Heat semigroup and singular PDEs

Transcription

1 Hea emigroup and ingular PDE I. BAILLEUL 1 and F. BERNICOT wih an Appendix by F. Bernico & D. Frey Abrac. We provide in hi work a emigroup approach o he udy of ingular PDE, in he line of he paraconrolled approach developed recenly by Gubinelli, Imkeller and Perkowki. Saring from a hea emigroup, we develop a funcional calculu and inroduce a paraproduc baed on he emigroup, for which commuaor eimae and Schauder eimae are proved, ogeher wih heir paraconrolled exenion. Thi machinery allow u o inveigae ingular PDE in poenially unbounded Riemannian manifold under mild geomeric condiion. A an illuraion, we udy he generalized parabolic Anderon model equaion and prove, under mild geomeric condiion, i well-poed characer in Hölder pace, in mall ime on a poenially unbounded -dimenional Riemannian manifold, for an equaion driven by a weighed noie, and for all ime for he linear parabolic Anderon model equaion in -dimenional unbounded manifold. Thi machinery can be exended o an even more ingular eing and deal wih Sobolev cale of pace raher han Hölder pace. Conen 1 Inroducion 1.1 Sae of he ar 1. Paraconrolled calculu A generalized parabolic Anderon model 6 Funcional calculu adaped o he hea emigroup 8.1 Hea emigroup on a doubling pace 8. Time derivaive and Carré du champ of he emigroup 13.3 Hölder and Beov pace hrough he hea emigroup 15 3 Paraproduc and commuaor eimae in Hölder pace Paraproduc baed on he emigroup Paraproduc eimae Commuaor eimae Paralinearizaion and compoiion eimae Schauder eimae 33 4 Paraconrolled calculu Paraconrolled diribuion Schauder eimae for paraconrolled diribuion 39 5 The generalized parabolic Anderon Model in dimenion Local well-poedne reul for generalized PAM Global well-poedne reul for linear PAM Renormalizaion for a weighed noie 46 A Hea kernel and echnical eimae 54 B Exenion of he heory 6 B.1 Regulariy aumpion 6 B. Funcional calculu and gradien eimae in Hölder and Sobolev pace 63 B.3 Paraproduc and commuaor eimae in Hölder-Sobolev pace 69 1 I.B. wa parly uppored by he ANR projec Reour Po-docoran, no. 11-PDOC-5; I.B. alo hank he U.B.O. for heir hopialiy, par of hi work wa wrien here. F. Bernico reearch i parly uppored by ANR projec AFoMEN no. 11-JS1-1-1 and HAB no. ANR-1-BS

2 B.4 Reoluion of PAM in uch a -dimenional eing 79 1 Inroducion 1.1 Sae of he ar Following he recen breakhrough of Hairer [35] and Gubinelli, Imkeller, Perkowki [3], here ha been recenly a remendou aciviy in he udy of parabolic ingular parial differenial equaion PDE, uch a he KPZ equaion x u = x u + ξ, he ochaic quanizaion equaion u = u 3 + ξ, or he generalized Parabolic Anderon Model equaion u = Fuξ in all of which ξ and for a pace or pace-ime whie noie. Each of hee equaion involve, under he form of a produc, a erm which doe no make ene a priori, given he expeced regulariy of he oluion in erm of he regulariy of he noie ξ. Hairer heory of regulariy rucure i buil on he inigh of earlier work [36, 4, 37] on dimenional pace-ime problem where he ued he framework of rough pah heory, under he form of Gubinelli conrolled pah, o make ene of previouly illpoed ingular PDE and give a meaningful oluion heory. Rough pah heory wa ued in hi approach a a framework for udying he properie in he 1-dimenional pace variable of poenial oluion. However, he very noion of a rough pah i inimaely linked wih he 1-dimenional ime axi ha paramerize pah. To by-pa hi barrier, boh he heory of regulariy rucure and he paraconrolled approach developed in [3] ake a a deparure poin he fac ha, like in rough pah heory, o make ene of he equaion, one need o enrich he noie ξ ino a finie collecion of objec/diribuion, and ha one hould ry and decribe he poenial oluion of a ingular PDE in erm of ha enriched noie. The laer depend on he equaion under udy and play in he heory of regulariy rucure he role plaid by polynomial in he uual C k world o give local decripion of funcion under he form of Taylor expanion a every pace-ime poin. The decripion of a oluion in he paraconrolled approach i of a differen naure and re on a global comparion wih he oluion o a linear equaion, u = ξ, in he above example, via he ue of Bony paraproduc. In boh approache, he ue of an anaz for he oluion pace allow for fixed poin argumen o give a robu oluion heory where he oluion become a coninuou funcion of all he parameer of he problem. So far, boh heorie have only been formulaed and eed on ome ingular PDE on he oru, o he excepion of he work [38, 39] on he parabolic Anderon model equaion in R and R 3, and our follow up work [7]. We inroduce in he preen work a funcional analyic eing in which we are able o exend he paraconrolled approach of [3] o inveigae ingular PDE of he form + L u = Fu, ξ for a econd order differenial operaor L, and a nonlinear erm Fu, ξ, on poenially unbounded, Riemannian or even ub-riemannian, manifold or graph. The change of

3 ign o + in he operaor i irrelevan. Thi i a priori far from obviou a he main analyic ool ued in he paraconrolled approach in he oru involve echnic from Fourier analyi ha do no make ene on manifold or graph. We develop o ha end a funcional calculu adaped o he hea emigroup aociaed wih he operaor + L, which we ue o define a paraproduc enjoying he ame regulariy properie a i Euclidean analogue. Such paraproduc adaped o a emigroup, a well a a paralinearizaion heory, have already been udied in recen work [1, 14]. However, he irregular characer of he noie ξ involved in he above moivaing equaion require u o improve he definiion of uch paraproduc o a o build a framework where o conider regulariy wih a negaive exponen; uch an exenion will be provided here. Building on hee ool, one can e up, a in [3], a framework where o inveigae he well-poed characer of a whole cla of parabolic ingular PDE. I i epecially nice ha all he objec in our framework are defined uniquely in erm of emigroup, unlike he noion of Hölder pace ued in he heory of regulariy rucure ha involve a meric rucure unrelaed o he equaion under udy. A a by-produc, we are able o handle ome general clae of operaor L whoe reamen eem o be beyond he preen-day cope of he heory of regulariy rucure, a illuraed in ome example given in ecion.1. I i unclear preenly how one can adap he differen noion and ool of he heory of regulariy rucure o exend hem o a Lipchiz manifold or graph eing, or o oher econd order operaor oher han he Laplace operaor, or how o work wih Sobolev pace inead of Hölder pace. Apar from he very definiion of a regulariy rucure on a manifold, he exience of he reconrucion operaor in hi eing eem in paricular challenging, a i proof in R d involve ome deep reul on wavele ha were no proved o far o hold rue on generic manifold, no even on all open e of R d. Their exenion o a non-mooh eing alo eem higly non-rivial. So i come a a good new ha one can ue ome reaonably elaborae heory of emigroup o implemen he alernaive machinery of paraconrolled calculu in ha eing; a decribed below, i alo allow u o have much flexibiliy on he operaor L and alo on he geomery of he ambian pace. Roughly peaking, we could ay ha he poin of view of he heory of regulariy rucure relie on he meric and differenial properie of he underlying pace, while he preen exenion of he paraconrolled calculu correpond o a funcional poin of view adaped o he operaor L involved in he parabolic ingular PDE. We link here hee wo ide of he medal by requiring from he hea emigroup e L > o have a kernel ogeher wih i gradien, ha aifie poinwie Gauian bound; hi decribe in ome ene he link beween he funcional calculu and he ambian pace, wih i meric and i differenial geomery. We explain in Appendix B how hi approach can be ued in he conex of Sobolev pace raher han Hölder pace. The former eing i lighly more difficul o handle, from a echnical poin of view, ince Sobolev pace involve imulaneouly all he frequencie, wherea for Hölder pace we can work a a fixed frequency cale. We do no know how uch exenion could be implemened wihin he eing of he regulariy rucure. The fir par of hi work i devoed o a precie udy of he o-called paraconrolled calculu in a very abrac eing, given by a doubling ambian pace, equipped wih a elf-adjoin operaor L generaing a emigroup wih Gauian bound for i kernel and i gradien. A uiable definiion of paraproduc i given, and he main rule of calculu for paraconrolled diribuion are decribed. Thi general heory i all we need o udy a number of parabolic ingular PDE on manifold. 3

