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Embryonic stem cell therapy applications for autoimmune, cardiovascular, and neurological diseases: A review

  • Parkinson’s disease, type 1 diabetes, and coronary artery disease are some of the few difficult diseases to control. As a result, there has been pressure in the scientific community to develop new technologies and techniques that can treat, or ultimately cure these life-threatening diseases. One such scientific advancement in bridging the gap is the use of stem cell therapy. In recent years, stem cell therapy has gained the spotlight in becoming a possible intervention for combating chronic diseases due to their unique ability to differentiate into almost any cell line. More precisely, embryonic stem cell therapy may hold the potential for becoming the ideal treatment for a multitude of diseases as embryonic stem cells are not limited in their ability to differentiate like their counterpart adult stem cells. Although there has been controversy around the usage of embryonic stem cells, there has been found a great deal of potential within the usage of these cells to treat a multitude of life-threatening diseases. In this article, we will break down the categories of diseases in which embryonic stem cell therapy can be applied into: autoimmune, neurological, and cardiovascular with three diseases relating to each category. Our aim is to provide a comprehensive review on the advantages of embryonic stem cells (ESCs) that can solve current obstacles and push advances towards stem cell therapies in the field for the most common diseases.

    Citation: Edgar J. Cubillo, Sang M. Ngo, Alejandra Juarez, Joshuah Gagan, Gisel D. Lopez, David A. Stout. Embryonic stem cell therapy applications for autoimmune, cardiovascular, and neurological diseases: A review[J]. AIMS Cell and Tissue Engineering, 2017, 1(3): 191-223. doi: 10.3934/celltissue.2017.3.191

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  • Parkinson’s disease, type 1 diabetes, and coronary artery disease are some of the few difficult diseases to control. As a result, there has been pressure in the scientific community to develop new technologies and techniques that can treat, or ultimately cure these life-threatening diseases. One such scientific advancement in bridging the gap is the use of stem cell therapy. In recent years, stem cell therapy has gained the spotlight in becoming a possible intervention for combating chronic diseases due to their unique ability to differentiate into almost any cell line. More precisely, embryonic stem cell therapy may hold the potential for becoming the ideal treatment for a multitude of diseases as embryonic stem cells are not limited in their ability to differentiate like their counterpart adult stem cells. Although there has been controversy around the usage of embryonic stem cells, there has been found a great deal of potential within the usage of these cells to treat a multitude of life-threatening diseases. In this article, we will break down the categories of diseases in which embryonic stem cell therapy can be applied into: autoimmune, neurological, and cardiovascular with three diseases relating to each category. Our aim is to provide a comprehensive review on the advantages of embryonic stem cells (ESCs) that can solve current obstacles and push advances towards stem cell therapies in the field for the most common diseases.


    This paper aims to extend the boundedness of martingale transforms to Banach function spaces. The martingale transform is one of the important topics in probability and martingale function spaces [1,2,3,4]. The main results of this paper use the extrapolation theory to obtain the boundedness of the martingale transform on Banach function spaces which includes rearrangement-invariant Banach function spaces, Orlicz spaces, Lorentz-Karamata spaces, Zygmund spaces, Lebesgue spaces with variable exponents and Morrey type spaces such as Morrey spaces with variable exponents and Lorentz-Karamata-Morrey spaces.

    We use two different methods, the Rubio de Francia extrapolation theory and the interpolation theorem by Zygmund.

    Roughly speaking, the Rubio de Francia extrapolation theory uses the weighted norm inequalities for the martingale transforms to obtain the mapping properties for those Banach function spaces which satisfy some conditions related with the boundedness of maximal function. The interpolation theorem extends the mapping properties to the Zygmund spaces. The Zygmund spaces are rearrangement-invariant Banach function spaces used to capture the mapping properties of operators on the limiting cases of the Lebesgue space Lp for p approaching 1.

    This paper is organized as follows. The definition of Banach function spaces on probability spaces and some assumptions for the probability spaces are given in Section 2. The main results are established in Section 3. The applications of the main results on some concrete function spaces such as the Orlicz spaces, the Lorentz-Karamata spaces, the Lebesgue spaces with variable exponents, the Lorentz-Karamata-Morrey spaces and the Morrey spaces with variable exponents are presented in Section 4.

    Let (Ω,F,P) be a complete probability space and M be the space of measurable functions on (Ω,F,P).

    Let F=(Fn)n0 be the filtration on (Ω,F,P) where (Fn)n0 is a nondecreasing sequence of sub-σ-algebras of. Let F1=F0.

