Hardy spaces for semigroups with Gaussian bounds

Transcription

1 Annali di Maemaica : hps://doi.og/1.17/s y Hady spaces fo semigoups wih Gaussian bounds Jacek Dziubański 1 Macin Peisne 1 Received: 3 July 217 / Acceped: 16 Ocobe 217 / Published online: 5 Novembe 217 The Auhos 217. This aicle is an open access publicaion Absac Le T = e L be a semigoup of self-adjoin linea opeaos acing on L 2,μ, whee, d,μis a space of homogeneous ype. We assume ha T has an inegal kenel T x, y which saisfies he uppe and lowe Gaussian bounds: C 1 μbx, exp c 1 dx, y 2 / C 2 T x, y μbx, exp c 2 dx, y 2 /. By definiion, f belongs o H 1 L if f H 1 L = sup > T f x L 1,μ <. We pove ha hee is a funcion ωx, < c ωx C, such ha H 1 L admis an aomic decomposiion wih aoms saisfying: supp a B, a L μb 1, and he weighed cancelaion condiion axωxdμx =. Keywods Hady space Maximal funcion Aomic decomposiion Gaussian bounds Hölde esimaes Mahemaics Subjec Classificaion Pimay 42B3; Seconday 42B25 42B35 35J1 1 Inoducion Le, d be a meic space equipped wih a nonnegaive Boel measue μ. We shall assume ha μ = and <μbx, <, >, whee Bx, = {y dx, y } denoes he closed ball ceneed a x and adius. Suppose, d,μis a space of homoge- The eseach was suppoed by he Polish Naional Science Cene Naodowe Cenum Nauki Gan No. DEC 212/5/B/ST1/672. B Macin Peisne macin.peisne@uw.edu.pl Jacek Dziubański jacek.dziubanski@mah.uni.woc.pl 1 Insyu Maemayczny, Uniwesye Wocławski, pl. Gunwaldzki 2/4, Wocław, Poland

2 966 J. Dziubański, M. Peisne neous ype in he sense of Coifman and Weiss [8], which means ha he doubling condiion holds, namely: Thee is C > such ha μbx, 2 CμBx, fo > andx. I is well known ha he doubling condiion implies ha hee ae q > andc d > such ha μbx, s C d s q μbx, fo x, >, and s Suppose ha L is a nonnegaive densely defined self-adjoin linea opeao on L 2,μ. Le T = e L, >, denoe he semigoup of linea opeaos geneaed by L.Weimpose ha hee exiss T x, y, such ha T f x = T x, y f y dμy. 1.2 Clealy, T x, y = T y, x fo > and a.e. x, y. Moeove, we assume he following lowe and uppe Gaussian bounds, ha is, hee ae consans c 1 c 2 > andc > such ha C 1 μbx, exp c 1dx, y 2 C T x, y μbx, exp c 2dx, y fo > and a.e. x, y. I is well known ha 1.3 implies ha fo evey n N we have n n T x, y C n n μbx, exp c ndx, y2, 1.4 fo > and a.e. x, y wih some C n, c n >. Fo his fac see, e.g., [2, 7.1], [1,27]. The Hady space H 1 L elaed o L is defined by means of he maximal funcion of he semigoup T, namely { } H 1 L := f L 1,μ f H 1 L := sup T f <. > L 1,μ On he ohe hand, we define aomic Hady spaces as follows. Suppose we have a space wih a doubling measue σ and a quasi-meic ρ. We call a funcion a an ρ, σ -aom if hee exiss a ball B = B ρ x, := {x ρx, x } such ha: supp a B, a σb 1,and ax dσx =. B By definiion, a funcion f L 1,σbelongs o Ha 1ρ, σ, ifheeexisρ, σ -aoms a k and complex numbes λ k, such ha f = λ k a k k=1 and λ k <. If such sequences exis, we define he nom f H 1 a ρ,σ o be he infimum of k=1 λ k in he above pesenaions of f. Noice ha in his paagaph we have changed he noaion. This is because in he aicle we will use diffeen meics and measues. Fo x le Φ x :,, be he nondeceasing funcion defined by The fis main esul of his pape is he following. k=1 Φ x = μbx,. 1.5

3 Hady spaces fo semigoups wih Gaussian bounds 967 Theoem 1 Suppose ha, d,μ is a space of homogeneous ype and assume ha fo each x he funcion Φ x is a bijecion on,. Le a nonnegaive self-adjoin linea opeao L on L 2,μ be given such ha he semigoup T = e L saisfies 1.3. Then hee exis a consan C > and a funcion ω on, < C 1 ωx C, such ha he spaces H 1 L and Ha 1 d,ωμcoincide and he coesponding noms ae equivalen, C 1 f H 1 a d,ωμ f H 1 L C f H 1 a d,ωμ. Moeove, ω is L-hamonic, ha is T ω = ω fo >. Le us noice ha he assumpion on Φ x implies ha μ = and μ is nonaomic. This will be used lae on. By definiion, we call a semigoup consevaive if T x, y dμy = fo > andx. Theoem 2 Suppose ha, d,μis a space of homogeneous ype and assume ha fo each x he funcion Φ x is a bijecion on,. Le a nonnegaive self-adjoin linea opeao LonL 2,μbe given such ha he semigoup T = e L saisfies 1.3 and 1.6. Then, he spaces H 1 L and Ha 1 d,μcoincide and he coesponding noms ae equivalen, i.e., hee exiss C > such ha C 1 f H 1 a d,μ f H 1 L C f H 1 a d,μ, I appeas ha Theoem 2 is equivalen o Theoem 1, see Sec. 3. Le us also emphasize ha we do no equie any egulaiy condiions on he kenels T x, y. Howeve,iuns ou ha 1.3 implies Hölde-ype esimaes on T x, y, which ae cucial fo his pape. This will be discussed in Sec. 4 see Theoem 5 and Coollay 14. In fac, Sec. 4 gives a sho, independen, and self-conained poof of Hölde-ype esimaes of he hea kenel ha saisfies 1.3, which can be ineesing in is own igh. Fuhemoe, alhough Theoems 1 and 2 ae saed fo he hea semigoup, we would like o emphasize ha he same heoems can be poved fo he Hady space H 1 L associaed wih he Poisson semigoup e L as well, see Theoem 4 and Remak 21. The heoy of he classical Hady spaces on he Euclidean spaces R n has is oigin in sudying holomophic funcions of one vaiable in he uppe half-plane. The eade is efeed o he oiginal woks of Sein and Weiss [32], Bukholde e al. [6], Feffeman and Sein [15]. Vey impoan conibuion o he heoy is he aomic decomposiion of he elemens of he H p spaces poved by Coifman [7] in he one-dimensional case and hen by Lae [23] fo H p R n. The heoy was hen exended o he space of homogeneous ype see, e.g., [8,24, 33]. Fo moe infomaion concening he classical Hady spaces, hei chaaceizaions and hisoical commens we efe he eade o Sein [31]. A vey geneal appoach o he heoy of Hady spaces associaed wih semigoups of linea opeaos saisfying he Davies Gaffney esimaes was inoduced by Hofmann e al. [2] see also[1,3]. Le us poin ou ha he classical Hady spaces can be hough as hose associaed wih he classical hea semigoup e. Finally we wan o emak ha he pesen pape akes moivaion fom [13,14], whee he auhos sudied H 1 spaces associaed wih Schödinge opeaos + V on R n, n 3, wih Geen bounded poenials V. In wha follows C and c denoe diffeen consans ha may depend on C d, q, C, c 1, c 2. By U V we undesand U CV and U V means C 1 V U CV.

