On n-dimensional Hilbert transform of weighted distributions

Transcription

1 O -dimesioal Hilbert trasform of weighted distributios MARTHA GUMÁN-PARTIDA Departameto de Matemáticas, Uiversidad de Soora, Hermosillo, Soora 83000, México Abstract We de e a family of cougate Poisso trasforms of distributios T i the optimal space of weighted distributios w :::w D 0 L, by meas of the S 0 -covolutio. We prove that their boudary values i the topology of this space of distributios are of the form H (T ), where H is the - dimesioal Hilbert trasform of T. Keywords: S 0 -covolutio, weighted distributio spaces, Poisso ad Hilbert trasform. 000 Mathematics Subect Classi catio: 46F05, 46F0, 46F. Itroductio ad otatio The -dimesioal Hilbert trasform H is a covolutio operator with the kerel p:v: x ::: p:v: x that is de ed for a certai class of tempered distributios. This class ca be easily ideti ed. I fact, the optimal space of tempered distributios for which the -dimesioal Hilbert trasform is de ed is the class of weighted itegrable distributios w :::w DL 0 (see [0],[]). The covolutio cosidered here, is the so-called S 0 -covolutio, a commutative operatio for tempered distributios developed by Y. Hirata ad H. Ogata [8] ad R. Shiraishi [] with the purpose of extedig the validity of the Fourier exchage formula F (S T ) = F (S) F (T ), where the product o the right-had side must be uderstood i a appropriate sese that we will precise later. The aim of this paper is to solve the followig boudary value problem: give a distributio T w :::w DL 0, we cosider a family of cougate Poisso trasforms of T ad prove that its boudary value i the topology of w :::w DL 0 is H (T ). Thus, we geeralize a result proved by Padey ad Sigh i ([9, Th. 4.3]) for the space of distributios DL 0 p, < p <. The paper is divided i three sectios. I Sectio we cosider two family of kerels: the product domai versio of the oe-dimesioal Poisso kerel Correspodig author. martha@gauss.mat.uso.mx

2 ad the product domai versio of the oe-dimesioal cougate Poisso kerel. Both versios correspod to the domai R +:::R + cosistig of copies of the upper half-plae ad they are related by meas of the oe-dimesioal Hilbert trasform. We preset here some results cocerig to the S 0 -covolutio with these kerels. I Sectio 3 we cosider the -dimesioal Hilbert trasform of distributios i w :::w DL 0 ad we prove the mai results of this work. Fially, cocerig otatio, partial derivatives will be deoted where is a multi-idex ( ; :::; ). We shall use the stadard abbreviatios = + ::: +, x = x :::x. For a fuctio g, we will idicate with _ g the fuctio x! g ( x) : Give a distributio T, we will deote with T the distributio '! T; ' _, where ' is a appropriate test fuctio. The Fourier trasform will be deoted as F. The letter C will idicate a positive costat, probably di eret at di eret occurreces. Poisso ad cougate Poisso trasform of weighted distributios I this sectio we will cosider the Poisso ad cougate Poisso trasforms of certai weighted itegrable distributios. These trasforms are de ed by meas of the S 0 -covolutio with the product domai versios of the oedimesioal Poisso ad cougate Poisso kerels, respectively. Let us rst review the otio of S 0 -covolutio that will be used. The spaces of fuctios ad distributios related to this otio ca be foud i [] ad [6]. For the sake of completess we give a brief accout of them. The space of itegrable distributios DL 0 is, by de itio, the strog dual of the space B _ of smooth fuctios ' : R! C such '! 0 as x!, for each multi-idex. _B is a closed subspace of the space B cosistig of those smooth fuctios ' with the property ' is bouded for every multi-idex, edowed with the topology of the uiform covergece o R of each derivative. The space C0 of compactly supported smooth fuctios de ed o R, is dese i B _ but ot i B. Each T DL 0 ca be represeted as T = P f, where f L. Cosequetly, we have the strict iclusios E 0 DL 0 S 0. It is possible to cosider DL 0 as the strog dual of the space B, provided that we edow B with a topology that gives rise to the followig otio of sequece covergece: a sequece f' g coverges to ' if, for each multi-idex, sup k@ ' k < ad the sequece f@ ' g coverges ' uiformly o compact sets. If we deote as B c the resultig topological space, it ca be proved that C0, ad so B, _ is dese i B c. Thus, it turs out that DL 0 is also the dual of B c. Other spaces that will be of iterest later are the followig: For p <, let D L p = f' C ' L p for every multi-idex g,