    For any martingale f=(fn)n0 on Ω, write dif=fifi1, i>0 and d0f=f0. Let f=(fn)n0 be a uniformly integrable martingale. We identify the martingale f with its pointwise limit f where the existence of the limit is guaranteed by the uniform integrability. For any integrable function f, the martingale generated by f is given by fn=Enf where En is the expectation operator associated with Fn, n0.

    The maximal function and the truncated maximal function of the martingale f is defined by

    Mf=supi0|fi|andMnf=sup0in|fi|,n0,

    respectively.

    For any predictable sequence v=(vn)n0 and martingale f, the martingale transform Tv is defined as

    (Tvf)n=nk=1vkdkf,(Tvf)0=0.

    The celebrated result on the convergence of martingale transform states that whenever f is a bounded L1 martingale, then Tvf=((Tvf)n)n0 converges almost everywhere on {xΩ:Mv(x)<}.

    Definition 2.1. A Banach space XM is said to be a Banach function space on (Ω,F,P) if it satisfies

    1. fX=0f=0a.e.,

    2. |g||f|a.e.gXfX,

    3. 0fnfa.e.fnXfX,

    4. LXL1.

    Item (4) of the above definition guarantees that for any measurable set E, χEX and E|f|dPCfX. In particular, Item (4) assets that for any fX, f=(fn)n0 is a bounded L1 martingale. Therefore, for any fX, the pointwise limit of Enf, limnEnf=f, exists and we identify the martingale f=(fn)n0 with f.

    Moreover, whenever vL=supn0vnL<, for any fX, the martingale transform Tvf=((Tvf)n)n0 converges almost everywhere on Ω. Thus, we are allowed to identify the martingale transform Tvf with its pointwise limit (Tvf).

    Furthermore, Item (4) guarantees that X is a Banach function space defined in [5,Chapter 1,Definitions 1.1 and 1.3]. Thus, the results in [5] apply to the Banach function space defined in Definition 2.1.

    The rearrangement-invariant Banach function spaces are examples of Banach function spaces. The reader is referred to [5,Chapter 2,Definition 4.1] for the definition of rearrangement-invariant Banach function spaces. Particularly, the Lorentz spaces, the Lorentz-Karamata spaces, the Orlicz spaces and the grand Lebesgue spaces are examples of Banach function spaces. The family of Banach function spaces also includes the Lebesgue spaces with variable exponents and the Morrey type spaces [1,6]. Notice that the Morrey type spaces on Rn are generally not Banach function spaces on Rn. For the definitions of Morrey type spaces on Rn, the reader is referred to [7,Definition 2.4].

    The reader is referred to [1,2,3,8] for the mapping properties of martingale transforms on Lebesgue spaces, Hardy spaces, Orlicz spaces, rearrangement-invariant Banach function spaces and Morrey spaces.

    We recall the definition of associate space from [5,Chapter 1,Definitions 2.1 and 2.3].

    Definition 2.2. Let X be a Banach function space. The associate space of X, X, is the collection of all measurable function f such that

    fX=sup{Ω|fg|dP:gX,gX1}<.

    According to [5,Chapter 1,Theorems 1.7 and 2.2], when X is a Banach function space, X is also a Banach function space. In addition, the Lorentz-Luxemburg theorem [5,Chapter 1,Theorem 2.7] yields

    X=(X). (2.1)

    We have the Hölder inequality for X and X, see [5,Chapter 1,Theorem 2.4].

    Theorem 2.1. Let X be a Banach function space. Then, for any fX and gX, we have

    Ω|fg|dPfXgX.

    For any Banach function space X, the weak type Banach function space wX consists of all fM satisfying

    fwX=supλ>0λχ{xΩ:|f(x)|>λ}X<.

    For any 0<r< and Banach lattice X, the r-convexification of X, Xr is defined as

    Xr={f:|f|rX}.

    The vector space Xr is equipped with the quasi-norm fXr=|f|r1/rX.

    We begin with the assumptions imposed on the filtration F. We assume that every σ-algebra Fn is regular and generated by finitely or countably many atoms, where BFn is called an atom if it satisfies the nested property. That is, any AB with AFn satisfying

    P(A)=P(B)orP(A)=0.

    Denote the set of atoms by A(Fn) and write A=n0A(Fn). Whenever F=(Fn)n0 satisfy the above condition, we say that F is generated by atoms.

    The notion of atoms were used in [9] for the studies of the martingale Hardy spaces with variable exponent and in [1,6] for the studies of the martingale Morrey and Campanato spaces.

    Suppose that F is generated by atoms. For any measurable function f, it is easy to see that

    Mf(x)=supAx1P(A)A|f|dP (2.2)

    where the supremum is taken over all AA containing x.

