Non Tangential Convergence for the Ornstein-Uhlenbeck Semigroup.

Transcription

1 Divulgacions Matmáticas Vol. 6 No. (008), pp Non Tangntial Convrgnc for th Ornstin-Uhlnbck Smigroup. Convrgncia no tangncial para l smigrupo d Ornstin-Uhlnbck Ebnr Pinda (pinda@uicm.ucla.du.v) Dpartamnto d Matmática, Dcanato d Cincia y Tcnología, UCLA Apartado 400 Barquisimto 300 Vnzula Wilfrdo Urbina R.(wurbina@ulr.cins.ucv.v) Dpartamnto d Matmáticas, Facultad d Cincias, UCV. Apartado 4795, Los Chaguaramos, Caracas 04-A Vnzula, and Dpartmnt of Mathmatics and Statistics, Univrsity of Nw Mxico, Albuqurqu, NM, 873, USA. Abstract In this papr w ar going to gt th non tangntial convrgnc, in an appropriatd parabolic gaussian con, of th Ornstin-Uhlnbck smigroup in providing two proofs of this fact. On is a dirct proof by using th truncatd non tangntial maximal function associatd. Th scond on is obtaind by using a gnral statmnt. This scond proof also allows us to gt a similar rsult for th Poisson-Hrmit smigroup. Ky words and phrass: Non tangntial convrgnc, Ornstin- Uhlnbck smigroup, Poisson-Hrmit smigroup, Hrmit xpansions. Rsumn En st artículo vamos a obtnr la convrgncia no tangncial, n un cono gaussiano parabólico apropiado, dl smigrupo d Ornstin- Uhlnbck dando dos prubas difrnts d llo. La primra s una pruba dircta usando la función maximal no tangncial truncada asociada. La sgunda pruba s obtin usando principios gnrals. Esta última pruba nos prmit obtnr un rsultado análogo para l smigrupo d Poisson-Hrmit. Palabras y frass clavs: Convrgncia no tangncial, smigrupo d Ornstin-Uhlnbck, smigrupo d Poisson-Hrmit, dsarrollos d Hrmit. Rcivd 006/03/05. Rvisd 006/07/0. Accptd 006/07/5. MSC (000): Primary 4C0; Scondary 6A99.

2 08 Ebnr Pinda, Wilfrdo Urbina Introduction Lt us considr th Gaussian masur γ d (x) x π d/ Ornstin-Uhlnbck diffrntial oprator with x R d and th L x x, x. () Lt β (β,..., β d ) N d b a multi-indx, lt β! d i β i!, β d i β i, i x i, for ach i d and β β... β d d. Lt us considr th normalizd Hrmit polynomial of ordr β, in d variabls d h β (x) ( ) β i β i x ( β β!) / i i x β ( x i i ), () i thn, sinc th on dimnsional Hrmit polynomials satisfis th Hrmit quation, s [7], thn th th normalizd Hrmit polynomial h β is an ignfunction of L, with ignvalu β, Lh β (x) β h β (x). (3) Givn a function f L (γ d ) its β-fourir-hrmit cofficint is dfind by ˆf(β) < f, h β > γd f(x)h β (x)γ d (dx). R d Lt C n b th closd subspac of L (γ d ) gnratd by th linar combinations of {h β : β n}. By th orthogonality of th Hrmit polynomials with rspct to γ d it is asy to s that {C n } is an orthogonal dcomposition of L (γ d ), L (γ d ) which is calld th Winr chaos. Lt J n b th orthogonal projction of L (γ d ) onto C n. If f is a polynomial, J n f ˆf(β)h β. β n Th Ornstin-Uhlnbck smigroup {T t } t 0 is givn by T t f(x) t ( x + y ) t x,y ( t ) d/ t f(y)γ d (dy) R d y t x π d/ ( t ) d/ t f(y)dy. (4) n0 R d C n Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4

