# Interpolation Methods for Stochastic Processes Spaces

Authors: E. Nursultanov; T. Aubakirov
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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.


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## Body

## 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(Ω,𝔊,P) is a complete probability space. A family G={𝔊n}n⩾1 of σ-algebras 𝔊n such that 𝔊1⊆⋯⊆𝔊n⊆⋯⊆𝔊 is called a filtration.LetG be a filtration, a sequence {Xn}n⩾1 of random variables Xn measurable function with respect to the σ-algebra 𝔊n. Then we say that the set X=(Xn,𝔊n)n⩾1 is a stochastic process.LetF={𝔉n}n⩾1 be a system of sets satisfying the condition 𝔉1⊆⋯⊆𝔉n⊆⋯⊆𝔊. We say that a stochastic process X=(Xn,𝔊n)n⩾1 is defined on a system F={𝔉n}n⩾1 if 𝔉n⊂𝔊n,n∈ℕ. For a stochastic process X, which is defined on a system F={𝔉n}n⩾1, we define the sequence of numbers X¯(F)={X¯n(F)}n, where
(1)X¯k(F)=supA∈𝔉k,P(A)>01P(A)|∫A‍XkP(dω)|.
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 setsF={𝔉n}n⩾1: 𝔉n=Ω, in this case the sequence X¯(F)={X¯n(F)}n is a sequence of averages of a process X={Xn}; for 𝔉n=𝔊n-1 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) 𝔉n=𝔊n∧τ, where τ=τ(ω) is the fixed stopping time, and 𝔊n∧τ={A∈𝔊:A∩{τ=n}∈𝔊n}; (2) 𝔉n=𝔊n∧τn, where τn=τn(ω), and n∈ℕ is the sequence of the stopping times.We consider the classes of stochastic processes defined onF, which characterize the speed of convergence of sequence {X¯n(F)/n}n to zero.ByNp,q(F), 0<p<∞,  0<q⩽∞ we denote the set of all stochastic processes X, defined on F for which
(2)∥X∥Np,q(F)=(∑k=1∞‍k-1-(q/p)X¯kq)1/q<∞
if 0<q<∞ and
(3)∥X∥Np,∞(F)=supkk-1/pX¯k<∞
if q=∞.Let us denote(4)Npα,q(F)={X=(Xn,𝔉n)n⩾1:(∑k=0∞‍(2αkΔXk¯)q)1/q<∞}
for 0<q<∞, and
(5)Npα,∞(F)={X=(Xn,𝔉n)n⩾1:supk2αkΔXk¯<∞},
for q=∞, where
(6)ΔXk¯=supA∈𝔊,P(A)>01(P(A))1-(1/p)|∫A‍(X2k-X2k-1)P(dω)|.We consider that(7)Xα(ω)={X[α](ω)ifα⩾10ifα<1,
where [α] is the integer part of the number α. In particular X1/2(ω)≡0.A random variableτ, which takes values in the set (1,2,…,∞), is called the Markov time of the filtration G={𝔊n}n⩾1, if {ω:τ(ω)=n}∈𝔊n for any n∈ℕ. The Markov time τ, for which τ(ω)<∞ (a.p.) [9], is called the stopping time.LetX=(Xn,𝔊n)n⩾1 be a stochastic process and τ be the Markov time. By Xτ we denote the stopped process Xτ=(Xn∧τ,𝔊n), where Xn∧τ(ω)=∑m=1n-1Xmχτ=m(ω)+Xnχτ⩾n(ω) and χA(ω) is the characteristic function of the set A.We assume also that(8)Xα={X[α]ifα⩾10=(0,𝔊n)n⩽1ifα<1,∑k=αβbk∶=∑k=[α][β]bk.TheNpα,q(F) spaces are spaces of converging stochastic processes, where parameters α,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 writeA≲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 SpacesNp,q(F) and Npα,q(F)

We say [9] that stochastic process (Xn,𝔊n)n⩾1 is a martingale if for every n∈ℕ the following conditions hold: (1) E|Xn|<∞; (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,𝔊n)n=1∞ is a submartingale (supermartingale).Definition 1.
LetF={𝔉n}n be a fixed system of sets, X=(Xn,𝔊n)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∈𝔉k(9)|∫A‍XkP(dω)|⩽C|∫A‍XmP(dω)|.This inequality implies thatX¯k(F)⩽cX¯m(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.
Letα>0, 0<q⩽∞, and 1<p<∞. If X=(Xn,𝔊n)n⩾1∈Npα,q(G)∩W(G), then there exists a random variable X∞ such that Xn→n→∞X∞ (a.p.).Proof.
LetLp,∞(Ω) be the Marcinkiewicz-Lorentz space and 2ν-1⩽n<2ν. Using the equivalent norm of Lp,∞(Ω) spaces (see [6]) and measurability of function Xn with respect to σ-algebra 𝔊n, we get the following:
(10)∥Xn∥Lp,∞[0,1]≍supA∈𝔊,P(A)>01(P(A))1/p′|∫A‍XnP(dω)|=supA∈𝔊n,P(A)>01(P(A))1/p′|∫A‍XnP(dω)|⩽CsupA∈𝔊2ν,P(A)>01(P(A))1/p′|∫A‍X2νP(dω)|⩽C∑k=0ν‍supA∈𝔊2k,P(A)>01(P(A))1/p′×|∫A‍(X2k-X2k-1)P(dω)|⩽C∥X∥Np0,1(G).
