Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 152043,12 pages http://dx.doi.org/10.1155/2013/152043

Research Article

Interpolation Methods for Stochastic Processes Spaces

E. Nursultanov1 and T. Aubakirov2

1 Lomonosov Moscow State University (Kazakh Branch) and Gumilyov Eurasian National University, Munatpasova 7, Astana 010010, Kazakhstan

2 Autonomous Organization of Education Nazarbayev Intellectual Schools, Astana 010010, Kazakhstan

Correspondence should be addressed to E. Nursultanov; er-nurs@yandex.ru Received 2 September 2013; Revised 4 November 2013; Accepted 18 November 2013 Academic Editor: Henryk Hudzik

Copyright © 2013 E. Nursultanov and T. Aubakirov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The scales of classes of stochastic processes are introduced. New interpolation theorems and boundedness of some transforms of stochastic processes are proved. Interpolation method for generously monotonous processes is entered. Conditions and statements of interpolation theorems concern the fixed stochastic process, which differs from the classical results.

1. Introduction

Interpolation methods of functional spaces are one of the basic tools to get inequalities in parametrical spaces. These methods are widely applied in the theory of stochastic processes (see [1-5] and other).

In this paper classes of stochastic processes are considered, which, in some sense, are analogues of the net spaces which were investigated in [6-8].

Assume that (Q, G, P) is a complete probability space. A family G = [Gn}n>1 of a-algebras Gn such that G1 C ■■■ C Gn C ■■■ C G is called a filtration.

Let G be a filtration, a sequence [Xn}n>1 of random variables Xn measurable function with respect to the o-algebra G„. Then we say that the set X = (Xn, Gn)n^1 is a stochastic process.

Let F = [Fn}n^1 be a system of sets satisfying the condition F C ■ ■■ C Fn C ■■■ C G. We say that a stochastic process X = (Xn, Gn)n^1 is defined on a system F = [Fn}nn if Fn c Gn, n e N. For a stochastic process X, which is defined on a system F = [Fn}n^1, we define the sequence of numbers X(F) = [Xn(F)}n, where

Xk (F)= sup -(—\\xkP(dw). (1)

We call this sequence a majorant of a process X on a system of sets F.

Let us give some examples of a choice of a system of sets P = {FJnzv Fn = in this case the sequence X(F) = [Xn(F)}n is a sequence of averages of a process X = {Xn}; for F„ = &n-i it is a majorant of sequence of conditional averages M(Xn | Gn-1); and for Fn = Gn it is a majorant of a process X = {Xn}. The following cases are interesting: (1) F„ = GnAr, where r = t(w) is the fixed stopping time, and GnAr = {A e G : A n{r = n}e G J; (2) F„ = G„ATn, where Tn = Tn(w), and n e N is the sequence of the stopping times.

We consider the classes of stochastic processes defined on F, which characterize the speed of convergence of sequence [Xn(F)/n}n to zero.

By Npq(F), 0 < p < >x>, 0 < q ^ >x> we denote the set of all stochastic processes X, defined on F for which

wx\K(P) = (lk-1-(qlp)< ™ (2)

if 0 < q < rn and

UXII*WF) = sup k-1/pxk < ™ (3)

if q = to.

Let us denote N"'q (F)

X = (Xn, FXn :(l{2ak^Xk)q) < œ for 0 < q < œ, and

Napm (F) =

X=(Xn, FXzi : sup 2akAXk <œ

(1) E\Xn\ < œ; (2) E(Xn+1 \ Gn) = Xn (a.p.). If instead of property (2) it is assumed that E(Xn+1 \ Gn) ^ Xn(E(Xn+1 \ Gn) ^ Xn), then we say that a process X = (Xn, ©J^ is a (4) submartingale (supermartingale).

Definition 1. Let F = {Fn}n be a fixed system of sets, X = (Xn, ©n)n be a stochastic process defined on F. We say that a process X belongs to the class W(F) if there exists a constant c such that for every k ^m and for every A e Fk

for q = œ, where

= sup - , ... . , .

aSGp(a)>0 (P(A))1-{1/p] Ua

We consider that

\ (X2, -X2k-i )P(da)

xa (<0) =

X[a] (w) if a^l 0 if a< l,

where [a] is the integer part of the number a. In particular

Xm(w) = 0.

A random variable t, which takes values in the set (1,2,..., ot), is called the Markov time of the filtration G = [Gn}nn, if [w : r(w) = n} e Gn for any n e N. The Markov time t, for which t(w) < ot (a.p.) [9], is called the stopping time.