3 edowed with the topology de ed by the family of orms 3 k'k m;p = 4 X k@ 'k p 5 p m =p, m = 0; ; ; :::. The space D L p is a Fréchet space ad we have the dese ad cotiuous strict iclusios C0 D L p D 0. The space D L is the space B de ed above. For p < q we have the cotiuous strict iclusios D L p D L q. Whe < p, we deote with DL 0 the topological dual of D p Lp0, edowed with the strog dual topology. We have the cotiuous strict iclusios DL 0 p DL 0 for p < q. Moreover, every distributio T q D0 Lp ca be represeted as T = P f, f L p. Thus DL 0 is cotiuously icluded i p S0. Now, we de e the otio of S 0 -covolutio. De itio ([8],[]) Give two tempered distributios T ad S, it is said that the S 0 -covolutio of T ad S exists if T S ' D 0 L for every ' S. Whe the S 0 -covolutio exists, the map S! C '! T S ' ; D 0 L ;Bc de es a tempered distributio which will be deoted by T S. This is a commutative operatio. Moreover, De itio coicides with the classical de itio i all the cases i which the latter makes sese. Remark Y. Hirata ad H. Ogata i [8] itroduced the S 0 -covolutio to exted the validity of the Fourier exchage formula (origially proved by L. Schwartz i [] for pairs of distributios i O 0 c S 0 ): give T; S S 0 such that the S 0 -covolutio T S is de ed, the F (T S) = F(T )F(S). () The product o the right had side of () is uderstood i the followig sese: for ay two -sequeces f' k g k= ad f kg k=, the sequeces f(f (T ) ' k) F (S)g k= ad ff (T ) (F (S) k )g k= coverge i D0 to the same distributio ad this commo limit is deoted by F (T ) F (S). As it is kow, a -sequece is a sequece f' k g k= of o-egative fuctios i C0 with the followig properties:. Supp ' k coverges to 0 whe k!.. R ' k = for every k. 3

4 We will cosider the followig two -dimesioal kerels, amely: The product domai versio of the oe-dimesioal Poisso kerel P (y) (x) = Y P y (x ), () = where x = (x ; :::; x ) R, y = (y ; :::; y ), (y) > (0), meaig that y > 0; :::; y > 0, ad P y (x ) = y P (x =y ), with P (x ) =, = ; :::;. +x This product domai versio of the Poisso kerel correspods to the domai R + ::: R + cosistig of copies of the upper half-plae. The product domai versio of the oe-dimesioal cougate Poisso kerel Q (y) (x) = Y Q y (x ) (3) = for x = (x ; :::; x ) R, (y) > (0) ad Q y (x ) = y Q (x =y ), with Q (x ) = x +x, = ; :::;. It is well kow the fact that for y > 0, = ; :::; Q y (x ) = H P y (x ) (4) where H P y is the classical oe-dimesioal Hilbert trasform of Py, that is, H P y = p:v: x P y. The optimal spaces for S 0 -covolutio with the kerels () ad (3) are described i the followig lies. De itio 3 ([0],[],[3]) Let w = + x =, = ; :::;. The w :::w D 0 L = T D 0 : w :::w T D 0 L with the topology iduced by the map w :::w D 0 L! D 0 L T! w :::w T. Similarly, is de ed the space w :::w D 0 L. It ca be proved the followig characterizatio for these spaces. Lemma 4 ([4], [5]) 8 9 < w :::w DL 0 = : T S0 : T = f, f L w :::w = ; fiite 8 9 < w:::w D L 0 = : T S0 : T = f, f L w :::w = ;. fiite 4