Taking into account thatNpα,q(G)↪Np0,1(G), for α>0, we have ∥Xn∥Lp,∞[0,1]⩽c∥X∥Npα,q(G).
ButM|Xn|⩽c∥Xn∥Lp,∞[0,1], therefore by the Doob theorem ([11]), the process Xn converges almost surely.Lemma 3.
Letα>0,  0<q⩽∞,  1<p<∞, and X=(Xn,𝔊n)n⩾1∈W(G). Then
(11)∥X∥Npα,q(G)≍(∑k=0∞‍(2αkXk~)q)1/q,
where
(12)Xk~=supA∈𝔊,P(A)>01(P(A))1/p′|∫A‍(X∞-X2k-1)P(dω)|.Proof.
The existence ofX∞ follows from Lemma 2. Further, we have
(13)Xk~=supP(A)>01(P(A))1/p′|∫A‍∑m=k∞‍(X2m-X2m-1)P(dω)|⩽∑m=k∞‍supA∈𝔊2m1(P(A))1/p′|∫A‍(X2m-X2m-1)P(dω)|=∑m=k∞‍ΔX¯m.
Therefore, using Lemma 8, we obtain
(14)(∑k=0∞‍(2αkXk~)q)1/q⩽(∑k=0∞‍(2αk∑m=k∞‍ΔX¯m)q)1/q⩽(∑m=0∞‍2mαqΔX¯mq)1/q≍∥X∥Npα,q(G).
The reverse inequality follows from the expression:
(15)ΔX¯k=supP(A)>01(P(A))1/p′|∫A‍(X2k-X2k-1)P(dω)|⩽supP(A)>01(P(A))1/p′|∫A‍(X2k-X∞)P(dω)|+supP(A)>01(P(A))1/p′|∫A‍(X∞-X2k-1)P(dω)|.
The lemma is proved.Lemma 4.
LetX∈W(F). Then(1)
for0<q⩽q1⩽∞,
(16)∥X∥Np,q1(F)⩽cp,q,q1∥X∥Np,q(F),(17)∥X∥Npα,q1(F)⩽cα,p,q,q1∥X∥Npα,q(F),(2)
forε>0, 0<q,q1⩽∞,
(18)∥X∥Np,q1(F)⩽cp,q,q1∥X∥Np+ε,q(F),(19)∥X∥Npα,q1(F)⩽cα,p,q,q1∥X∥Npα+ε,q(F),where cp,q,q1,  cα,p,q,q1>0 depend only on the indicated parameters.Proof.
Let us prove inequalities (16), (18). The proof of inequalities (17), (19) is similar. Let ε>0. By Minkowski’s inequality and by the generalized monotonicity of a process X=(Xn,𝔊n)n⩾1 we get the following:
(20)∥X∥Np,q1(F)=(∑k=1∞‍kεq1-1k-εq1-(q1/p)X¯kq1)1/q1≲(∑k=1∞‍kεq1-1(∑r=k∞‍r-q(ε+(1/p))-1X¯rq)q1/q)1/q1≲(∑r=1∞‍r-q(ε+(1/p))-1X¯rq(∑k=1r‍kεq1-1)q/q1)1/q≲(∑r=1∞‍r-(q/p)-1X¯rq)1/q=∥X∥Np,q(F).
To prove the second statement it is enough to show that ∥X∥Np1,q1(F)⩽c∥X∥Np,∞(F) and apply the first statement. Since p1<p, we have the following:
(21)∥X∥Np1,q1(F)=(∑k=1∞‍k-(q1/p1)-1X¯kq1)1/q1⩽supkk-1/pX¯k(∑k=1∞‍k(q1/p)-(q1/p1)-1)1/q1=c∥X∥Np,∞(F).Remark 5.
Properties of theNp,q(F) spaces given in Lemma 4 show that the second parameter q is weak with respect to the first p. These properties of the spaces are important in the interpolation.Lemma 6.
Let0<p<∞,  a>1. If X∈W(F), then for 0<q<∞(22)∥X∥Np,q(F)≍(∑k=0∞‍(a-k/pX¯ak)q)1/q,
and for q=∞(23)∥X∥Np,∞(F)≍supk∈ℕa-k/pX¯ak.Proof.
Using the generalized monotonicity of a processX, we have the following:
(24)∥X∥Np,q(F)=(∑k=1∞‍k-(q/p)-1X¯kq)1/q=(∑k=0∞‍∑l=akak+1-1‍l-(q/p)-1X¯lq)1/q≳(∑k=0∞‍a-kq/pX¯akq∑l=akak+1-1‍1l)1/q≳(∑k=0∞‍(a-k/pX¯ak)q)1/q.
One can prove the reverse estimate in a similar way.Lemma 7 (Hölder inequality).
Let0<p1,p2,q<∞,  0<t,s1,s2⩽∞ and (1/q)=(1/p1)+(1/p2), (1/t)=(1/s1)+(1/s2). If stochastic processes X=(Xn,𝔊n)∈Np1,s1(F) and Y=(Yn,𝔊n)∈Np2,s2(G), then XY=(XnYn,𝔊n)∈Nq,t(F) and
(25)∥XY∥Nq,t(F)⩽∥X∥Np1,s1(F)∥Y∥Np2,s2(G).Proof.