    Write M0f=|f|. For any kN, let Mk be the k-iteration of M.

    Next, we recall the definition of the Muckenhoupt weight functions on probability space. The Muckenhoupt weight functions on Rn were introduced by Muckenhoupt and the extension of this class of weight functions on probability space was given by Izumisawa and Kazamaki in [10].

    In the followings, we recall the definition of the Muckenhoupt weight functions when F is generated by atoms.

    Definition 2.3. Let (Ω,F,P) be a complete probability space where F is generated by atoms. A nonnegative integrable function ω is said to be an Ap weight if it satisfies

    [ω]Ap=supAA(1P(A)AωdP)(1P(A)AωppdP)pp<

    where p=pp1. A nonnegative integrable function ω is said to be an A1 weight if there is a constant C>0 such that for any AA

    1P(A)AωdPCω,a.e.onA. (2.3)

    The infimum of all such C is denoted by [ω]A1. We define A=p1Ap.

    We have ApAq, 1pq. It is well known that ωAp, p(1,), if and only if for all n0

    (ωn)((ωpp)n)pp<C.

    When p=1, we have ωA1 if and only if P(Mnωcωn)=1 for all n0.

    Here we give a brief outline on the proof of the above characterization of A1. When P(Mnωcωn)=1 for all n0, by applying limn on both sides of Mnωcωn, we obtain (2.3). Whenever ω satisfies (2.3), for any 1kn, we have ωkCωn. By taking supremum over 1kn, we get MnωCωn.

    We now present the weighted norm inequalities for martingale transforms from [8] and [11,Theorem 5.3].

    Theorem 2.2. Let p(1,), ωA1 and (Ω,F,P) be a complete probability space. If F is generated by atoms and the predictable sequence v=(vn)n0 satisfies vL=supn0vnL<, then for any fLp(ω),

    TvfLp(ω)C[ω]A1fLp(ω). (2.4)

    for some C>0 and there is a constant C>0 such that for any fL1(ω),

    supλ>0λχ{xΩ:|Tvf(x)|>λ}L1(ω)C[ω]A1(1+log(2β[ω]A1))fL1(ω). (2.5)

    Definition 2.4. Let X be a Banach function space, (Ω,F,P) be a complete probability space. Suppose that F is generated by atoms. For any Banach function space X, we write XM if the maximal function M is bounded on X. We write XM if the maximal function M is bounded on X.

    It is well known that the Lebesgue space LpMM, p(1,). According to the Boyd interpolation theorem [5,Chapter 3,Thereom 5.16], the maximal function is bounded on the rearrangement-invariant Banach function space X whenever the Boyd's indices of X are located in (0,1). For brevity, the reader is referred to [5,Chapter 3,Definitions 5.12] for the definition of the Boyd's indices. Consequently, some Lorentz spaces, Orlicz spaces, Lorentz-Karamata spaces and grand Lebesgue spaces belong to M, see [2,Section 6] for the Boyd's indices of the above mentioned function spaces.

    The main result on the boundedness of martingale transform on Banach function spaces is presented and established in this section. We obtain this result by using the Rubio de Francia extrapolation theory and the Zygmund interpolation theorem. We begin with the definition of an operator used in the Rubio de Francia extrapolation theory.

    Definition 3.1. Let X be a Banach function space and (Ω,F,P) be a complete probability space. Suppose that F is generated by atoms and XM. Define

    RXh=k=0Mkh2kMkXX (3.1)

    where MXX denote the operator norm of M.

    As XM, we see that

    RXhXk=0MkhX2kMkXX2hX.

    That is, RX is bounded on X.

    The following proposition gives an embedding result for Banach function space into weighted Lebesgue spaces.

    Proposition 3.1. Let (Ω,F,P) be a complete probability space and X be a Banach function space. Suppose that F is generated by atoms. If there exists a p(1,) such that X1/p and X1/pM, then

    Xh(X1/p)Lp(R(X1/p)h). (3.2)

    Proof: As X1/pM, we find that R(X1/p) is bounded on (X1/p).

    Let fX. For any h(X1/p), Theorem 2.1 yields

    Ω|f|pR(X1/p)hdP|f|pX1/pR(X1/p)h(X1/p)CfpXh(X1/p).

    Therefore, fLp(R(X1/p)h).

    With the above result, we can get rid of the density or the approximation arguments and obtain the mapping properties of the martingale transform on the entire Banach function space.

    We are now ready to present and establish the main result of this paper, the mapping properties of the martingale transforms on Banach function spaces. We follow the ideas from [7] to obtain the following result.