5 Cosequetly, both spaces of distributios are closed uder di eretiatio. They also cotai kow spaces of distributios, amely: Propositio 5 For p <, D 0 L p w :::w D 0 L w :::w D 0 L ad the iclusios are strict ad cotiuous. Proof. Accordig to Lemma 4 ad the characterizatio for the space DL 0 p, the iclusios are immediate sice L p,! L w :::w,! L w :::w. That the iclusios are strict, is show by the followig examples. Let T be the tempered distributio de ed by the fuctio f (x) = + x = ::: + x = where 0 < <. Sice w :::w T L (R ), this shows that T w:::w D L 0. However, T = w :::w DL 0 because if we cosider the sequece (x) = (x =) ::: (x =), = ; :::;, where C0 (R), 0, (t) = for t < ad (t) = 0 for t >, we ca easily see that! i B c but :::w T; = (x =) S 0 ;S ( + x dx ::: )( )= w 0 ( + x )( )= dx :::! as! 0 (x =) ( + x ) dx ( )= ( + x ) ( )= dx sice 0 < <. This shows that the secod iclusio is strict. To see the rst iclusio is proper, let us cosider rst the case p =. We ca take the distributio T = w :::w which clearly belogs to w :::w DL 0 sice w :::w T L (R ), however, takig a sequece ( ) = as above, we have that! i B c but ht; i 0 ( + x )= dx :::! as!. 0 ( + x ) = dx Now, cosider the case < p <. Let q ad be such that =p + =q =, 0 < <, ad pick such that =q < <. Let T be the distributio T = + x = ::: + x =. Sice w :::w T = + x (+)= ::: + x (+)= L (R ) is clear that T w :::w DL 0. For C0 (R) as above, let us cosider the sequece ( ) = de ed by (x) = (x =) ::: (x =). 5

6 The sequece ( ) = coverges to 0 i D L q sice for every multi-idex k@ k q q (t (+)q ) q dt (t ) q dt = k q )+q k@ q ::: k q k@ q (q! 0 as! because q >. Thus k k m;q! 0 as!. However, for > we have ht; i ( + x dx ::: )= " dx C ::: dx x = C! as! x # ( + x ) = dx sice > 0. This cocludes the proof of the Propositio. The authors of [3] have determied the optimal spaces of tempered distributios S 0 -covolvable with the kerel P (y). I fact, they prove: Theorem 6 ([3]) Give T S 0, the followig statemets are equivalet: i) T w :::w D 0 L. ii) T is S 0 -covolvable with P (y) for each (y) > (0). Also, L. Schwartz i [0] ad Alvarez ad Carto-Lebru i [] characterized the optimal space of tempered distributios S 0 -covolvable with the - dimesioal Hilbert kerel p:v: x ::: p:v: x, amely, Theorem 7 ([]) Let T S 0. The, the followig statemets are equivalet: a) T w :::w D 0 L. b) T is S 0 -covolvable with p:v: x ::: p:v: x. I view of (4) ad Theorem 7, it seems that the optimal space for S 0 - covolutio with the kerel Q (y) is w :::w D 0 L.This ca be easily show ust appealig to the followig geeralizatio of Theorem 7 proved i [, Th. 9]. Theorem 8 ([]) Let T S 0. The, the followig statemets are equivalet: a) T w :::w D 0 L. 6

