi h he weak∗ opology is sepa able. By de ini ion o he weak∗ opology, he maps b∗→ | ⟨b∗, F(x)⟩ | a e con inuous in such a opology, meaning ha we can ake a coun able dense se in {b∗:∥b∗∥B∗≤1}, say {b∗ n}n∈N o w i e ∥F(x)∥B= supn∈N| ⟨b∗ n, F(x)⟩ | and conclude ha ∥F(x)∥Bis a measu able unc ion o xowing o he sup emum o e a coun able se and he assump ion ha | ⟨b∗ n, F(x)⟩ | a e measu able unc ions o x µ-a.e. 3.1 In eg a ion o ec o - alued unc ions 41 ( he con en ion o p=∞is L∞,∞ B(X) = L∞ B(X)) be he se o (equi alence classes o ) ec o - alued measu able unc ions such ha such sup emum is ini e. Name hem ec o - alued weak Lebesgue spaces, o simply weak Lebesgue spaces i he e is no isk o con usion. As an example, ake ∈Lp(X)(a classic complex- alued Lebesgue unc ion) and b∈Ban elemen o some complex Banach space. Conside , o e e y x∈X, he p oduc (x)b∈Ba.e. x∈X. Le us check De ini ion 3.2 o his elemen . ZX ∥ (x)b∥p Bdµ(x) = ZX | (x)|p∥b∥p Bdµ(x) =ZX | (x)|pdµ(x)∥b∥p B=∥ ∥p p∥b∥p B<∞ F om he e, we conclude ha (x)b∈Lp B(X). Mo eo e , ∥ b ∥Lp B(X)=∥ ∥p∥b∥B. Le Lp(X)⊗B:= span{ b : ∈Lp(X), b ∈B}be he se o all ini e linea combina ions o elemen s o he o m jus ea ed. P oposi ion 3.4. Lp(X)⊗Bis dense in Lp B(X) o 1≤p < ∞. P oo . We ind help in he σ- ini eness o he measu e space X. Le F∈Lp B(X) and le Xj↗Xbe a coun able sequence o inc easing measu able se s ending o he o al space such ha 0< µ(Xj)<∞ ∀ j∈N. By assump ion (i) in De ini ion 3.1, we can ake a coun able se {bn}n∈N⊂Bwhich is dense in he image o F, excep o a se o measu e 0which we may igno e. Fix a global ϵ > 0. By densi y, o each j∈Nand ϵj>0( ha we a e going o choose la e on) ∀x∈Xj∃n∈Nsuch ha ∥F(x)−bn∥B< ϵj. Nex , we cons uc a measu able unc ion o associa e wi h each xand F(x)a bn∈ {bn}n∈N ha is close enough o F(x). De ine2 Φj:Xj−→ N x−→ min{n∈N:∥F(x)−bn∥< ϵj}.(3.3) A his poin , we ha e g ouped all x∈Xjin o he measu able disjoin se s Xn j:= {x∈Xj: Φj(x) = n},Xj=Fn≥1Xn j. Wi h he will o cons uc ing an 2The unc ions Φja e measu able because one can induc i ely exp ess he p eimages o single ons as Φ−1 j(n) = {x∈Xj:∥F(x)−bn∥B< ϵj}Tn−1 i=1 (Φ−1 j(i))c∀n≥1. O cou se, he maps x→ ∥ F(x)−bn∥Ba e measu able in he es ic ed measu e space (Xj,Σj, µ), in alignmen wi h De ini ion 3.1. 42 Vec o - alued ex ensions elemen o Lp(X)⊗Bwhich esembles F, de ine, o each n∈N, he measu able unc ions n(x) := (1x∈Xn j 0x /∈Xn j. In he emaining o he p oo , we check ha Fis app oxima ed by he sequence o elemen s PN m=1 mbm∈Lp(X)⊗Bindexed by N∈N. Fi s , es ima e he in eg als o e each Xn jwi h n≤N. ZXn jF(x)− N X m=1 m(x)bm p B dµ(x) = ZXn j ∥F(x)−bn∥p Bdµ(x) ≤ϵp jµ(Xn j)=2−jµ(Xn j) µ(Xj)ϵ once we ha e chosen ϵp j:= 2−j µ(Xj)ϵ. Spli ing he global in eg al acco ding o j and N, we each ZXF(x)− N X m=1 m(x)bm p B dµ(x)≤X j≥1ZXjF(x)− N X m=1 m(x)bm p B dµ(x) =X j≥1Z∪N n=1Xn jF(x)− N X m=1 m(x)bm p B dµ(x) +X j≥1Z∪∞ n=N+1Xn jF(x)− N X m=1 m(x)bm p B dµ(x) | {z } E(N) ≤X j≥1 N X n=1 2−jµ(Xn j) µ(Xj)ϵ+E(N)≤X j≥1 2−jϵ+E(N) = ϵ+E(N). P o ing ha he e o e m E(N) ends o ze o as N→ ∞ would conclude he p oo . A e ealizing ha he inne sum anishes in he in eg a ion domain o E(N), ha E(N) = Z∪j≥1∪∞ n=N+1Xn j ∥F(x)∥p Bdµ(x)→0as N→ ∞ is ue can be shown by he domina ed con e gence heo em and he ac ha F∈Lp B(X). The e is a easonable way o de ining in eg a ion o elemen s in Lp(X)⊗B making he mos o he Lp(X)in eg a ion s uc u e. 3.1 In eg a ion o ec o - alued unc ions 43 De ini ion 3.5. Le b be a enso p oduc o a unc ion ∈L1(X)and some Banach space elemen b∈B. De ine i s in eg al o e he measu e space X: ZX (x)b dµ(x) := ZX (x)dµ∀b∈B. This de ini ion ex ends o all L1(X)⊗Bby imposing linea i y o his in eg al ope a ion. Resembling he iangle inequali y, we ge he ollowing inequali y o he in eg al in he p e ious de ini ion. I is going o become essen ial o he p oo o Theo em 3.9. P oposi ion 3.6. I F ∈L1(X)⊗B, hen RXF dµ B≤ ∥ F ∥L1 B(X). P oo . The s a emen is p e y ob ious o simple unc ions: Le F (x) = PN n=1 PK k=1 ak n1Xk(x)bnbe a linea combina ion o in eg able simple unc ions mul