SinceYn is measurable with respect to an algebra 𝔊n, we have XkYk¯(F)⩽Xk¯(F)Yk¯(G) and hence
(26)∥XY∥Nq,t(F)=(∑k=1∞‍(k-1/pXkYk¯)t1k)1/t⩽(∑k=1∞‍(k-1/p1Xk¯(F))s11k)1/s1×(∑k=1∞‍(k-1/p2Yk¯(G))s21k)1/s2=∥X∥Np1,s1(F)∥Y∥Np2,s2(G).We will need the following Hardy-type inequalities.Lemma 8.
Lets⩾1, ν>0, α>0, β>0, and γ>0; then for a nonnegative sequence a={ak}k the following inequalities hold:
(27)(∑k=1∞‍k-αs-1(∑l=1(γk)‍lβ-1al)s)1/s⩽Cα,β,s,γ(∑k=1∞‍k(β-α)s-1aks)1/s,(∑k=1∞‍kαs-1(∑l=(γk)∞‍lβ-1al)s)1/s⩽Cα,β,s,γ(∑k=1∞‍k(β+α)s-1aks)1/s,(∑k=0∞‍(2-αk∑m=0γk‍2βmam)s)1/s⩽Cα,β,s,γ(∑k=1∞‍(2(β-α)kak)s)1/s,(∑k=0∞‍(2αk∑m=γk∞‍2βmam)s)1/s⩽Cα,β,s,γ(∑k=1∞‍(2(β+α)kak)s)1/s.


## 3. Interpolation Method for Stochastic Processes

LetT={Tn}n=1∞ be a transform that transforms a stochastic process X, which is defined on the system F={𝔉}n=1∞, to the stochastic process T(X)={Tn(X),Φn}n=1∞, which is defined on the system R={ℜ}n=1∞. We say that the transform T is quasilinear if there exists a constant C>0 such that for any n∈ℕ the following inequality holds almost surely:
(28)|Tn(X)-Tn(Y)|⩽C|Tn(X-Y)|.It is known ([9]) that if a process X=(Xn,𝔊n)n⩾1 is a martingale (submartingale), then the process Xτ=(Xn∧τ,𝔊n)n⩾1 is also a martingale (submartingale).DenoteXn*(ω)=max1⩽k⩽n|Xk(ω)| and X*=(Xn*,𝔊n)n⩾1.The transformsXτ and X* of the stochastic process X are examples of quasilinear transforms.LetA=(A0(F),A1(F)) be a pair of quasinormed own subspaces of linear Hausdorff stochastic processes spaces 𝔑(F), which is defined on a probability space (Ω,𝔉,P) with a filtration F={𝔉n}n⩾1. Obviously, this pair is compatible pair and hence the scale of interpolation spaces is defined with respect to the real method ([12]).Moreover, let for0<θ<1,0<q<∞(29)(A0,A1)θ,q={∫0∞‍X∈𝔑(F):∥X∥(A0,A1)θ,qq=∫0∞‍(t-θK(t,X))qdtt<∞}
and for q=∞(30)(A0,A1)θ,∞={sup0<t<∞X∈𝔑(F):∥X∥(A0,A1)θ,∞=sup0<t<∞t-θK(t,X)<∞},
where
(31)K(t,X;A0,A1)=infX=X0+X1(∥X0∥A0+t∥X1∥A1)
is the Peetre functional.LetR={τk(ω)}k=1∞ be a sequence of stopping times with respect to a filtration F and A(F)=(A0(F),A1(F)) be a pair of quasinormed own subspaces 𝔑(F). Let X∈𝔑(F) and t∈(0,∞). We define the following:
(32)KR(t,X)=K(t,X;A0,A1,R)=infτ∈R∪{0}(∥X-Xτ∥A0+t∥Xτ∥A1).
Here the infimum is taken over all stopping times from R. Moreover for 0<q<∞(33)(A0,A1)θ,qR={∫0∞‍X∈𝔑(F):∥X∥(A0,A1)θ,qq=∫0∞‍(t-θKR(t,X))qdtt<∞},
and for q=∞(34)(A0,A1)θ,∞R={sup0<t<∞X∈𝔑(F):∥X∥(A0,A1)θ,∞=sup0<t<∞t-θKR(t,X)<∞}.Theorem 9.
Let(A0(F),A1(F)),(B0(Φ),B1(Φ)) be two compatible pairs of stochastic processes and let R={τ(ω)} be some fixed family of Markov times with respect to a filtration F. If T is a quasilinear map for stochastic processes X=(Xn,𝔉n)n⩾1 and
(35)∥T(X-Xτ)∥B0⩽M0∥X-Xτ∥A0,∥T(Xτ)∥B1⩽M1∥Xτ∥A1
for all stopping times τ∈R, then
(36)∥T(X)∥Bθ,q⩽CM01-θM1θ∥X∥Aθ,qR,
where the constant C is from the definition of quasilinearity of the operator T.Proof.