Let X = (Xn, Gn)n>l be a stochastic process and t be the Markov time. By Xr we denote the stopped process Xr =

(Xn^ Gn\ where Xn^A^ = Tm^l XmXr=m(u>) + XnXr^n(^) and xa(w) is the characteristic function of the set A. We assume also that

if a^l

0 = (0, &n)nxi if a < l,

Zk=a h := I,

■k=[a] Uk

The Np'q(F) spaces are spaces of converging stochastic processes, where parameters a,q, and p characterize the speed and the metrics, in which a given process converges.

In this paper we prove a Marcinkiewicz-type interpolation theorem for the introduced space. An interpolation method, essentially related to the properties of the Markov stopping times, is introduced. In the last paragraph the given interpolation method is applied to Besov type space with variable approximation properties. Part of the results were announced in [10].

We write A < B (or A > B) if A ^ cB (or cA ^ B) for some positive constant c independent of appropriate quantities involved in the expressions A and B. Notation A — B means that A< B and A> B.

2. Properties of the Spaces Np>q(F) and Nc^'q(F)

We say [9] that stochastic process (X^ ©n)n^i is a martingale if for every ne N the following conditions hold:

\ XkP(dw) ^clf XmP(dw)

JA I JA

This inequality implies that Xk(F) ^ cXm(F) for every k ^ m. The class W(F) contains martingales, nonnegative submartingales, and nonpositive supermartingales. The stochastic process from W(F) we call generalized monotone.

Lemma 2. Let a > 0, 0 < q ^ œ, and l < p < œ. If X =

), then ther Xm (a.p.).

(Xn, Gn)n>1 e N"'q(G) n W(G), then there exists a random

variable Xm such that Xn

Proof. Let LPtm(Q) be the Marcinkiewicz-Lorentz space and 2]-1 ^ n < 2]. Using the equivalent norm of Lp^(Q.) spaces (see [6]) and measurability of function Xn with respect to o-algebra Gn, we get the following:

\\Xn\\Lp^[0,l]

- sup -I \ XnP(du>)

A€&,P(A)>0(P(A))l/p IJa

Am„P(A)>O (P (A))

^ C sup ■

Ai&2r,P(A)>0 (P(A))

i)^ Ua xnp(d")

¡WIL ^ Pid")

/ sup -—

t=0A^G2k ,P(A)>0 (P(A))1/P

x I \ (X2t

^ cyxyN0I(G).

) P (dw)

Taking into account that №"'q(G)

N0p1(G),for a > 0,

wehave ^Jl^i] ^ ^X^^.

But M\Xn\ ^ cHXnHL ra[0>l], therefore by the Doob theorem ([11]), the process Xn converges almost surely. □

Lemma 3. Let a > 0, 0 < q ^ ot, 1 < p < ot, and X = (Xn, Gn W e W(G). Then

llxlk*(G) - ( Z(2akXk)

sup , ,

AeG,P(A)>0 (P(A))1/p I JA

) P (dw)

Proof. The existence of Xm follows from Lemma 2. Further, we have

Xk = sup

P(A)>0 (P (A))1/pl

/• TO

X (X2m - X2m-\ )P(dw)

JA „tl

to 1 I f

^ V sUP -— I (X2rn X2m-1 )P(dw)

£:kAJim (P(A))1/p'\ Ja 2 2 '

Therefore, using Lemma 8, we obtain

^ (to( to \1\1/cl

k=0\ m=k

£ l^AXl

■ m=0

- ||X|k?(G)-

The reverse inequality follows from the expression:

(2) for e > 0, 0 < q, q1 ^ to,

^(p) ^ Cpmi llXlk+e,,(F)>

»n1;^1(F) ^ ca,p,q,qi p^+^w

AX* = sup TT^J? I f (X# - X*-1 )p(dw)

P(A)>0 (P(A))1/p I J A

^ sup , ,LUp> \ f -Xm)P(du)

P(A)>0 (P(A))1/p I J A

+ sup , * 1/pl I f (Xm -X^ )P(dco) P(A)>0 (P(A))1/p I Ja

The lemma is proved. □

Lemma 4. Let X e W(F). Then

(1) for 0 < q^ q1 ^ to,

yxyNMl(F) ||X||Nm(F), (16)

\\x\\n?" (F) W^WN^F)' (17)

(18) (19)

where cpqqi, capqqi > 0 depend only on the indicated parameters.