7 b) T is S 0 -covolvable with the class P. The class P above deotes the family of distributios p:v:k (x ; :::; x ) where k is a kerel i L loc (R fx = 0 or ::: or x = 0g) with the followig cacellatio ad size properties: k (x ; :::; x ; :::; x ) dx = 0 for each 0 < a < b, = ; :::; (5) ad, a<x <b k (x ; :::; x ) C x ::: x for x 6= 0, = ; :::;, (6) for some positive costat C. The family of kerels Q (y) with (y) > 0 is icluded i the class P. Thus, we ca state: Theorem 9 Let T S 0. The, the followig statemets are equivalet: a) T w :::w D 0 L. b) T is S 0 -covolvable with Q (y) for each (y) > (0). ;Bc The S 0 -covolutio of a distributio T w :::w DL 0 with the kerel Q (y) is a fuctio whose value at x is give by 0 Y Y T Q (y) (x) w (t )T t ; w (t )Q y (x t ) (7) = for each (y) > 0. Ideed, writig Y w = with f L (R ), for ' S we have = T = f fiite T Q (y) ; ' S 0 ;S = w :::w T w :::w Q(y) ' ; D 0 L ;Bc. (8) The fuctio w :::w Q(y) ' belogs to B sice Q (y) ' is a C fuctio 7

8 such that Q(y) ' (x) C C R = R = R = Y x t = y y ' (t) dt + ((x t ) =y ) Y + ((x t ) =y ) = y + ((x t ) =y ) ' (t) dt Y + ((x t ) =y ) = ' (t) dt y R = = Y y + (x =y ) = + (t =y ) = + t dt Y + (x =y ) = y + (t =y ) = + t dt, (9) hece w :::w Q(y) ' is bouded. w :::w Q(y) ' = X C w :::@ w ' ad takig ito accout w is w or a C bouded fuctio, proceedig as above we ca show agai w :::w Q(y) ' is bouded. Thus, we ca expad the right had side of (8) as w :::w T; w :::w Q(y) ' D 0 L ;Bc ad usig Fubii s theorem we ca ally obtai (7). We have see i Lemma 4 that w :::w DL 0 is closed uder di eretiatio. Moreover we ca also obtai the followig results. Lemma 0 If T w :::w DL 0 T Q (y) = (@ T ) Q (y) for each multi-idex. Proof. The proof of this Lemma follows exactly the same scheme tha that of [5, Lemma 9] but this time usig the the fact that for every = ; :::;, each ' S (R) ad every k = 0; ; ; ::: k ' Cy w, which ca be proved as we did i estimate (9). Lemma Give T w :::w DL 0 space w :::w D L for each (y) > 0. the S 0 -covolutio T Q (y) belogs to the 8

9 Proof. By Lemmas 4 ad 0, it su ces to prove that T Q (y) L w :::w for every T w :::w DL 0. Oce agai, followig step by step the scheme of the proof of [5, Lemma 0] ad oticig that for = ; :::; we w (t ) C + t Q y (x t )! (+( C ; + (x t ) )) y + y, we ca ally prove our assertio. Remark As a cosequece of Lemma, for each (y) > 0, the S 0 -covolutio with the kerel Q (y) preserves the space L w :::w. The proofs of Lemmas 0 ad rely o the same argumets give to prove the correspodig results for the kerel P (y). Moreover, for this kerel it ca be proved ([5, Th. ]): Theorem 3 [5] Give T w :::w D 0 L, the S 0 -covolutio T P (y) coverges to T i w :::w D 0 L as (y)! (0) +. It is also possible to cosider the family K of kerels that ca be writte as P y :::P yk Q yi :::Q yi k where < ::: < k, i < ::: < i k ad f ; :::; k g \ fi ; :::; i k g =?. The, for T w :::w DL 0, the S 0 -covolutio T P y :::P yk Q yi :::Q yi k is de ed. I fact, it is a fuctio whose value at x is give by 0 w (t )T t ; w (t )P y (x t ):::w i k (t i k )Q yi k (x i k t i k ) A = D 0 L ;Bc (0) Remark 4 Give T w :::w DL 0 the S 0 -covolutio of T ad ay of the kerels of the class K belogs to the space w :::w D L for each (y) > 0. Ideed, give K K, all that we eed to otice is that T K L w :::w ad this ca be see as i Lemma by usig (0) ad the w (t ) C + P y (x t ) C! (+( ; + (x t ) )) y + y, 9