iplied by some elemen s in B, we e ak na e hei complex alues and Xk a e measu able se s. Then, ZX F dµ B =ZX N X n=1 K X k=1 ak n1Xk(x)bndµ(x)B = K X k=1 N X n=1 ak nµ(Xk)bnB ≤ K X k=1  N X n=1 ak nbnB µ(Xk) = K X k=1 ZXk N X n=1 ak nbnB dµ(x) =ZX N X n=1 K X k=1 ak n1Xk(x)bnB dµ(x) = ∥F ∥L1 B(X). The las s ep is legi due o he indica o unc ions o disjoin se s. I we deno e by S(X) he space o in eg able simple unc ions o e X, aking in o accoun ha he inclusion S(X)⊗B⊂L1(X)⊗Bis a dense inclusion, we a e done by con inui y. Wha P oposi ion 3.6 ells us is ha he linea map o Banach spaces ZX ·dµ :L1(X)⊗B⊂L1 B(X)−→ B F −→ ZX F dµ (3.4) is con inuous (equi alen ly bounded). The e o e, by P oposi ion 3.4, we can ex end he ope a o (3.4) by con inui y uniquely o an ope a o ac ing now 44 Vec o - alued ex ensions on all o he L1 B(X)space. We shall use he same no a ion o he ex ended ope a o ZX ·dµ :L1 B(X)−→ B F−→ ZX Fdµ. (3.5) P oposi ion 3.7. Fo F∈L1 B(X), he elemen RXFdµ ∈Bis cha ac e ised by b∗,ZX F dµ=ZX ⟨b∗, F(x)⟩dµ(x)∀b∗∈B∗.(3.6) P oo . Checking ha RXFdµ is he only elemen in B ha sa is ies (3.6) is immedia e by he cus oma y way o assuming he e exis s a di e en elemen b∈B ha also e i ies (3.6) and eaching he con adic ion ha RXFdµ −b= 0∈B. To p o e ha RXFdµ buil as in (3.5) sa is ies (3.6), le F∈L1 B(X)and w i e F=F +G, whe e F ∈L1(X)⊗Band G∈L1 B(X)is such ha , ixed ϵ > 0,∥G∥L1 B(X)< ϵ (which is pe mi ed hanks o P oposi ion 3.4). Checking (3.6) o F is s aigh o wa d by linea i y. The densi y a gumen eads as ollows. b∗,ZX F dµ=b∗,ZX F +ZX G dµ =b∗,ZX F dµ+b∗,ZX G dµ=ZX ⟨b∗, F (x)⟩dµ(x) + b∗,ZX G dµ =ZX ⟨b∗, F(x)⟩dµ(x)−ZX ⟨b∗, G(x)⟩dµ(x) + b∗,ZX G dµ E en ually, i emains o show ha he second and hi d e ms a e a bi a ily small. Fo he second e m, le us use duali y o no ms. ZX ⟨b∗, G(x)⟩dµ(x)≤ZX ∥b∗∥B∗∥G(x)∥Bdµ(x) =∥b∗∥B∗∥G∥L1 B(X)≤ ∥ b∗∥B∗ϵ Fo he hi d e m, we also ely on he inequali y in P oposi ion 3.6. b∗,ZX G dµ≤ ∥ b∗∥B∗ZX G dµ B ≤ ∥ b∗∥B∗∥G∥L1 B(X)<∥b∗∥B∗ϵ This ou e led o (3.6). 3.1 In eg a ion o ec o - alued unc ions 45 Ha ing de eloped a g ounding o in eg a ing ec o - alued unc ions, we a e in a posi ion o shoo o an ex ension o ou main heo em, Theo em 2.12, now in his new as e se ing. We a e going o showcase how o make he mos o he Calde ón-Zygmund heo y de eloped in Chap e 2. The p oblem o , gi en a linea Lp(Rn)-bounded ope a o , a emp ing o show ha i has a ec o - alued ex ension was aced a couple decades be o e he bi h o he Calde ón-Zygmund heo y. E o s we e made o y o show es ima es o he kind: Gi en a pa icula known linea bounded ope a o on Lp(Rn), H (say, o ins ance, he Hilbe ans o m, i n= 1), choose B=ℓq(R) and p o e ha he ex ended linea ope a o ˜ H:Lp ℓq(R)(R)→Lp ℓq(R)(R)gi en by ˜ H({ j}j∈N) := {H j}j∈Nis bounded, namely  X j≥1 |H j|q!1 qp ≤Cp,q  X j≥1 | j|q!1 qp .(3.7) E en hough, wi h enough imagina ion, one can come up wi h a my iad o ope a o s ac ing on ec o - alued unc ions, i was wildly di icul o shel e a comple e amoun o hem unde he same heo y. I was no un il he machine y o Calde ón-Zygmund was disco e ed ha we go a gene al enough heo y o akin ope a o s. Re ie ing he idea o ope a o s gi en by in eg a ion agains , in some sense, a ke nel, since we would like o wo k wi h ec o - alued unc ions, he ke nel K(x, y)is no longe going o be a unc ion, bu a linea ope a o mapping Banach spaces in o Banach spaces. Le L(A, B)deno e he Banach space o bounded linea ope a o s mapping he Banach space Ain o he Banach space B. In o de o ca e ully selec he ope a o s we a e homing in on, we conside K wi h domain in he p oduc measu e space X×X,K(x, y)being ill-de ined along he diagonal x=y, and aking alues K(x, y)∈ L(A, B). In simila i y wi h he hypo heses o Theo em 2.12, le us assume K o be measu able (in he sense o De ini ion 3.1) and locally in eg able away om he diagonal. Mo eo e , whene e F∈L∞ A(X)has compac suppo and x /∈supp(F), he objec o s udy Tis gi en by TF(x) = ZX K(x, y)F(y)dµ(y). Unde such assump ions, o x /∈supp(F), ∥T (x)∥B≤Zsupp(F)⊆K′⊆X ∥K(x, y)F(y)∥Bdµ(y) ≤ ∥ F∥L∞ A(X)Zsupp(F)⊆K′⊆X ∥K(x, y)∥L(A,B)dµ(y)<∞ 46 Vec o - alued ex ensions meaning ha T (x)is a well de ined elemen o Bi x /∈supp(F). I is no o be o go en ha we should impose some kind o Hö mande condi ion. Analogously o De ini ion 2.11: De ini ion 3.8. A ke nel Kon he p oduc measu e space ((X, d),Σ, µ)× ((X, d),Σ, µ) aking alues in L(A, B)is said o sa is y he Hö mande con- di ion i sup y,y0∈XZd(x,y)≥Cd(y,y0) ∥K(x, y)−K(x, y0)∥L(A,B)dµ(x) = D < ∞(3.8) o some cons an s C > 1and D > 0. As onishingly, he na u al gene aliza ion o Theo em 2.12 u ns ou o wo k in his se ing as well! Theo em 3.9. Le ((X, d),Σ, µ)be a measu e me ic space wi h he doubling p ope y. Le A, B be Banach spaces and le Tbe a linea ope a o which is ep esen ed by TF(x) = ZX K(x, y)F(y)dµ(y) whene e F∈L∞ A(X)wi h compac suppo and x /∈supp(F), whe e he ec o - alued ke nel K∈ L(A, B)is measu able in X×Xand locally in eg able away om he diagonal. Assume ha (i) Tis bounded om Lq A(X) o Lq B(X) o a ixed 1< q ≤ ∞,∥TF ∥Lq B(X)≤ Cq∥F∥Lq A(X), and (ii) he ope a o ke nel Ksa is ies he Hö mande condi ion in (3.8)wi h con- s an s Cand D. Then, (a) he ope a o Thas a bounded ex ension mapping Lp A(X) o Lp B(X), wi h 1< p < q. Fu he mo e, ∥TF ∥Lp B(X)≤Cp∥F∥Lp A(X),1<p<∞ o F∈Lp A(X)and Cponly depending on p, q, Cq, C and D. (b) The ope a o Thas a bounded weak- ype (1,1) ex ension ha sa is ies λµ({x∈X:∥TF(x)∥B> λ})≤C1∥F∥L1 A(X)∀λ > 0(3.9) o F∈L1 A(X)and C1only depending on q, Cq, C and D. 3.1 In eg a ion o ec o - alued unc ions 47 P oo . The key new ema k, in compa ison wi h he p oo o Theo em 2.12 is ha , when le ing F∈L1 A(X)in o de o p o e he weak- ype (1,1) es ima e, we can apply he Calde ón-Zygmund decomposi ion o he unc ion x→ ∥ F(x)∥A, which lies in L1(X). We ske ch he p oo emphasizing he sligh di e ences. The goal is, once again, showing Tis weak- ype (1,1) so ha Ma cinkiewicz in e pola ion applies. Le λ > 0and le F∈L1 A(X)∩L∞ A(X). By Theo em 2.3 and Rema k 2.6, he e exis s a pa i ion o he space X= FX⊔Ω,Ω = Fk∈NQksuch ha ∥F∥A≤λa.e. x∈Ωand 1 µ(Qk)ZQk ∥F(x)∥Adµ(x)≤CCZλ, ∀k∈N. I is sui able o de ine G(x) := (F(x)x∈FX 1 µ(Qk)RQkF(x)dµ(x)x∈Qk⊂Ω,∀k∈N. Consequen ly, BF(x) := F(x)−G(x),∀x∈X3. As in he sibling p oo , we educe ma e s o showing he weak- ype (1,1) p ope y o each o he wo e ms (see (2.10)): µ({x∈X:∥TF(x)∥B> λ})≤µx∈X:∥TG(x)∥B>λ 2 +µx∈X:∥TBF(x)∥B>λ 2.(3.10) Gis weak- ype (1,1): This is ca ied ou in he same way, using hypo hesis (a) and he ac ha , his ime, ∥G(x)∥q−1≤(CCZ λ)q−1a.e. x∈X. No e ha , in he case q=∞, we would p oceed as ema ked in he o me p oo . Bis weak- ype (1,1): Le us check how he new Hö mande condi ion helps us ind he way ou his ime. S ep (2.11) wo ks he same way using he maximal heo em in he scala se ing: µx∈Ω∗∗ :∥TBF(x)∥B>λ 2≤AHLc∗∗ λ∥F∥L1 B(X). 3We keep on using he no a ion Gand BFs anding o “good” and “bad” o p ese e he adi ion, bu please do no mis ake he unc ion BF(x) o he Banach space B. Simila ly, make a dis inc ion o he ec o - alued unc ion Fand he se FX om he Calde ón- Zygmund decomposi ion. 48 Vec o - alued ex ensions To deal wi h he second e m in (3.10), p oceed as in he scala - alued case o each λ 2µx∈Ω∗∗c:∥TBF(x)∥B>λ 2 ≤X kZΩ∗∗cZQk K(x, y)Bk(y)dµ(y)B dµ(x) and hen in oduce a second cons an ke nel ope a o e m elying on he ac ha Bk(which is he es ic ion o BF o Qk) in eg a es ze o on Qk. A e using iangle inequali y, Fubini-Tonelli, he ope a o no m inequali y, and an o e es ima ion o he in eg a ion domain, we each he s ep whe e o use he Hö mande condi ion (3.8). X kZΩ∗∗cZQk K(x, y)Bk(y)dµ(y)B dµ(x) ≤X kZQkZd(x,y)>Cd(y,yk) ∥K(x, y)−K(x, yk)∥L(A,B)dµ(x)∥Bk(y)∥Adµ(y) ≤D∥BF∥L1 B(X)≤D∥F∥L1 F(X) Once we know Tis weak- ype (q, q)(in ac , ype (q, q)) and weak- ype (1,1), bo h o cou se in he ec o alued se ing, we wan o use Macinkiewicz in e - pola ion. E en hough Theo em 1.16 is exposed in he scala - alued na u e, i u ns ou o be ue o ec o - alued unc ions (wi h he ob ious modi ica ions, in pa icula eplacing absolu e alues by no ms o he Banach spaces). In ac , he p oo in bo h cases