Proof. Let us prove inequalities (16), (18). The proof of inequalities (17), (19) is similar. Let e > 0. By Minkowski's

inequality and by the generalized monotonicity of a process X = (Xn, Gn)n^1 we get the following:

to /to

,k=l \r=k

< ( Vk^H Vr-q(e+(1/p)yiXq

< ( Vr-q(e+(l/p)yixq( y^1-1

q/%\ 1/t

<(Yr-(^-1X) = M

To prove the second statement it is enough to show that y^lNp (f) ^ cH^k (f) andapplythe first statement. Since p1 < p, we have the following:

TO \ 1/(ii

^=(l^x1

ITO \1/«1 (21)

^ sup k-1/Pxk(lk(qi/p)-(qi/pi)-1 k \k=i

= CllXlk,ra (F)'

Remark 5. Properties of the Npq(F) spaces given in Lemma 4 show that the second parameter q is weakwith respect to the first p. These properties of the spaces are important in the interpolation.

Lemma 6. Let 0 < p < to, a > 1. If X e W(F), then for 0 < q < to

llXlk^) - ( l(*k/PXa>)

and for q = to

llXk ^(F) - sup a k/PXa* ■

Proof. Using the generalized monotonicity of a process X, we have the following:

llxK,F) = (ik-(q/ph1 n)

= (YY r^-1^

\k=0 l=ak ,

' *+i 1 \ 1/<i m a -1 , \

I«"^ I 1 )

^=0 l=a* 1 )

One can prove the reverse estimate in a similar way. □

Lemma 7 (Holder inequality). Let 0 < pl,p2,q < ot, 0 < t,si,s2 ^ ot and (1/q) = (1/pi) + (1/p2), (1/t) = (1/si) + (1/s2). If stochastic processes X = (Xn, Gn) e Np s (F) and Y = (Yn, Gn) e NpisSi(G), then XY = (XnYn, Gj'e' Nq/F) and

HXYIk,(P) ^hxhNm wlim^ (G). (25)

Proof. Since Yn is measurable with respect to an algebra Gn, we have X^Yk(F) ^ X~k(F) Tk(G) and hence

= ( I{k-1/PXkYkf1

I(k-1/PlX-k (F))

»(I^2 Yk (G))21

= "^k, (F)WYWNp2,S2 (G)'

We will need the following Hardy-type inequalities.

Lemma 8. Let s ^ 1, v > 0, a > 0, ß > 0, and y > 0; then for a nonnegative sequence a = {a^k the following inequalities hold:

'm /(yk) \S\1/S /m \1/s

Ik—Hll^aA ) ^Ca,Jlk(^)s-1 aSk

,k=1 \'l=1

f m ( m \S\l/S (m \1/S

Ik-1! It-1 A ) ^Ca,jlk(^)s-1ask

,k=1 \l=(yk) J J \k=1

S 1/ S 1/ S

m y k m 1/ S

I(2-ak I2ß"

,k=0\ m=0

^( I(2(^\y

S 1/ S 1/ S

I(2ak I2ßmc,

^ k=0\ m=yk

C.ßJ I(2(ß+a)\y

3. Interpolation Method for Stochastic Processes

Let T = [Tn}jj=l be a transform that transforms a stochastic process X, which is defined on the system F = {F}^, to the stochastic process T(X) = [Tn(X), ^n}jj=l, which is defined on the system R = {R} . We say that the transform T is quasilinear if there exists a constant C > 0 such that for any n e N the following inequality holds almost surely:

\Tn (X)-Tn (7)1 ^C\Tn (X-Y)|.

It is known ([9]) that if a process X = (Xn, Gn)n>l is a martingale (submartingale), then the process XT = (XnM, Gn)n^l is also a martingale (submartingale).

Denote X*n(w) = maxKkin|Xk(«)| and X* =

(K, GnW.

The transforms Xr and X* of the stochastic process X are examples of quasilinear transforms.

Let A = (A0(F),Al(F)) be a pair of quasinormed own subspaces of linear Hausdorff stochastic processes spaces N(F), which is defined on a probability space (Q, F, P) with a filtration F = [Fn}n^l. Obviously, this pair is compatible pair and hence the scale of interpolation spaces is defined with respect to the real method ([12]).