10 ad = ; Q y (x t )! (+( C ; + (x t ) )) y + y, 3 Hilbert trasform of distributios i w :::w D 0 L For T w :::w DL 0, let us deote by H (T ) the -dimesioal Hilbert trasform of T, that is, H (T ) = p:v: ::: p:v: T. () x x I [9, Th. 4.3], Padey ad Sigh prove that for T DL 0 the p S0 -covolutio T Q (y) coverges to H (T ) i DL 0 i the weak sese as (y)! p (0)+. I view of this remark ad results of Sectio, it is atural to thik that give T w :::w DL 0 the S 0 -covolutio T Q (y) coverges to H (T ) i w :::w DL 0 as (y)! (0) +. The rst step to prove this assertio is the followig result: Propositio 5 If T is a distributio i w :::w DL 0 the, for every > 0; :::; > 0 we have that H (T ) w :::w DL 0. Moreover, H w :::w DL 0 is ot icluded i DL 0. I particular, H w :::w DL 0 w :::w DL 0. Proof. We rst otice that for T w :::w D 0 L ad every ' S we have hh (T ) ; 'i S 0 ;S = ( ) ht; H (')i w:::w D 0 L ;w :::w B c () = ( ) w :::w T; w :::w H (') D 0 L ;Bc (3) Ideed, by de itio of S 0 -covolutio ad usig the fact that w :::w p:v: ::: p:v: _ '! B x x (see [, Th. 9]) we obtai * hh (T ) ; 'i S 0 ;S = T p:v: ::: p:v: + _ '! ; x x D 0 L ;Bc (4) * = T; p:v: ::: p:v: _ '+ x x w :::w D 0 L ;w = ( ) ht; H (')i w:::w D 0 L ;w :::w B c :::w B c 0

11 ad also, the right had side of (4) ca be writte as * w :::w T w :::w p:v: ::: p:v: + _ '!! ; x x = ( ) w :::w T; w :::w H (') D 0 L ;Bc. This proves our claim. Now, assume without loss of geerality that w :::w L (R ). Cosider rst the case = 0. To prove that H (T ) w :::w D 0 L D 0 L ;Bc T f with f it su ces to show that the tempered distributio w :::w H (T ) is cotiuous i C0 with the topology of _B. Let ' C0, the D E w :::w H (T ) ; ' S 0 ;S = D E ( ) w :::w T; w :::w H w :::w ' D 0 L ; B _ h i f w :::w H w :::w ' (x) dx. (5) R Deote by = w :::w ' C0. We eed to [w :::w H ( )] = X w :::@ w (6) We have that H ( ) (x) = = lim "!0;:::;"!0 lim "!0;:::;"!0 k= y >" ::: X I k, y >" (x y ; :::; x y ) y :::y dy :::dy where I k = ::: ad A = f" < y < g or A = f < y g, = ; :::;. A A Let us cosider ay of these itegrals, say, for example I = ::: y > y > " <y < (the other itegrals ca be aalyzed i a similar way).

12 Thus I y > ::: y > " <y < (x y ; :::; x y ; x y ) (x y ; :::; x y ; x ) y :::y y (x y ; :::; x y ; x ty ) ::: dtdy :::dy. y > " <y < 0 y :::y y > Usig the iequalities + x = C + (x y ) = y for y >, = ; :::;, ad + x = C + (x ty ) = if 0 t ad y <, we obtai for such that < < +, = ; :::; w (x ) :::w (x ) I C ::: y > y > " <y (x y ; :::; x y ; x ty = C w (x y ) ::: y > 0 y > w (x y ) :::w (x y ) w (x ty ) dtdy :::dy w (x y ) h w (x ty ) w (x y ) :::w (x y ) w (x ty ) " <y < (x y ; :::; x y ; x ty dtdy :::dy C + ( )= ::: + ( w (x y ) dy ::: y w (x y ) dy " <y < y > C + ( )= ::: + C sup + ( )= ::: + ( X )= k@ 'k R C X k@ 'k. 0 w (x ty ) dtdy