ollows exac ly he same ou e, see Theo em 1.18 in [5], Chap e 5, Sec ion 1. Finally, he ac ha we wo ked wi h F∈L1 A(X)∩L∞ A(X)due o he echnical issues su ounding he ke nel is no an obs acle since his space is dense in L1 A(X), hus once he weak- ype es ima e is p o ed, a densi y a gumen yields he es ima e o he whole space. All hese machine y we buil u ns ou use ul o bound ec o - alued ope - a o s, o e en scala - alued ones, h ough a a ie y o di e en echniques. We a e s a ing some mo e c ucial esul s ha ake pa in hose echniques. La e on in his chap e , we a e going o expe imen wi h hem. He e is an ex ension esul o ope a o s ac ing on scala - alued unc ions o ope a o s ac ing on ec o - alued unc ions. Theo em 3.10 (Ma cinkiewicz-Zygmund).Le (Xi,Σi, µi)be wo σ- ini e mea- su e spaces, i= 1,2. Le T:Lp(X1)→Lq(X2),1≤p, q < ∞, be a linea 3.2 Li lewood-Paley heo y 49 bounded ope a o wi h no m ∥T∥. Then,  X j∈N |T j(x)|2!1 2q ≤Cp,q ∥T∥ X j∈N | j(x)|2!1 2p (3.11) o ( j)j∈N∈Lp ℓ2(Rn)and some cons an Cp,q >0depending on pand q. Wha unde pins he co esponding p oo a e andomisa ion echniques, ha we a e going o discuss la e on. The p oo can be ound in [5], Chap e 5, Theo em 2.7. This esul s pa ially answe s he ques ion we men ioned abou ex ending he Hilbe ans o m o ℓp. Theo em 3.10 sol es he p oblem o p= 2. In gene al, i is no possible o ind ex ensions o unc ions aking alues in Banach spaces ha a e no Hilbe spaces; such a s uc u e is eally help ul. Likewise, i ins ead o a single linea bounded ope a o we ha e a collec ion o hem, he co esponding ec o - alued ope a o is in gene al unlikely o be ameable, unless in e y nice si ua ions like he ollowing one. Theo em 3.11. Le {Ij}j∈Nbe an a bi a y coun able collec ion o in e als in Rnwi h sides pa allel o he coo dina e axis, and deno e by TIj he mul iplie ope a o s associa ed o he indica o unc ion o each in e al on he equency side. Then, o all 1<p<∞and ( j)j∈N∈Lp ℓ2(Rn),  X j∈N |TIj j(x)|2!1 2p ≤Cp X j∈N | j(x)|2!1 2p (3.12) whe e Cp>0is a cons an depending on pand n. The de ails o he p oo can be ound in [5], Chap e 5, Co olla y 2.13. Le us explain he main idea hough. Since he in e als ha e sides pa allel o he coo dina e axis, i is possible o gi e a enso p oduc s uc u e o he ope a o ha educes ma e s o he one dimensional case. In his posi ion, one ealises ha he indica o unc ion o an in e al 1[a,b]can be w i en as he sum o wo sign unc ions, wi h he in en ion o make he mul iplie o he Hilbe ans o m show up (use De ini ion 1.33). The p oo concludes by an applica ion o he Ma cinkiewicz-Zygmund heo em (Theo em 3.10) and he ac ha he Hilbe ans o m is a bounded ope a o on Lp(Rn), o 1<p<∞. 3.2. Li lewood-Paley heo y The Li lewood-Paley heo y has i s o igin in he 1930’s, i s ly de eloped by J. E. Li lewood and R. Paley wi h he main in en ion o s udying Fou ie 56 Vec o - alued ex ensions and by Fubini-Tonelli, BpZRnZ[0,1] |P (x)|pd dx1 p =BpZ[0,1] ZRn |P (x)|pdx d 1 p =BpZ[0,1] ∥P ∥p pd 1 p . Finally, we make use o es ima e (3.23) o each BpZ[0,1] ∥P ∥p pd 1 p ≤BpZ[0,1] Cp p∥ ∥p pd 1 p =BpCp∥ ∥p. One could wonde why we made an e o o build a smoo h Li lewood-Paley squa e unc ion ins ead o chopping he equency space wi h ough cu o s. Well, he answe is in wha comes nex . Le Aj:= {ξ∈Rn: 2j−1≤ |ξ| ≤ 2j}be he annuli con aining he equen- cies wi h magni ude be ween 2j−1and 2j,j∈Z. De ine he ollowing ough mul iplie s. d Pj (ξ) := 1Aj(ξ)ˆ (ξ)∀j∈Z As in he smoo h case, one has a decomposi ion o he iden i y, (x) = X j∈Z Pj (x), as well as a squa e unc ion. De ini ion 3.19 (Rough Li lewood-Paley squa e unc ion). S (x) := X j∈Z |Pj (x)|2!1 2 (3.24) By i s cons uc ion, i is na u al o s a he s udy in L2(Rn). By Planche el and Fubini-Tonelli, S 2 2=ZRnX j∈Z |Pj (x)|2dx =X j∈ZZRn |Pj (x)|2dx =X j∈ZZRn |1Aj(ξ)ˆ (ξ)|2dξ =ZRnX j∈Z 1Aj(ξ)|ˆ (ξ)|2dξ =ZRn |ˆ (ξ)|2dξ =∥ ∥2 2. 