Moreover, let for 0 < 0 < 1, 0 < q < ot

"uoaiv,

(Ao,^ = {xe N (F):

^ (t-9 K(t,X)f f <m}

and for q = >x>

(A0^1)^ ={X6 N (f):\\x\\{aoai v

= sup t-9K(t,X) < ml,

0<t<m J

K(t,X;Ao,Ai) = xf (||Xo||A0 + ¿INL,) (31)

is the Peetre functional.

Let R = [Tk(w)}jj=l be a sequence of stopping times with respect to a filtration F and A(F) = (A0(F),Al(F)) be a pair of quasinormed own subspaces N(F). Let X e N(F) and t e (0, ot). We define the following:

KR (t,X) = K(t,X;A0,A1,R)

T6RU{0)

■WA )•

Here the infimum is taken over all stopping times from R. Moreover for 0 < q < ot

(A o.Af^ = {lf N (F):

\q(A0,Ai)e,q jdt

(t-eKR (t,X))^<m\

^m II ^

and for q = to

(*0.A1 )ITO = N (F) : |X|(Ao^I)9jK

= sup t KR (t, X) < to

0<t<TO

Theorem 9. Let (A0(F),A 1(F)), (B0(O),B1(O)) be two compatible pairs of stochastic processes and let R = {t(o>)} be some fixed family of Markov times with respect to a filtration F. IfT is a quasilinear map for stochastic processes X = (Xn, Fn)n»1 and

\\T(X-r)\\Bo < Mo\\X-r HKX')^ \\Xr\\Ai

for all stopping times t e R, then

IIIWUb^ ^ CMl-eMiillXllAL, (36)

lT(X)l,

f Kf -INk ))qdif f (WT(X)-T(X

+I№T)IU)) j

+I№1Ibi ))qfUq

CM0 (T (t-9 inf (||X - XT

^Jo V T6RU{0^11

9 dt\ 1q

= CMyM'jxtA.

The theorem is proved.

Lemma 10. Let a > 1 and R = be stopping times. Then

for 0 < q < to

(ao,aI)ö

V (a9nKR(an,X)y

where the constant C is from the definition of quasilinearity of the operator T.

Proof. Consider the following:

\n=-iy:)

l|X|l(A„^i)Jro - suP a-6"KR (a"' X) -

yXy(A„,Ai),ra - supa-9"K(an,X).

The proof is similar to the proof of the Lemma 4.

4. Interpolation Properties of the Spaces NM(F)

Theorem 11. Let 1 < p0 < p1 < to, 1 < q0,q1tq < to, 0 < e < 1, (1/p) = ((1- 6)/p0) + (6/P1), and R = {k}kcN be the stopping times. Then for any stochastic process X =

(Xn, Fn)n>1,

yXyNM(F) ^ ^^(N^o (F),NM1 (F))e,, (39)

where the constant c depends only on parameters pt, qt, 6, i = 0,1.

IfX=(Xn, Fn)n>A e W(F), then

p^n^o(f),nm1 (p))lq ^ c||x||nm(f)>

where the constant c also depends only on parametres pt, qt, 6, i = 0,1.

Proof. Let X = Y + Z be any representation of a process X, where Y e Npo^ (F), Z e Np (F). To prove the first statement of the theorem, we use the following inequality:

~v <Y +Z

^m < 1 m + Zm'

For any a > 1 we have the following:

an/pXn „

^ a-(n/p)+(n/Po) (a-n/Po Ya„ + a-(n/Po)+(n/Pi)a-n/Pi ^ ^ a-(n/p)+(n/po)

x (sup a-n/poYa„ + a-(n/po)+(n/pi)sup a-n/piZa„

= a9((1/po)-(1/pi ))n

n-((1/po)-(1/pi))ny

By putting a = 2P„Pi/(Pi-P„\ we get an/pXn < 2nBK(2-n, X). Therefore, using (22) and Lemma 8, we have the following:

- ( V (a-n/P*nT

^ ( V (2n9K(2-n,X)f

\n=-TO

= ( V (2-n9K(2n,X)f

lffll(N,

po,™ (F).Npl^(F))e„'

Let us prove the second statement of the theorem. Let X = (Xn, Fn)n>1 e W(F). By using Lemmas 8 and 3 we have the following:

yXy(N (F)N (F))n (1 PoAo(r)'1 Pirn (r))e,a

V (29nKR (2-n,X))C

= ( Y (29nn(II«

+2-n\\XK

1/q (43)

V (29nnd"

11NPO,1

+2-n\\XK

11np,.i

Let k = [2ny], then taking into account that X2* = 0, n < -1, we obtain the following:

V (29n (||x-X:

+ 2-H|X2"1

n„„, II IIn„,

= (V(29n (\\X-Xrr I +2-n\\x

r2"r\\

= (Y(29n (|X-X2"'|

Mlv V2-9nA

+ 2-n\\X

Thus, we obtain the following:

(NPom (F)'NMi (PK

V(29n (\\X-Xrr | + 2-n|x:

-n\\v2nr\\

+ llXllqN.