13 Thus, (5) ca be estimated as f (x) w :::w H w :::w ' (x) dx C X R k@ 'k kfk. The above computatios show that T L w :::w implies HT w :::w DL 0. (7) Now, let us suppose that 0. We rst otice that if f L w :::w the H (@ f) H (f). (8) The relatio (8) ca be easily proved ust usig (), (3) ad the fact that (8) is valid for every ' S. Therefore, give T w :::w DL 0, let us say T f with f L w :::w, accordig to (7) we have that H (T ) H (f) w :::w DL 0 sice this last space is closed uder di eretiatio because w :::w D 0 L = 8 < : T S0 : T = X 9 f, f L w :::w ; fiite (see [4]). Fially, to show that H w :::w DL 0 is ot icluded i D 0 L, it su ces to otice that the distributio T = DL 0 ad H () = p:v: x ::: p:v: x = DL 0 (see [, Prop. 8]). The followig result is crucial. Propositio 6 Give T w :::w DL 0, the S 0 -covolutio T P (y) coverges to T i w :::w DL 0 as (y)! (0) +. Proof. It will be eough to prove that for every f L w :::w we have P (y) f! f i L w :::w as (y)! (0) +, sice L w :::w is cotiuously icluded i w :::w DL 0 ad the latter space is closed uder di eretiatio. Let us assume rst that f = ' (x ) :::' (x ) with ' C c (R) for = ; :::;. Thus 3 P(y) f f X 4 Y L k' (R ) i k 5 Py ' ' L! 0 (R) = i6= as y ; :::; y! 0 + sice ' L (R), = ; :::;. As a cosequece, whe f is ay fuctio i C c (R) ::: C c (R) we ca easily show that P(y) f f L! 0 as (y)! (R (0)+ ) 3

14 ad sice L (R ) is cotiuously icluded i L w :::w, the assertio is proved for this kid of fuctios. Now, let us cosider ay fuctio f L w :::w ad let " > 0. Sice C c (R) ::: C c (R) is dese i L w :::w, we ca d a fuctio g C c (R) ::: C c (R) such that kf gk L (w :::w ) < ". Hece P (y) f f L (w :::w ) P (y) (f g) L (w :::w ) Sice f (t) Y g (t) 4 = + P(y) g g L (w Y f (t) g (t) 4 R :::w = + P (y) g g L (w Y = f (t) g (t) 4 R ) + " :::w = P y (x t ) ) + " P y w 3 (t ) 5 dt dx + x + P(y) g g L (w :::w ) + ". 3 P y w Y (t ) 5! f (t) g (t) w (t ) as (y)! (0) + ad f g L w :::w 3, if we were able to show that the family Y f (t) g (t) 4 P y w (t ) 5, 0 < y <, = ; :::;, is domiated by = a itegrable fuctio, we would have for y ; :::; y small eough that P(y) f f L (w :::w ) < 4" = ad the Propositio would be proved. 3 Y To show that f (t) g (t) 4 P y w (t ) 5 is cotrolled by a itegrable fuctio, we proceed as follows. Fix = ; :::; ad write d (x ) = P y (x ) dx, (u) = u =, u 0. We observe that (R) = ad is a = = 3 5 dt 4