4 968 J. Dziubański, M. Peisne The pape is oganized as follows. In Sec. 2 we pove ha esimae 1.3 implies he exisence of an L-hamonic funcion ω such ha < C 1 ωx C. The equivalence of Theoems 1 and 2 is a consequence of he Doob ansfom, see Sec. 3. Then a poof of Theoem 2 is given in a few seps. Fis, in Sec. 4 we pove Hölde-ype esimaes fo a consevaive semigoup. Then, in Sec. 5 we inoduce a new quasi-meic d and sudy is popeies. In Sec. 6 we apply a heoem of Uchiyama on he space, d,μo complee he poof of Theoem 2. Finally, in Sec. 7 we povide some examples of semigoups ha saisfy assumpions of Theoem 1. 2 Gaussian esimaes and bounded hamonic funcions In his secion we assume ha he semigoup T saisfies 1.3. Clealy, C 1 T x, ydμy C fo all > and a.e. x. Fo a posiive inege n define ω n x = 1 n T s x, ydμy ds. n Then, C 1 ω n x C. Recall ha a meic space wih he doubling condiion is sepaable, hen so is L 1,μ. Using he Banach Alaoglu heoem fo L,μ hee exiss a subsequence n k and ω L,μ, such ha ω nk ω in -weak opology. Obviously, C 1 ωy C. Ou goal is o pove ha T ωx = ωx fo >. To his end, we wie T x, yωy dμy = lim T x, yω nk y dμy k 1 nk = lim T +s x, z dμz ds k n k = lim ω 1 nk + n k x + lim T s x, z dμz ds k k n k n k 1 lim T s x, z dμz ds. k n k Since he las wo limis end o zeo, as k, we obain T x, yωy dμy = lim ω n k x. 2.1 k Fom 2.1 we ge ha lim k ω nk x exiss fo a.e. x and he limi has o be ωx. Moeove, T ωx = ωx. Thus we have poved he following poposiion. Poposiion 3 Assume ha a semigoup T saisfies 1.3. Then hee exiss a funcion ω and C > such ha < C 1 ωx C and T ωx = ωx fo evey >.

5 Hady spaces fo semigoups wih Gaussian bounds Doob ansfom In his secion we wok in a slighly moe geneal scheme. Le,μbe a σ -finie measue space and L be a self-adjoin opeao on L 2,μ. We assume ha he songly coninuous semigoup T = exp L admis a nonnegaive inegal kenel T x, y, soha T f x = T x, y f y dμy. Moeove, assume ha hee exiss ω saisfying < C 1 ωx C, haisl-hamonic. Namely, fo evey > one has T x, yωy dμy = ωx fo a.e. x. Obviously, his implies ha sup x,> T x, y dμy C. Define a new measue dνx = ω 2 x dμx and a new kenel K x, y = T x, y ωxωy. 3.1 The semigoup K given by K f x = K x, y f y dνy is a songly coninuous semigoup of self-adjoin inegal opeaos on L 2,ν. The mapping L 2,μ f ω 1 f L 2,ν is an isomeic isomophism wih he invese L 2,ν g ωg L 2,μ. Clealy, K gx = ωx 1 T ωgx. Moeove, he posiive inegal kenel K x, y is consevaive, ha is, K x, y dνy = ωx 1 T ωx = 1. Thus he above change of measue and opeaos, which is called Doob s ansfom see, e.g., [16], conjugaes he semigoup T wih he consevaive semigoup K. I is clea ha he opeaos K ae conacions on L 1,ν. Consequenly, K is a songly coninuous semigoup of linea opeaos on L 1,ν. To see his, i suffices o show ha lim K χ A χ A L 1,ν = fo any measuable se A of finie measue. We have ha lim K χ A χ A dν lim K χ A χ A L 2 A,ννA 1/2 =. On he ohe hand K χ A x dνx = νa = χ A L 1,ν. Hence, K χ A x dνx = K χ A xdνx = K χ A xdνx K χ A x dνx A c A c A = χ A xdνx K χ A x dνx as, A A which complees he poof of he song coninuiy of K on L 1,ν.