3 Non Tangntial Convrgnc for th Ornstin-Uhlnbck Smigroup. 09 {T t } t 0 is a strongly continuous Markov smigroup of contractions on L p (γ d ), with infinitsimal gnrator L. Also, by a chang of variabl w can writ, T t f(x) f( t u + t x)γ d (du). (5) R d Dfinition.. Th maximal function for th Ornstin-Uhlnbck smigroup is dfind as T f(x) sup T t f(x) t>0 sup 0<r< π d/ ( r ) y rx d/ r f(y)dy. (6) R d In [4] C. Gutiérrz and W. Urbina obtaind th following inquality for th maximal function T f, T f(x) C d M γd f(x) + ( x ) d x f,γd, (7) whr M γd f is th Hardy-Littlwood maximal function of f with rspct to th gaussian masur γ d, M γd f(x) sup f(y) γ d (dy). (8) r>0 γ d (B(x, r)) B(x,r) Unfortunatly, this inquality only allows to gt th wak (,) continuity of T f in th on dimnsional cas, d, but allows to gt a pointwis convrgnc rsult. Svral rsults of this papr, s Lmma. and Thorm., us tchniqus containd in that papr. If f L (γ d ), u(x, t) T t f(x) is a solution of th initial valu problm { u (x, t) Lu(x, t) t u(x, 0) f(x) whr u(x, 0) f(x) mans that lim u(x, t) f(x), a.. x t 0 + W want to prov that this convrgnc is also non-tangntial in th following sns. Lt { } Γ p γ(x) (y, t) R d+ + : y x < t x (9) Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4

4 0 Ebnr Pinda, Wilfrdo Urbina b a parabolic gaussian con. W want to prov that lim T t f(y) f(x), a.. x (y,t) x,(y,t) Γ p γ(x) Using th Bochnr subordination formula (s [6]), λ π 0 u u λ /4u du, w dfin th Poisson-Hrmit smigroup {P t } t 0 as P t f(x) π 0 u u T t /4uf(x)du. (0) {P t } t 0 is also a strongly continuous smigroup on L p (γ d ), with infinitsimal gnrator ( L) /. From (4) w obtain, aftr th chang of variabl r t /4u, P t f(x) π (d+)/ R d 0 t xp ( t /4 log r ) ( ) xp y rx r dr f(y)dy. () ( log r) 3/ ( r ) d/ r Dfinition.. Th maximal function for th Poisson-Hrmit smigroup is dfind as P f(x) sup P t f(x). () t>0 If f L (γ d ), u(x, t) P t f(x) is solution of th initial valu problm u (x, t) Lu(x, t) t u(x, 0) f(x) whr u(x, 0) f(x) mans that lim u(x, t) f(x), a.. x t 0 + W want to prov that this convrgnc, for th Poisson-Hrmit smigroup, is also non-tangntial in th following sns. Lt { Γ γ (x) (y, t) R d+ + : y x < t } x, (3) Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4

5 Non Tangntial Convrgnc for th Ornstin-Uhlnbck Smigroup. b a gaussian con. Also w want to prov that lim P tf(y) f(x), a.. x (y,t) x,(y,t) Γ γ(x) In ordr to study th non-tangntial convrgnc for th Ornstin-Uhlnbck smigroup w ar going to considr th following maximal function, that was dfind by L. Forzani and E. Fabs [3]. Dfinition.3. Th non tangntial maximal function associatd to th Ornstin-Uhlnbck smigroup is dfind as Tγ f(x) sup T t f(y). (4) (y,t) Γ p γ(x) Using an inquality for a gnralizd maximal function, obtaind by L. Forzani in [] (for mor dtails s [8] pag and 88 9), it can b provd that Tγ f is wak (, ) and strong (p, p) for < p <, with rspct to th Gaussian masur. Actually for th non-tangntial convrgnc for th Ornstin-Uhlnbck smigroup it is nough to considr a truncatd maximal function. Lt Γ p (x) { (y, t) R d+ + : y x < t, 0 < t < x }, (5) 4 b a truncatd parabolic gaussian con. Dfinition.4. Th truncatd non-tangncial maximal function associatd to th Ornstin-Uhlnbck smigroup is dfind as T f(x) sup T t f(y). (6) (y,t) Γ p (x) In th nxt lmma w ar going to gt a inquality bttr than (8) for th truncatd non tangntial maximal function T f, which implis, immdiatly, that T f is wak (, ) and strong (p, p) for < p <, with rspct to th gaussian masur. Lmma.. T f(x) C d M γd f(x), (7) for all x R d Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4