4 4 1. Paraconrolled calculu The mechanic of paraconrolled calculu [3] i elemenary and eay o ue; we decribe i here a he preen work only exend i cope while keeping i rucure unouched unlike our work [7]. We ue a omewha informal yle in hi ecion and ake he ochaic PDE given by he parabolic Anderon model equaion PAM u = u ξ a an illuraion. The ymbol ξ and here for a -dimenional paial whie noie, of Hölder regulariy 1. A he beginning of hi ory i he fac ha one expec a oluion u o ha equaion o be 1 +-parabolic Hölder regular, a a conequence of he regularizing properie of he hea emigroup, while hi regulariy i no ufficien o make ene of he produc u ξ, a he um of heir parabolic Hölder regulariy i no poiive. The poin, however, i ha u i no expeced o be any kind of 1 -Hölder funcion, raher i i expeced o behave, a mall pace-ime cale, like he oluion X of he elemenary well-poed equaion + X = ξ, wih null iniial condiion. The paraconrolled approach o olving he -dimenional PAM equaion coni in he following hree ep proce. 1 Se yourelf a ξ-dependen anaz for he oluion pace, made up of funcion ha behave like X a mall pace-ime cale, and equipped wih a Banach pace rucure. Show ha he produc u ξ i well-defined for any elemen u of he anaz pace. 3 Solve he equaion via a fixed poin argumen. The uble poin here i ha he daa of he diribuion ξ ielf i no ufficien o give ene o he produc u ξ, and ha we really need ha ξ be random o build on he probabiliy pace where i i defined anoher diribuion ogeher wih which one can make ene of all he produc u ξ, for any u in he anaz pace. Once hi enrichmen of ξ ha been done by purely probabiliic mean, he above hree ep proce i run in a deerminiic eing. From a echnical poin of view, a 1 -Hölder funcion u will be aid o behave like X a mall cale if i i of he form u Π v X, wih v bounded, up o ome erm more regular han X; wrie u = Π v X+u, wih a remainder u of Hölder regulariy ricly greaer han 1. The bilinear operaor Π, which will be a generalized paraproduc, ha appear here ha good coninuiy properie on large clae of diribuion and aifie he ideniy ab = Π a b + Π b a + Πa, b for any bounded funcion a, b, for a coninuou operaor Π, on L L ha happen o exend coninuouly o pair of Hölder regular diribuion for which he um of heir regulariy i poiive. In he oru, he Lilewood-Paley decompoiion of a and b a an infinie um of mooh funcion whoe Fourier ranform have uppor in dyadic annuli can be ued o define Π and Π,, by wriing ab = a i b j = a i b j + b j a i + a i b j. i,j i<j 1 j<i 1 i j 1 Thi definiion juifie ha we call Π, he diagonal operaor. The following formal analogy wih he rule of ochaic calculu will enlighen he core echnical ool

5 of paraconrolled calculu decribed in a econd. coninuou maringale one ha dmn = MdN + NdM + d M, N. Recall ha if M and N are wo The above pace of funcion u = Π v X + u, can be urned ino a Banach pace. Once hi anaz for he oluion pace ha been choen, remark ha he produc u ξ can formally be wrien a u ξ = Π u ξ + Π ξ u + Πu, ξ = Π u ξ + Π ξ u + Π Π v X, ξ + Πu, ξ. Since u ha Hölder regulariy ricly bigger han 1 and ξ i 1 -regular, he um of heir regulariy indice i poiive, and he erm Πu, ξ i perfecly well-defined. Thi live u wih Π Π v X, ξ a he only undefined erm. The following fac i he workhore of paraconrolled calculu. The rilinear map Ca, b, c := Π Π a b, c a Πb, c happen o depend coninuouly on a, b and c provided hey are Hölder diribuion, wih he um of heir Hölder exponen poiive. Noe he paralell beween he coninuiy of hi commuaor and he rule for ochaic differenial, for which, given anoher coninuou maringale P, we have d MdN, P = M d N, P. The formal produc u ξ can hu be wrien a a um of well-defined erm plu he formal produc v ΠX, ξ, wih a diagonal erm ΠX, ξ ill undefined on a purely analyic bai. Thi i where probabiliy come ino play. If one regularize ξ ino ξ ε, wih X ε defined accordingly, one can prove ha here exi a funcion/conan C ε uch ha he renormalized quaniy ξ,ε := Π X ε, ξ ε C ε converge in probabiliy o ome limi diribuion ξ of Hölder regulariy = ; hi i enough o make ene of he produc v ξ on an analyical bai; bu replacing Π X ε, ξ ε by Π X ε, ξ ε C ε in he decompoiion of u ξ ε amoun o looking a he produc u ξ ε C ε. The enhancemen ξ := ξ, ξ of ξ i called a rough, or enhanced, diribuion, and one can ue i o define he produc u ξ from he above formula. A ha poin, i doe no come a a urprie ha one can hen e PAM equaion a a fixed poin problem in he anaz pace, and ha he unique oluion o he problem a i happen o be i he limi of he oluion o he elemenary problem + u ε = u ε ξ ε C ε. The PAM equaion 1.1 i aid o have been renormalized. More complicaed problem are reaed along hee line of reaoning in [3, 18, 64, 65], o cie bu a few. All ue he above paraproduc machinery in he eing of he oru, where i i defined via Paley-Lilewood decompoiion, uch a decribed in hi ecion. We inroduce in hi work a far more flexible paraproduc, defined inrinically in erm of he emigroup aociaed wih he operaor L ha play he role of in he equaion, in a very general geomerical eing. Thi offer he poibiliy o inveigae ochaic PDE in a manifold eing, which i our primary moivaion. We gain much flexibiliy along he way, in erm of operaor ha can be ued in place of, and even in Euclidean domain, he cope of he preen work eem o be beyond he preen day knowledge provided by he heory of regulariy rucure. 5