6 97 J. Dziubański, M. Peisne Fuhe, we easily see ha he semigoup T is songly coninuous on L 1,μ. Indeed, if f L 1,μhen g = ω 1 f L 1,νand T f f L 1,μ = K gx gx ωx 1 dνx K gx gx dνx as. Now we discuss he equivalence of Theoems 1 and 2. LeT and K be he semigoups elaed by 3.1 wih geneaos L and R, especively. I easily follows fom he Doob ansfom ha f H 1 L if and only if ω 1 f H 1 R and f H 1 L ω 1 f H 1 R.In ohe wods H 1 L f ω 1 f H 1 R is an isomophism of he spaces. Assume ha he space H 1 R admis an aomic decomposiion wih aoms ha saisfy he cancelaion condiion wih espec o he measue ν,haiseveyg H 1 R can be wien as g = λ j a j wih j λ j g H 1 R and a j ae aoms wih he popey a j dν =. Then evey f H 1 L admis aomic decomposiion f = λ j b j wih aoms b j ha saisfy cancelaion condiion b j ω dμ =. 4 Hölde-ype esimaes on he semigoups In his secion we conside a consevaive semigoup T having an inegal kenel T x, y ha saisfies uppe and lowe Gaussian bounds 1.3. Le P x, y = π 1/2 e u T 2 /4ux, y du 4.1 u be he kenels of he subodinae semigoup P = e L. Theoem 4 Le L be a nonnegaive self-adjoin linea opeao on L 2,μsuch ha he semigoup T = e L saisfies 1.3 and 1.6. Then hee is a consan α>such ha dy, z α P x, y P x, z P x, y 4.2 wheneve dy, z. Theoem 5 Unde he assumpions of Theoem 4 hee ae consans α, c > such ha T x, y T x, z μbx, dy, z α 1 cdx, y2 exp 4.3 wheneve dy, z. Hölde egulaiy of semigoups saisfying Gaussian bounds was consideed in vaious seings by many auhos. We efe he eade o Gigo yan [16], Hebisch and Saloff-Cose [19], Saloff-Cose [28], Gyya and Saloff-Cose [17], Benico, Coulhon, and Fey [2], and efeences heein. Hee we pesen a sho alenaive poof of 4.3. To his end we shall fis pove some auxiliay poposiions and hen Theoem 4. Finally, a he end of his secion, we shall make use of funcional calculi deducing Theoem 5 fom Theoem 4.

7 Hady spaces fo semigoups wih Gaussian bounds 971 Poposiion 6 Fo x, y and > we have P x, y 1 μ B x, + dx, y + dx, y. 4.4 Poof Conside fis he case dx, y. Thendx, y +. The uppe bound follows by 1.3and1.1. Indeed, P x, y 1/4 1 du μ B x, 2 + μbx, 1 u u 1/4 1/4 μbx, 1 du + μbx, 1 u μbx, 1. Also, 1.3 implies lowe bounds, since P x, y 1 1/4 1/4 e u μbx, du μ B x, u e u 2 u q du u e u exp 4c 1 u du μbx, 1. μ B x, u 2 u 2 u Le us now un o he case dx, y.thendx, y + dx, y.using1.3, we have P x, y Moeove, by 1.1, e u exp 4c 2 udx, y 2 /2 du = μ B x, u 2 u 2 /dx,y 2 + = J 1 + J 2. 2 /dx,y J 1 2 /dx,y 2 2 /dx,y 2 μ B μ B 1 x, 1 du 2 u u x, dx,y 2 du u 1 μ Bx, dx, y dx, y 4.6 and J 2 1 exp 4c 2 udx, y 2 / 2 2dx, y μ Bx, dx, y 2 /dx,y 2 1 μ Bx, dx, y u q du u dx, y. 4.7 Esimaes give uppe esimae. Fo he lowe esimae, ecall ha dx, y and obseve ha

8 972 J. Dziubański, M. Peisne P x, y 2 2 /dx,y 2 2 /dx,y 2 e u exp 4c1 udx, y 2 / 2 μ B x, 2 u /dx,y 2 du μbx, dx, y/2 2 /dx,y 2 u 1 μbx, dx, y dx, y. du u Coollay 7 Thee is a consan C > such ha if dy, z, hen C 1 P x, y C. 4.8 P x, z Poof The coollay is a simple consequence of 4.4. Fo he poof one may conside wo cases: dx, y 2 hen dx, z 3anddx, y >2 hen dx, y dx, z. Poposiion 8 Thee exiss a consan γ, 1 such ha he following saemen holds: If hee ae y, z,, a 1, b 1 > given such ha dy, z < and fo all x we have a 1 P x, y P x, z b 1, 4.9 hen hee is a subineval [a 2, b 2 ] [a 1, b 1 ] such ha b 2 a 2 = γb 1 a 1 and fo all 2 and all x one has a 2 P x, y P x, z b 2. Poof The poof, which akes some ideas fom [21], is an adaped vesion of he poof of [13, Pop. 3.1]. Fo he eade convenience we pesen he deails. Le m = a 1 + b 1 /2and θ = b 1 a 1 /a 1 + b 1, 1, sohaa 1 = 1 θm and b 1 = 1 + θm. Define { Ω + = x m P } x, y P x, z b 1, Ω = \Ω +. Obviously, eihe μω Bz, μbz, /2oμΩ + Bz, μbz, /2. Hee we shall assume ha he lae holds. The poof in he opposie case is simila. Denoe B = Bz,, s =.Fox,wehave P x, y m1 θ P s x,wp w, z dμw + m P s x,wp w, z dμw Ω Ω + = m1 θp x, z + mθ P s x,wp w, z dμw Ω + m1 θp x, z + mθ P s x,wp w, z dμw Ω + B m1 θp x, z + mθ μb 2 Noice ha dw, z s and, by Coollay 7, inf P sx,w inf P w, z = J. w B w B inf w B P sx,w P s x, z.