3.2 Li lewood-Paley heo y 57 No only is he ough squa e unc ion bounded in L2(Rn)bu also i is an isom- e y. F om he e, wi h L2(Rn)as a oo hold, one would expec some a gumen ha p o es Lp(Rn)boundedness. Well, wi h g ea su p ise, i u ns ou ha he beha iou o he squa e unc ion depends on he dimension o Rn, as he ollowing esul s depic . Theo em 3.20. Le n= 1. The ough Li lewood-Paley squa e unc ion (3.24) is bounded on Lp(R) o 1<p<∞. | -1 4 | -1 2 | -1 | -2 | 1 4 | 1 2 | 1 | 2 – 1 Figu e 3.3: The mul iplie unc ions 1A0and ψ0in he p oo o Theo em 3.20 compa e like his. P oo . I 1Aj(ξ),Aj:= {ξ∈Rn: 2j−1≤ |ξ| ≤ 2j} ∀ j∈Z, a e he mul iplie s co esponding o he ough squa e unc ion, le us co e hem by mul iplie s esembling he smoo h ones. Le ψj(ξ)∈C∞ c(R)such ha ψj(ξ)=1on 2j−1≤ |ξ| ≤ 2jand ψj(ξ) = 0 on |ξ| ≤ 2j−2and |ξ| ≥ 2j+1. No happy wi h his, equi e 2 = X j∈Z ψj(ξ)∀ξ∈R ∖ {0}. Such unc ions can be cons uc ed simila ly as we did in Lemma 3.12. As a consequence we can ela e he smoo h and ough mul iplie s by 1Ajψj=1Aj=⇒Pj◦P′ j=Pj, in he unde s anding ha d P′ j (ξ) := ψj(ξ)ˆ (ξ)∀j∈Z. This allows us o w i e S p= X j∈Z |PjP′ j (x)|2!1 2p ≤Cp X j∈Z |P′ j (x)|2!1 2p =Cp∥S′ ∥p. 58 Vec o - alued ex ensions He e, we used Theo em 3.11, bu no ice ha i only wo ks because in R he mul iplie s 1Ajcan be seen as indica o unc ion o in e als, no only annuli. We educed he p oblem o bounding he smoo h Li lewood Paley squa e unc- ion, so he p oo is comple e by Theo em 3.18 ( alid bo h o Sand S′, i does no ma e he a ian o he cons uc ion o he smoo h squa e unc ion). Theo em 3.21. Le n≥2. The ough Li lewood-Paley squa e unc ion (3.24) is bounded on Lp(Rn)i and only i p= 2. P oo . We ha e al eady seen ha Sde ines an isome y on L2(Rn). F om he simple bound |P0 (x)| ≤ S (x) = X j∈Z |Pj (x)|2!1 2 we deduce ha i Sis bounded on Lp(Rn), hen so is |P0 (x)|. Bu |P0 (x)|is he es o wo ball mul iplie ope a o s, so by he ball mul iplie heo em6, o n≥2 his can only happen i p= 2. In wha p ecedes, we ha e exposed a clea mani es a ion o an ubiqui ous phenomenon in ha monic analysis: In he mul idimensional case n≥2, a mul- iplie whose suppo has smoo h cu ed bounda y is going o beha e nas ie he oughe he decay o he mul iplie unc ion is nea he bounda y. Such a bounda y is degene a e in he n= 1 case, so hese heu is ics do no apply. 6The ball mul iplie heo em is a deep esul in ha monic analysis, o en linked o he s udy o con e gence o pa ial Fou ie in eg als and pa ial Fou ie sums. I was a majo open p oblem in he 1960’s; in ac , many people belie ed he ball mul iplie would be bounded in a la ge ange o alues o p. The b eaking pape [4] o Cha les Fe e man sol ed he p oblem in 1971. The p oo is based, in i s u n, on he exis ence o Kakeya se s. CHAPTER 4 Applica ions and examples 4.1. Hö mande mul iplie s In his sec ion, we a e dealing wi h a pa icula amily o mul iplie s ha a ise, o ins ance, in pa ial di e en ial equa ions. We p esen hem as a se ing whe e o apply he Calde ón-Zygmund heo y. In he unde s anding o he mul i-index no a ion α∈Nnas in Chap e 1, he e is how we de ine hem. De ini ion 4.1. A mul iplie unc ion m∈L∞(Rn)∩C∞(Rn∖{0})is called aHö mande mul iplie i |∂αm(ξ)| ≤ Cα |ξ||α|∀ξ∈Rn∖{0},∀α∈Nn.(4.1) We will also conside he case when hese mul iplie s do no enjoy ull eg- ula i y bu jus up o a ce ain o de . Theo em 4.2. Le mbe a Hö mande mul iplie . Then, i s associa ed dis- ibu ional ke nel Kag ees wi h a smoo h unc ion away om he o igin ha e i ies |∂αK(x)| ≤ Aα |x|n+|α|∀α∈Nn(4.2) o some cons an s Aα. P oo . We s a wi h applying he smoo h Li lewood-Paley decomposi ion (3.14) o he mul iplie . W i e m(ξ) = X j∈Z mj(ξ)∀ξ= 0 whe e each mjcon ains he in o ma ion o he mul iplie ma ound he equen- cies 2j. Tha is, mj(ξ) := m(ξ)ψ2−jξ∀j∈Z. 59 60 Applica ions and examples Now de ine hei co esponding ke nels using he Fou ie in e sion o mula, Kj(x) := ZRn mj(ξ)e2πixξdξ. Since mis a smoo h bounded unc ion, so is mjas well as compac ly suppo ed. Thus, each mjis Lebesgue in eg able (as well as hei de i a i es), om whe e Kjis a well de ined smoo h unc ion. No e ha his conclusion would be desi - able o m, bu a p io i his may no be he case. Le us shoo o bounds o each piece o he ke nel. Begin wi h he basic iden i y: (−2πix)γ∂αKj(x) = ZRn ∂γ(mj(ξ)(2πiξ)α)e2πixξdξ The ollowing c ude inequali y ollows om he assump ion (4.1) and he ac ha mjis suppo ed on he annulus o adii 2j−1and 2j+1. (2π|x|)γ|∂αKj(x)| ≤ (2π)αZRn |∂γ(mj(ξ)ξα)|dξ ≤(2π)α|B1(0)|2n(j+1)C1(α, γ, Cα)2j(|α|−|γ|)≤C2(n, α, γ, Cα)2j(n+|α|−|γ|) A e his compu a ion, γas a pa ame e is eed. Se M:= |γ|and conclude ha |∂αKj(x)| ≤ C2(n, α, M, Cα)2j(n+|α|−M)|x|−M∀M∈N.