In contrast to existing works on stochastic averaging on finite intervals, we establish an averaging principle on the whole real axis, i.e. the so-called second Bogolyubov theorem, for semilinear stochastic ordinary differential equations in Hilbert space with Poisson stable (in particular, periodic, quasi-periodic, almost periodic, almost automorphic etc) coefficients. Under some appropriate conditions we prove that there exists a unique recurrent solution to the original equation, which possesses the same recurrence property as the coefficients, in a small neighborhood of the stationary solution to the averaged equation, and this recurrent solution converges to the stationary solution of averaged equation uniformly on the whole real axis when the time scale approaches zero.
Citation: David Cheban, Zhenxin Liu. Averaging principle on infinite intervals for stochastic ordinary differential equations[J]. Electronic Research Archive, 2021, 29(4): 2791-2817. doi: 10.3934/era.2021014
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In contrast to existing works on stochastic averaging on finite intervals, we establish an averaging principle on the whole real axis, i.e. the so-called second Bogolyubov theorem, for semilinear stochastic ordinary differential equations in Hilbert space with Poisson stable (in particular, periodic, quasi-periodic, almost periodic, almost automorphic etc) coefficients. Under some appropriate conditions we prove that there exists a unique recurrent solution to the original equation, which possesses the same recurrence property as the coefficients, in a small neighborhood of the stationary solution to the averaged equation, and this recurrent solution converges to the stationary solution of averaged equation uniformly on the whole real axis when the time scale approaches zero.
Let T be the Calderón-Zygmund singular integral operator and b be a locally integrable function on Rn. The commutator generated by b and T is defined by [b,T]f=bT(f)−T(bf). The investigation of the commutator begins with Coifman-Rochberg-Weiss pioneering study and classical result (see [6]). The classical result of Coifman, Rochberg and Weiss (see [6]) states that the commutator [b,T]f=T(bf)−bTf is bounded on Lp(Rn) for 1<p<∞ if and only if b∈BMO(Rn). The major reason for considering the problem of commutators is that the boundedness of commutator can produces some characterizations of function spaces (see [1,6]). Chanillo (see [1]) proves a similar result when T is replaced by the fractional integral operator. In [11], the boundedness properties of the commutators for the extreme values of p are obtained. In recent years, the theory of Herz space and Herz type Hardy space, as a local version of Lebesgue space and Hardy space, have been developed (see [8,9,12,13]). The main purpose of this paper is to establish the endpoint continuity properties of some multilinear operators related to certain non-convolution type fractional singular integral operators on Herz and Herz type Hardy spaces.
First, let us introduce some notations (see [8,9,10,12,13,15]). Throughout this paper, Q will denote a cube of Rn with sides parallel to the axes. For a cube Q and a locally integrable function f, let fQ=|Q|−1∫Qf(x)dx and f#(x)=supQ∋x|Q|−1∫Q|f(y)−fQ|dy. Moreover, f is said to belong to BMO(Rn) if f#∈L∞ and define ||f||BMO=||f#||L∞; We also define the central BMO space by CMO(Rn), which is the space of those functions f∈Lloc(Rn) such that
||f||CMO=supr>1|Q(0,r)|−1∫Q|f(y)−fQ|dy<∞. |
It is well-known that (see [9,10])
||f||CMO≈supr>1infc∈C|Q(0,r)|−1∫Q|f(x)−c|dx. |
For k∈Z, define Bk={x∈Rn:|x|≤2k} and Ck=Bk∖Bk−1. Denote by χk the characteristic function of Ck and ˜χk the characteristic function of Ck for k≥1 and ˜χ0 the characteristic function of B0.
Definition 1. Let 0<p<∞ and α∈R.
(1) The homogeneous Herz space ˙Kαp(Rn) is defined by
˙Kαp(Rn)={f∈Lploc(Rn∖{0}):||f||˙Kαp<∞}, |
where
||f||˙Kαp=∞∑k=−∞2kα||fχk||Lp; |
(2) The nonhomogeneous Herz space Kαp(Rn) is defined by
Kαp(Rn)={f∈Lploc(Rn):||f||Kαp<∞}, |
where
||f||Kαp=∞∑k=02kα||f˜χk||Lp. |
If α=n(1−1/p), we denote that ˙Kαp(Rn)=˙Kp(Rn), Kαp(Rn)=Kp(Rn).
Definition 2. Let 0<δ<n and 1<p<n/δ. We shall call Bδp(Rn) the space of those functions f on Rn such that
||f||Bδp=supd>1d−n(1/p−δ/n)||fχQ(0,d)||Lp<∞. |
Definition 3. Let 1<p<∞.
(1) The homogeneous Herz type Hardy space H˙Kp(Rn) is defined by
H˙Kp(Rn)={f∈S′(Rn):G(f)∈˙Kp(Rn)}, |
where
||f||H˙Kp=||G(f)||˙Kp. |
(2) The nonhomogeneous Herz type Hardy space HKp(Rn) is defined by
HKp(Rn)={f∈S′(Rn):G(f)∈Kp(Rn)}, |
where
||f||HKp=||G(f)||Kp. |
where G(f) is the grand maximal function of f.
The Herz type Hardy spaces have the atomic decomposition characterization.
Definition 4. Let 1<p<∞. A function a(x) on Rn is called a central (n(1−1/p),p)-atom (or a central (n(1−1/p),p)-atom of restrict type), if
1) Suppa⊂B(0,d) for some d>0 (or for some d≥1),
2) ||a||Lp≤|B(0,d)|1/p−1,
3) ∫a(x)dx=0.