6 6 1.3 A generalized parabolic Anderon model A an illuraion of our machinery, we udy he ochaic PDE given by he generalized parabolic Anderon model equaion gpam u + Lu = Fu ξ, u = u, on ome poibly unbounded -dimenional Riemannian manifold M aifying ome mild geomeric condiion. One can ake a operaor L he Laplace-Belrami operaor or ome ub-ellipic diffuion operaor; ee ecion.1 for example. The nonlineariy F i C 3 b, and ξ and here for a weighed Gauian noie wih weigh in L L ee he definiion in ecion 5.3. The deerminiic counerpar of he gpam equaion can be e once we are given a rough diribuion ζ = ζ, ζ ; we wrie informally u ζ for he produc operaion in he anaz pace a lighly differen and more precie noaion will be adoped laer on. The following reul involve ome parabolic Hölder pace C γ, wih negaive exponen γ, ha are defined in ecion.3 in erm only of he emigroup P generaed by L. We refer o ecion 5 for a full aemen and proof of hee reul. The geomerical aumpion aed here are inroduced and explained in he nex ecion. The nex wo aemen hold rue provided he hea kernel of he emigroup generaed by L, ogeher wih i gradien, aify ome Gauian bound given in he begining of ecion.1; condiion on L enuring ha hee bound hold are alo given here. The leer C α and for a paial ha i ime-independen Hölder pace. Theorem 1.1. Aume ha he meaured meric manifold M, d, µ i equipped wih a volume doubling meaure, and ha he hea emigroup generaed by L ha a kernel ha aifie ome Gauian bound UE, ogeher wih i gradien Lip. Le α 3, 1, an iniial daa u C α, a nonlineariy F Cb 3, and a poiive ime horizon T. Le ζ = ζ, ζ be a rough diribuion, wih ζ C α and ζ C T C α. a Local well-poedne for gpam. For a mall enough ime horizon T, he generalized PAM equaion 1. u + Lu = Fu ζ, u = u ha a unique oluion. b Global well-poedne for PAM. Under he aumpion ha he rough diribuion ake value in ome pace of weighed diribuion, he PAM equaion u + Lu = u ζ, u = u ha a unique global in ime oluion in ome funcion pace. The implemenaion of hi reul in he cae where ζ = ξ i a random Gauian paial noie ake he following form, for a precie verion of which we refer o heorem 5.5; i hold in he ame geomerical eing a he above reul. The propery of he meaure µ pu forward in he aemen i called Ahlfor regulariy; wrie B r x for a meric ball of cener x and radiu r. Theorem 1.. Suppoe, in addiion o he Gauian bound UE and Lip aified by he hea kernel and i derivaive, ha he reference meaure µ on he manifold M i doubling and aifie he uniform lower bound µ B r x c 1 r ν, for all x M, for ome poiive conan c 1 and he homogeneou dimenion ν. Le ξ and for a ime-independen weighed noie in pace, and e ξ ε := P ε ξ, and X ε = P ξ ε d.

7 [ a There exi a ime-independen funcion C ε := E Π L 1 ξ ε, ξ ε ] on M uch ha he pair ξ ε, X ε C ε converge in probabiliy o a random rough diribuion ξ. b If u ε and for he oluion of he renormalized equaion u ε + Lu ε = F u ε ξ ε C ε F u ε Fu ε, u ε = u i converge in probabiliy o he oluion u of equaion 1. driven by ξ. Noe ha one canno expec he renormalizing funcion C ε o be conan unle he manifold M i homogeneou and he operaor L commue wih he group acion which hold in he oru when working wih he Laplacian. Noe alo ha we do no aume M o be bounded. Working wih a weighed noie raher han wih whie noie allow u o by-pa he omewha heavy ue of weighed Hölder pace, uch a done in [38, 39] and [7]; he laer work deal, among oher hing, wih paraconrolled calculu in weighed Hölder pace. We have organized our work a follow. Secion preen he funcional eing in which our heory i e. The main geomerical aumpion on he geomeric background are given in ecion.1, where example are given; hee aumpion involve he properie of he hea kernel of he emigroup e L generaed by L. A family of operaor i inroduced in ecion., which will play in he equel he role played by Fourier projecor in he claical Lilewood-Paley heory. We inroduce in ecion.3 a cale of Hölder pace, defined uniquely in erm of he emigroup e L. A paraproduc i inroduced in ecion 3.1 and i hown in ecion 3. o enjoy he ame coninuiy properie a i Euclidean analogue. A crucial commuaor eimae beween paraproduc and reonan erm i proved in ecion 3.3, ogeher wih ome paralinearizaion and compoiion eimae in ecion 3.4. Following [3], we hen inroduce in ecion 4.1 wha play he role in our eing of paraconrolled diribuion, and prove ome fundamenal Schauder eimae in ecion 3.5. Secion o 4 give u all he maerial needed o inveigae a number of ingular PDE on manifold from he poin of view of paraconrolled diribuion. Secion 5 i dedicaed o he proof of heorem 1.1 and 1.. We end hi work by Appendix B, joinly wrien wih Dorohee Frey, in which we explain how we can weaken our aumpion of Lipchiz regulariy of he hea kernel Lip, which we make in he main body of hi work, in erm of more geomerical properie. We alo how ha one can prove reul in Sobolev pace imilar o hoe proved in Hölder pace in he main body of ha work. 7 We collec here a number of noaion ha will be ued hroughou ha work. For a ball B of radiu r and a real λ >, denoe by λb he ball concenric wih B and wih radiu λr. We hall ue u v o ay ha here exi a conan C independen of he imporan parameer uch ha u Cv and u v o ay ha u v and v u. We alo adop he non-convenional noaion γ a for he claical gamma funcion, defined for a > by he formula γ a := x a e x dx x ;