(4.3) Nex , we would like o es ima e he comple e sum Pj∈Z|∂αKj(x)|by s a e- gically spli ing i in o wo sums and using he inequali y (4.3) wi h wo di e en alues o M. Wi h M= 0 and by a geome ic summa ion, X 2j≤|x|−1 |∂αKj(x)| ≤ C2(n, α, Cα)X 2j≤|x|−1 2j(n+|α|)≤C3(n, α, Cα)|x|−n−|α|. Analogously, selec M=n+|α|+ 1: X 2j>|x|−1 |∂αKj(x)| ≤ C2(n, α, Cα)|x|−(n+|α|+1) X 2j>|x|−1 2−j≤C4(n, α, Cα)|x|−n−|α|. All in all, we ha e jus shown X j∈Z |∂αKj(x)| ≤ C5(n, α, Cα)|x|−n−|α|(4.4) which esembles wha we a e aiming o show. He e is he limi ing a gumen ha leads o he end o he p oo . 4.1 Hö mande mul iplie s 61 We see ha P|j|≤NKj ends o Kas N→ ∞ in he sense o dis ibu ions. Take any φ∈S(Rn)and compu e he ollowing limi . lim N→∞  X |j|≤N Kj−K (φ) = lim N→∞ ZRn X |j|≤N mj(ξ)−m(ξ) ˇφ(ξ)dξ = 0 In he las s ep, we used he ac s ha , by cons uc ion, P|j|≤Nmj(ξ)→m(ξ) poin wise as N→ ∞ (p o iding ξ= 0); and ha hei di e ence is bounded by 2m(ξ)which is a bounded unc ion. domina ed con e gence heo em applies since ˇφis a Schwa z unc ion. Recalling ha we showed (4.4) ( ocus on α= 0), we know ha Pj∈ZKj(x)is poin wise and absolu ely con e gen away om he o igin. Since i con e ges o Kin he sense o dis ibu ions, we conclude ha Kis a dis ibu ion which ag ees wi h a unc ion away om he o igin. Mo eo e , ca ying ou an analogous compu a ion o he de i a i es ∂αKj, we deduce ha he unc ion wi h which Kag ees is smoo h. E en ually, once we know Kcan be iewed as a smoo h unc ion away om he o igin, (4.2) s ems om (4.4) by iangle inequali y. F om a wide pe spec i e, wha Theo em 4.2 ells us is ha a Hö mande mul iplie is no only a bounded unc ion ( hus gi ing an associa ed bounded ope a o on L2(Rn)) bu also i e i ies he g adien condi ion (2.8) (which, ecall, is s onge han he Hö mande condi ion). By he main Theo em 2.9 (say, in he se ing o Rn), a Hö mande mul iplie de ines, on he spacial side, an ope a o ha is bounded on Lp(Rn)and is weak- ype (1,1)! Co olla y 4.3. Hö mande mul iplie s de ine bounded ope a o s on Lp(Rn) o 1<p<∞. No e ha in he p oo o Theo em 4.2, we hea ily elied on he hypo hesis m∈C∞(Rn)a he ime o deducing he c ude es ima e (4.4). As soon as we d op he ull smoo hness hypo hesis, we loose Theo em 4.2. Ne e heless, i we assume m o be su icien ly egula (al hough no smoo h), we can show ha he associa ed ke nel s ill sa is ies he Hö mande condi ion! This is one o he easons why we ca e abou such a cumbe some condi ion in heo ems like Theo em 2.9 ins ead o jus assuming a mo e handy condi ion like he g adien one: The Hö mande mul iplie s wi hou ull egula i y, bu wi h enough, a e s ill emb aced by he Hö mande condi ion (no by he g adien one hough), as he ollowing heo em showcases. Theo em 4.4. Le m∈L∞(Rn)∩Cℓ(Rn∖{0})be a mul iplie unc ion such ha |∂αm(ξ)| ≤ Cα |ξ||α|∀ξ∈Rn∖{0},∀α∈Nn,0≤ |α| ≤ ℓ. (4.5) 62 Applica ions and examples wi h ℓ:= ⌈n 2⌉ he smalles in ege g ea e han o equal o n 2. Then, i s associa ed dis ibu ional ke nel Kag ees wi h a unc ion away om he o igin ha e i ies he Hö mande condi ion: sup |y|>0Z|x|≥2|y| |K(x−y)−K(x)|dx =B < ∞. o some cons an B > 0. As he p oo o Theo em 4.4 in ol es simila echnicali ies as he one o The- o em 4.2, we jus no e ha he p oo can be ound in [12], Chap e 6, Sec ion 4.4. Essen ially, ins ead o deducing a c ude es ima e as (4.4), a nea e applica- ion o Planche el leads o a simila es ima e con empla ing less egula i y ye u ning ou use ul. Co olla y 4.5. Hö mande mul iplie s wi h limi ed egula i y, as in he p e i- ous heo em, de ine bounded ope a o s on Lp(Rn) o 1<p<∞. 