6 Ebnr Pinda, Wilfrdo Urbina Proof. Lt us tak u(y, t) T t f(y) and without loss of gnrality lt us assum f 0. Lt a o 0 and a j j, j N, thn a j < a j+ j N, and lt us dnot A j (y, t) {u R d : a j ( t ) t y u < a j ( t ) }, th annulus with cntr t y. Now considr for ach j N th ball with cntr t y, and radius a j ( t ) and lt us dnot it by B j (y, t) B( t y, a j ( t ) ), thn A j (y, t) B j (y, t) \ B j (y, t) u(y, t) π d ( t ) d π d ( t ) d R d j t f(u)du A j (y,t) t f(u)du Now if (y, t) Γ p (x) and t y u < a j ( t ) thn, t x u t x t y + t y u t (x y) + t y u < t t + a j ( t ) < t + a j ( t ) < ( + a j )( t ), sinc t < t if t < 0.8 Considring C j (x, t) B( t x, ( + a j )( t ) ), w hav u(y, t) π d ( t ) d a j j C j (x,t) f(u)du Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4

7 Non Tangntial Convrgnc for th Ornstin-Uhlnbck Smigroup. 3 Now, f(u)du f(u) u u du C j (x,t) C j (x,t) f(u) u t x + t x.(u t x)+ t x u du C j (x,t) (+aj) ( t )+(+a j)( t ) t x + t x f(u) u du C j(x,t) but, t x u < ( + a j )( t ) and thrfor, x u x t x + t x u < ( t ) x +( + a j )( t ). w gt Taking C j(x,t) D j (x, t) B(x, ( t ) x +( + a j )( t ) ), f(u) u du D j(x,t) f(u) u du M γd f(x) u du M γd f(x) u x +x(x u) x du D j (x,t) D j (x,t) M γd f(x) x D j (x,t) u x + x x u du M γd f(x) x + x (( t ) x +(+a j )( t ) ) M γd f(x) x + x (( t ) x +(+a j)( t ) ) D j (x,t) E j(x,t) u x du w dw whr E j (x, t) B(0, ( t ) x +( + a j )( t ) ). Sinc γ d is a d dimnsional masur, and using that t < x 4, w gt Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4

8 4 Ebnr Pinda, Wilfrdo Urbina C j (x,t) f(u) u du C d M γd f(x) x + x (( t ) x +(+a j )( t ) ) (( t ) x +( + a j )( t ) ) d C d M γd f(x) x + x (( t ) x +(+a j )( t ) ( t ) d (( t ) x +( + aj)( + t ) ) d C d M γd f(x) x + ( t ( t ) d t ) ( ( t ) t +(+a j ) ( t ) t + ( + a j )( + t ) ) d. Thrfor C j (x,t) f(u) du (+a j ) ( t )+(+aj ) ( t ) + t x t C j (x,t) f(u) u du (+a j ) ( t )+(+aj ) ( t ) t + x C d M γd f(x) x + ( t ) t +(+a j ) ( t ) t ( t ) d ( ( t ) t + ( + a j )( + t ) ) d t ) (+a j ) ( )+4(+aj ) ( + ( t ) t t ( t ) d ( ) ( t ) d + ( + a j)( + t ) C d M f(x) γd t (+a j ) ( )+4(+a j ) +.( t ) d ( + ( + aj ) ) d C d M γd f(x), sinc 0 < t < 4 and t t <, + t <, if t > 0. Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4

9 Non Tangntial Convrgnc for th Ornstin-Uhlnbck Smigroup. 5 Thus, u(y, t) π d ( t ) d j a j C j (x,t) f(u)du C d M γd f(x) π d ( + t ) d ( t ) d a j (+a j ) ( )+4(+a j ) + ( t ) d ( + ( + aj) ) d j C d M γd f(x) π a j +(+a j ) ( )+4(+a j ) + ( + ( + a j ) ) d, j sinc + t. Now it is asy to s that a j + ( + a j) ( ) + 4( + aj) [ (( ) + 4 ) + j] j, which is ngativ for j sufficintly big, thn a j +(+a j ) ( )+4(+a j ) +.( + ( + a j). ) d <. j Thus u(y, t) C d M γd f(x) and sinc (y, t) Γ p (x) is arbitrary T f(x) sup u(y, t) C d M γd f(x). (y,t) Γ p (x) Now w ar rady to stablish th convrgnc rsult for th Ornstin-Uhlnbck smigroup. Thorm.. Th Ornstin-Uhlnbck smigroup {T tf} convrgs in L (γ d ) a. if t 0 +, for any function f L (γ d ), lim u(x, t) f(x), a.. x (8) t 0 + Morovr, if u(y, t) T t f(y) thn u(y, t) tnds to f(x) non tangntially,i.. Proof. W hav, lim T tf(y) f(x), a.. x. (9) (y,t) x,(y,t) Γ p γ (x) u(y, t) π d ( t ) d R d t f(u)du, Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4