Further, we have the following:

11NPo,i

= Vk-1/Po(x - X2"r)k - V2-k/Po (X - X2"r)2

= V 2-k/po (X - X2"r)2k ^ (1 + C) V 2-k/p0X2k,

= Vk-1(1/Pi)Xf - V2-k/PiX

k=1 ny

2"V 2k

V2-k/PiX2k +X2„y Y 2-k/p,

C ( Y2-k/piX2, + 2ny((1/Po)-(1/Pi)) Y 2-k/poX2,

By using Lemma 8 for (1/y) = (1/p0)-(1/p1),(46), and (47), we have the following:

Y(29n\\X-X2"rl1

"NP0,1

TO t TO

q\ 1/q

Y(29n Y2-k/p0X2

y n=0\ k=ny

< ( Y(2-k/PX2*)q ) - 1X1

m ,, ,,q y2-(1-d)nq\\x2"V\\q

rn / ny

Z2-nq(1-e) ( Z2-k/plX2,

, n=0 V k=0

+2»y((1/po)-(1/pi)) Z 2-k/paX2k

q 1/ q

< ( Z (2 k/pX2k )q

\\xKq (p)-

By applying Minkowski's inequality to (45) and using Lemma 4, estimates (48), we obtain

(49) □

Corollary 12. Let 0 < p0 < p1 < œ, 0 < q0 < q1 < œ, 0 < 8<l, l$s$œ, (l/q) = ((l - 0)/qo) + (d/q^, (Hp) = ((l-d)/p0) + (d/p1 ), X e W(F), and T = {Tn}™=1 be a quasilinear transform. If for any k e N U {0} the following conditions hold:

\\T(X-Xk)\\

$ M0 X-Xk

lNP«,i(Fy

\T(X)\Nps(R) ^ Cp>s\X\Nps(F), X 6 NPS (F) nW(F)

for any p and s such that p 6 (a,b), 1 ^ s ^ m.

5. Boundedness of Some Operators in Class Np>q (F)

Let Y = (Yn, Gn)n>0 be a stochastic sequence and V = (Vn, Gn-1) be a predicted sequence (G^ = G0). A stochastic sequence V-Y = ((V ■ Y)n, Gn) such that

(V-Y)n = V0Y0 + ZVAYt,

where AYt = Yt -Yt-l is called the transform of Y with respect to V.IfY isamartingalethenwesay that V-Y is the martingale transform.

Theorem 15. Let 0 < q < p < ot, 1 ^ t ^ ot and 1/r = 1/q -1/p. Let V-Y bea martingale transform ofa martingale Y by predicted sequence V = (Vn, Fn-l). If

kra(G) + \\(nAVn)\\Nrm(G) $ B (57)

HV-YHn^G) ^HYIIN^(G), (58)

(50) where a constant c depends only on parametres p, q, and t.

Proof. Let V-Y be a martingale transform of a martingale Y by predicted sequence V = (Vn, Fn-l); that is,

IIT(X)Hn,№) $ cm^M'I

»N , S(F),

where C > 0 depends only on p0, p1, q0, q1, and d.

Taking into account that the measurable function may be considered as a martingale, by corollary we may receive Marcinkiewicz-Calderon interpolation theorem (see [13]).

Corollary 13 (Marcinkiewicz-Calderon theorem). Let 0 < p0 < pl < ot, 0 < q0 < ql ^ ot, d e (0,1), 1/p = (1 - 9)/p0 + 9/pl, and 1/q = (1 - 9)/q0 + 9/ql. IfT is a quasilinear map and

T:L pi1 (D, v)-^LqtJX with the norm M{, i = 0, l,

T : LpiT (D, v) Lq,r (Q, , ||r|| ^ M^M9,. (53)

