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8/2/2019 AADM Template

1/18

v l le onl ne t http: pefm th etf rs

. . . x (xxxx) xxx{xxx d 0.2298/AADMxxxxxxxx

So e We hted Besov S aces n Quantu Ca cu us

Ak a NEMRI & Bel ace SELMI

n th s p per su sp ces ofLp re ene us ng q tr nsl t ons Tq;x oper torn erences c lle q Besov sp ces We prov e ch r cter z t on of these

sp ces y the q convolut on

Introduction

M h r nt r r h t v t h f d n th th r nd ppl t n f Q ntC l l . Th br n h f th t nt n t nd n nd f l ppl -t n . F r x pl th - ll d - n l f p l f n t n nd h p r tr

r , ll d - r h v n ppl t n n r th t th r , b n t r ,nt ph , r p th r [2], nd th r r f n nd th t .

Appl t n f th th t n l d p p l t n b l [4], tr n l-[5], nt ll nt r b t ntr l [8], ppr x t n th r [20], nd n n l

n n r n [ 9], n th r . O r nt r t n th p p r t h r t r zht d B v p n Q nt C l l , ll d ht d -B v p . In

th l l th r r n t d n B v p ([3],[ 8],[23],[22]).In th p p r xpr -B v p n t r f nv l t n q ' t th d r nt

nd f th f n t n ' :Th p n b d r b d b n f d r nd r nt l p r t r xq ( [ 3], [ 4], [2 ]).

Thr h t th p p r q- ht w : Rq + ! R+ ll b q- r bl f n -t n, w > 0 . ., nd ll v h r t r z t n f q- ht d q-B v p .L t w b r bl p t v f n t n nd 1 p m nd p

0m

0t nd f r

th nj t xp n nt . W h ll d n t b w q th p f v n f n t n

20 0 M hem c ub ec C c n. 33D05, 33D 5, 33D90.e gh ed, Be v ce , u n um C cu u .

1

NON-ACTIVATEDVERSIONwww.avs4you.com


2/18

2 Akr m N M R Belg cem S M

: Rq ! C h th t

k k p;m ;w;q=n Z

+1

0

k q k

w(x)qxx

om

+ (1 m )

r

k k p; ;w;q= nfn

> 0; k q k w(x) a: :x 2 Rq +

o(m = + )

h r

(1) q (y) = q (y) (y):

P t A q nd A q f r th f ll n l

(2) A q =n' 2 S q 0;

Z +10

(Fq' (t))2 qt

t= 1 f r 2 Rq +

o

nd

(3) A q =n' 2 A q; supp' [0 1]

Z +10

x' (x) qx = 0o

h r S q 0 = f 2 S q;

Z +10

(x) qx = 0g:

L t w ' b n S q 0 nd 1 p m + h ll d n t b w q

th p f th v n f n t n : Rq ! C b l n n L (Rq +qx

(1 + x2)) nd

t f n

k kBp;m

;w;' ;q=

n Z +10

k ' t q k

w(t)

qt

t

om

+ (1

m + )

nd

k kBp; ;w;' ;q= nfn

> 0; k ' t q k k w(x); a: : t 2 Rq +

o(m = + )

h r ' t(x) = t ' (t x) t 2 Rq + nd x 2 R+:

O r bj t v t nd ht ( h h xt nd th t ! t ) h rn ( t ll) t h h r t r z t n f q- ht d B v p :

w q = w q ( th v l nt n r ):

F r th nd nl th q-C ld n' f r l [ 7] nd t l nt r S hur

L . Ob rv th t th q-B v r nd p nd nt f th h f ' n A q:Th nt nt f th p p r r f ll . In S t n 2, ll t b d n -t n nd r lt b t q-h r n n l . In S t n 3, v nd t n b tq- ht nd pr v th nn t n b t n th p w q nd w q:



3/18

Some We ghte Besov Sp ces n u ntum lculus 3

2 Pr liminari s

In ll th l, 0 q 1 nd d pt th n t t n n [7].