10 6 Ebnr Pinda, Wilfrdo Urbina considring Ωf(x) lim α 0 + [ sup u(y, s) f(x) (y,s) Γ p γ (x),0<s<α and lt us st f(x) f(x)χ (0,k) + f(x)(i χ (0,k) ) f (x) + f (x), for k N fix. Lt us prov that Ωf(x) C d M γd f (x), a., for x k. Lt us considr x a Lbsgu s point for f L (γ d ), i.. x vrifis lim f(u) f(x) γ d (du) 0 r 0 + γ d (B(x; r)) B(x;r) Thn givn ɛ > 0 thr xists 0 < δ < such that f(u) f(x) γ d (du) < ɛ, γ d (B(x; r)) B(x;r) for 0 < r < δ. Lt us dfin g as g(u) dpnds on x and M γd g(x) < ɛ. On th othr hand, sinc whr thn w gt, { f(u) f(x) if u x δ 0 if u x > δ u(y, t) f(x) u (y, t) f (x) + u (y, t) f (x) u i (y, t) π d ( t ) d R d u (y, t) f (x) π d ( t ) d t f i(u)du i,, R d t (f (u) f (x))du ], Thus g π d ( t ) d x u δ t (f (u) f (x))du + π d ( t ) d x u >δ t (f (u) f (x))du. Now w hav that if x k and (y, t) Γ p γ(x) with t < x (y, t) Γ p (x). Thus u x δ implis u u x + x u x + x < δ + k < + k k 4, thn Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4

11 Non Tangntial Convrgnc for th Ornstin-Uhlnbck Smigroup. 7 and thn, f (u) f(u) f (x) f(x). Thrfor π d ( t ) d π d ( t ) d π d ( t ) d x u δ x u δ R d T g(x) C d M γd g(x) C d ɛ. t (f (u) f (x))du t (f(u) f(x))du t g(u)du Now obsrv that if (y, t) Γ p γ(x) and t u x u y + y x and thus δ thn, u x > δ implis δ < δ < u y + y x < u y +t u y + δ, thus u y > δ. Thrfor, π d ( t ) d u x >δ t (f (u) f (x))du π d ( t ) d π d ( t ) d u x >δ + f (x) π d ( t ) d u y > δ + f π d ( t ) d (x) t f (u) du u x >δ t f (u) du u x >δ t du t du. Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4

12 8 Ebnr Pinda, Wilfrdo Urbina Now, w hav π d ( t ) d π d ( t ) d π d ( t ) d π d ( t ) d u y > δ u y > δ, u <k u y > δ, u <k u y > δ, u <k t f (u) du t f (u) du t f(u) du t u f(u) u du Thn for 0 < t < log k π d ( t ) d u y > t y u t f(u) u du. δ, u <k ( 4k + δ 4k + δ ) ; u y > δ, u < k implis that t y u t y t u + t u u t (y u) (u t u) but 0 < t < log t y u u t u t y u ( t ) u ( ) t δ δ k( t ) t + k k, ( ) 4k + δ and thrfor t > 4k + δ 4k + δ 4k + δ, thn, t ( δ + k ) k > 4k + δ 4k + δ ( δ + k ) k 4k + δ (k + δ) k 4(k + δ) 4k + δ 4 k 4k + δ 4k 4 δ 4. Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4