Consider the following:(37)∥T(X)∥Bθ,q=(∫0∞‍(t-θinfT(X)=Y0+Y1(∥Y0∥B0+t∥Y1∥B1))qdtt)1/q⩽(∫0∞‍(t-θinfτ∈R∪{0}(M1M0∥T(X)-T(Xτ)∥B0+t∥T(Xτ)∥B1M1M0))qdtt)1/q⩽C(∫0∞‍(t-θinfτ∈R∪{0}(M1M0∥T(X-Xτ)∥B0+t∥T(Xτ)∥B1M1M0))qdtt)1/q⩽CM0(∫0∞‍(t-θinfτ∈R∪{0}(M1M0∥X-Xτ∥A0+tM1M0∥Xτ∥A1))qdtt)1/q=CM01-θM1θ∥X∥Aθ,qR(F).
The theorem is proved.Lemma 10.
Leta>1 and R={k}k∈ℕ be stopping times. Then for 0<q<∞(38)∥X∥(A0,A1)θ,qR≍(∑n=-∞∞‍(a-θnKR(an,X))q)1/q,∥X∥(A0,A1)θ,q≍(∑n=-∞∞‍(a-θnK(an,X))q)1/q,∥X∥(A0,A1)θ,∞R≍supn∈ℤa-θnKR(an,X),∥X∥(A0,A1)θ,∞≍supn∈ℤa-θnK(an,X).The proof is similar to the proof of the Lemma4.


## 4. Interpolation Properties of the SpacesNp,q(F)

Theorem 11.
Let1<p0<p1<∞, 1⩽q0,q1,q⩽∞,0<θ<1, (1/p)=((1-θ)/p0)+(θ/p1), and R={k}k∈ℕ be the stopping times. Then for any stochastic process X=(Xn,𝔉n)n⩾1,
(39)∥X∥Np,q(F)⩽c∥X∥(Np0,q0(F),Np1,q1(F))θ,q,
where the constant c depends only on parameters pi,  qi,  θ, i=0,1.
IfX=(Xn,𝔉n)n⩾1∈W(F), then
(40)∥X∥(Np0,q0(F),Np1,q1(F))θ,qR⩽c∥X∥Np,q(F),
where the constant c also depends only on parametres pi,qi,θ,i=0,1.Proof.
LetX=Y+Z be any representation of a process X, where Y∈Np0,q0(F), Z∈Np1,q1(F). To prove the first statement of the theorem, we use the following inequality: X¯m⩽Y¯m+Z¯m.
For anya>1 we have the following:
(41)a-n/pX¯an⩽a-(n/p)+(n/p0)(a-n/p0Y¯an+a-(n/p0)+(n/p1)a-n/p1Z¯an)⩽a-(n/p)+(n/p0)×(supna-n/p0Y¯an+a-(n/p0)+(n/p1)supna-n/p1Z¯an)=aθ((1/p0)-(1/p1))n×(∥Y∥Np0,∞(F)+a-((1/p0)-(1/p1))n∥Z∥Np1,∞(F)).
By putting a=2p0p1/(p1-p0), we get a-n/pX¯n⩽2nθK(2-n,X). Therefore, using (22) and Lemma 8, we have the following:
(42)∥X∥Np,q(F)≍(∑n=-∞∞‍(a-n/pX¯n)q)1/q⩽(∑n=-∞∞‍(2nθK(2-n,X))q)1/q=(∑n=-∞∞‍(2-nθK(2n,X))q)1/q≍∥X∥(Np0,∞(F),Np1,∞(F))θ,q.
Let us prove the second statement of the theorem. Let X=(Xn,𝔉n)n⩾1∈W(F). By using Lemmas 8 and 3 we have the following:
(43)∥X∥(Np0,q0(F),Np1,q1(F))θ,qR≍(∑n=-∞∞‍(2θnKR(2-n,X))q)1/q=(+2-n∥Xk∥Np1,1)infk⩾0)q∑n=-∞∞‍(2θninfk⩾0(∥X-Xk∥Np0,q0+2-n∥Xk∥Np1,q1)infk⩾0)q∑n=-∞∞+2-n∥Xk∥Np1,1)infk⩾0)q‍)1/q≲(+2-n∥Xk∥Np1,1)infk⩾0)q∑n=-∞∞‍(2θninfk⩾0(∥X-Xk∥Np0,1+2-n∥Xk∥Np1,1)infk⩾0)q∑n=-∞∞‍)1/q.
Let k=[2nγ], then taking into account that X2nγ≡0,  n⩽-1, we obtain the following:
(44)∥X∥(Np0,q0(F),Np1,q1(F))θ,qR≲(∑n=-∞∞‍(2θn(∥X-X2nγ∥Np0,1+2-n∥X2nγ∥Np1,1))q)1/q=(∑n=0+∞‍(2θn(∥X-X2nγ∥Np0,1+2-n∥X2nγ∥Np1,1))q)1/q+(∑n=-∞-1‍(2θn(∥X∥Np0,1))q)1/q=(∑n=0∞‍(2θn(∥X-X2nγ∥Np0,1+2-n∥X2nγ∥Np1,1))q+∥X∥Np0,1q∑n=1+∞‍2-θnq)1/q.
Thus, we obtain the following:(45)∥X∥(Np0,q0(F),Np1,q1(F))θ,qR≲(∑n=0∞‍(2θn(∥X-X2nγ∥Np0,1+2-n∥X2nγ∥Np1,1))q+∥X∥Np0,1q∑n=0∞‍)1/q.