Corollary 14. Let T = [Tn}jj=l be a quasilinear transform such that for any p e (a,b) and for any X e Npl(F) n W(F) the following weak inequality holds:

ivwiin^r) $cpi

in„1(F)-

(V ■ Y)n =ZVkAYk,

where Y0 = 0, AYk = Yk-Yk-1.ByAbel'stransform (V ■ Y)n = Xl-1 AVkYk + VnYn, we get the following:

l|v^y|l (g) = sup n-1iq(v^y)n

$ sup n-1/q ( ZAVkYk + VnYn

neN \k=1

Taking into account that AVk,Yk are measurable functions with respect to the algebra Gk, we have AVkYk ^

AVk Tk,VX ^Wn and

sup nllq(V~Y)n ^ sup Z (k-llqkAVk) (k-(l/p)-%)

+ n-1/rVnn-1/p Yn

$ supk1-(1/rm -IIYIIn^g) (61)

+ rnHNr,ra(G)^ UN^G)

$ BUYUnr1(g)-

Nqo,^(R)

Hence the weak inequality is proved as follows:

If 2s Zn<2s+1, then

for 1 < q < p < to.

Let 0 < q < p < to, (1/r) = (1/q) - (1/p), 0 < q0 < q < q1 < to and 0 < p0 < p < p1 < to. Let a pair of numbers (p0, p1) and (q0, q1) satisfy the following condition:

1 1 = 1 1 = 1 1

% P0 h P1 1 P

Then from that is proved above it follows that

ll^l^o < Blin^o, W^k^o <BWYWNpul(G)

n-1/P sup XTP(dw)

A£&„,P(A)>0P(A) Ua

= n 1/p sup

AeG„,P(A)>0

Y1 Janwr

XrP(dw)

1 s 2'+1-1 I ,

^2-S/P «up —Y Y I XrP(du)

Ae&„ + , P (A)t-r, —it \JAnWr

^2-s/p sup

t=0 r=2' s 2t+i — 1

1 S 2 —1

-—Y Y P(Anwr)Xr

' t=0 r=2'

2 —1

^^ sup P7A-)lX2'+i — 1 YP(A^Wr)

AeG2s+i P (A) t=0

s 2 —1

= 2—s/p sup ^»i—1 YP(Wr \A)

AeG2s+i_lt=0 r=2t

for 0 < p < q < to.

Taking into account that for any stopping time k e N processes Yk and Y-Yk are martingales, it is possible to apply Theorem 9. Then

\\v-y\in_ ^CB\\Y\\n„

1 = 1-9 + 9

11-9 9 — =-(66)

% % h Pe P0 P1

Note that there exists 6 e (0,1) such that (1/pe) = (1/p). Then it follows from (63) that (1/qe) = (1/q). □

Theorem 16. Let 0 < p < to, 1 < q < to and X = (Xn, Gn)n>1 e W(G) and t(w) be the Markov time and let XT = (XnAr, Gn)n^1 be a stopped process. Then

Z Y2—t/pX2t+i—1 Z 21/pY2—m)/pX2t+i

t=0 t=0

= 21/pY2—k/pX2* Z c\\x\\n (g)-

Now, using Corollary 12 we get the statement of the Theorem 16. □

Corollary 17. Let 0 < p < to, 1 ^ q ^ to and a process X = (Xn, Gn)n>1 be a nonnegative submartingale. Then the process X* = (X*n, Gn)n>1 is also submartingale and

IKIkjG) ~ \\XKq(G).

Proof. It follows from Theorem 16 that

\\X*L (G) ^c\\X\\NM(Gy

m»«* z c\\x\\nm(g).

(67) The reverse inequality is trivial.

Proof. Denote

Wr = [w : t (w) = r} 6 Gr, r = I,n-1, Wn = [w:t (w) Zn} 6 Gn.

6. Interpolation Properties of the Space Np'q(G), the Embedding Theorems

Theorem 18. Let 1 < p,q,q0,q1 < to, a0 < a1, 0 < 6 < (68) 1, a = (1 - 6)a0 + 6a1, and R = {r}reN. Then

Let us sllOW that PIn^(G) Z C\\X\\Ni,ii (G).

(N^0 (G), Napi'qi (G))e = Np'q (G). (72)

Proof. Using Lemmas 10 and 4 we have the following:

UXU(№'0'îo N"1^1 )§

\Np >Np )0,q

Z (29nKR (2-n,X)f

i œ / \q\l1i

= ( Z (2enn (H^IIn- +2-n\\Xr\\Nri )) )

\n=-œ ^ '

/ œ / \q\1/q

<( Z{29nn (|IX-X1N;0'1 +2-n\\xr\\Nr )) ) .