13 Non Tangntial Convrgnc for th Ornstin-Uhlnbck Smigroup. 9 Thrfor u y > δ, u < k implis t y u > δ 4 and thus π d ( t ) d k π d ( t ) d u y > δ t f (u) du u y > δ, u <k δ 6( t ) f(u) u du δ 6( t ) +k f(u) u δ 6( t ) +k du f π d ( t ) d R d π d ( t ) d,γd On th othr hand, taking th chang of variabl s u t y, w hav π d ( t ) d f (x) π d ( t ) d f(x) π d ( t ) d f (x) u x >δ t du x s t y >δ x s t y >δ s t ds s t ds, sinc, f (x) f(x) as x ( k < k. ) k δ/ Thus taking 0 < t < log, x s t y > δ implis k 3δ/4 But s s x + t y + x t y s x + t y ( t y x) s x + t y t y x. t y x t y t x + t x x t y x +( t ) x Thus, sinc t δ, t t + ( t )(k ). s x + t y t y x > δ t t ( t )(k ) δ t δ (k )( t ) δ (k ) + (k δ ) t, Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4

14 0 Ebnr Pinda, Wilfrdo Urbina ( ) k δ/ and as 0 < t < log, thn t > k 3δ/4 k 3δ/4 k δ/. Hnc, s > δ (k ) + (k δ/) t > δ (k ) + (k δ/) k 3δ/4 k δ/ δ (k ) + k 3δ/4 δ 3δ/4 δ 4. Thn x s t y > δ implis s > δ ( ) k δ/ 4 if 0 < t < log. k 3δ/4 s Thrfor, taking w, t π d ( t ) d f(x) π d ( t ) d f(x) π d u y > δ t f (u) du s t ds s > δ 4 w dw. w > δ 4 t Now sinc, x k < k, thnf (x) 0. Hnc u (y, t) f (x) u (y, t) T f (x) C d M γd f (x) for (y, t) Γ p (x). Thrfor, u(y, t) f(x) u (y, t) f (x) + u (y, t) f (x) u (y, t) f (x) + u (y, t) C d ɛ + if (y, t) Γ p γ(x) and δ 6( t ) +k ( t ) d { 0 < t < min log f,γd + f(x) π d w dw w > δ 4 t +C d M γd f (x), ( ) ( ) 4k + δ k δ/, log, 4k + δ k 3δ/4 x } : a. 4 Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4

15 Non Tangntial Convrgnc for th Ornstin-Uhlnbck Smigroup. Thus taking suprmum on (y, t) Γ p γ(x), 0 < t < α < a and thn taking α 0 + w obtain, Ωf(x) C d (ɛ + M γd f (x)) for all ɛ > 0 and almost vry x with x k. Givn ɛ > 0, lt us tak k sufficintly larg such that f,γd C d ɛ, thn by th stimation of Ω and th wak continuity of M γd w gt γ d ({x R d : x k, Ωf(x) > ɛ}) ɛ and that implis that Ωf(x) 0 a.. A similar proof for th Poisson-Hrmit smigroup, using th non-tangntial maximal function dfind as P γ f(x) sup P t f(y), (0) (y,t) Γ γ (x) and its analogous truncatd vrsion, should b possibl but it has som tchnical difficultis that w hav bn unabl to ovrcom so far. Lt us now prov a gnral statmnt for familis of linar oprators that will allow us to gt a simplr proof of th non-tangntial convrgnc, both for th Ornstin-Uhlnbck smigroup and also for th Poisson-Hrmit smigroup. It is a gnralization of Thorm. of J. Duoandikotxa s book []. Thorm.3. Lt {T t} t>0 b a family of linar oprators on L p (R d, µ) and for any x R d, lt Γ(x) b a subst of R d+ + such that x is in (Γ(x)), that is to say x is an accumulation point of Γ(x). Lt us dfin T f(x) sup{ T tf(y) : (y, t) Γ(x)}, for f L p (R d, µ) and x R d. If T is wak (p, q) thn th st { } S f L p (R d, µ) : lim Ttf(y) f(x) a.. is closd in L p (R d, µ). Proof. Lt us considr a squnc (f n) in S such that f n f in L p (R d, µ), thn T tf(y) f(x) T tf n(y) f n(x) T t(f f n)(y) (f(x) f n(x)), this implis that for ach n N, for almost vry x, Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4