4.1.1. Ellip ic di e en ial ope a o s We a e now ocusing on pa ial di e en ial ope a o s wi h cons an coe icien s o he kind Du =X |α|≤k cα∂αu(4.6) wi h k, n ∈Nand unde s anding he mul i-index no a ion α∈Nnas in Chap e 1. De ini ion 4.6. We say ha an ope a o o he ype (4.6)is ellip ic i he cha ac e is ic polynomial o he associa ed homogeneous di e en ial ope a o o deg ee k Du =X |α|=k cα∂αuF T −−→ P(ξ) = X |α|=k cαξαξ∈Rn(4.7) only anishes a ξ= 0:P(ξ)= 0 ∀ξ= 0. Le us i s wo k di ec ly wi h a homogeneous di e en ial ope a o o deg ee k, as in Eq. (4.7). Conside he PDE Du(x) = (x) o a gi en unc ion :Rn→R(le us wo k ou o mal compu a ions o he momen ). Equi alen ly, in he equency domain, P(ξ)ˆu(ξ) = ˆ (ξ) 4.1 Hö mande mul iplie s 63 wi h ξ∈Rnand Pbeing he associa ed cha ac e is ic polynomial o he di e - en ial ope a o D. The compu a ion (∂αu)ˆ(ξ) = (2πiξ)αˆu(ξ) = (2πiξ)α P(ξ)P(ξ)ˆu(ξ) = (2πiξ)α P(ξ)ˆ (ξ)(4.8) is alid whene e ξ= 0. Hence, (4.8) wishes o be in e p e ed in he sense o mul iplie s. Indeed, neglec ing he cons an 2πi, de ine he mul iplie mh(ξ) := ξα P(ξ)|α|=k. (4.9) Since we assumed D o be homogeneous, Pis a homogeneous polynomial o deg ee k, meaning ha Eq. (4.9) de ines a singula homogeneous mul iplie o deg ee 0: mh∈L∞(Rn)mh(λξ) = mh(ξ)∀λ > 0, ξ = 0. No happy wi h his, we ealise ha a Hö mande mul iplie has a isen! mh in (4.9) is clea ly smoo h and bounded away om he o igin. Fu he mo e, because o he ana omy o he de i a i es o quocien s o polynomials, he pa - ial de i a i es o mhsa is y (4.1). Thanks o his, by aking in e se Fou ie ans o ms and Lpno ms in (4.8) and accoun ing o Co olla y 4.3, we ge ∥∂αu∥p≤ ∥ Tmh∥ ∥ Du ∥p,|α|=k, 1<p<∞.(4.10) P ecisely, i Dis an ellip ic di e en ial ope a o wi h cons an coe icien s, hen any monomial pa ial di e en ial ope a o o he same o de as Dis bounded, in he Lpno m, by D. We lea e he ask o explaining he meaning o (4.10) o a e discussing he nonhomogeneous case. The nex s ep is conside ing a nonhomogeneous ellip ic di e en ial ope a o o he kind (4.6). In such scena io, le us spli i s cha ac e is ic polynomial in o a sum o he homogeneous e ms o deg ee k,Ph, and he emaining lowe o de e ms, Pl:P(ξ) = Ph(ξ) + Pl(ξ). We show ha he polynomial Pis bounded below away om he o igin. |P(ξ)|=X α≤k cαξα=|ξ|kPhξ |ξ|+Pl(ξ) No e ha he es ic ion o he uni sphe e o he homogeneous pa Phappea s, bu since Ph(ξ)does no anish o ξ= 0, such a es ic ion is bounded below in modulus by some cons an ch>0. Fu he mo e, Plis a polynomial o deg ee a mos k−1, hus he e exis s ano he cons an Cl>0such ha |Pl(ξ)| ≤ Cl|ξ|k−1 o |ξ| ≥ 1. We hen each he bound |ξ|kPhξ |ξ|+Pl(ξ)≥ch|ξ|k−Cl|ξ|k−1≥Cl|ξ|k 64 Applica ions and examples o |ξ| ≥ R, whe e R:= max{1, }and > 0is such ha ch k= 2Cl k−1, so =2Cl ch. All in all, Pis bounded below away om he o igin: |P(ξ)| ≥ ClRk o |ξ|> R. Ha ing ema ked his, le ϕ∈C∞ c(Rn)be a bump unc ion such ha ϕ(ξ) = 1 o ξ∈BR(0) and ϕ(ξ)=0 o ξ /∈B2R(0). The adap a ion o he compu a ion in (4.8) is as ollows. (∂αu)ˆ(ξ) = (2πiξ)αˆu(ξ) = (2πiξ)αˆu(ξ)ϕ(ξ) + (2πiξ)αˆu(ξ)(1 −ϕ(ξ)) = (2πiξ)αϕ(ξ)ˆu(ξ) + (2πi)αξα P(ξ)(1 −ϕ(ξ))P(ξ)ˆu(ξ)(4.11) I is sa e o di ide by P(ξ)in he second e m because (1 −ϕ(ξ)) = 0 in BR(0) and we saw ha |P|is bounded below elsewhe e. Again, neglec ing 2πi ac o s om now on, de ine he mul iplie mnh(ξ) := ξα P(ξ)(1 −ϕ(ξ)) |α|=k(4.12) which is no homogeneous a all his ime. Howe e , i clea ly li es in L∞(Rn)∩ C∞(Rn∖{0})and, hank ully, i is a he end o he day a Hö mande mul iplie : o |ξ|< R,mnh(ξ) = 0; o R≤ |ξ| ≤ 2R,mnh(ξ)is bounded; and o |ξ|>2R, mnh(ξ) = ξα P(ξ). E en hough Pis no homogeneous his ime, since in he la e case we a e away om he o igin, he bound (4.1) holds o mnh. To sum up, mnh unmasks as a Hö mande mul iplie . Thus, by Co olla y 4.3, he mul iplie de ines an ope a o Tmnh which is bounded on Lp(Rn) o 1<p<∞. Take in e se Fou ie ans o ms and hen Lpno ms in Eq. (4.11) o ob ain ∥∂αu∥p≤ ∥ (ϕ(ξ)ξα)ˇ∗u∥p+∥Tmnh (Du)∥p,|α|=k. (4.13) Do no be a aid o he unc ion (ϕ(ξ)ξα)ˇ. I is a Schwa z unc ion i we ake in o accoun ha ϕ∈S(Rn)also is. U ilising P oposi ion 1.5 o he i s e m and he boundedness o Tmnh (Co olla y 4.3) o he second one, we each ∥∂αu∥p≤ ∥ (ϕ(ξ)ξα)ˇ∥1∥u∥p+∥Tmnh ∥ ∥ Du ∥p,|α|=k, 1<p<∞. (4.14) This kind o es ima es ((4.10) and (4.14)) a e use ul in he con ex o s udying egula i y o solu ions o pa ial di e en ial equa ions. Imagine ha we ake he ini ial da a =Du o li e in Lp(Rn), and suppose ha we we e capable o show ha he solu ion o he equa ion uis also an Lp(Rn) unc ion. Then, wha es ima e (4.14) ells us is ha , in ac , he solu ion ubelongs o he Sobole space Wp,k(Rn). 