\n=-œ "" /

Putting r = [2ny], y = a1 - a0 we get the following:

UX^N-o«) Nxi'qi )§

\Np >Np )0,q

Z(29n (\\X-Xrr IL, + 2" №

Wn;1,1

+uxuqr„o,i ) ■

It follows from the definition that (AX22 )k = AXk for k = 0,1...,r-1; (AX2r)k = 0 for k ^ r, A(X-X2r)k = 0 for k = 0,1...,r-1; A(X-X2r )k = AXk for k^ r; therefore,

\\X21 , = T2akAXk, 11 ll<(G) k=0

IIX-X2! , =Y2akAX

IN1 (G)

Substituting these equalities in (74) and applying Lemmas 4 and 8, we get the following:

m(Nro»o N"iqi )%

œ / œ

Z28nq( Z 2"okAXk + 2-n Z 2"ikAXk

/ œ \1/q

< ( Z(2(ao+ey)nAxk)q )

\n=0 )

Z(2("i-(1-e^nÄXk )q) + ||X||n'

= 3iixiin*

For the proof of reverse estimate we use the fact that for any and k the following equality holds:

AXk = A (X - X2')k + (AX22)k-

Then we have the following:

= ( Z(2akAxk)q

Z ¡2ak-aok (sup 2aonA(X-X2' )n

1 \ n»0

+2"ok-"ik ■ sup 2ain(AXr)n)))

= (Z i2'

uk-a-ok ^2r

+2"o k-"i k ^2r I

WN1™

Substituting 2"1 a<0 = a and using Lemma 4, we have the following:

IIXINq $ ( Z(a ■K(a-k,X,Napo'œ,Napi'œ))q

$U X||(n*O,to N^i'™)* ,

\Np Np )0,q

since (Np0'q", Np1'^ )*q ^ {Nap0'm,Napi'm)lq, the proof is complete. □

Theorem 19. Let 1 < r ^ p < œ, 1 ^ q ^ œ and a = (l/r)-(\/p) and let the filtration G = {©n}n>1 besuch that for every k = 1,2... and for all A e ©k the following condition holds:

where the constant C > 0 does not depend on k. Then

Nuq (G)^NM (G)

no,™

Proof. Let us show that

K'1 (G) ^ Npm (G).

N^G) ~ suP2 k/PX2k k

sup2 k/p sup —I [ X2kP(dw) k Am.iP(A) IJa

sup2 k/p sup —-—-

/ aJtP(Ä)

Y \ (X2m -Xim-1 )P(dw)

- V 1-m/p 1 1 ^ sup Y 2 r sup -—r-~r

kto AJ^ (P (A))1/r (P(A))1/r

\ (X2m -X2rn-! )P(dw) IJa

According to the condition (80), for A e F2m we have that P(A) > (C/2m). Therefore for a = (1/r) - (1/p) we get the following:

IIxIIn№(g)

^ CsupY2-(m/p)+(m/r)

sup -1-— I \ (X2m - X2m-1 ) P (dw)

(P(A))1/r IV 2 2

AeG2m (P(A)Y" ¡JA IIXIU,!.

Thus, (82) is proved.

Now, let a0 < a1, 1 < p0 < p1 < to, and 6 e (0,1) such that

--+ a0 =--+ a1 = —,

P0 P1 r

a= (l-d)o0 + da1,

- = --L LL

P Po Pi '

Then using interpolation Theorems 18 and 11 we obtain the following:

(N"0'1 (G), N"1'1 (G))e = N"'q (G),

(Npom (G),Npim (G))eq = NM (G).

It follows from (82) that (N^^„'1(G),N^^i'1(G))e (Npom(G),Npim(G))e,r Hence, N?'q(G) -where a = (1/r) - (1/p). The proof is complete. □

NM(G),

7. Spaces with Variable Approximation Properties by Haar System

In this paragraph we consider some applications of the introduced interpolation method to Besov type spaces with variable approximation properties.

Let Q = [0,1] and let F be a a-algebra of Borel subsets of set Q, P a linear Lebesgue measure on F, F = [Fn}n^1 the Haar filtration, and R = [Tk}'j=0 a sequence of stopping times such that for any k ^ 0 the following conditions hold: To = 0,Tk + 1 ^ Tk+1 (a.p.) and

lim Tk (w) = œ (a.p.).