16 Ebnr Pinda, Wilfrdo Urbina lim sup T t f(y) f(x) lim sup T t(f f n)(y) (f(x) f n(x)) lim sup T t (f f n )(y) + lim sup f(x) f n (x) T (f f n )(x) + f(x) f n (x). On th othr hand, if w know that a b + c thn a > λ implis b > λ c > λ. Thn, givn λ > 0 and n N, lim sup T t f(y) f(x) > λ implis and this implis that, givn λ > 0, ({ T (f f n )(x) > λ f(x) f n(x) > λ a.. µ x : lim sup T tf(y) f(x) > λ }) µ ({ x : T (f f n )(x) > λ }) ({ +µ x : f(x) f n (x) > λ }) for all n N. Thrfor, ({ µ x : ( C λ f ) q fn p + ( ) p f fn p, λ lim sup T tf(y) f(x) > λ and sinc this is tru for all λ > 0, w gt that ({ as { x : µ x : lim sup T tf(y) f(x) > 0 lim sup T tf(y) f(x) > 0 { n x : } }) }) 0 0, lim sup T t f(y) f(x) > n }. Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4

17 Non Tangntial Convrgnc for th Ornstin-Uhlnbck Smigroup. 3 Thus lim Ttf(y) f(x) a.. and thn f S. Thrfor S is a closd st in L p (R d, µ). Finally, as a consqunc of this rsult, w gt th non-tangntial convrgnc for th Ornstin-Uhlnbck smigroup {T t } t>0 and th Poisson-Hrmit smigroup {P t } t>0. Corollary.4. Th Ornstin-Uhlnbck smigroup {T t } t>0 and th Poisson-Hrmit smigroup {P t} t>0 vrify lim T tf(y) f(x) a.. x, (y,t) x,(y,t) Γ p γ (x) lim P t f(y) f(x) a.. x. (y,t) x,(y,t) Γ γ (x) Proof. Lt us discuss th proof for th th Ornstin-Uhlnbck smigroup {T t } t>0. Th proof for th Poisson-Hrmit smigroup {P t} t>0 is totally similar. It is immdiat that for any givn polynomial f(x) n ( k0 J kf(x), sinc T t f(y) T n t k0 J kf(y) ) n k0 tk J k f(y), w hav th non-tangntial convrgnc, for all x R d. Now considring th st S { lim T t f(y) f(x), (y,t) x,(y,t) Γ p γ (x) f L p (γ d ) : lim T tf(y) f(x) a.. (y,t) x,(y,t) Γ p γ (x) corrsponding to th Ornstin-Uhlnbck smigroup, thn th polynomials ar in S. From th prvious rsult, sinc non-tangntial maximal function for th Ornstin- Uhlnbck smigroup Tγ f is wak (, ) with rspct to th Gaussian masur, w gt that th st S is closd in L p (γ d ) and sinc th polynomials ar dns in L p (γ d ) thn S L p (γ d ). }, Acknowldgmnt W want to thank th rfrs for thir suggstions and/or corrctions that improvd th prsntation of this papr. Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4

18 4 Ebnr Pinda, Wilfrdo Urbina Rfrncs [] Duoandikotxa, J. Fourir Analysis, Graduatd Studis in Mathmatics, Volum 9, AMS, R. I., 00. [] Forzani, L. Lmas d cubriminto d tipo Bsicovitch y su aplicación al studio dl oprador maximal d Ornstin-Uhlnbck. Tsis d Doctorado. Univrsidad Nacional d San Luis, Argntina, 993. [3] Forzani, L., Fabs, E. Unpublishd manuscript, 994. [4] Gutiérrz, C., Urbina, W., Estimats for th maximal oprator of th Ornstin- Uhlnbck smigroup. Proc. Amr. Math. Soc. 3 (99), [5] Sjögrn P., Oprators associatd with th Hrmit Smigroup-A Survy, J. Fourir Anal. Appl., (3)(997), [6] Stin E., Singular Intgrals and Diffrntiability Proprtis of Functions, Princton Univ. Prss, Princton, Nw Jrsy, 970. [7] Szgö, G., Orthogonal polynomials, rv. d., Amr. Math. Soc. Colloq. Publ., vol. 3, Amr. Math. Soc., Providnc, R. I., 959. [8] Urbina W.m Análisis Armónico Gaussiano: una visión panorámica, Trabajo d Ascnso, Facultad d Cincias, UCV, 998. Availabl in [9] Zygmund, A., Trigonomtric Sris, nd. d., Cambridg Univ. Prss., Cambridg, 959. Divulgacions Matmáticas Vol. 6 No. (008), pp. 07 4