8 8 he capial leer Γ will be ued o denoe he carré du champ operaor of ome oher operaor. For p [1, ] and every f L p, he L p -norm, wih repec o he meaure µ, i denoed by f p. For p, q [1, ], we wrie T p q for he operaor norm of an operaor T from L p o L q. For an ineger k, we wrie Cb k for he e of funcion coninuouly differeniable k-ime f : R R, equipped wih he norm f C k := f + up b f i. 1 i k Funcional calculu adaped o he hea emigroup A announced in he inroducion, hi ecion i dedicaed o decribing he funcional framework where we hall e our udy. Secion.1 e he geomerical framework needed for wha we wan o do, in erm of a emigroup. We inroduce in ecion. ome operaor ha will play he role of localizer in frequency pace. Thee operaor are ued in ecion.3 o define a cale of Hölder pace which will be inrumenal in he equel..1 Hea emigroup on a doubling pace Le denoe by M, d be a locally compac eparable meriable pace, equipped wih a Radon meaure µ, ricly poiive on any non-empy open e. Given a ball Bx, r of cener x and radiu r, he noaion V x, r will and in he equel for µ Bx, r. To make hing concree, he pace M, d will mainly be for u mooh Riemannian manifold or a poibly infinie meric graph. We hall aume ha he meric meaure pace M, d, µ aifie he following volume doubling propery VD V x, r V x, r, for all x M and poiive r, which can be aed equivalenly under he form r ν.1 V x, r V x,, for ome poiive caling facor ν, for all x M, and all < r; i implie he inequaliy dx, y + r ν V x, r V y,, for any wo poin x, y in M and < r. Anoher eay conequence of he volume doubling propery i ha ball wih a non-empy inerecion and comparable radii have comparable meaure. Le alo be given a non-negaive elf-adjoin operaor L on L M, µ wih dene domain D L L M, µ. Denoe by E i aociaed quadraic form, defined by he formula Ef, g := flg dµ, on a domain F which conain D L. We hall aume ha he Dirichle form E i rongly local and regular; we refer he reader o he book [6, 34] of Fukuhima & co. and Gyrya Saloff-Coe for precie definiion and background on Dirichle form. The reader unfamiliar wih hi eing may hink of he Laplace operaor in a compac Riemannian manifold. Thee wo properie will be obviouly aified in he M

9 example we hall work wih. I follow from hee condiion ha he operaor L generae a rongly coninuou emigroup e L > of conracion on L M, µ which i conervaive, in he ene ha e L 1 = 1, for all ; ee e.g. Subecion..7 in he book [34]. We hall alo aume ha he emigroup e L ha a kernel, given for > all poiive ime by a non-negaive meaurable real-valued funcion p on M M, uch ha e L f x = p x, yfy dµy, M for µ-almo all x in M, and every f D L. The kernel p i called he hea kernel aociaed wih L. We aume ha i aifie for all < 1 and µ-almo all x, y, he following ypical upper eimae 1 p x, y V x, V y,. 9 Under he volume doubling condiion VD, he previou eimae elf-improve ino a Gauian upper eimae UE for he hea kernel and i ime derivaive UE a a p x, y c V x, V y, exp dx, y. ha hold for a fixed poiive conan c, for all ineger a, all ime < 1, and µ- almo every x, y M; ee for inance he aricle [8, Theorem 1.1] for he Riemannian cae, and he work [, Secion 4.] for a meric meaure pace eing. We alo aume ha he hea kernel aifie he following Lipchiz condiion Lip p x, y p z, y dx, z 1 V x, V y, exp c dx, y. Le ini here ha inequaliie UE and Lip are aumed o hold only for < 1, raher han for all poiive ime. I follow claically from he Gauian eimae UE and he volume doubling propery ha he hea emigroup e L > i uniformly bounded on L p M, µ for every p [1, ], and rongly coninuou for p [1,. La, noe ha e L i, under hee condiion, bounded analyic on < 1 L p M, µ, for every 1 < p < +, which mean in paricular ha he ime-derivaive L n e L < 1 are bounded on Lp M, µ uniformly in < 1, for every ineger n ; ee [57]. A commen i in order here, abou our wo aumpion UE and Lip. In he heory of regulariy rucure or Euclidean heory of paraconrolled calculu, regulariy a any order may be conidered becaue of he implici ue of he very nice differenial geomery of Euclidean pace, or he oru. In our curren and far more general framework, ince we only have a poinwie aumpion on he hea kernel and i gradien, i i naural o expec ha one canno quanify he regulariy of ome objec o an order greaer han 1. Tha i why in he differen aemen proved in he nex ecion ome exra mild condiion on he regulariy exponen will appear, a compared wih heir Euclidean analogue. Since we aim o work wihin he preen opimal / minimal eing, hee new limiaion canno be removed wihou addiional aumpion, and we hall be rericed o udy regulariy properie a order a mo 1, including negaive order; hi i no rericive a far a applicaion are concerned in he preen work. Here are four repreenaive clae of example of doubling meric meaure pace and Dirichle form aifying he above condiion. Thi li of example emphaize

10 1 ha we have much flexibiliy in erm of he operaor L a well a in erm of he underlying pace M, d, µ. a Markov chain. Le X be a counable e equipped wih a Markov chain, pecified by a ymmeric Markov kernel k : X X R +, and le m be a non-negaive funcion on X, ued o define a meaure m on X, wih deniy m wih repec o he couning meaure µ. Denoe by, m he calar produc on l m. Conider alo for ineger n 1 he ieraed kernel k n defined recurively by k n x, y := k n 1 x, zkz, y µdz. Denoing by K he ymmeric Markov operaor wih kernel k wih repec o µ, he formula Ef, g = 1 k xy fx f y gx g y = x X x,y X 1 f x g x k xy g y m x m x y X = f, Lg m aociaed wih he non-negaive elf-adjoin operaor 1 Lg x = g x k xy g y = 1 gx Kg m x m x x, y X define a rongly local regular Dirichle form and allow u o generae he coninuou hea emigroup e L. The above um in x i implicily rericed o hoe x for which m x >, o here i no lo of generaliy in auming ha m >. The map k induce a diance d on X by leing be equal o min { n 1 ; z,..., z n, wih z = x, z n = y and k z i, z i+1 >, for i =..n 1 }, for y x. Following Grigor yan reul [9], one can give growh condiion on he m-volume of d-ball ha enure he conervaive characer of he emigroup generaed by L in l m. Then i i claical ha geing Gauian upper eimae for he emigroup e L i very cloely relaed o geing dicreeime verion of Gauian eimae for he ieraed Markov chain K n n 1, and imilarly for he Lipchiz regulariy of heir kernel. Uually, given uch a dicree framework, one prefer o work wih he dicree-ime Markov chain raher han he coninuou hea emigroup. To obain upper Gauian eimae and a Lipchiz regulariy for he ieraed Markov chain on a graph i he opic of a huge lieraure o which we refer he reader; ee for inance work by Hebich and Saloff-Coe [41] for dicree group and by Ichiwaa [44] for an exenion o nilpoen covering graph and more recenly [45] for a perurbaion of hee previou reul. For example, he regular graph Z d and Z/NZ d have hea emigroup aifying he Gauian eimae UE and he Lipchiz propery Lip. Needle o ay, for a large finie graph X, E, wih edge e E, and b xy = 1 if x, y E, and m x = y X b xy, he previou reul hold wih he graph diance in he role of d. b Second order differenial operaor on Riemannian manifold. Le M, d, µ be a doubling, poibly non-compac, complee Riemannian manifold wih Ricci curvaure bounded from below. Then he hea emigroup e generaed by he Riemannian Laplace operaor aifie boh he upper Gauian eimae UE and he Lipchiz regulariy Lip for mall ime < 1, and for every ime > if he Ricci curvaure i nonnegaive; ee [61] and [48] for