For a function f(x) e L[0,1] we denote by {ck(f)}k>1 the Fourier coefficients by Haar functions system {Hk(x)}k>1 ([14]). For the given stopping time Tk(w) we denote

S(f,Tk)(w) =

Y Cm (f)Hm (w) if kzl

0 if k = 0,

which we call the Fourier-Haar partial sum of a function f, corresponding to the Markov time Tk.

Let 1 < p < to, 0 < q < to, a e R. By Bap'q [£] we denote the set of functions f e L[0,1],for which

Ê?m = (l2akq\\f-S(f,Tk)\\Lp ) < œ (89)

for 0 < q < œ,

B;nm = sup2" \\f-S(f'Tk)\\Lt < œ,

for q = to.

Conceptually, the introduced spaces are close to spaces with variable smoothness. Here we mention works ofLeopold [15], Cobos and Fernandez [16], and Besov [17-20].

Lemma 20. Let 1 < p < to, f e L^[0,1], andletS(f,r) be the Fourier-Haar partial sum with respect to the Markov time t. Then

llS(A)||, <4f\\P- (91)

Proof. Denote

Ft = (Ae F ■■An{T = n}e Fn for every n^1}. (92)

Let Lpm[0,1] be the Marcinkiewicz-Lorentz space. Using the equivalent norm of Lpm[0,1] spaces (see [6]) and

martingale properties of Fourier-Haar partial sums we get the following:

sup -r-^-~7\\AS(f,r)P(dw)

AeF,P(A)>0 (P(A))1/pl I Ja 1

sup - ,

AeFr,P(A)>0 (P(A))1/p 1

r) P (dw)

AeFr,P(Ä)>0 (P(A))1/pl I Ja

t\\ S(f

7| f f(w)P(dw)

sup -I f f(w)P (dw)

Äe%,P(Ä)>0 (P(A))1/p \JA

IIW0.1]-

Now, applying the interpolation theorem (see [12]), we obtain the statement of the lemma. □

Lemma 21. Let 1 < p < ot, 0 < q ^ ot, a e R, and

^ ( I29qn ( l2«°k\\A(frk)

\n=1 \k=ny

-n V"1 ~aik

ny-1 \q\1/q

+2-n l2«lk\\A(f,rk)\\Lp ) )

q 1/ q

*( I(26n l2a°k\\A(f,rk)\\Lp )

\n=1\ k=ny )

(m / ny-1

I(2-(1-d)n l2«lk\\A(f,rk)

n=1\ k=0

By applying Lemma 8 we get the following:

llaF® [R],B™ [R])lq ^ C\\f\\B;-q [R]'

q 1/ q

Let us prove the reverse embedding. Let f e (Bap0,q0 [£], BPp'qi [^])gq, f = f0 + fl be an arbitrary representation of a function, f0 e BPp°'q° [£] and fl e BPp ,qi [£]. Then

2ak\\A(f,Tk)l

A(frk)= I cr (f)Hr (w). (94)

r=Tt + 1

IIbCTr]

I2«kq\\A(f,rk)\\

Proof is similar to the proof of Lemma 3.

Theorem 22. Let 1 ^ q0, q1, q ^ m, 0 < a0 < a1, 0 < d < 1, and a= (1 - d)a0 + da1. Then

(Bap°q [Ä], Bapiq [Ä]) = Bapq [Ä]. (96)

Proof. By using Lemma 21 we have the following:

bp [R]>Bp

^ c\\f kB^mB^m)^

I2eqnmf (I2P0k\\A(f,rk)

r-1 \q\Vq

+2-n I2Plk\\A(f,rk)\[

2Pk (\\A(fo,rk)l +\\A(f1,rk)\\L

^2(P-P)k (sup2P°r\\A(f0,r,

r> HL»

+2(P°-Pi)ksup2Pir\\A(f1,rr,

( P- P

°)k (life

0|Ib"»'~[r]

(Pq -Pi)k

1|Ib;;i'~[r]

Since the representation f = f0 + fl is arbitrary, we have the following:

2Pk\\A(f,rk)\\Lp

< 2(P-Po)kg (f 2(Po-Pi)k. Ba0'm [ß] bPi'm [_£])

Hence, putting a = 2pi-pq we get the following:

IIb^[r]

^ I(2(Pi-P°)SkKR (f 2(P"-Pi)k))q

= (I(a9kKR (f,ak))q

kB^mB^m)^

^cllfll(B»*> [RIB;1*1 [r])«q

The theorem is proved.

Acknowledgment

This research was partially supported by Ministry of Education and Science of the Republic of Kazakhstan (0112RK02176,

0112RK00608).

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