9 Hady spaces fo semigoups wih Gaussian bounds 973 Since s, Poposiion 6 implies and Theefoe P s x, z P x, z inf w B P w, z μb 1. J m 1 θ+ κθ P x, z = m 1 θ1 κ P x, z, 4.1 whee κ, 2. Moeove, fom 4.9 and he semigoup popey we easily ge P x, y b 1 P s x,wp w, z dμw b 1 P x, z Defining γ = 1 κ/2, 1, b 2 = b 1 and a 2 = m 1 θ+ κθ, wehave b 2 a 2 = 2mθ 1 κ/2 = b 1 a 1 1 κ/2 = γb 1 a 1. Now 4.1 ogehe wih 4.11gives a 2 P x, y P x, z b 2 fo x. Poof of Thoem 4 Having Coollay 7 and Poposiion 8 poved, we follow agumens of [13] o obain he heoem. By Coollay 7 hee ae b 1 > a 1 > such ha fo y, z and > saisfying dy, z < we have a 1 P x, y P x, z b 1 fo all x. Fom Poposiion 8 we deduce ha hee exiss ωy, z such ha P x, y lim = ωy, z unifomly in x P x, z I follows fom 4.1 ha P x, y dμy = 1. Recall ha P x, y = P y, x. Using 4.12, P y, x 1 = P y, x dμx = P z, x P z, x dμx ωy, z. Thus ωy, z = 1. Assume ha dy, z <. Len N be such ha dy, z 2 n < 2dy, z. Se = 2 n. Clealy, dy, z and a 1 P x, y P x, z b 1. Obseve ha n log/dy, z. Applying Poposiion 8 n-imes we aive a P x, y P x, z 1 dy, z α γ n γ c log/dy,z, wih α>and he poof of Theoem 4 is finished.

10 974 J. Dziubański, M. Peisne Finally, we devoe he emaining pa of his secion fo deducing Theoem 5 fom Theoem 4. This is done by using a funcional calculi. Fis, we need some pepaaoy facs. Recall ha q is a fixed consan saisfying 1.1. By W 2,σ R we denoe he Sobolev space wih he nom f W 2,σ R = 1 + ξ 2 σ f ξ 1/2 2 dξ. R Le ξde L ξ be he specal esoluion fo L. Fo a bounded funcion m on [, he fomula m L = mξ de L ξ defines a bounded linea opeao on L 2,μ. Fuhe we shall use he following lemma, whose poof based on finie speed popagaion ofhewaveequaionsee[9] can be found in [11]. Lemma 9 [11, Lemma 4.8] Le κ>1/2, β>. Then hee exiss a consan C > such ha fo evey even funcion m W 2,β/2+κ R and evey g L 2,μ, supp g By,, we have dx,y >2 fo j Z. m2 j Lgx 2 dx, y β dμx C2 j β m 2 W 2,β/2+κ R g 2 L 2,μ Poposiion 1 Le β>qand κ>1/2. Thee is a consan C > such ha fo evey F W 2,β/2+κ R wih supp F 1/2, 2 he inegal kenels F2 j Lx, y of he opeaos F2 j L saisfy F2 j Lx, y dμx C F W 2,β/2+κ R fo j Z. Poof Fo y and j Z se U = By, 2 j, U k = By, 2 k j \ By, 2 k j 1, k = 1, 2,...Defineg j,k,y x = T 2 2 j x, yχ Uk x, k =, 1, 2,... Then, using 1.1and 1.3, we have g j,y,k L 2,μ a k μu 1/2, 4.13 whee a k = C C q 2 kq exp c 2 2 2k 2 is a apidly deceasing sequence. Le mξ be he even exension of e ξ 2 Fξ. Obviously, m W 2,β/2+κ R F W 2,β/2+κ R. Then, F2 j L = m2 j LT 2 2 j, and, consequenly, F2 j Lx, y = m2 j Lg j,y,k x k= By he Cauchy Schwaz inequaliy, 1.1, and 4.13 wege m2 j Lg j,y,k L 1 By,2 k j+1,μ μby, 2k j+1 1/2 m2 j Lg j,y,k L 2,μ μby, 2 k j+1 1/2 μby, 2 j μby, 2 j 1/2 m L R g j,y,k L 2,μ 2 kq a k m W 2,β/2+κ R. 4.15

11 Hady spaces fo semigoups wih Gaussian bounds 975 We un o esimae m2 j Lg j,y,k L 1 By,2 k j+1 c,μ. Uilizing he Cauchy Schwaz inequaliy and Lemma 9, we obain, dx,y>2k j+1 [ [ m2 j Lg j,y,k x dμx dx,y>2k j+1 dx,y>2k j+1 m2 j Lg j,y,k x 2 dx, y β 1/2 dμx] dx, y β 1/2 dμx] 2 k j 2 k j dx, y ] β 1/2 dμx 2 k j. [ 2 kβ/2 m W 2,β/2+κ R g j,y,k L 2,μ dx,y>2k j Recall ha β>qand hence is no difficul o check ha 1.1 leads o dx, y β dμx 2 kq μu dx,y>2k j+1 2 k j Thus, by 4.13, 4.16, and 4.17, we ge m2 j Lg j,y,k x dμx 2 kβ/2 2 kq/2 m W 2,β/2+κ Ra k, dx,y>2k j+1 which, combined wih 4.15and4.14, complees he poof of he poposiion. Fo > se ψ ξ = exp ξ ξ 2. Lemma 11 The opeaos ψ L have inegal kenels Ψ x, y ha saisfy Ψ x, y dμy C. sup x, > Poof Denoing θ ξ = exp ξ 2 exp ξ 1, whee,ξ >, we have ψ L = T + θ L. Clealy, by 1.3, sup,x> T x, y dμy C. Thus we concenae ou aenion on θ L. Le η Cc 1/2, 2 be a paiion of uniy such ha θ ξ = θ ξη2 j ξ = θ, j ξ. j Z j Z Denoe θ, j ξ = θ, j 2 j ξ = ηξθ 1 2 j ξ. One can easily veify ha fo n N {} hee ae consans C n, c n > such ha dn dξ θ { L 2 j n, j C fo j log 2 n R exp c n 2 2 j. fo j log 2