    Theorem 3.2. Let (Ω,F,P) be a complete probability space and X be a Banach function space. Suppose that F is generated by atoms and the predictable sequence v=(vn)n0 satisfies vL=supn0vnL<.

    1. If there exists a p(1,) such that X1/p is a Banach function space and X1/pM, then there exists a constant C>0 such that for any fX

    TvfXCfX. (3.3)

    2. If XM, then there exists a constant C>0 such that for any fX

    TvfwXCfX. (3.4)

    Proof: We first prove (3.3). For any h(X1/p) with h(X1/p)1, the definition of R(X1/p), we see that

    MR(X1/p)hk=0Mk+1h2kMTkXX2MXXR(X1/p)h.

    According to (2.3), we have R(X1/p)hA1 with [R(X1/p)h]A12MXX.

    In addition, we have

    R(X1/p)h(X1/p)k=0Mkh(X1/p)2kMk(X1/p)(X1/p)k=0Mk(X1/p)(X1/p)h(X1/p)2kMk(X1/p)(X1/p)2h(X1/p).

    Thus, R(X1/p) is bounded on (X1/p).

    For any fX, as Theorem 2.2 guarantees that Tv is bounded on Lp(R(X1/p)h), the embedding (3.2) assures that Tvf is well defined and

    Ω|Tvf|pR(X1/p)hdPCΩ|f|pR(X1/p)hdP

    for some C>0 independent of h.

    Theorem 2.1 and the boundedness of R(X1/p) give

    Ω|Tvf|pR(X1/p)hdPC|f|pX1/pR(X1/p)h(X1/p)CfpXh(X1/p).

    Obviously, the definition of R(X1/p) guarantees that

    Ω|Tvf|p|h|dPΩ|Tvf|pR(X1/p)hdPCfpXh(X1/p).

    By taking supremum over h(X1/p) with h(X1/p)1 on both sides of the above inequalities, Definition 2.2. and (2.1) assert that

    TvfpX=|Tvf|pX1/p=sup{Ω|Tvf|p|h|dP:h(X1/p)1}CfpX

    which gives the boundedness of the martingale transform Tv:XX.

    Next, we prove (3.4). Let fX. In view of Theorem 2.2, for any λ>0, Fλ=λχ{xΩ:|Tvf(x)|λ} is well defined. Moreover, for any hX, Theorems 2.1, 2.2 and the boundedness of RX yield a constant independent of λ and h such that

    ΩFλ|h|dPΩFλRXhdPCΩ|f|RXhdPCfXRXhXCfXhX.

    By taking supremum over hX with hX1 on both sides of the above inequalities, Definition 2.2 and (2.1) assert that

    FλXCfX

    for some C>0 independent of λ>0. By taking supremum over λ>0, we obtain

    TvfwX=supλλχ{xΩ:|Tvf(x)|λ}X=supλ>0FλXCfX

    which is (3.4).

    Next, we use the Zygmund interpolation theorem to study the mapping properties of the martingale transforms on Zygmund spaces. The Zygmund spaces are used to capture the mapping properties of the martingale transform for the limiting cases of Lebesgue spaces Lp when p is approaching to 1.

    Definition 3.2. Let αR and (Ω,F,P) be a complete probability space. The Zygmund space L1(logL)α consists of all Lebesgue measurable functions f satisfying

    fL1(logL)α=inf{λ>0:Ω(|f|λ)(log(e+|f|λ))αdP}<.

    When α=0, we have L1(logL)0=L1. The reader is referred to [5,Chapter 4,Section 6] for more information of Zygmund spaces.

    Theorem 3.3. Let α(1,). If T is a linear operator, T:LpLp for some p(1,) and T:L1w-L1 are bounded, then T:L1(logL)α+1L1(logL)α is bounded.

    For the proof of the above result, see [5,Corollary 6.15]. Notice the result given in [5,Corollary 6.15] are presented with the assumption that T is of joint weak type (1,1,p,p). In view of [5,Chapter 4,Proposition 4.2,Definition 4.9 and Theorem 4.11], whenever T:LpLp for some p(1,) and T:L1w-L1 are bounded, T is of joint weak type (1,1,p,p).

    In view of Theorems 2.2, the martingale transform Tv satisfies the conditions in Theorem 3.3, therefore, we obtain the following result.

    Theorem 3.4. Let (Ω,F,P) be a complete probability space. If F is generated by atoms and the predictable sequence v=(vn)n0 satisfies vL=supn0vnL<, then for any α>1, the martingale transform Tv:L1(logL)α+1L1(logL)α is bounded.

    The above result yields and extends the boundedness of Tv:L1(logL)L1.