Further, we have the following:(46)∥X-X2nγ∥Np0,1=∑k=1∞‍k-1/p0(X-X2nγ)¯k≍∑k=0∞‍2-k/p0(X-X2nγ)¯2k=∑k=nγ∞‍2-k/p0(X-X2nγ)¯2k⩽(1+C)∑k=nγ∞‍2-k/p0X¯2k,(47)∥X2nγ∥Np1,1=∑k=1∞‍k-1(1/p1)Xk2nγ¯≍∑k=0∞‍2-k/p1X2k2nγ¯⩽∑k=0nγ‍2-k/p1X¯2k+X¯2nγ∑k=nγ∞‍2-k/p1⩽C(∑k=0nγ‍2-k/p1X¯2k+2nγ((1/p0)-(1/p1))∑k=nγ∞‍2-k/p0X¯2k).
By using Lemma 8 for (1/γ)=(1/p0)-(1/p1), (46), and (47), we have the following:
(48)(∑n=0∞‍(2θn∥X-X2nγ∥Np0,1)q)1/q≲(∑n=0∞‍(2θn∑k=nγ∞‍2-k/p0X¯2k)q)1/q≲(∑k=0∞‍(2-k/pX¯2k)q)1/q≍∥X∥Np,q(F),(∑n=0∞‍2-(1-θ)nq∥X2nγ∥Np1,1q)1/q≲(+2nγ((1/p0)-(1/p1))∑k=nγ∞‍2-k/p0X¯2k)q∑n=0∞‍2-nq(1-θ)(+2nγ((1/p0)-(1/p1))∑k=nγ∞‍2-k/p0X¯2k)q∑k=0nγ‍2-k/p1X¯2k+2nγ((1/p0)-(1/p1))∑k=nγ∞‍2-k/p0X¯2k)q)1/q≲(∑k=0∞‍(2-k/pX¯2k)q)1/q≍∥X∥Np,q(F).
By applying Minkowski’s inequality to (45) and using Lemma 4, estimates (48), we obtain
(49)∥X∥(Np0,q0,Np1,q1)θ,qR⩽c∥X∥Np,q(F).Corollary 12.
Let0<p0<p1<∞, 0<q0<q1<∞, 0<θ<1, 1⩽s⩽∞, (1/q)=((1-θ)/q0)+(θ/q1), (1/p)=((1-θ)/p0)+(θ/p1), X∈W(F), and T={Tn}n=1∞ be a quasilinear transform. If for any k∈ℕ∪{0} the following conditions hold:
(50)∥T(Xk)∥Nq1,∞(R)⩽M1∥Xk∥Np1,1(F),∥T(X-Xk)∥Nq0,∞(R)⩽M0∥X-Xk∥Np0,1(F),
then
(51)∥T(X)∥Nq,s(R)⩽CM01-θM1θ∥X∥Np,s(F),
where C>0 depends only on p0,  p1,  q0,  q1, and θ.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).
Let0<p0<p1<∞,  0<q0<q1⩽∞, θ∈(0,1),  1/p=(1-θ)/p0+θ/p1, and 1/q=(1-θ)/q0+θ/q1. If  T is a quasilinear map and
(52)T:Lpi,1(D,ν)⟶Lqi,∞(Ω,μ)withthenormMi,i=0,1,
then
(53)T:Lp,τ(D,ν)⟶Lq,τ(Ω,μ),∥T∥⩽M01-θM1θ.Corollary 14.
LetT={Tn}n=1∞ be a quasilinear transform such that for any p∈(a,b) and for any X∈Np,1(F)∩W(F) the following weak inequality holds:
(54)∥T(X)∥Np,∞(R)⩽Cp∥X∥Np,1(F).
Then
(55)∥T(X)∥Np,s(R)⩽Cp,s∥X∥Np,s(F),X∈Np,s(F)∩W(F)
for any p and s such that p∈(a,b),  1⩽s⩽∞.


## 5. Boundedness of Some Operators in ClassNp,q(F)

LetY=(Yn,𝔊n)n⩾0 be a stochastic sequence and V=(Vn,𝔊n-1) be a predicted sequence (𝔊-1=𝔊0). A stochastic sequence V·Y=((V·Y)n,𝔊n) such that
(56)(V·Y)n=V0Y0+∑i=1n‍ViΔYi,
where ΔYi=Yi-Yi-1 is called the transform of Y with respect to V. If Y is a martingale then we say that V·Y is the martingale transform.Theorem 15.
Let0<q<p<∞,  1⩽τ⩽∞ and 1/r=1/q-1/p. Let V·Y be a martingale transform of a martingale Y by predicted sequence V=(Vn,𝔉n-1). If
(57)∥V∥Nr,∞(G)+∥(nΔVn)∥Nr,∞(G)⩽B,
then
(58)∥V·Y∥Nq,τ(G)⩽cB∥Y∥Np,τ(G),
where a constant c depends only on parametres p,q, and τ.Proof.
LetV·Y be a martingale transform of a martingale Y by predicted sequence V=(Vn,𝔉n-1); that is,
(59)(V·Y)n=∑k=1n‍VkΔYk,
where Y0=0,  ΔYk=Yk-Yk-1. By Abel’s transform (V·Y)n=∑k=1n-1‍ΔVkYk+VnYn, we get the following:
(60)∥V·Y∥Nq,∞(G)=supn∈Nn-1/q(V·Y)n¯⩽supn∈Nn-1/q(∑k=1n‍ΔVkYk¯+VnYn¯).