12 976 J. Dziubański, M. Peisne In ohe wods, fo abiay N, θ, j W 2,N R CN. j Z Using Poposiion 1 wih a fixed N > q/2+1 we ge ha he inegal kenels Θ, j x, y of he opeaos θ, j L = θ, j 2 j L saisfy Θ, j x, y dμy θ, j W 2,N R. Theefoe, Θ x, y = j Z Θ, j x, y is he inegal kenel of θ L and i saisfies Θ x, y dμy C. sup,x> Poof of Theoem 5 By he specal heoem T = ψ LP and T x, y = Ψ x,wp w, y dμw. Using Theoem 4 ogehe wih Lemma 11 and Poposiion 6, fo dy, z, wehave T x, y T x, z = Ψ x,wp w, y P w, z dμw dy, z α Ψ x,w P w, y dμw 4.18 μby, dy, z α 1. We claim ha fo dy, z one has T x, y T x, z μby, 1 dy, z α/2 exp cdx, y To pove he claim we conside wo cases. Case 1: 2dy, z dx, y. Recall ha dy, z; hus, dx, y 2 and 4.19 follows diecly fom Case 2: 2dy, z dx, y. In his case dx, y dx, z, soby1.3 we obain exp c 2dx,y 2 exp c 2dx,z 2 T x, y T x, z μby, + μbz, μby, cdx, y2 exp, whee in he las inequaliy we have uilized ha μbz, μby,, since dy, z <. By aking he geomeic mean of 4.18 and4.2 we obain To finish he poof obseve ha 4.19 implies 4.3. Indeed, μbx, = μby, μbx, μby, μby, μby, + dx, y μby,

13 Hady spaces fo semigoups wih Gaussian bounds 977 μby, 1 + dx, y q and using he exponen faco we can eplace μby, by μbx,. Remak 12 Le us emak ha Lemma 11, which is cucial in ou poof of Theoem 5, can be poved by applying funcional calculus of Hebisch [18]. Thus, Theoem 5 can be also obained wihou using he finie popagaion speed of he soluion of he wave equaion 2 + Lu, x =, u, x = u x, u, x =. As a consequence of 1.4andTheoem5 we ge wha follows. Coollay 13 The funcion T x, y is coninuous on,. As a diec consequence of Theoem 5 and Doob ansfom see 3.1 we ge he following coollay. Noice ha in Coollay 14 we do no assume ha T x, y is consevaive. Coollay 14 Assume ha he semigoup T saisfies 1.3. Then hee ae consans α, c > such ha T x, y ωxωy T x, z μbx, dy, z 1 ωxωz wheneve dy, z. α exp cdx, y2 5 Measues and disances To pove Theoem 2 we inoduce a new quasi-meic d on, which is elaed o d and μ. To his end, se dx, y = inf μb, whee he infimum is aken ove all closed balls B conaining x and y see, e.g., [8,25]. Denoe Bx, = { y dx, y }. In he lemma below we sae some popeies of d, which ae known among specialiss, and which we shall need lae on. Since hei poofs ae vey sho and i is difficul fo us o indicae one efeence which conains all of hem, we povide he deails fo he convenience of he eade. Lemma 15 The funcion d has he following popeies: a hee exiss C b > such ha fo x, y we have Cb 1 μbx, dx, y dx, y μbx, dx, y. 5.1 b d is a quasi-meic, namely hee exiss A 1 such ha dx, y A 1 dx, z + dz, y. Moeove, if he measue μ has no aoms and μ =, hen: c he measue μ is egula wih espec o d, namely fo x and >, μ Bx, ;

14 978 J. Dziubański, M. Peisne d fo x and > hee exiss R > such ha Bx, Bx, R and μbx, R μ Bx, ; e fo x and R > hee exiss > such ha Bx, R Bx, and μ Bx, μbx, R. Poof a Se R = dx, y. Clealy, dx, y μbx, R, asx and y belong o Bx, R. On he ohe hand, if x and y belong o a ball B = Bz,, henr 2; hence, Bx, R Bz, 3 and μbx, R μbz, 3 μbz,. By aking he infimum ove all balls B conaining boh x and y, we conclude ha μbx, R C dx, y. b Fo evey x, y, z,wehavedx, y dx, z+dz, y. Assume ha = dx, z dz, y. Thenx, y Bz,. By using a, we deduce ha dx, y μbz, dz, x dx, z + dz, y. c Given x, by ou addiional assumpions, he funcion, μbx, is inceasing and { μbx, as, μbx, + as +. Le x and >. Fo evey y Bx,, wehaveμbx, dx, y dx, y. Hence R = sup {dx, y y Bx, } < +. Le y Bx, such ha dx, y R 2.Then Bx, Bx, R Bx, 2dx, y. 5.2 Hence μ Bx, μbx, R μbx, 2dx, y μbx, dx, y dx, y. 5.3 On he ohe hand, T = inf{ > μbx, } >. As μbx, T/2 <, wehave dx, y <, foeveyy Bx, T/2; hence, Bx, T/2 Bx,. Consequenly, μbx, 2T μbx, T/2 μ Bx,, which ogehe wih 5.3 complees he poof of c. d is a simple consequence of 5.2, 5.3, and c. e Se = μbx, R. Ify Bx, R, hen dx, y and, consequenly, Bx, R Bx,. Clealy, by c, μ Bx, = μbx, R. Le us ecall ha in Theoems 1 and 2 we assume ha Φ x is a bijecion on,. This obviously implies ha μ = and ha μ is nonaomic. As a consequence of d and e of Lemma 15 we obain he following coollay. Coollay 16 Suppose haμ hasno aoms andμ =. Then, he aomic Hady spaces H 1 a d,μand H 1 a d,μcoincide and he coesponding aomic noms ae equivalen. We finish his secion by Lemma 17, which is used lae on. Define A 2 := C b C d 2 q 3, whee C d, q, andc b ae as in 1.1and5.1.