4.2 Ma cinkiewicz mul iplie s 65 4.2. Ma cinkiewicz mul iplie s He e, we p esen ano he in e es ing amily o mul iplie s. E en hough one can de ine Ma cinkiewicz mul iplie s in he se ing o Rn, i will be mo e il- lus a i e and p ac ical o wo k in he eal line R. Le I= [2j+1,2j]o I= [−2j,−2j+1], j ∈Z, be a dyadic in e al. De ini ion 4.7. A mul iplie unc ion mon he eal line is a Ma cinkiewicz mul iplie i m∈L∞(R)∩C1(R ∖ {0})and he e exis s B≥0such ha ZI |m′(ξ)|dξ ≤B < ∞(4.15) o all dyadic in e als I= [2j,2j+1]and I= [−2j+1,−2j], j ∈Z. Rema k 4.8. The C1 egula i y condi ion may be elaxed so ha ins ead o equi ing (4.15), mo e gene ally we demand ha mhas uni o mly bounded a i- a ion1on dyadic in e als. Rema k 4.9. The ollowing s aigh o wa d compu a ion shows ha a Hö man- de mul iplie in Ris a Ma cikiewicz mul iplie in Ras well. Z2j+1 2j |m′(ξ)|dξ ≤C1Z2j+1 2j dξ |ξ|=C1log 2j+1 2j=C1log(2) ∀j∈Z Indeed, Ma cinkiewicz mul iplie s gene alise Hö mande mul iplie s. Ou na u al conce n now is he Lpboundedness o his b oade class o mul iplie s. Theo em 4.10. A Ma cinkiewicz mul iplie m, as in De ini ion 4.7 de ines an ope a o Tmwhich is bounded on Lp(R) o 1<p<∞. So a , e e y ime we ha e s a ed a heo em he hesis o which is ha a ce ain ope a o is bounded on Lp(X) o 1< p < ∞, he s a egy o he p oo has been in oking he Calde ón-Zygmund heo y. This ime, howe e , Ma cinkiewick mul iplie s do no e i y he Hö mande condi ion. To be accu- a e, he in e se Fou ie ans o m o a Ma cinkiewick mul iplie may no e en be a unc ion, meaning ha he Hö mande condi ion makes no sense in his con ex . The e o e, such a s a egy is au oma ically disca ded. In spi e o his, we a e going o succeed by making wise use o he ec o - alued heo y and he Li lewood-Paley heo y. The ollowing is a ske ch o a p oo ; we ha e no in oduced enough machine y o explain he ull p oo . P oo . We deno e by TI he mul iplie ope a o whose mul iplie unc ion is he indica o unc ion o he in e al I. Likewise, deno e by TmI he mul iplie 1To expand knowledge on his space o unc ions, see [14], Chap e 2, Sec ion 1. 72 Beyond he pa adigm also p esen s his p ope y, as i can be simila ly shown. Un o una ely, he dyadic e sion o he sphe ical maximal ope a o ˜ S (x) := sup k∈ZZSn−1 (x−2kω)dσ(ω) is no compa able o he con inuous sup emum e sion S . The only ue inequali y is he i ial ˜ S ≤ S . The eason o his p ope y o ail is ha , in he case o he Ha dy-Li lewood maximal unc ion, he se o cen ed solid balls a e a nes ed collec ion o se s, whe eas he hollow cen ed sphe es o di e en adii a e no nes ed by any means. Pe haps su p isingly, his issue leads o di e en beha iou s o S and ˜ S . He e is wha we can say abou his pai o ope a o s. Theo em 5.3. The dyadic sphe ical maximal ope a o ˜ S is bounded in Lp(Rn) o 1< p ≤ ∞. This is, o ∈Lp(Rn), ˜ S p≤Cp∥ ∥p, o some cons an Cp>0depending on pand n. Conjec u e 5.4. The dyadic sphe ical maximal ope a o ˜ S is weak- ype (1,1). So o any λ > 0and ∈L1(Rn), λ|{x∈Rn:˜ S (x)> λ}| ≤ C1∥ ∥1, o some cons an C1>0depending on n. Theo em 5.5. The sphe ical maximal ope a o Sis bounded in he ollowing cases: (a) Fo any dimension n≥2,n n−1< p ≤ ∞ and ∈Lp(Rn), ∥ S ∥p≤Cp∥ ∥p, o some cons an Cp>0depending on pand n. (b) Fo n≥3,Sis o es ic ed ype5a he endpoin . By his, we mean: ∥ S ∥Ln n−1,∞(Rn)≤Cn∥ ∥Ln n−1,1(Rn), whe e Cnis a cons an depending on n. Howe e , (b) is alse o n= 2: (c) Sdoes no map L2,1(R2) o L2,∞(R2). See S ein and Bou gain’s wo k in his ega d, in [11] and [1]. 5Lea n mo e abou Lo en z spaces and hei in e pola ing ole in [6], Chap e 1. 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