Taking into account that ΔVk,Yk are measurable functions with respect to the algebra 𝔊k, we have ΔVkYk¯⩽ΔVk¯Yk¯,VnYn¯⩽Vn¯Yn¯ and
(61)supn∈Nn-1/q(V·Y¯)n⩽supn∈N∑k=1n‍(k-1/qkΔVk¯)(k-(1/p)-1Yk¯)+n-1/rVn¯n-1/pYn¯⩽supkk1-(1/r)ΔVk¯·∥Y∥Np,1(G)+∥Vn∥Nr,∞(G)∥Y∥Np,∞(G)⩽B∥Y∥Np,1(G).
Hence the weak inequality is proved as follows:
(62)∥V·Y∥Nq,∞(G)⩽B∥Y∥Np,1(G)
for 1<q<p<∞.
Let0<q<p<∞, (1/r)=(1/q)-(1/p), 0<q0<q<q1<∞ and 0<p0<p<p1<∞. Let a pair of numbers (p0,p1) and (q0,q1) satisfy the following condition:
(63)1q0-1p0=1q1-1p1=1q-1p.
Then from that is proved above it follows that
(64)∥V·Y∥Nq0,∞(G)⩽B∥Y∥Np0,1(G),∥V·Y∥Nq1,∞(G)⩽B∥Y∥Np1,1(G)
for 0<p<q<∞.
Taking into account that for any stopping timek∈N processes Yk and Y-Yk are martingales, it is possible to apply Theorem 9. Then
(65)∥V·Y∥Nqθ,τ⩽cB∥Y∥Npθ,τ,
where
(66)1qθ=1-θq0+θq1,1pθ=1-θp0+θp1.
Note that there existsθ∈(0,1) such that (1/pθ)=(1/p). Then it follows from (63) that (1/qθ)=(1/q).Theorem 16.
Let0<p<∞,  1⩽q⩽∞ and X=(Xn,𝔊n)n⩾1∈W(G) and τ(ω) be the Markov time and let Xτ=(Xn∧τ,𝔊n)n⩾1 be a stopped process. Then
(67)∥Xτ∥Np,q(G)⩽c∥X∥Np,q(G).Proof.
Denote(68)Wr={ω:τ(ω)=r}∈𝔊r,r=1,n-1¯,Wn={ω:τ(ω)⩾n}∈𝔊n.
Let us show that ∥Xτ∥Np,∞(G)⩽c∥X∥Np,1(G).
If2s⩽n<2s+1, then
(69)n-1/psupA∈𝔊n,P(A)>01P(A)|∫A‍XτP(dω)|=n-1/psupA∈𝔊n,P(A)>01P(A)|∑r=1n‍∫A∩Wr‍XrP(dω)|⩽2-s/psupA∈𝔊2s+1-11P(A)∑t=0s‍∑r=2t2t+1-1‍|∫A∩Wr‍XrP(dω)|⩽2-s/psupA∈𝔊2s+1-11P(A)∑t=0s‍∑r=2t2t+1-1‍P(A∩Wr)Xr¯⩽2-s/psupA∈𝔊2s+1-11P(A)∑t=0s‍X¯2t+1-1∑r=2t2t+1-1‍P(A∩Wr)=2-s/psupA∈𝔊2s+1-1∑t=0s‍X¯2t+1-1∑r=2t2t+1-1‍P(Wr∣A)⩽∑t=0s‍2-t/pX¯2t+1-1⩽21/p∑t=0s‍2-(t+1)/pX¯2t+1=21/p∑k=0s+1‍2-k/pX¯2k⩽c∥X∥Np,1(G).
Now, using Corollary 12 we get the statement of the Theorem 16.Corollary 17.
Let0<p<∞,  1⩽q⩽∞ and a process X=(Xn,𝔊n)n⩾1 be a nonnegative submartingale. Then the process X*=(Xn*,𝔊n)n⩾1 is also submartingale and
(70)∥X*∥Np,q(G)~∥X∥Np,q(G).Proof.
It follows from Theorem16 that
(71)∥X*∥Np,q(G)⩽c∥X∥Np,q(G).
The reverse inequality is trivial.


## 6. Interpolation Properties of the SpaceNpα,q(G), the Embedding Theorems

Theorem 18.
Let1⩽p,q,q0,q1⩽∞,α0<α1,0<θ<1,α=(1-θ)α0+θα1, and R={r}r∈ℕ. Then
(72)(Npα0,q0(G),Npα1,q1(G))θ,qR=Npα,q(G).Proof.
Using Lemmas10 and 4 we have the following:
(73)∥X∥(Npα0,q0,Npα1,q1)θ,qR≍(∑n=-∞∞‍(2θnKR(2-n,X))q)1/q=(∑n=-∞∞‍(2θninfr⩾0(∥X-Xr∥Npα0,q0+2-n∥Xr∥Npα1,q1))q)1/q≲(∑n=-∞∞‍(2θninfr⩾0(∥X-Xr∥Npα0,1+2-n∥Xr∥Npα1,1))q)1/q.