15 Hady spaces fo semigoups wih Gaussian bounds 979 Lemma 17 Suppose ha we have a space of homogeneous ype, d, μ such ha he funcion Φ x defined in 1.5 is a bijecion on,. Assume ha x,, > ae elaed by = μbx, and saisfy: dy, z, A 2 dy, z <. Then dx, y 2dx, z. Poof Suppose, owad a conadicion, ha dx, y <.Fom5.1wege = μbx, μby, 2 C d 2 q μby, C d 2 q μby, dy, z C d 2 q C b dy, z <, so he fis inequaliy is poved. Similaly, assume dx, z <dx, y/2. Then dx, y/2 dy, z 2dx, y. Thus, using 5.1, dy, z Cb 1 μby, dy, z C 1 b C d2 q 1 μby, dx, y Cb 1 C d2 q 2 μbx, dx, y Cb 1 C d2 q 3 μbx, = A 1 2 and we come o a conadicion. 6 Poof of Theoem 2 In ode o pove Theoem 2 we shall use a esul of Uchiyama [33], which we sae below in Theoem 18. Denoe by, d,μ he space equipped wih a quasi-meic d and a nonnegaive measue μ, wheeμ =. Assume moeove ha μ Bx,, 6.1 whee x, > and Bx, is a ball in he quasi-meic d.lea 1 be a consan in he quasi-iangle inequaliy, i.e., dx, y A 1 dx, z + dz, y, x, y, z. 6.2 Addiionally, we impose ha hee exis consans γ 1,γ 2,γ 3 >, A A 1 and a coninuous funcion T, x, y of vaiables x, y and > such ha T, x, x 1, U 1 T, x, y dx, 1 γ1 y, U 2 if dy, z < + dx, y/4a, hen U 3 T, x, y T, x, z γ2 1 dy, z 1 + dx, 1 γ3 y, fo all x, y, z and >. As in [33], we conside he maximal funcion f + x = sup T, x, y f y dμy > { and he Hady space H 1, T = f L 1,μ f H 1, T := f + L }. 1,μ < Recall ha he aomic space Ha 1 d,μis defined in Sec. 1.

16 98 J. Dziubański, M. Peisne Theoem 18 [33], Coollay 1 Suppose ha, d,μ, T saisfy 6.1, 6.2, U1, U2, and U3. Then he spaces H 1, T and H 1 a d,μcoincide and f H 1, T f H 1 a d,μ. Assume ha he kenel T x, y saisfies 1.3 and he semigoup T is consevaive. Recall ha in Sec. 4 we poved Hölde-ype esimae 4.3 fo T x, y. Define T, x, y by T, x, y := T x, y, whee = x, is such ha μbx, =. 6.3 In wha follows, > and x ae always elaed by 6.3. Le us noice ha fom Coollay 13 and by he assumpion ha Φ x is a coninuous bijecion on, we have ha T is a coninuous funcion on,. Theoem 19 Suppose ha T x, y saisfies uppe and lowe Gaussian bounds 1.3 and he semigoup T is consevaive. Then he kenel T, x, y saisfies U1, U2, and U3. Poof On-diagonal lowe esimae U1 is an immediae consequence of lowe Gaussian bound 1.3. Fo evey fixed δ>, he uppe esimae T, x, y dx, 1 δ y 6.4 follows fom he uppe esimaes fo T x, y, moe pecisely by combining wih T, x, y 1 exp 1 + dx, 1+δ y cdx, y2 μbx, dx, y 1+δ μbx, dx, y q1+δ cdx, y 2 exp The lae esimae holds fo any δ>. Thus U2 is poved wih any γ 1 >. To finish he poof we need Hölde-ype esimae U3. This is poved in Poposiion 2 below. Poposiion 2 Le α be a consan as in Theoem 5. Thee exiss A A 1 such ha fo δ>we have T, x, y T, x, z dx, 1 δ y dy, z α q 6.7 if dy, z + dx, y/4a. Poof Se A = maxa 1, A 2,see6.2 and Lemma 17. Assuming ha dy, z + dx, y/4a le us begin wih some obsevaions. Fisly, we claim ha i suffices o pove 6.7fo dy, z </2A. Indeed, if dy, z > /2A, hen dy, z dx, y/2a and, consequenly, dx, y A 1 dx, z + A 1 dz, y A 1 dx, z + dx, y/2.

17 Hady spaces fo semigoups wih Gaussian bounds 981 So, dx, y dx, z and 6.7 follows fom 6.4 by using he iangle inequaliy. Fom now on we assume ha dy, z </2A. Secondly, if dy, z, hen using 1.1and5.1, dy, z μby, dy, z 1/q μby, μby, dy, z 1/q μby, + dx, y 1/q 6.8 μbx, dy, z 1/q dx, y 1 +. Thidly, if dy, z, hen using 1.1and5.1, 1 μby, dy, z μby, = dy, z μbx, μby, dy, z μby, + dx, y μby, dy, z dx, y q Le us un o he poof of 6.7. Case 1: dy, z dx, y/2. Then dx, y dx, z. Thus, accoding o 1.3and4.3 combined wih 6.8and6.9 we obain T, x, y T, x, z 1 exp 1 exp dx, y cdx, y2 dy, z α min, 1 dy, z cdx, y2 1 δ dy, z α/q 1 + α/q, dx, y α whee in he las inequaliy we have used 6.6. Case 2: dy, z >dx, y/2and > dy, z.then dx, y μbx, =.Using 4.3and6.8, T, x, y T, x, z 1 dy, z α α/q dy, 1 z. Case 3: dy, z >dx, y/2 and < dy, z. Then, dy, z /2A /2A 2 and by Lemma 17 we have 2dx, z >dx, y. Hence, using 6.5and6.9, T, x, y T, x, z 1 cdx, y2 exp 1 cdx, y2 dy, z α/q dx, y α exp dy, z α/q 1 + dx, y 1 δ, whee in he las inequaliy we have used 6.6. This finishes he poof of Poposiion 2.