A q- h ft d f t r l d n d b

(4) (a; q)0 = 1 (a; q) ==0

(1 aq ); n = 1 2

nd r n r ll

(a ar; q) =

r

=

(a ; q) :(5)

Th b h p r tr r r q-h p r tr r r v n f r r,nt r b

r' s(a ar; b bs; q x) =1X=0

[( 1) qn(n 1

2 ] +s r(a ar; q)

(b bs; q)

x

(q; q):

Th q-d r v t v q f f n t n n n p n nt rv l v n b

(6) q (x) =(x) (qx)

(1 q)xx 6= 0

nd ( q )(0) =0(0) pr v d d

0(0) x t . Th q- h ft p r t r r d n d

b

(7) ( q )(x) = (qx) ( q )(x) = (q x):

Rq = f q k 2 Zg; Rq + = f+q k 2 Zg; eRq + = f+q k 2 Zg [ f0g: Th q-J n nt r l fr 0 t ( r p t v l t + ) d n d b

0

(x) qx = (1 q)a+1X

=0

(aq )q1

0

(x) qx = (1 q)+1X1

(q )q :(8)

R mark 0 0 O erve th t ( ee [ ]) the q-inte r l (8) i Riem nn-tieltje inte r l with re pe t to tep n tion h vin innitely m ny point

o in re e t the point q with the j mp t the pointq ein q : I we ll thi tep n tion q(t) then q(t) = qt:

N t th t f r n 2 Z nd a 2 Rq, h v(9)

1

0

(q x) qx =1

q

1

0

(x) qx0

(q x) qx =1

q

qn

0

(x) qx:



4/18


Th q2- nt r t n b p rt v n f r t bl f n t n nd g b :

(10) (x) q g(x) qx = h (x)g(x)i (qx) q g(x) qx: Th f ll n d n t n f th q- n v n ( [9]) b

(x; q2) = ' (0 q q2; (1 q)2x2) =1X=0

( 1) b (x; q2)(11)

h r , h v p t

b (x; q2) = b (1; q2)x2 = q ( )(1 q)2

(q; q)2x2 :(12)

Th f n t n b nd d nd f r v r x 2 Rq h v

j (x; q2)j1

(q; q2)21:(13)

W r r th t f r 2 C, th f n t n x ! (x; q2) th nl t n f th q-d r nt l t

(14)

8 0 h th t

(34)s

0

w(t)

tqt w(s); a: : s 2 Rq +:

A q- ht w d t b (b q)- ht f th r x t > 0 h th t

(35)+1

s

w(t)

t2qt

w(s)

s; a: : s 2 Rq +:

W l p t W0 ;q th p f (b q)- ht t f q-D n nd t n.

W l th n t t n A q nd A q f r th f ll n l

A q = ' 2 S 0 q;+1

0

(Fq' (t))2 qt

t= 1 or 2 Rq +

o(36)

A q = ' 2 A q; supp' [0 1]+1

0

x' (x) qx = 0o:(37)



8/18


nition 0 0 3 Let" > 0, 0 ndw be q-weight.w i id to be ( "

;q)-weight if there exi t > 0 ch th t:

(38)s

0

t"w(t)qt

t s"w(s); .e. s > 0

w i id to be (b;q) weight if there exi t > 0

(39)+1

s

w(t)

tqt

t

w(s)

s; .e. s > 0:

We write W" ;q = ( ";q) \ (b;q)

Proposition 0 0 3 1: Ifw 2 ( ";q ) thenw 2 ( "0 ;q) for ny"0> ".

2: Ifw 2 (b;q) thenw 2 (b0;q) for ny

0> .

Proposition 0 0 4 1:Ifw(t) = w(t ); thenw 2 (b"; q) if nd only if, w 2 ( "; q ):

2: Ifw 2 W" ;q; thenw(t) n(t" t ); > 0:

L t n v n x pl

ampl 0 0 Ifw(t) = t ; > 0 thenw 2 W" ;q for ny > nd" > :

L mma 0 0 Let 2 S q then we h ve:

(40) (x) =+1

0

q ' t q ' t(x)qt

t; for l l ' 2 A q:

Proof. L t g(x) =+1

0q ' t q ' t(x) qt

tth n

Fqg() =+1

0

+1

0

q ' t q ' t(x)t (x; q2) qx qt

=+1

0

Fq( q ' t q ' t)()t qt

=+1

0

Fq( )()(Fq(' t)())2t qt

= Fq( )()+1

0

(Fq(' )(t))2 qt

t= Fq( )():

Fr th f t th t ' 2 A q nd th f Th r 0.0.2 bt n th r lt. Proposition 0 0 5 Let' 2 A q nd 2 S q then for 2 Rq +

(41) Fq () =+1

0

Fq(' t q ' t q )()qt

t:



9/18


Ind d, n ' 2 A q

1 =+1

0

(Fq' (t))2 qt

t

=+1

0

Fq(' t q ' t)()qt

t

th n fr L 0.0.1, nd th r t r l t n n Pr p t n 0.0.2, n lth t

Fq () =+1

0

Fq (' t q ' t)()Fq ()qt

t

=+1

0

Fq (' t q ' t q )()qt

t

:

R mark 0 0 2 The l t propo ition how th t

" ;q =

"

' t q ' t qqt

tconverge to inS q " ! 0 nd ! :

On th r ll th q-C ld rn' f r l v n n [ 7]:

Th or m 0 0 4 Let 2 L (Rq +q

( + 2)) nd' 2 A q:For 0 " dene

(42) " ;q(x) =

"

(' t q ' t q )(x)qt

t:

Then " ;q converge to inS0q

0" ! 0 nd ! :

T n h th pr l n r t n l t t t q-v r n f S h r l th tll b f l f r r p rp .