Putting r=[2nγ],γ=α1-α0 we get the following:
(74)∥X∥(Npα0,q0,Npα1,q1)θ,qR≲(∑n=0∞‍(2θn(∥X-X2nγ∥Npα0,1+2-n∥X2nγ∥Npα1,1))q+∥X∥Npα0,1q∑n=0∞‍)1/q.
It follows from the definition that (ΔX2r)¯k=ΔX¯k for k=0,1…,r-1;(ΔX2r)¯k=0 for k⩾r,Δ(X-X2r)¯k=0 for k=0,1…,r-1;Δ(X-X2r)¯k=ΔX¯k for k⩾r; therefore,
(75)∥X2r∥Npα,1(G)=∑k=0r-1‍2αkΔX¯k,∥X-X2r∥Npα,1(G)=∑k=r∞‍2αkΔX¯k.
Substituting these equalities in (74) and applying Lemmas 4 and 8, we get the following:
(76)∥X∥(Npα0q0,Npα1q1)θ,qR≲(∑n=0∞‍2θnq(∑k=nγ∞‍2α0kΔX¯k+2-n∑k=0nγ-1‍2α1kΔX¯k)q+∥X∥Npα0,1q∑k=nγ∞‍)1/q≲(∑n=0∞‍(2(α0+θγ)nΔX¯k)q)1/q+(∑n=0∞‍(2(α1-(1-θ)γ)nΔX¯k)q)1/q+∥X∥Npα,q=3∥X∥Npα,q.
For the proof of reverse estimate we use the fact that for any r and k the following equality holds:
(77)ΔX¯k=Δ(X-X2r)¯k+(ΔX2r)¯k.
Then we have the following:
(78)∥X∥Npα,q=(∑k=0∞‍(2αkΔX¯k)q)1/q⩽(∑k=0∞‍(2αk-α0k(supn⩾02α0nΔ(X-X2r)¯n+2α0k-α1k·supn⩾02α1n(ΔX2r)¯n))q∑k=0∞‍)1/q=(∑k=0∞‍(2αk-α0k(∥X-X2r∥Npα0,∞+2α0k-α1k∥X2r∥Npα1,∞))q∑k=0∞‍)1/q.
Substituting 2α1-α0=a and using Lemma 4, we have the following:
(79)∥X∥Npα,q⩽(∑k=0∞‍(aθk·K(a-k,X,Npα0,∞,Npα1,∞))q)1/q⩽∥X∥(Npα0,∞,Npα1,∞)θ,qR,
since (Npα0,qo,Npα1,q1)θ,qR↪(Npα0,∞,Npα1,∞)θ,qR, the proof is complete.Theorem 19.
Let1<r⩽p<∞,  1⩽q⩽∞ and α=(1/r)-(1/p) and let the filtration G={𝔊n}n⩾1 be such that for every k=1,2… and for all A∈𝔊k the following condition holds:
(80)P(A)⩾Ck,
where the constant C>0 does not depend on k. Then
(81)Nrα,q(G)↪Np,q(G).Proof.
Let us show that(82)Nrα,1(G)↪Np,∞(G).(83)∥X∥Np,∞(G)≍supk2-k/pX¯2k=supk2-k/psupA∈𝔊2k1P(A)|∫A‍X2kP(dω)|≍supk2-k/psupA∈𝔊2k1P(A)×|∑m=0k‍∫A‍(X2m-X2m-1)P(dω)|⩽supk∑m=0k‍2-m/psupA∈𝔊2m1(P(A))1/r′1(P(A))1/r×|∫A‍(X2m-X2m-1)P(dω)|.
According to the condition (80), for A∈𝔉2m we have that P(A)>(C/2m). Therefore for α=(1/r)-(1/p) we get the following:
(84)∥X∥Np,∞(G)⩽Csupk∑m=0k-1‍2-(m/p)+(m/r)×supA∈𝔊2m1(P(A))1/r′|∫A‍(X2m-X2m-1)P(dω)|≍∥X∥Nrα,1.
Thus, (82) is proved.
Now, letα0<α1,  1<p0<p1<∞, and θ∈(0,1) such that
(85)1p0+α0=1p1+α1=1r,α=(1-θ)α0+θα1,1p=1-θp0+θp1.
Then using interpolation Theorems 18 and 11 we obtain the following:
(86)(Nrα0,1(G),Nrα1,1(G))θ,q=Nrα,q(G),(Np0,∞(G),Np1,∞(G))θ,q=Np,q(G).
It follows from (82)  that (Nrα0,1(G),Nrα1,1(G))θ,q↪(Np0,∞(G),Np1,∞(G))θ,q. Hence, Nrα,q(G)↪Np,q(G), where α=(1/r)-(1/p). The proof is complete.


## 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.LetΩ=[0,1] and let 𝔉 be a σ-algebra of Borel subsets of set Ω, P a linear Lebesgue measure on 𝔉, F={𝔉n}n⩾1 the Haar filtration, and R={τk}k=0∞ a sequence of stopping times such that for any k⩾0 the following conditions hold: τ0≡0,τk+1⩽τk+1 (a.p.) and
(87)limk→∞τk(ω)=∞(a.p.).