18 982 J. Dziubański, M. Peisne Poof of Theoem 2 Assuming 1.3 and1.6 we obain Hölde-ype esimae 4.3. Recall once moe ha he assumpion on Φ x implies μ = and μ is nonaomic. Then we define a new quasi-meic d. By Coollay 16 we ge ha H 1 a d,μ = H 1 a d,μ.weapply Theoem 18 o he space, d,μ. The assumpions of Theoem 18 ae veified in Theoem 19 and Poposiion 2. In his way we ge f H 1, T f H 1 a d,μ. Using once again he assumpion on Φ x and he definiion of T we easily obseve ha f H 1, T = f H 1 L, which finishes he poof of Theoem 2. Le us ecall ha Theoem 1 follows fom Theoem 2. This is elaboaed a he end of Sec. 3. Remak 21 Unde he assumpions of Theoem 1 one can pove, by he same mehods, ha he Hady space H 1 { L = f L 1,μ } f H 1 L := sup P f x < > L 1,μ coincides wih Ha 1 d,ωμ and he coesponding noms ae equivalen. To his end, one uses: Poposiion 6, Doob s ansfom, Theoem 4, andtheoem18 applied o he kenel P, x, y = P x, y, whee = x, is defined by he elaion μbx, =. 7 Examples In his secion we give examples of self-adjoin semigoups wih he wo-sided Gaussian bounds. 7.1 Laplace Belami opeaos Le, g be a complee Riemannian manifold wih he Riemannian disance dx, y and he Riemannian measue μ saisfying he doubling popey and he Poincaé inequaliy f f B 2 dμ C 2 f 2 dμ fo all >, Bx, Bx,2 whee denoes he gadien on. I is well known ha he kenel of he hea semigoup geneaed by he Laplace Belami opeao saisfies wo-sided Gaussian bounds 1.3 and Hölde esimaes 4.3. Fo deails and moe infomaion concening he hea equaion on Riemannian manifolds we efe he eade o [28] and efeences heein. 7.2 Schödinge opeaos On = R d wih he Euclidean meic and he Lebesgue measue we conside he Schödinge opeao L = + V, whee is he sandad Laplacian and V is a locally inegable funcion.

19 Hady spaces fo semigoups wih Gaussian bounds 983 I is well known see, e.g., [29] ha fo V, d 3, he semigoup T = e L admis kenels T x, y wih he uppe and lowe Gaussian bounds 1.3 if and only if V is a Geen bounded poenial, ha is, sup x y 2 d V y dy <. 7.1 x R d R d Hady spaces associaed wih Schödinge opeaos on R d saisfying 7.1 wee sudied in [13]. Acually, as we have aleady menioned, his wok moivaed us o sudy he poblem of H 1 spaces wih he Gaussian bounds in he genealiy as in Theoems 1 and 2. Ou second example concens Schödinge opeaos L = + V wih nonposiive poenials. Fo d 3fixp 1, p 2 > 1 saisfying p 1 < d/2 < p 2. Then hee is a consan cp 1, p 2, d > such ha if V and V L p 1 R d + V L p 2 R d cp 1, p 2, d, hen he inegal kenel T x, y of he semigoup T = e L exiss and saisfies wo-sided Gaussian bounds T x, y d/2 x y 2 exp The esul can be found in Zhang [35]. A slighly diffeen poof of 7.2, based on bidges of he Gauss Weiesass semigoup, can be obained by using Lemma 1.2 ogehe wih Poposiion 2.2 of [4]. 7.3 Bessel Schödinge opeao Le α> 1 and conside =, and dμx = x α dx. Noice ha he space,μ wih he Euclidean meic d e x, y = x y is a space of homogeneous ype. We conside he classical Bessel opeao B f x = f x α x f x, which is self-adjoin posiive on L 2,μ, and he associaed semigoup of linea opeaos S = e B. I is well known ha S is consevaive and has he inegal kenel S x, y = 2 1 exp x2 + y 2 xy I α 1/2 xy α 1/2, see, e.g., [5, Chape 6]. Hee I α 1/2 denoes he modified Bessel funcion of he fis kind, see, e.g., [34]. The kenel S x, y saisfies wo-sided Gaussian esimaes C 1 μ Bx, exp c 1 x y 2 C S x, y μ Bx, exp c 2 x y Fo a sho poof of 7.4 see[12, poof of Lemma 4.2]. Theefoe, using Theoem 2 we obain aomic chaaceizaion of H 1 B ha was peviously poved in [3]. In his subsecion we conside peubaions of B of he fom L = B + V,

20 984 J. Dziubański, M. Peisne whee a poenial V is nonnegaive and locally inegable. Moe pecisely, on L 2,μwe define he quadaic fom Q f, g = f xg x dμx + V x f xgx dμx, wih he domain Dom Q = cl { f Cc 1 [, f + = } { f L 2,μ } Vf L 2,μ, whee cla sands fo he closue of he se A in he nom f L 2,μ + f L 2,μ.The fom Q is posiive and closed. Thus, i coesponds o he unique self-adjoin opeao L on L 2,μwih he domain Dom L = { f DomQ h L 2,μ g DomQ Q f, g = } h g dμ. By definiion, Lf = h when f, h ae elaed as above. Le T = exp L be he semigoup geneaed by L. The Feynman Kac fomula saes ha T f x = E exp x V b s ds f b, 7.5 whee b s is he Bessel pocess on, associaed wih S.Using7.5 one ges ha he semigoup T has fom 1.2, whee T x, y S x, y. 7.6 Theefoe he uppe Gaussian esimaes fo T x, y follows simply fom 7.4 fo any locally inegable V. On he ohe hand, he elaion beween S x, y and T x, y is given by he peubaion fomula S x, y = T x, y + S s x, zv zt s z, y dμz ds 7.7 Fom now on we conside only α>1. We ae ineesed in poving he lowe Gaussian esimaes, bu his can be done only fo some poenials V. Fo ohe poenials Hady spaces may have a local chaace, see, e.g., [22]. Le Ɣx, y = S x, y d be he fomal kenel of B 1. In addiion o V Lloc 1 and V, we shall need one moe assumpion, a vesion of he global Kao condiion, cf. 7.1, V Kao := sup Ɣx, yv y dμy <. 7.8 x Fomally, we can ephase his as B 1 V L,μ. Le us poin ou ha Ɣx, y x + y α+1. This can be easily obained fom 7.3 and well-known asympoics fo he Bessel funcion I α 1/2, see also [26, Sec.2]. In Lemmas 22 and 23 we pove ha unde assumpion 7.8 lowe Gaussian esimaes 1.3 hold fo T x, y. The esimaes ae poved in a simila way as in he case of he Schödinge opeao on he Euclidean space. Fo he convenience of he eade we povide he deails.