11 reference. Paricular example are mooh compac Riemannian manifold, or unbounded Riemannian manifold wih pinched negaive Ricci curvaure, uch a hyperbolic pace. Even on he Euclidean pace R d, we may conider a econd order divergence form operaor L = diva given by a map A aking value in real ymmeric marice and aifying he uual ellipiciy/accreiviy condiion. Then if A i Hölder coninuou, i i known ha L generae a elf-adjoin emigroup aifying he properie UE and Lip; ee [4]. Similarly, conider an open bounded ube Ω R d wih Lipchiz boundary for example o enure he Ahlfor regulariy and conider he elf-adjoin Laplace operaor L aociaed wih Dirichle or Neumann boundary condiion. There i an exenive lieraure o decribe aumpion on Ω uch ha UE and Lip are aified. The preen eing may well be beyond he preen cope of regulariy rucure, for which he Green funcion of he operaor need o aify ome regulariy aumpion ha were no proved o hold rue under a ole Hölder coninuiy aumpion for A, and whoe formulaion on a manifold i a real problem ouide he realm of Lie group or homogeneou pace. On he oher hand, he heory developed here work well in ha relaively minimal eing. The eimae UE and Lip alo hold when working on a convex or C - regular bounded ube of he Euclidean pace, wih L given by Laplace operaor wih Neumann boundary condiion; ee [63]. c Sub-ellipic lef invarian diffuion on group. Le G be a unimodular conneced Lie group, endowed wih i lef-righ Haar meaure µ. Conider a family X := {X 1,..., X l } of lef-invarian vecor field on G aifying Hörmander condiion. They define a cla of admiible pah γ, characerized by he exience, for each of hem, of meaurable funcion a 1,..., a k uch ha one ha k γ = a i X i l. i=1 The lengh of uch a curve i defined a γ := 1 1 l 1 a i i=1 and he Carno-Caraheodory diance dx, y beween any wo poin x, y of G i defined a he infimum of he lengh of all admiible curve joining x o y. We hen conider he ublaplacian defined by k := Xi. i=1 Then he operaor generae a hea emigroup aifying boh he upper Gauian eimae UE and he Lipchiz regulariy Lip for mall ime, 1]; ee for inance Chaper 8 in he book [6]. If he group i nilpoen hen i i alo globally doubling [33] and o he hea emigroup aifie he Gauian upper bound UE and enjoy he Lipchiz propery Lip for every > ; ee [6, 55]. Paricular example of uch group, are raified Lie group, and o Heienberg group. For uch Heienberg-ype Lie group, a kind of Fourier ranform may be defined involving heir irreducible uniary repreenaion, which can be ued o define an analog of he Euclidean paraproduc d, 11

12 1 / paradifferenial calculu, uch a done i [7]. We hall ee, a a by-produc of he preen work, ha he rucure of hea emigroup i ufficien o conruc imilar ool wih greaer cope. d The general cae given by a ubellipic operaor i more difficul. Le M, d, µ be a complee and mooh conneced manifold endowed wih a elf-adjoin mooh locally ubellipic diffuion operaor L aifying L1 =. Then Baudoin and Garofalo inroduced in [9] a propery, called a generalized curvauredimenion inequaliy, which ha o be hough of a a lower bound on a ub- Riemannian generalizaion of he Ricci enor. Under uch a condiion, he hea kernel generaed by L aifie UE a well a Lip; ee [54]. We refer he reader o [9] for ome example of uch ub-ellipic eing and he fac ha he hea kernel alo aifie in ha cae ome Gauian lower bound. Throughou ha work, a poin o M will be fixed, which we hall ue o define a cla of e funcion, ogeher wih i dual cla of diribuion. Definiion. We define a Fréche pace of e funcion eing { S o := f D L n ; a 1, a N, 1 + do, } a 1 L a f <, n wih f := up do, a 1 L a f. a 1,a N A diribuion i a coninuou linear funcional on S o ; we wrie S o for he e of all diribuion. We poin ou ha he arbirary choice of poin o M i only relevan in he cae of a unbounded ambian pace M; even in ha cae, he e S o doe no depend on o, for o ranging inide a bounded ube of M. Every bounded funcion define for inance an elemen of S o. Example of e funcion are provided by he p x,, for every fixed x M and < 1. Indeed for ineger a 1, a, he upper bound UE wih he analyiciy of he emigroup yield ha L a p x, aifie he ame upper Gauian eimae han he hea kernel ielf and o we deduce ha 1 + do, y a 1 L a p x, y a V x, 1 + do, ya 1 e c dx,y V x, 1 + do, xa 1 e c dx,y a for ome poiive conan c and c. Noe ha he hea emigroup ac no only on funcion, bu alo on diribuion, by eing e L φ, f := φ, e L f for φ S o and f S o. We refer he reader o [17] and [49] for more deail on he exenion of he emigroup o diribuion. For a linear operaor T acing from S o o S o, i will be ueful below, o denoe by K T i Schwarz kernel, characerized by he ideniy T f, g = K T x, yfygx µdyµdx, giving an inegral repreenaion for every f, g S o.

13 . Time derivaive and Carré du champ of he emigroup 13 Le u inroduce here a family of operaor ha will play he role in our eing of he Fourier muliplier ued in he claical Lilewood-Paley heory, ha localize a funcion in frequency pace. Thee will be he building block ued o define a convenien paraproduc for our need, uch a done below in ecion 3.1. Definiion. Given a fixed poiive ineger a, e. Q a := L a e L and.3 P a := φ a L, where φ a x := 1 γ a for every >. x a d e, x, So we have for inance P 1 = e L, and Q 1 = Le L. The wo familie of operaor P a > and Q a are defined o a o have he relaion >.4 P a = Lφ al = γa 1 Q a, o Q a = 1 a a a e L, and P a = p a Le L, for ome polynomial p a of degree a 1, wih p a = 1. The analyiciy of he emigroup provide a direc conrol of he operaor P a and Q a. Propoiion.1. For any ineger a, he operaor P a and Q a have kernel aifying he Gauian eimae UE, and he Lipchiz regulariy propery Lip; a a conequence, hey are bounded in every L p pace for p [1, ], uniformly wih repec o, 1]. Following he above inerpreaion of he operaor Q a and P a, he following Calderón reproducing formula provide a decompoiion of a funcion f in L p M, µ ino a low frequency par and a high frequency par very imilar o he Lilewood-Paley decompoiion of a diribuion in erm of frequencie; ee e.g. [8]. Propoiion. Calderón reproducing formula. Given p 1, + and f L p M, µ, we have lim P a + f = f in L p M, µ for every poiive ineger a, and o.5 f = γ 1 a Q a f d + P a 1 f. Proof One know from heorem 3.1 in [3], ha he operaor L ha a bounded H funcional calculu in L p M, µ under he volume doubling condiion on M, d, µ, and he aumpion ha he hea kernel aifie he upper eimae UE. Since hi implie in paricular ecorialiy of L in L p M, µ, Theorem 3.8 in [1] yield he decompoiion of L p M, µ ino nullpace and range of L. Uing hi decompoiion, he Convergence Lemma implie for every f L p M, µ f = lim P a = γ 1 a f = Q a f d + P a 1 f, P a f d + P a 1 f