    In this section, we apply Theorem 3.2 to some concrete Banach function spaces, namely, the rearrangement-invariant Banach function spaces, the Orlicz spaces, the Lorentz-Karamata spaces, the Lebesgue spaces with variable exponents and the Morrey type spaces built on Banach function spaces.

    For any fM, define

    df(λ)=P({xΩ:|f(x)|>λ}),λ>0.

    The decreasing rearrangement of f is defined as

    f(t)=inf{λ:df(λ)t},t0.

    Two measurable functions f and g are equimeasurable if for any λ0, df(λ)=dg(λ). We say that a Banach function space X is rearrangement-invariant if for every pair of equimeasurable functions f,g, we have fX=gX.

    If (Ω,P) is a nonatomic measure space, in view of the Luxemburg representation theorem [5,Chapter 2,Theorem 4.10], for any rearrangement-invariant Banach function space X, there exists a norm ρX satisfies Items (1)–(3) of Definition 2.1 such that fX=ρX(f). We write ˉX for the Banach function space on [0,1] endowed with the norm ρX.

    The reader may consult [2,3] for studies of the martingale function spaces built on rearrangement-invariant function spaces.

    For any s0 and Lebesgue measurable function f on [0,1], define (Dsf)(t)=f(st), t(0,). Let DsˉXˉX be the operator norm of Ds on ˉX. We recall the definition of Boyd's indices for rearrangement-invariant Banach function spaces from [5,Chapter 3,Definition 5.12].

    Definition 4.1. Let (Ω,P) be a nonatomic measure space and X be a rearrangement-invariant Banach function space. Define the lower Boyd index of X, α_X, and the upper Boyd index of X, ˉαX, by

    α_X=inft(1,)logD1/tˉXlogt=limtlogD1/tˉXlogt,ˉαX=supt(0,1)logD1/tˉXlogt=limt0+logD1/tˉXlogt,

    respectively.

    According to [12,Theorem B], we have the following boundedness result of maximal function M on rearrangement-invariant Banach function spaces.

    Theorem 4.1. Let (Ω,P) be nonatomic and X be a rearrangement-invariant Banach function space. The maximal function M is bounded on X if and only if ˉαX<1.

    The results in [12,Theorem B] are for continuous martingale, the proof of the discrete martingale is similar, see also [5,Chapter 3,Theorem 5.17].

    The boundedness of the martingale transform for rearrangement-invariant Banach function space X satisfying 0<α_XˉαX<1 follows from Boyd's interpolation theorem [5,Chapter 3,Theorem 5.16], see also [3]. It is also a special case of [2,Theorem 4.4].

    Theorem 4.2. Let (Ω,F,P) be a complete probability space and X be a rearrangement-invariant Banach function space. Suppose that F is generated by atoms and the predictable sequence v=(vn)n0 satisfies vL=supn0vnL<.

    If 0<α_XˉαX<1, then there exists a constant C>0 such that for any fX,

    TvfXCfX.

    The reader is referred to [2,Theorem 4.4] for a generalization of the preceding result.

    Next, we consider the mapping properties of the martingale transform Tv in the limiting case, 0<α_XˉαX1.

    Theorem 4.3. Let (Ω,F,P) be a complete probability space and X be a rearrangement-invariant Banach function space. Suppose that F is generated by atoms and the predictable sequence v=(vn)n0 satisfies vL=supn0vnL<.

    If 0<α_XˉαX1, then there exists a constant C>0 such that for any fX

    TvfwXCfX.

    Proof: As F is generated by atoms, (Ω,P) is a nonatomic measure space. In addition, [5,Chapter 3,(5.33)] shows that ˉαX=1α_X<1, therefore, Theorem 4.1 asserts that the Hardy-Littlewood maximal function is bounded on XM. Consequently, Theorem 3.2 yields a constant C>0 such that for any fX,

    TvfwXCfX.

    We apply Theorem 4.3 to two concrete rearrangement-invariant Banach function spaces, the Lorentz-Karamata spaces and the Orlicz spaces.

    We first consider the Lorentz-Karamata spaces. We recall the definition of the Lorentz-Karamata spaces from [13].

    Definition 4.2. A Lebesgue measurable function b:[1,)(0,) is called as a slowly varying function if for any ϵ>0

    1. the function ttϵb(t) is equivalent to a non-decreasing function on [1,), and

    2. the function ttϵb(t) is equivalent to a non-increasing function on [1,).

    For any slowly varying function b, define

    γb(t)=b(t1),0<t1.