21 Hady spaces fo semigoups wih Gaussian bounds 985 Lemma 22 Suppose ha V Kao <. If x y, hen T x, y C 1 l μbx, 1. Poof Fis we shall pove Lemma 22 wih an addiional assumpion ha V Kao ε fo a fixed small ε>. By 7.7and7.6, S x, y T x, y = S s x, zv zt s z, ydμz ds = μbx, /2 1 V zt s z, y dμz ds + μby, 1 /2 μbx, 1 V Kao. /2 S s x, zv z dμz ds... ds +... ds /2 By choosing pope ε>wededuce he hesis fom lowe esimae 7.4foS x, y. Now assume ha he nom V Kao < is abiay. Fix q > 1, such ha V Kao = qε. Se V q x = V x/q, Vq Kao = ε, andlet q be he semigoup elaed o L q = B + V q. Using 7.5 and Hölde s inequaliy, T χby, μby, x = E x exp [E x exp [ = T q χby, μby, V b s q χ By, b ds q μby, V b s χby, b ds q μby, q x] [ S χby, μby, ] q [ E x χby, b μby, x ] q/q ] q q 7.9 Le us noice ha T x, y = lim μ By, 1 T x, z dμz By, fo a.e. x, y. By leing in7.9, using 7.4 and he fis pa of he poof, fo a.e. x, y we obain ha T x, y T q x, y q S x, y q/q μbx, 1. Lemma 23 Suppose ha V Kao <. Then T x, y μbx, 1 c x y 2 exp. Poof Assume ha x y 2 / 1andsem = 4 x y 2 / 4. Fo i =,...,m, le x i = x + iy x/m, sohax = x, x m = y, and x i+1 x i = x y /m. Denoe B i = Bx i, /4 m and obseve ha y i+1 y i y i x i + x i x i+1 + x i+1 y i+1 4 m + 2 m + 4 m = m

22 986 J. Dziubański, M. Peisne fo y i B i and y i+1 B i+1. Now we use he semigoup popey, Lemma 22, andhe doubling popey of μ o obain T x, y =... T /m x, y 1 T /m y 1, y 2...T /m y m 1, y dμy 1... dμy m 1 c1 m 1... μbx, /m 1...μBy m 1, /m 1 dμy 1... dμy m 1 B 1 B m 1 c2 m 1 μbx, /m 1 μb 1...μB m 1 μbx 1, /m...μbx m 1, /m c3 m μbx, 1 = μbx, 1 m ln c 1 e 3 μbx, 1 e c x y 2. Noice ha c 1, c 2, c 3,andc in his esimae depend only on he consan C l fom Lemma 22 and he doubling consan of μ. Obviously, he space, d e,μsaisfies he assumpions of Theoem 1. Since we have he wo-sided Gaussian esimaes fo T see Lemma 22,7.6, and 7.4 we obain he following coollay. Coollay 24 Suppose ha α>1and V saisfies 7.8. Thenheeexissω such ha < C 1 ωx C, T ω = ω fo > and H 1 L = Ha 1d e,ωμ. Moeove, f H 1 L f H 1 a d e,ω μ. Acknowledgemens We wan o hank Pascal Auche, Fédéic Benico, Li Chen, and Gzegoz Plebanek fo hei emaks. We ae gealy indeb o Jacek Zienkiewicz fo convesaions concening Schödinge opeaos. The auhos ae gaeful o he efeee fo he/his helpful commens which impoved he pesenaion he pape. Open Access This aicle is disibued unde he ems of he Ceaive Commons Aibuion 4. Inenaional License hp://ceaivecommons.og/licenses/by/4./, which pemis unesiced use, disibuion, and epoducion in any medium, povided you give appopiae cedi o he oiginal auhos and he souce, povide a link o he Ceaive Commons license, and indicae if changes wee made. Refeences 1. Ausche, P., Duong,.T., McInosh, A.: Boundedness of banach space valued singula inegal opeaos and hady spaces. Unpublished pepin Benico, F., Coulhon, T., Fey, D.: Gaussian hea kenel bounds hough ellipic Mose ieaion. J. Mah. Pues Appl , Beanco, J.J., Dziubański, J., Toea, J.L.: On Hady spaces associaed wih Bessel opeaos. J. Anal. Mah. 17, Bogdan, K., Dziubański, J., Szczypkowski, K.: Shap Gaussian esimaes fo Schödinge hea kenels: L p inegabiliy condiions. Aiv e-pins Boodin, A.N., Salminen, P.: Handbook of Bownian Moion Facs and Fomulae. Pobabiliy and is Applicaions, 2nd edn. Bikhäuse, Basel Bukholde, D.L., Gundy, R.F., Silvesein, M.L.: A maximal funcion chaaceizaion of he class H p. Tans. Am. Mah. Soc. 157, Coifman, R.R.: A eal vaiable chaaceizaion of H p. Sud. Mah. 51, Coifman, R.R., Weiss, G.: Exensions of Hady spaces and hei use in analysis. Bull. Am. Mah. Soc. 834, Coulhon, T., Sikoa, A.: Gaussian hea kenel uppe bounds via he Phagmén-Lindelöf heoem. Poc. Lond. Mah. Soc , Davies, E.B.: Hea Kenels and Specal Theoy, Cambidge Tacs in Mahemaics, vol. 92. Cambidge Univesiy Pess, Cambidge Dziubański, J., Peisne, M.: Remaks on specal muliplie heoems on Hady spaces associaed wih semigoups of opeaos. Rev. Un. Ma. Agenina 52,

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