For a function f(x)∈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 τk(ω) we denote
(88)S(f,τk)(ω)={∑m=1τk(ω)‍cm(f)Hm(ω)ifk⩾10ifk=0,
which we call the Fourier-Haar partial sum of a function f, corresponding to the Markov time τk.Let1<p<∞, 0<q⩽∞, α∈ℝ. By Bpα,q[R] we denote the set of functions f∈L[0,1], for which
(89)∥f∥Bpα,q[R]=(∑k=0∞‍2αkq∥f-S(f,τk)∥Lp)1/q<∞
for 0<q<∞,
(90)∥f∥Bpα,∞[R]=supk2αk∥f-S(f,τk)∥Lp<∞,
for q=∞.Conceptually, the introduced spaces are close to spaces with variable smoothness. Here we mention works of Leopold [15], Cobos and Fernandez [16], and Besov [17–20].Lemma 20.
Let1<p<∞,f∈Lp[0,1], and let S(f,τ) be the Fourier-Haar partial sum with respect to the Markov time τ. Then
(91)∥S(f,τ)∥p⩽c∥f∥p.Proof.
Denote(92)𝔉τ={A∈𝔉:A∩{τ=n}∈𝔉nforeveryn⩾1}.
Let Lp,∞[0,1] be the Marcinkiewicz-Lorentz space. Using the equivalent norm of Lp,∞[0,1] spaces (see [6]) and martingale properties of Fourier-Haar partial sums we get the following:
(93)∥S(f,τ)∥Lp,∞[0,1]≍supA∈𝔉,P(A)>01(P(A))1/p′|∫A‍S(f,τ)P(dω)|=supA∈𝔉τ,P(A)>01(P(A))1/p′|∫A‍S(f,τ)P(dω)|=supA∈𝔉τ,P(A)>01(P(A))1/p′|∫A‍f(ω)P(dω)|⩽supA∈𝔉,P(A)>01(P(A))1/p′|∫A‍f(ω)P(dω)|=∥f∥Lp,∞[0,1].
Now, applying the interpolation theorem (see [12]), we obtain the statement of the lemma.Lemma 21.
Let1<p<∞, 0<q⩽∞, α∈ℝ, and
(94)Δ(f,τk)=∑r=τk+1τk+1‍cr(f)Hr(ω).
Then
(95)∥f∥Bpα,q[R]≍(∑k=0∞‍2αkq∥Δ(f,τk)∥Lpq)1/q.Proof is similar to the proof of Lemma3.Theorem 22.
Let1⩽q0,q1,q⩽∞,0<α0<α1,0<θ<1, and α=(1-θ)α0+θα1. Then
(96)(Bpα0,q0[R],Bpα1,q1[R])θ,qR=Bpα,q[R].Proof.
By using Lemma21 we have the following:
(97)∥f∥(Bpα0,q0[R],Bpα1,q1[R])θ,qR⩽c∥f∥(Bpα0,1[R],Bpα1,1[R])θ,qR≍(∑n=1∞‍2θqninfr(∑k=r∞‍2α0k∥Δ(f,τk)∥Lp+2-n∑k=0r-1‍2α1k∥Δ(f,τk)∥Lp)q)1/q⩽(∑n=1∞‍2θqn(∑k=nγ∞‍2α0k∥Δ(f,τk)∥Lp+2-n∑k=0nγ-1‍2α1k∥Δ(f,τk)∥Lp)q)1/q⩽(∑n=1∞‍(2θn∑k=nγ∞‍2α0k∥Δ(f,τk)∥Lp)q)1/q+(∑n=1∞‍(2-(1-θ)n∑k=0nγ-1‍2α1k∥Δ(f,τk)∥Lp)q)1/q.
By applying Lemma 8 we get the following:
(98)∥f∥(Bpα0,q0[R],Bpα1,q1[R])θ,qR⩽c∥f∥Bpα,q[R].
Let us prove the reverse embedding. Let   f∈(Bpα0,q0[R],Bpα1,q1[R])θ,qR, f=f0+f1 be an arbitrary representation of a function, f0∈Bpα0,q0[R] and f1∈Bpα1,q1[R]. Then
(99)2αk∥Δ(f,τk)∥Lp⩽2αk(∥Δ(f0,τk)∥Lp+∥Δ(f1,τk)∥Lp)⩽2(α-α0)k(supr∈ℕ2α0r∥Δ(f0,τr)∥Lp+2(α0-α1)ksupr∈ℕ2α1r∥Δ(f1,τr)∥Lp)=2(α-α0)k(∥f0∥Bpα0,∞[R]+2(α0-α1)k∥f1∥Bpα1,∞[R]).
Since the representation f=f0+f1 is arbitrary, we have the following:
(100)2αk∥Δ(f,τk)∥Lp⩽2(α-α0)kKR(f,2(α0-α1)k;Bpα0,∞[R],Bpα1,∞[R]).
Hence, putting a=2α1-α0 we get the following:
(101)∥f∥Bpα,q[R]⩽(∑k=0∞‍(2(α1-α0)θkKR(f,2(α0-α1)k))q)1/q=(∑k=0∞‍(aθkKR(f,a-k))q)1/q⩽∥f∥(Bpα0,∞[R],Bpα1,∞[R])θ,qR⩽c∥f∥(Bpα0,q0[R],Bpα1,q1[R])θ,qR.
The theorem is proved.



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*Source: 152043-2013-12-26.xml*