    Definition 4.3. Let 0<r,p<, b be a slowly varying function and (Ω,F,P) be a complete probability space. The Lorentz-Karamata space Lr,p,b consists of all fM satisfying

    fLr,p,b=(10tpr1(γb(t)f(t))pdt)1p<.

    The reader is referred to [13,14,15] for the studies of martingale functions spaces built on Lorentz-Karamata spaces. In addition, it had been further extended to the Orlicz-Karamata spaces in [16].

    When 1<r,p<, the Lorentz-Karamata space Lr,p,b is a rearrangement-invariant Banach function space. Furthermore, according to [17,Proposition 6.1], we have α_Lr,p,b=ˉαLr,p,b=1r. Notice that the Boyd's indices defined in [17,Proposition 6.1] are reciprocals of the ones defined in Definition 4.1.

    When b=1, the Lorentz-Karamata space Lr,p,b becomes the Lorentz space Lr,p.

    We now apply Theorem 3.2 to obtain the mapping properties of the martingale transform on L1,p,b.

    Theorem 4.4. Let p(1,), b be a slowly varying function and (Ω,F,P) be a complete probability space. If F is generated by atoms and the predictable sequence v=(vn)n0 satisfies vL=supn0vnL<, then there exists a constant C>0 such that for any fL1,p,b,

    TvwL1,p,bCfL1,p,b.

    When b=1. the above theorem gives the mapping properties of Tv on the Lorentz space L1,p.

    Next, we consider the Orlicz spaces. A function Φ:[0,][0,] is called a Young's function if it is convex, left-continuous and Φ(0)=0.

    Definition 4.4. Let Φ be a Young's function. The Orlicz space LΦ consists of all fM satisfying

    fLΦ=inf{λ:ΩΦ(|f|/λ)dP}<.

    For any Young's function, we write Φ2 if there exists a constant C>0 such that for any t0

    Φ(2t)CΦ(t).

    The complementary function (the conjugate function) of Φ is defined as

    ˜Φ(t)=sup{stΦ(s):s0},t0.

    It is well known that the complementary function of ˜Φ is Φ and

    (LΦ)=L˜Φ.

    Theorem 3.2 yields the mapping properties of the martingale transform on the Orlicz space LΦ.

    Theorem 4.5. Let Φ be a Young's function and (Ω,F,P) be a complete probability space. Suppose that Φ2. If F is generated by atoms and the predictable sequence v=(vn)n0 satisfies vL=supn0vnL<, then there exists a constant C>0 such that for any fLΦ,

    TvwLΦCfLΦ. (4.1)

    Proof: As Φ2, [18,Volume Ⅱ,Corollary 2.b.5] asserts that α_Lϕ>0. Notice that the Boyd's indices defined in [18,Volume Ⅱ,Corollary 2.b.5] are reciprocals of the ones defined in Definition 4.1. Since 1ˉαL˜Φ=α_LΦ>0. Thus, Theorem 4.1 guarantees that the maximal function M is bounded on L˜Φ. That is, LΦM. Theorem 3.2 yields (4.1).

    For the martingale transform on exponential Orlicz spaces, the reader is referred to [19].

    We apply Theorem 3.2 to Lebesgue spaces with variable exponents. Let P(Ω) be the collection of all measurable functions p():Ω(0,). For any measurable set AΩ, write

    p+(A)=supxAp(x),p(A)=infxAp(x),

    p+=p+(Ω) and p=p(Ω).

    Definition 4.5. Let p():Ω[1,] be a measurable function. The Lebesgue space with variable exponent Lp() consists of all measurable functions f satisfying

    fLp()=inf{λ>0:ρ(|f(x)|/λ)1}<

    where

    ρ(f)={xΩ:p(x)}|f(x)|p(x)dx+fχ{xΩ:p(x)=}L.

    We call p(x) the exponent function of Lp().

    According to [20,Theorem 3.2.13], Lp() is a Banach function space. Furthermore, we have (Lp())=Lp() where 1p(x)+1p(x)=1 [20,Theorem 3.2.13].

    The martingale theory with variable exponents was begun with the fundamental paper [9] by Jiao et al. Now we recall a condition that guarantees the boundedness of the maximal function M on Lp(), see [9,Theorem 3.5] and [21].

    Theorem 4.6. Let (Ω,F,P) be a complete probability space. Suppose that F is generated by atoms. If p()P(Ω) satisfies 1<pp+< and

    P(A)p(A)p+(A)Kp(),AA (4.2)

    for some Kp()>0, then there exists a C>0 such that for any fLp(),

    MfLp()CfLp().