15 cocave fuctio. Now, usig Jese s iequality we obtai for 0 < y < Thus, f (t) P y w (t ) = Y g (t) 4 = + (t x ) d = (x )! = + (t x ) d (x )! + (t x ) d (x ) = = C P (t x ) P y (x ) dx = C P +y (t ) = C + t =. 3 P y w Y (t ) 5 C f (t) g (t) ad the right had side of (9) is a itegrable fuctio o R. This completes the proof. = + t = (9) Lemma 7 H (T ) P (y) = T Q (y) for every T w :::w D 0 L ad (y) > (0). Proof. Let ' S, the takig ito accout that H (T ) w :::w DL 0 w :::w P (y) ' B c we have that ad H (T ) P (y) ; ' S 0 ;S = H (T ) P(y) ' ; D 0 L ;Bc = H (T ) ; P (y) ' w :::w D 0 L ;w :::w B c. (0) O the other had, sice w :::w Q (y) ' B c T Q (y) ; ' S 0 ;S = T Q (y) ' ; D 0 L ;Bc = T; Q (y) ' w :::w D 0 L ;w :::w B c = T; ( ) H P (y) ' w :::w D 0 L ;w = T; ( ) p:v: ::: p:v: x x :::w B c P (y) ' w :::w D 0 L ;w :::w B c = H (T ) ; P (y) ' w :::w D 0 L ;w :::w B c. () From (0) ad () we obtai that H (T ) P (y) = T Q (y). As a immediate cosequece of the previous results we ca obtai: 5

16 Theorem 8 Give T w :::w D 0 L, the S 0 -covolutio T Q (y) coverges to H (T ) i w :::w D 0 L as (y)! (0) +. Proof. Sice H (T ) w :::w D 0 L, accordig to Propositio 6 H (T ) P (y)! H (T ) i w :::w D 0 L as (y)! (0) +. Now, by Lemma 7 we get the desired coclusio. Refereces [] Alvarez, J. ad Carto-Lebru, C., 999, Optimal spaces for the S 0 - covolutio with the Marcel Riesz kerels ad the -dimesioal Hilbert kerel, I: W.O. Bray ad µc.v. Staoević (Ed.) Aalysis of Divergece: Cotrol ad Maagemet of Diverget Processes (Bosto: Birkhäuser), [] Alvarez, J. ad Guzmá-Partida, M., 00, The S 0 -covolutio with sigular kerels i the euclidea case ad the product domai case. Joural of Mathematical Aalysis ad Applicatios, 70, [3] Alvarez, J., Guzmá-Partida, M. ad Skórik, U., 003, S 0 -covolvability with the Poisso kerel i the Euclidea case ad the product domai case, Studia Mathematica, 56 (), [4] Alvarez, J., Guzmá-Partida, M. ad Pérez-Esteva, S., Harmoic extesios of distributios, To appear i Mathematische Nachrichte. [5] Alvarez, J., Guzmá-Partida, M. ad Pérez-Esteva, S., 006, N-harmoic extesios of weighted itegrable distributios, Joural of Fuctio Spaces ad Applicatios, 4 (3), [6] Barros-Neto, J., 973, A itroductio to the theory of distributios (New York: Marcel Dekker). [7] Dierolf, P. ad Voigt, J., 978, Covolutio ad S 0 -covolutio of distributios, Collectaea Mathematica, 9, [8] Hirata, Y. ad Ogata, H., 958, O the exchage formula for distributios, Joural of Scieces of the Hiroshima Uiversity, Series A,, [9] Padey, J.N. ad Sigh, O.P., 994, Characterizatio of fuctios with Fourier trasform supported o orthats, Joural of Mathematical Aalysis ad Applicatios, 85, [0] Schwartz, L., 96, Causalité et aalyticité, Aais da Academia Brasileira de Ciêcias, 34, 3-. 6

17 [] Schwartz, L., 966, Théorie des distributios (Paris: Herma). [] Shiraishi, R., 959, O the de itio of covolutios for distributios, Joural of Scieces of the Hiroshima Uiversity, Series A, 3, 9-3. [3] Sigh, O.P. ad Padey, J.N., 990, The -dimesioal Hilbert trasform of distributios, its iversio ad applicatios, Caadia Joural of Mathematics, 4 (),