14 14 where he limi i aken in L p M, µ and where we have ued ideniy.4; ee e.g. [3, Theorem D] or [46, Lemma 9.13]. We hall alo make an exenive ue in he equel of he quare-roo of L, given by i carré du champ operaor Γ, defined for all f, g D L D L a a bilinear operaor aifying he ideniy Ef, g := flg dµ = glf dµ = Γf, g dµ. M M I i alo given by he explici formula Γf, g = 1 Lfg flg glf ; we hall wrie D Γ L for i domain, which conain D L. A a horhand, we wrie Γf for Γf, f 1 in he equel, which can be hough a he lengh of he inrinic gradien of f. I follow from he conervaive propery of L and i non-negaive characer, ha he bilinear map Γ i poiive and aifie he ideniy Γf = Γf, f dµ = flf dµ = Ef, f. L M M From i poiive propery, a generalized Cauchy-Schwarz inequaliy yield ha for every f, g D L hen.6 Γf, g Γf, fγg, g = ΓfΓg. According o he Beurling-Deny-Le Jan formula, he carré du champ aifie a Leibniz rule.7 Γfg, h = f Γg, h + g Γf, h, for all f, g, h D Γ, and a chain rule.8 L F f = F f Lf + F f Γf, f. for every funcion F C b R and every f D L; he funcion F f i auomaically in D L ee e.g. [6, Secion 3.] and [58, Appendix] for hee poin. The following poinwie and L p -eimae for he inrinic gradien of he emigroup will be ued everal ime in a crucial way; i proof i given in Appendix A. I ay ha he carré du champ of he emigroup aifie alo ome Gauian poinwie eimae, a given by he following claim. Propoiion.3. The following inequaliy hold.9 Γ e L f 1 x c V x, V y, exp dx, y fy dµy, M for every >, every funcion f L, and almo every x M. Conequenly, we have Γ e L p p <, up > for every p [1, ]. We may replace he emigroup e L in he above equaion by any of he operaor P a, for any a. M

15 .3 Hölder and Beov pace hrough he hea emigroup pace Λ σ if f Λ σ := f + up <dx,y 1 15 Le u recall a a ar ha given a parameer σ, 1], a bounded funcion f L i aid o belong o he Hölder fx fy dx, y σ <. Recall on he oher hand he definiion of he inhomogeneou Beov pace aociaed o a emigroup; hey were preciely udied in everal work, uch a [17] or [3], o name bu a few. We hall make an exenive ue of hee pace. Definiion.4. Fix a poiive ineger a, an exponen p, q 1,, and σ R. A diribuion f S o, i aid o belong o he Beov pace Bp,q σ if f B σ p,q := e L f p + q σ Q a f d q <. p Thi definiion of he pace doe no depend on he ineger a 1, provided a i big enough. We refer he reader o [17] for deail abou uch pace and a proof of he fac ha hey do no depend on he parameer a ued o define hem, provided a i ufficienly large wih repec o σ. The limiing cae p = q = lead o he following definiion. Definiion. Le a poiive ineger a be given. For σ,, a diribuion f S o i aid o belong o he pace C σ if f C σ := e L f + up Q a f σ <. < 1 Thi definiion of he pace doe no depend on he ineger a 1. We give in Appendix A a imple and elf-conained proof ha he pace C σ doe no depend on a, and ha any wo norm C σ, defined wih wo differen value of a, are equivalen. The following propoiion juifie ha we call he pace C σ Hölder pace, for all σ <, poibly non-poiive. Propoiion.5. For σ, 1, he pace Λ σ and C σ are he ame and he wo correponding norm are equivalen. We give here a complee proof of hi propoiion a i provide an elemenary illuraion of how he properie of he operaor Q a are ued o make acual compuaion. Thi kind of reaoning and compuaion will be ued repeaedly in he equel, when working wih our paraproduc. Recall ha he operaor Q a have kernel K a Q aifying Gauian poinwie eimae, by propoiion.1. Proof We divide he proof in wo ep, by howing ucceively ha Λ σ i coninuouly injeced in C σ, and ha, converely, C σ i coninuouly injeced in Λ σ. Sep 1. Λ σ C σ. Noe fir ha ince he Hölder pace Λ σ i made up of bounded funcion, i i included in S o. Fix an ineger a 1; hen for every, 1, we have Q a f x = Q a f fx x = For he poin z M, wih dx, z < 1, we have fz fx dx, z σ f Λ σ σ f Λ σ K a Q x, z fz fx µdz.

16 16 o ha dx,z K Q a x, z fz fx µdz σ f Λ σ σ f Λ σ, KQ a x, z µdz ince Q a ha a kernel aifying Gauian poinwie bound. The ame bound how ha dx,z 1 K Q a x, z fz fx µdz f Λ σ σ f Λ σ σ f Λ σ. KQ a dx,z 1 dx,z 1 1 V x, e c Similarly, we have K a Q x, z fz fx µdz f 1 dx,z o i come ha he inequaliy Q a e c f Λ σ σ f Λ σ, f x σ f Λ σ x, z dx, z σ µdz 1 dx,z dx,z dx, z σ µdz K a Q x, z µdz hold uniformly in, 1, and for every x M, which prove ha f C σ f Λ σ. Sep. C σ Λ σ. Le f C σ be given. Uing he decompoiion of he ideniy provided by Calderón reproducing formula f = e L f + we fir deduce ha f i bounded, wih f f C σ 1 + σ Q 1 f d, d f C σ. Moreover, for any wo poin x, y, wih < dx, y 1, we have { e fx fy = L f x e L f } y + { } { = e L fx e L fy + + Q 1 { Q 1 1 x f { f Q f x } 1 y f f Q 1 x Q Q 1 } d f y y } d.

17 One can ue he Lipchiz regulariy Lip of he hea kernel o bound he fir erm in he above um, giving e L fx e L p1 fy x, z p 1 y, z fz µdz dx, y f. A imilar bound hold for Le L, by analyiciy of he hea kernel, he econd erm admi a imilar upper bound. Le now focu on he hird erm, uing a imilar reaoning and noing ha Q = 16 Q Q. So, for dx, y, we can wrie Q f x Q KQ f y x, z K Q y, z K Q fz µdz dx, y KQ f dx, y σ f C σ. If dx, y, hen we direcly have f x f y Hence, { Q Q f x Q f y Q } d dx,y Q σ dx, y σ f C σ, ince σ, 1. Conequenly, we have obained fx fy dx, y σ f C σ f σ f C σ. d 1 dx, y + σ dx,y 17 d f C σ uniformly for every x y wih dx, y 1, o indeed f Λ σ f C σ. Our main example of a C σ diribuion wih negaive Hölder exponen σ will be given by ypical realizaion of a poibly weighed noie over M, µ ee Propoiion 5.4. To prove ha regulariy propery, i will be convenien o aume ha he meric meaure pace M, d, µ ha he following propery, called Ahlfor regulariy. There exi a poiive conan c 1 uch ha V x, 1 c 1, for all x M, which, by he doubling propery, implie ha we have.1 V x, r c 1 r ν, for ome poiive exponen ν, all x M and all < r 1. The conan ν i d on a d-dimenional manifold equiped wih a mooh meaure. Thi i a relaively weak aumpion ha i eenially aified in a Riemannian eing for cloed manifold wihou boundary and injeciviy radiu bounded below by a poiive conan or in a ub-domain of he Euclidean pace provided ha he boundary i Lipchiz. Under ha addiional non-degeneracy aumpion on he volume meaure, we have he following Beov embedding, proved in Appendix A.