    The condition (4.2) is very crucial for the studies of maximal function on Lebesgue spaces with variable exponents defined on probability spaces, it replaces the log-Hölder continuity [20,Defintion 4.1.1] for the study of Hardy-Littlewood maximal function for Lp() on Rn.

    The reader is referred to [9,22,23] for more information on the maximal function on martingale function spaces built on Lebesgue spaces with variable exponents.

    Lemma 4.7. Let (Ω,F,P) be a complete probability space. Suppose that F is generated by atoms. If p()P(Ω) satisfies 1<pp+< and (4.2), then p() also satisfies (4.2).

    Proof: For any AA, since p(A)=(p+(A)) and p+(A)=(p(A)), we find that

    P(A)p(A)p+(A)=P(A)(p+(A))(p(A))=P(A)p+(A)p+(A)1p(A)p(A)1=P(A)p(A)p+(A)(p+(A)1)(p(A)1)P(A)p(A)p+(A)(p1)2K1(p1)2p().

    Thus, p() satisfies (4.2).

    We are now ready to establish the boundedness of the martingale transform Tv on Lp().

    Theorem 4.8. Let (Ω,F,P) be a complete probability space. Suppose that F is generated by atoms and the predictable sequence v=(vn)n0 satisfies vL=supn0vnL<.

    If p()P(Ω) satisfies 1<pp+< and (4.2), then there exists a constant C>0 such that for any fLp()

    TvfLp()CfLp(). (4.3)

    Proof: Take r(1,p). Write q()=p()/r. Thus, Lp()/r is a Banach function space.

    We find that

    P(A)q(A)q+(A)=P(A)p(A)p+(A)rK1rp().

    Therefore, q() satisfies (4.2). Lemma 4.7 guarantees that q()=(p()/r) also satisfies (4.2). In addition, 1<(p()/r)(p()/r)+<. Theorem 4.6 assures that the maximal function is bounded on L(p()/r)=(Lp()/r). Consequently, we are allowed to apply Theorem 3.2 and obtain the boundedness of Tv on Lp().

    We apply Theorem 3.2 to study the martingale transform on Morrey type spaces [1,6]. We recall the definition of Morrey type spaces from [1,Definition 2.4].

    Definition 4.6. Let (Ω,F,P) be a complete probability space, X be a Banach function space. Suppose that F is generated by atoms and u:A(0,). The Morrey space MX,u consists of all fM satisfying

    fMX,u=supBA1u(B)χBfX<.

    When X=Lp, p(1,) and u(B)=P(B)λ+1p, λR, the Morrey space MX,u becomes the Morrey space Lp,λ introduced and studied in [6].

    As the Morrey space MX,u is defined on F which is generated by atoms, it satisfies the assumptions that F={Fn} are generated by at most countable families of atoms given in [1,6]. Thus, the results in [1,6] apply to MX,u.

    A result on the boundedness of martingale transform on Morrey spaces MX,u is given in [1,Theorem 3.4]. It requires the assumption that the martingale transform Tv is bounded on X. As the boundedness of the martingale transform Tv on Banach function space X is obtained in Theorem 3.2, the result in [1,Theorem 3.4] can be refined as follows.

    Theorem 4.9. Let (Ω,F,P) be a complete probability space. Let X be a Banach function space. Suppose that F is generated by atoms and u:T(0,). If there exists a p(1,) such that XM, X1/p is a Banach function space, X1/pM and there exists a constant C>0 such that for any m0 and BA(Fm),

    mj=0u(Bj)u(B)χBXχBjXC (4.4)

    where Bj is the unique element in A(Fmj) containing B, then Tv:MX,uMX,u is bounded.

    Proof: As X1/p is a Banach function space and X1/pM, Theorem 3.2 assures that Tv:XX is bounded. Therefore, [1,Theorem 3.4] yields the boundedness of Tv:MX,uMX,u.

    We apply Theorem 4.9 to the Lorentz-Karamata-Morrey spaces and the Morrey spaces with variable exponent.

    We first consider the Lorentz-Karamata-Morrey spaces. Let, θ[0,1), r,p(1,), b be a slowly varying function and uθ(A)=χAθLr,p,b, AA. We denote the Lorentz-Karamata-Morrey space MLr,p,b,uθ by Mθr,p,b.

    We first state some results on the Lorentz-Karamata space from [24]. In view of [24,Proposition 3.4.33], we have

    (a0tpr1(γb(t))pdt)1pa1rγb(a).

    Thus, for any BF, we get

    χBLr,p,b(P(B))1rγb(P(B)).

    We state a condition used to study the Morrey spaces. Let β>0. If for any I,JA with IJ, we have

    P(I)P(J)12β, (4.5)

    then we say that F is generated by β-atoms.