L mma 0 0 2 Let 1 p nd1

p+

1

p0 = 1:Let ( ) nd ( 2 2 2)

be two -nite me re p ce nd let K: 2 ! R+ be me r ble f nctionnd write ( ) for

( )(w2) = 1

K(w w2) (w ) (w ):

If there exi t > 0 nd me r ble f nctionhi : i ! R+(i = 1 2) ch th t

1

K(w w2)h0(w ) (w ) h

02 (w2); 2 a:(43)

2

K(w w2)h2(w2) 2(w2) h (w ); a: :(44)

Then dene bo nded oper tor fromL ( ) into L ( 2 2):



10/18


Proof. Fr (43) nd H ld r n l t h v

j ( )(w2)j h2(w2)( 1

K(w w2)h (w )j (w )j (w ))1p :

Appl n (44) nd F b n ' th r t t

k ( )k (( 1 2

K(w w2)h2(w2) 2(w2))h (w )j (w )j (w ))1p

2k k :

3 haract rization ofq-B sov spac s

W b n r t b t bl h n r l t hn l r lt th t n b d t r l tpr p rt b t q-d r n q nd q- nv l t n ' t q :

L mma 0 0 3 For ll jyj 1 we h ve

(45) k yq'k q (1 + q)jyj+1

0

j q y( q y' (x))j qx:

Proposition 0 0 6 Let1 p , % 0 nd' 2 A q:Then there exi t q > 0

ch th t if 2 L (Rq +q

( + 2)) then we h ve th t

(46) k ' t q k q q

1

0

n((x

t) (

t

x) ) k q k q

qx

x

nd

(47) k q k q q

1

0

n((x

t) 1) k ' t q k q

qx

x:

Proof. S n+1

0

' (x) qx = 0 t f ll th t

' t q (y) =+1

0

' t(x) q (y) qx

fr q-M n ' n l t n t

k' t q k q = (

+1

0j

+1

0' t(x) q (y) qxj qy)

1p

+1

0

j' t(x)j(+1

0

j q (y)j qy)1p

qx

+1

0

jxj

tj' (

x

t)jk q k q

qx

x:



11/18


H n , th r l t n (46) f ll fr th tr v l t t

jyj + j' (y)j q f jyj 1

j' (y)j q f jyj 1:

T pr v (47) r ll th t f r 0 "

(48) q " (y) =

"

( q' t) q ' t q (y)qt

t

H n q-M n ' n l t nd q-Y n ' n l t v

k q " k q

"

k q' tk qk' t q k qqt

t

S n

kyq' k q 2k'k q jyj 1

nd L 0.0.3 n h

k q' tk q = kt

q 'k q q n(1x

t):

Th r f r , n th pr v t t r l t n (48), L 0.0.3, L 0.0.2nd pl l t n r nt h (47).

Alth h f r th p rp f th p p r nl p rt l r f th f ll npr p t n ll b d, h ll t t n r l v r n f t th t nd nt r t nn t n r ht.

Proposition 0 0 7 Given 0 " 1 p nd w q-weight, letcon ider

(49) " (s t) =w(s)

w(t)n((

s

t)" (

t

s) ):

If w(s) = 1

p0

(s)1p (s ) for ome p ir of q-weight 2 W" ;q then there

exi t q > 0 ndg : Rq + ! R+ q-me r ble ch th t

(50)

1

0 " (s t)g

0(s)

qs

s qg

0(t)

nd

(51)1

0

" (s t)g (t)qt

t qg (s):



12/18


Proof. L t t g(t) = 0(t)

0(t ):Th n g

0(s) = (s)=w(s) nd g (t) =

w(t)(t ).