18 18 Lemma.6 Beov embedding. Given < σ <, and 1 < p <, we have he following coninuou embedding. B σ p,p B σ p, B σ ν p, = C σ ν p Beov embedding can be ued in a very efficien way o inveigae he regulariy properie of random Gauian field, a will be illuraed in ecion 5.3. Remark. Le u poin ou here ha our Hölder pace C σ, wih σ <, coincide in he Euclidean eing wih hoe ued by Hairer [35], and, roughly peaking, defined in erm of caling properie of he pairing of a diribuion wih a one parameer family of recaled funcion. Indeed, on he Euclidean pace i i known ha o define Beov pace or Hölder pace hrough Lilewood-Paley funcional, we may choe any good Fourier muliplier aifying uiable condiion; he laer are aified by he derivaive Q a of he hea emigroup. So our pace correpond o he andard inhomogeneou pace defined by any Lilewood-Paley funcional. From wavele or frame characerizaion [5], we can conclude ha our Hölder pace coincide wih hoe ued in [35] or [38]. Before urning o he definiion of our paraproduc, we cloe hi ecion wih wo coninuiy properie involving he Hölder pace C σ, which we hall ue in he equel. Propoiion.7. For any σ,, and every ineger a, we have P a f C σ. 1 f Proof We have by conrucion P a 1 = 1 + α 1 L + + α a 1 L a 1 e L, for ome coefficien α 1,..., α a 1. A we have by definiion e L f f C σ, and L l e L = Q l 1, for l = 1... a 1, we ee ha L l e L f f C σ, ince we have een above ha we can chooe he parameer a in he definiion of he Hölder pace. Propoiion.8. For σ, 1, we have σ Γ e f L f C σ. up,1] The ame concluion hold wih any of he operaor P a in he role of e L. Proof Given, 1], ue Calderón reproducing formula o wrie Γ e L f Γ e L Q 1 f d + Γ e 1+L f. We divide he inegraion inerval in he above-righ hand ide ino, and [, 1] o bound ha erm. For <, we have e L Q 1 = + e L Q 1, o we can ue + Propoiion.3 o ge Γ e L Q 1 f Γ e L Q 1 f + σ f C σ.

19 19 Similarly for, hen e L Q 1 = e L + Q 1 +, and we have Γ e L Q 1 f 1 Γ e L Q 1 +f 1 σ f C σ. Similar compuaion give he eimae Γ e 1+L f f C σ. We conclude by inegraing wih repec o, 1, uing here he fac ha σ < 1. 3 Paraproduc and commuaor eimae in Hölder pace 3.1 Paraproduc baed on he emigroup Bony paraproduc machinery ha i roo in he Lilewood-Paley decompoiion of any diribuion f a a um of mooh funcion i f localized in he frequency pace, o a produc fg of any wo diribuion can formally be decompoed a 3.1 fg = i f j g = i f j g + i f j g =: 1 + Πf, g i,j i j i j 1 ino a um of produc of wo funcion ocillaing on differen cale, and an a priori reonan erm Πf, g. Thi decompoiion draw i uefulne from ome relaively elemenary a priori eimae ha how ha he erm 1 above make ene and i well-conrolled under exremely general condiion, while he reonan erm Πf, g can be hown o define a coninuou map from C α C β o C α+β, provided α + β >. Thee eimae rely crucially on ome properie inheried from he very definiion of he Lilewood-Paley block a Fourier projecor. Thee properie canno be graped o eaily in our emigroup eing; however, we hall ue he operaor P a, Q a and Γ or LP a a frequency projecor, wih P a projecing on frequencie lower han or equal o 1, and Q a, Γ or LP a a localizing a frequencie of order 1. Indeed, in he oru, and working wih he Euclidean Laplacian, he operaor Q a ha for inance a Fourier ranform equal o Q a λ = λ a e λ ; i i eenially localized in an annulu λ 1. Similar explici Fourier picure for he oher operaor can be given in he eing of he oru. Thi frequency inerpreaion of hee operaor will be our main guide in he definiion of our paraproduc given below. Thi paraproduc will depend on a choice of a poiive ineger-valued parameer b ha can be uned on demand in any given problem; he bigger b i, he more we can do ome inegraion by par i will be fixed a ome poin, and he reader i invied no o boher abou i value. To clarify noaion, we hall repeaedly ue below he noaion f g for he uual produc of wo funcion.

20 Raher han aring wih Bony decompoiion 3.1, we ake a a aring poin Calderon reproducing formula 3. fg = lim P b = 1 { 1 γ b where P b P b + 1 f, g, f P b g = Q b f P b g { + P b P b P b P b f Q b g 1 f, g := P b 1 P b 1 f P b 1 g } f P b d g + 1f, g + Q b P b f P b g } d and for he low-frequency par of he produc of f and g, and where we implicily make he neceary aumpion on f and g for he above formula o make ene. Guided by he above heuriic argumen abou he role of he operaor P a, Q a, ec. a frequency projecor, we decompoe he erm involving he produc of P a f and P a g, by uing he definiion of he carré du champ operaor Γ and wrie Q b P b = Q b 1 f P b g LP b L φ 1 φ = Lφ1 φ + Lφ φ 1 Γφ 1, φ f P b g =: B g f + B f g + Rf, g. + Q b 1 P b If one rewrie ideniy 3. under he form fg =: f LP b g Q b 1 Γ { } d + 1f, g P b wih obviou noaion, hi ugge o decompoe i a {1 fg = + Bg f } + { + B f g } d + Rf, g + 1f, g f, P b g and o idenify he inegral of he erm ino bracke in he above formula a paraproduc, and by defining he reonan erm a he inegral of Rf, g. Thi i wha wa done in [14] where hi noion of paraproduc, inroduced in [1], wa hown o have nice coninuiy properie in Hölder pace C α, provided one deal wih poiive exponen α. Given our need o deal wih negaive exponen, a refinemen of hi decompoiion eem o be needed o ge ome coninuiy properie for negaive exponen a well. We hu ue he carré du champ formula in each erm 1 and, and decompoe 1 under he form LP b Q b 1 =: A g f + Sf, g, f P b g { + P b Γ Q b 1 f, P b g wih Sf, g he um of he wo erm ino bracke, and = A f g + Sg, f. P b Q b 1 } f LP b g Noe ha he funcion A f g, Sf, g,... all depend implicily on ime. Thi decompoiion lead o he following definiion.