    Consequently, for any B,DA with BD, there exists a constant C>0 such that

    χBLr,p,bχDLr,p,bC(P(B))1rγb(P(B))(P(D))1rγb(P(D)).

    Since b satisfies Item (1) of Definition 4.2, for any ϵ(0,1r), there exists a constant C>0 such that

    χBLr,p,bχDLr,p,bC(P(B)P(D))1rϵ. (4.6)

    As a result of the above inequality, we find that

    mj=0uθ(Bj)uθ(B)χBLr,p,bχBjLr,p,b=mj=0(χBLr,p,bχBjLr,p,b)1θCmj=0(P(B)P(D))(1θ)(1rϵ)

    for some C>0. Whenever F is generated by β-atoms, (4.5) yields

    mj=0uθ(Bj)uθ(B)χBLr,p,bχBjLr,p,bCmj=0(12β)(mj)(1θ)(1rϵ)<C

    for some C>0. The above inequality shows that uθ satisfies (4.4) with X=Lr,p,b. We are allowed to use Theorem 4.9 to obtain the boundedness of the martingale transform on the Lorentz-Karamata-Morrey space in the following corollary.

    Corollary 4.10. Let θ[0,1), β0, r,p(1,), b be a slowly varying function and (Ω,F,P) be a complete probability space. If F is generated by β-atoms and the predictable sequence v=(vn)n0 satisfies vL=supn0vnL<, then the martingale transform Tv is bounded on Mθr,p,b.

    Proof: As r(1,), in view [17,Proposition 6.1], we have α_Lr,p,b=ˉαLr,p,b=1r. Theorem 4.1 guarantees that Lr,p,bM. In addition, for any q(1,min(r,p)), we have (Lr,p,b)1/q=Lr/q,p/q,bq and ˉαLr/q,p/q,bq=1qr<1. Thus, Lr/q,p/q,bq is a Banach function space and Theorem 4.1 yields Lr/q,p/q,bqM. The boundedness of the martingale transform Tv:Mθr,p,bMθr,p,b is guaranteed by Theorem 4.9.

    We now turn to the Morrey spaces with variable exponents. Let p()P(Ω) with 1<pp+< satisfying (4.2). Let θ[0,1). Define uθ(B)=χBθLp(), BA. We denote the Morrey space with variable exponent MLp(),uθ by Mθp().

    For any B,DA with BD In view of (2.2), we have

    χDP(B)P(D)MχB. (4.7)

    For any r(1,p), Lp()/r is a Banach function space and p()/r satisfies (4.2). Theorem 4.6 assures that the maximal function M is bounded on Lp()/r. By applying the norm Lp()/r on both sides of (4.7),

    χDLp()/rP(B)P(D)MχBLp()/rCχBLp()/r.

    That is,

    P(B)P(D)CχBLp()/rχDLp()/r=C(χBLp()χDLp())r.

    As p() satisfies (4.2), the maximal function is bounded on Lp(). According to [1,Lemma 2.8], we find that for any AA, we have

    P(A)χALp()χALp()CP(A).

    Consequently,

    χBLp()χDLp()C(P(B)P(D))11r. (4.8)

    For any θ[0,1), we get

    mj=0uθ(Bj)uθ(B)χBLp()χBjLp()=mj=0(χBLp()χBjLp())1θCmj=0(P(B)P(Bj))(1θ)(11r).

    Whenever F is generated by β-atoms with β>0, we find that

    mj=0uθ(Bj)uθ(B)χBLp()χBjLp()Cmj=0(12β)(mj)(1θ)(11r)<C (4.9)

    for some C>0. Thus, uθ fulfills (4.4) with X=Lp(). Consequently, we obtain the following result for martingale transform on Mθp().

    Corollary 4.11. Let θ[0,1), β0 and (Ω,F,P) be a complete probability space. Let p()P(Ω) satisfy 1<pp+< and (4.2). If F is generated by β-atoms and the predictable sequence v=(vn)n0 satisfies vL=supn0vnL<, then the martingale transform Tv is bounded on Mθp().

    Proof: Since 1<pp+< and p() satisfies (4.2), Theorem 4.6 guarantees that Lp()M. For any r(1,p), Lp()/r is a Banach function space and Lemma 4.7 assures that Lp()/rM. In addition, (4.9) asserts that uθ fulfills (4.4) with X=Lp(). Therefore, Theorem 4.9 yields the boundedness of Tv on Mθp().

    The author thanks the reviewers for their valuable suggestions, especially, correcting a mistake on Theorem 4.6.

    The authors declare there is no conflicts of interest.

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