Th r f r ,

+1

0

" (s t)g0(s)

qs

s=

1

w(s)

+1

0

(s) n((s

t)" (

t

s) )

qs

s

=1

t"w(t)

t

0

s"(s)qs

s+

t

w(t)

+1

t

(s)

sqs

s

q(t)

w(t)= qg

0(t):

On th th r h nd,

+1

0 " (s t)g (t)

qt

t = w(s)

+1

0(t ) n((

s

t )" (

t

s ) )

qt

t

=w(s)

s

s

0

t(t )qt

t+ s"w(s)

+1

s

(t )

t"qt

t

=w(s)

s

1

s 1(t)

tqt

t+ s"w(s)

s 1

0

t"(t)qt

t

q(s )w(s) = qg (s):

W n d th f ll n L th t ll ft r n v r l f th r n npr f .

L mma 0 0 4 For ll x y 2 Rq we h ve

(52) q y(1

1 + x2)

1

1 + x2(1 + q

1 q

12 y):

Proof. L t x y 2 Rq , fr (27) pl r rr n n n 2 l d

(53) q (1

1 + x2)

1 + q1 q

11 + x2

:

Appl n n (11), n d d th r lt. Proposition 0 0 8 Let1 p nd let be q-me r ble f nction.

If k q k q2 L (Rq +qx

(1 + x2)) then 2 L (Rq +

qx

(1 + x2)):

Proof. L t 2 L0(Rq + qx) h th t > 0 . . Th n q-H ld r' n l t nd

q-F b n ' th r v

+1

0

+10

j q (y) (y)j

(1 + x2)qx

(y) qy :



13/18


Th r f r ,

+1

0

j q (y) (y)j(1 + x2)

qx f r a: : y 2 Rq +:

S n x ! (1 + x2) 2 L (Rq + qx) t h n

+1

0

j q (y)j

(1 + x2)qx f r a: : y 2 Rq +:

F n ll , fr L 0.0.4 nd r l t n (13) th r x t q = (q q2)2 > 0 h

th t f r ll y 2 Rq +, q y(1

(1 + x2)) q

1

(1 + x2)th n n h

+1

0

j q (y)j

(1 + x2)qx =

+1

0

j (x)j q y(1

(1 + x2)) qx

q

+1

0

j (x)j

(1 + x2)qx :

N t rt th n r lt f th p p r th th q = h h l

f ll fr Pr p t n 0.0.6.

Th or m 0 0 5 Let 1 p ' 2 A q ndw 2 W0 ;q:Then

(54) 1 w q =1

w q with e iv lent eminorm:

Proof. L t 2 1 w q th n n h

+1

0

k q k q

(1 + x2)qx q

1

0

w(x)(1 + x2)

qx

q

0

w(x)qx

x+

1

w(x)qx

x2

h t b n d th Pr p t n 0.0.8 v

1

0

j (x)j

(1 + x2)qx :

L t pr v th t k' t q k q qw(t):Fr (46) n Pr p t n 0.0.6 f r % = 1n h

k' t q k q q1t

t

0

k q k q qx + t1

t

k q k qqx

x2

q t

0

x

tw(x)

qx

x+ t

1

t

w(x)qx

x2

qw(t):



14/18


C nv r l f 2 1 w q th n fr (47) n h

k q k q q0

k' t q k qqtt

+ x1

k' t q k qqtt2

q

0

w(t)

tqt + x

w(t)

t2qt

qw(x):

W pr v n th n Th r n th q = 1:

Th or m 0 0 6 Let 1 p ' 2 A q nd w 2 W0 ;q ch th t (t) =w (t ):Then

(55) w q = w q with e iv lent eminorm:

Proof. A 2 w q . L t r t pr v th t1

0

j (x)j

(1 + x2)qx :

Fr Pr p t n 0.0.4 h v

1

xw(x)q

1

xn(1

1

x) q

1

xn(x

1

x)

q

(1 + x2):

H n1

0

k q (x)k q

(1 + x2)qx q

1

0

k q k q

w(x)qx

x

nd ppl Pr p t n 0.0.8 n.

W ll pr v th t k kBp;1 ;w;' ;q qk k p;1 ;w;q:

U n (46) n Pr p t n 0.0.6 th % = 1 h v

1

0

k' t q k q

w(t)qt q

1

0

10

n(x

t

t

x)k q k q

w(t)qx

x

qtt

= q

1

0

k q k q 1

0

n(x

t

t

x)(t )

qt

t

qxx

= q1

0

k q k q

0

t(t )

x

qt

t+

1 x(t )

t

qt

t

qxx

q

1

0k q k q

1x

1

1(t) qt

t2+

1

0

(t) qtt

qxx

q

1

0

k q k q(x )qx

x

q

1

0

k q k q

w(x)qx

x:



15/18


L t 2 w q fr (47) n Pr p t n 0.0.6 nd q-F b n ' th rt

1

0

k q k q

w(x)qx

x q

1

0

k' t q k q 1

0

(x ) n(1x

t)

qx

x

qtt

= q1

0

k' t q k q 1

0

(s) n(11

st)

qs

s

qtt

= q1

0

k' t q k q t 1

0

(s)

s+

1

t 1(s)

s2qs qt

t

q

1

0

k' t q k q(t )qt

t

= q

1

0

k' t q k q

w(t)qt:

Th or m 0 0 7 Let 1 p 1 m ' 2 A q nd w be q-weight

ch th tw(t) =

1

m0 (t)

1m (t )

for ome p ir ofq-weight 2 W0 ;q:Then

(56) w q = w q with e iv lent eminorm:

Proof. A th t 2 w q:W ll r t h th t

1

0

j (x)j

(1 + x2) qx :

W d n t

(x) =x

(1 + x2)w(x)

nd r th pt n 2 W0 ;q n h 2 L0(Rq +

q ):Ind d,

1

0

0(t)

qt

t

1

0

(t)m

0m (t )

t0

(1 + t20)

qt

t:

U n Pr p t n 0.0.4 h v (s) q n(1 s):Th r f r

1

0

0(t)

qt

t

1

0

(t) x(1 t0

)t

0

(1 + t20)

qt

t

q

0

(t)qt

t+

1

0

(t)

t

qt

t

:



16/18


17/18


h r

(x t) = w(t)w(x)

n(1 xt

):

N t

( ) = (Rq +qx

x)

nd

( 2 2) = (Rq +qx

x):

C b n n n Pr p t n 0.0.7 nd L 0.0.2 t t th b nd dn f

fr L (Rq + qxx

) nt L (Rq + qtt

):Th r f r ,

k k p;m ;w;q qk (k' t q k q

w(t))k

m(Rq;+dqx

x)

qkk' t q k q

w(t)k

m(Rq;+dqt

t)

qk kBp;m ;w;' ;q:

REFERENCES

A reu unct ons q orthogon l w th respect to the r own zeros Proc AmerM th Soc 134( 6) 6 5 7

An rews q Ser es: the r evelopment n n lys s num er theory com ntor cs phys cs n computer lge r BMS Ser es Amer M th Soc Prov enceR 66( 86) 3 4

3 O Besov On f m ly of funct on sp ces n connect on w th em e ngs n extent ons Tru y M th nst Steklov 60 ( 66) 4 8

4 M B ohr M n n Zh ng Per o c ty of sc l r yn m c equ ton n ppl c t onto popul t on mo e ls M th An l ppl 330 ( 7)

5 M Bohr T u son uler typ e oun ry v lue pro lems n qu ntum c lculus ntern t on l ourn l of Appl e M them t cs n St t st cs 9( 7) 3

6 h ou l k mel n A touh Pos t v ty of q even tr nsl t on n nequ l tyn q our er n lys s PAM nequ l Pure Appl M th 171( 6) 4

7 sper n M R hm n B s c ypergeometr c ser es ncyclope of m them t csn ts ppl c t ons 35 m r ge un vers ty press



18/18


8 r v gne v s n R M rks ow eterm n st c must re l t me controllere Procee ngs of 5 RS ntern t on l onference on ntell gent Ro ots n

Systems Al ert Aug 6 ( 5) 3856 386

A touh n Bouzeour q cos ne our er Tr nsform n q e t qu t on Rm nju n n press

A touh h ou n l K mel nequ l t es n q our er n lys snequ l Pure Appl M th 171( 6) 4

A touh M mz n Bouze our The q J Bessel funct on Approx Theory115( ) 4 6

A touh n A Nemr str ut on An onvolut on Pro uct n u ntum lculus Afr spor M th 7( 8) 3 58

3 T M lett psch tz sp ces of funct ons on the c rcle n the sc M th An l

n ppl 39( 7 ) 5 58

4 T M lett Temp er tures Bessel p otent ls n psch tz sp ces Pro c on onM th Soc 20( 7 ) 74 768

5 ckson on q e n te ntegr ls u rt Pure Appl M th 41( ) 3 3

6 A Nemr n B Selm So olev Type Sp ces n u ntum lculus M th An lAppl 359( ) 588 6

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8 Peetre New Thoughts on Besov Sp ces uke Un v M th Ser es N ( 76)

S S ny l Stoch st c yn m c qu t on Ph ssert t on M ssour Un vers ty of Sc ence n Technology ( 8)

Sheng M g en erson n v s An xplo r t on of om ne yn m c er v t ves on t me Sc les n The r Appl c t ons Nonl ne r An lys s: Re lWorl Appl c t ons 7( 6) 3 5 4 3

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