21 1 Definiion. Given an ineger b and f,1 C and g L, we define heir paraproduc by he formula Π b g f = 1 { } d A g f + B g f γ b = 1 { LP b γ b Q b 1 f P b g } + Q b 1 LP b f P b d g. The well-defined characer of hi inegral i proved in propoiion 3. below. Wih hi noaion, Calderon formula become wih he low-frequency par fg = Π b g f + Π b f g + Πb f, g + 1 f, g P b 1 f, g := P b 1 and he reonan erm Π b f, g = 1 { } d Sf, g + Sg, f + Rf, g γ b = 1 { Q b 1 f LP b g γ b + 1 { γ b 1 γ b P b Q b 1 Γ LP b P b f Q b 1 f, P b P b 1 f P b 1 g + P b Γ g d g. + P b Γ Q b 1 P b f, } P b d g f, Q b 1 } d g Noe ha we have Π b 1 = Id 1, 1, a a conequence of our choice or renormalizing conan. 3. Paraproduc eimae We prove in hi paragraph he baic coninuiy eimae aified by he map defined by he low frequency par, he paraproduc and he reonan erm. Propoiion 3.1. Fix an ineger b. For any real number α, β and every poiive γ, we have f, g C γ f C α g C β. for every f C α and g C β. Proof Conider he collecion Q a Q a < 1 1 f, g = Q a P b 1 for a large enough ineger a γ. Then P b 1 f P b 1 g. Since 1, we have Q a P b = a e L L a P b 1, wih he operaor La P b 1 bounded on L. We obain he concluion from Propoiion.7 a we have Q a 1 f, g a P b 1 f P b 1 g γ f C α g C β.

22 The coninuiy properie of he paraproduc are given by he following aemen; hey are he exac analogue of heir claical counerpar, baed on Lilewood-Paley decompoiion, a can be found for inance in he exbook [8] of Bahouri, Chemin and Danchin. Propoiion 3.. Fix an ineger b. For any α, 1 and f C α, we have for every g L 3.4 Π b f g f C α C α for every g C β wih β < and α + β, Π b f g C α+β C β f C α. g g The range, 1 for he regulariy exponen can appear a unuual ince in he andard Euclidean heory uch coninuiy properie hold for every α R. However, a explained in ecion.1, he rericion α < 1 come from our opimal / minimal eing where we only aume a gradien eimae on he hea kernel. The rericion α > can be explained a follow. In he Euclidean heory, nice Fourier muliplier can be ued o have a perfec frequency decompoiion and he udy of paraproduc mainly relie on he following rule: he pecrum of he produc of wo funcion i included ino he um of he wo pecrum; hi come from he group rucure hrough he Fourier repreenaion of he convoluion. In our eing, our frequency decompoiion involving he hea emigroup i no o perfec and he previou rule on he pecrum doe no hold, a lea no in uch a perfec ene. Tha i why he new limiaion α > appear; i i inheren o he emigroup approach developed here. No uch limiaion hold in he more rericed eing developed in [7]. Proof Recall ha Π g b f = 1 LP b γ b Q b 1 Given, 1], conider Q b 1 b 1 L b P b Q b 1 LP b = and for ha Q b 1 LP b = Qb P b f P b g + Q b 1 LP b Π g f. For, we ue ha e L and Q b 1 Q b 1 = and Q b 1 Q b 1 f P b d g. b 1 Q b 1 e L, = Qb Q b. Hence, wih he uniform L -boundedne of Q, P operaor, we have Q b 1 Π b g f Q b 1 P b f g + LP b P b d f g b 1 Q b 1 P b b 1 LP b P b d + f g + f g. Since f C α we have Q b 1 Moreover, if g L hen f + LP b f α f C α. P b 1 g g

23 and if g C β wih β < hen P b g du P u + b 1 g du + 1 g C β β g C β. Q b u g u 1 β We deduce he following bound a a conequence. If g L hen Q b 1 f Π b g α α f C α g, d + c α d f C α g ince α, 1 and c 1. Thi hold for every > which yield 3.4. If g C β wih α + β, 1 hen Q b 1 Π b g f α+β d 1 b 1 + α+β d f C α g C β α+β f C α g, ince b 1 1 > α + β >. Thi hold for every >, which yield 3.5. Propoiion 3.3. Fix an ineger b >. For any α, β, 1 wih α + β >, for every f C α and g C β, we have he coninuiy eimae Π b f, g f C α g C α+β C β. Proof We recall ha Π b f, g = 1 γ b + 1 γ b + 1 γ b P b Q b 1 P b LP b Q b 1 Γ P b f LP b g f Q b 1 g + P b Γ + P b Γ f, P b d g. Q b 1 P b 1 f, P b d g f, Q b g d Conider he funcion Q b 1 Π b f, g, for every, 1]. I i given by an inegral over, 1, which we pli ino I an inegral over,, and II an inegral over, 1. Since f C α, he ue of Propoiion.8, wih α < 1, yield he eimae Q b 1 f + Γ Q b 1 f + LP b f + Γ Q b 1 f + Γ P b f α f C α; a imilar eimae hold wih g in place of f, and β in place of α. Uing he uniform L -boundedne of he differen approximaion operaor, we ge for he fir par Q b 1 I α+β d f C α g C β α+β f C α g C β, 3

24 4 where we ued he ric inequaliy α + β >. For he econd par, we oberve ha for > hen c Q b 1 P = e L L b 1 P and Q b 1 Q b 1 b 1 b 1 = Q e L. So we ge for he econd par Q b 1 II α+β α+β f C α g C β, uing he fac ha b 1 > α + β. b 1 d f C α g C β 3.3 Commuaor eimae Recall he dicuion of he paraconrolled calculu approach o he udy of he parabolic Anderon model equaion given in he inroducory ecion 1.. We have a ha poin he paraproduc and diagonal operaor in hand; we deal in hi ecion wih he fundamenal commuaor eimae inroduced in [3]. Reader familiar wih he baic of ochaic analyi will noice he imilariy of hi coninuiy reul and he rule aified by he bracke operaor in Iô heory d MdN, P = P d N, P ; hi i obviouly no a coincidence. Propoiion 3.4. Conider he a priori unbounded rilinear operaor Cf, g, h := Π b Π b g f, h g Π b f, h, on S o. Le α, β, γ be Hölder regulariy exponen wih α 1, 1, β, 1 and γ, 1]. If < α + β + γ and α + γ < hen, eing δ := α + β 1 + γ, we have 3.6 Cf, g, h C δ f C α g C β h C γ, for every f C α,g C β and h C γ ; o he commuaor define a coninuou rilinear map from C α C β C γ o C δ. Proof Noe fir ha he paraproduc Π g f i given, up o a muliplicaive conan, by he um of wo erm of he form d Af, g = Q 1 Q f P g, and he reonan par Πf, g by he um of five erm of he following form 3.7 Rf, g = P 1 Γ P f, d P 3 g, or Rf, g = P LP f Q g d, or Rf, g = P 1 Q f LP g d, where he operaor