Lemma 1. (see [9,13]) Let 1<p<∞. A temperate distribution f belongs to H˙Kp(Rn)(or HKp(Rn)) if and only if there exist central (n(1−1/p),p)-atoms(or central (n(1−1/p),p)-atoms of restrict type) aj supported on Bj=B(0,2j) and constants λj, ∑j|λj|<∞ such that f=∑∞j=−∞λjaj (or f=∑∞j=0λjaj)in the S′(Rn) sense, and
||f||H˙Kp( or ||f||HKp)≈∑j|λj|. |
In this paper, we will consider a class of multilinear operators related to some non-convolution type singular integral operators, whose definition are following.
Let m be a positive integer and A be a function on Rn. We denote that
Rm+1(A;x,y)=A(x)−∑|β|≤m1β!DβA(y)(x−y)β |
and
Qm+1(A;x,y)=Rm(A;x,y)−∑|β|=m1β!DβA(x)(x−y)β. |
Definition 5. Fixed ε>0 and 0<δ<n. Let Tδ:S→S′ be a linear operator. Tδ is called a fractional singular integral operator if there exists a locally integrable function K(x,y) on Rn×Rn such that
Tδ(f)(x)=∫RnK(x,y)f(y)dy |
for every bounded and compactly supported function f, where K satisfies:
|K(x,y)|≤C|x−y|−n+δ |
and
|K(y,x)−K(z,x)|+|K(x,y)−K(x,z)|≤C|y−z|ε|x−z|−n−ε+δ |
if 2|y−z|≤|x−z|. The multilinear operator related to the fractional singular integral operator Tδ is defined by
TAδ(f)(x)=∫RnRm+1(A;x,y)|x−y|mK(x,y)f(y)dy; |
We also consider the variant of TAδ, which is defined by
˜TAδ(f)(x)=∫RnQm+1(A;x,y)|x−y|mK(x,y)f(y)dy. |
Note that when m=0, TAδ is just the commutators of Tδ and A (see [1,6,11,14]). It is well known that multilinear operator, as a non-trivial extension of commutator, is of great interest in harmonic analysis and has been widely studied by many authors (see [3,4,5]). In [7], the weighted Lp(p>1)-boundedness of the multilinear operator related to some singular integral operator are obtained. In [2], the weak (H1, L1)-boundedness of the multilinear operator related to some singular integral operator are obtained. In this paper, we will study the endpoint continuity properties of the multilinear operators TAδ and ˜TAδ on Herz and Herz type Hardy spaces.
Now we state our results as following.
Theorem 1. Let 0<δ<n, 1<p<n/δ and DβA∈BMO(Rn) for all β with |β|=m. Suppose that TAδ is the same as in Definition 5 such that Tδ is bounded from Lp(Rn) to Lq(Rn) for any p,q∈(1,+∞] with 1/q=1/p−δ/n. Then TAδ is bounded from Bδp(Rn) to CMO(Rn).
Theorem 2. Let 0<δ<n, 1<p<n/δ, 1/q=1/p−δ/n and DβA∈BMO(Rn) for all β with |β|=m. Suppose that ˜TAδ is the same as in Definition 5 such that ˜TAδ is bounded from Lp(Rn) to Lq(Rn) for any p,q∈(1,+∞) with 1/q=1/p−δ/n. Then ˜TAδ is bounded from H˙Kp(Rn) to ˙Kαq(Rn) with α=n(1−1/p).
Theorem 3. Let 0<δ<n, 1<p<n/δ and DβA∈BMO(Rn) for all β with |β|=m. Suppose that ˜TAδ is the same as in Definition 5 such that ˜TAδ is bounded from Lp(Rn) to Lq(Rn) for any p,q∈(1,+∞) with 1/q=1/p−δ/n. Then the following two statements are equivalent:
(ⅰ) ˜TAδ is bounded from Bδp(Rn) to CMO(Rn);
(ⅱ) for any cube Q and z∈3Q∖2Q, there is
1|Q|∫Q|∑|β|=m1β!|DβA(x)−(DβA)Q|∫(4Q)cKβ(z,y)f(y)dy|dx≤C||f||Bδp, |
where Kβ(z,y)=(z−y)β|z−y|mK(z,y) for |β|=m.
Remark. Theorem 2 is also hold for nonhomogeneous Herz and Herz type Hardy space.
To prove the theorem, we need the following lemma.
Lemma 2. (see [5]) Let A be a function on Rn and DβA∈Lq(Rn) for |β|=m and some q>n. Then
|Rm(A;x,y)|≤C|x−y|m∑|β|=m(1|˜Q(x,y)|∫˜Q(x,y)|DβA(z)|qdz)1/q, |
where ˜Q(x,y) is the cube centered at x and having side length 5√n|x−y|.
Proof of Theorem 1. It suffices to prove that there exists a constant CQ such that
1|Q|∫Q|TAδ(f)(x)−CQ|dx≤C||f||Bδp |
holds for any cube Q=Q(0,d) with d>1. Fix a cube Q=Q(0,d) with d>1. Let ˜Q=5√nQ and ˜A(x)=A(x)−∑|β|=m1β!(DβA)˜Qxβ, then Rm+1(A;x,y)=Rm+1(˜A;x,y) and Dβ˜A=DβA−(DβA)˜Q for all β with |β|=m. We write, for f1=fχ˜Q and f2=fχRn∖˜Q,
TAδ(f)(x)=∫RnRm+1(˜A;x,y)|x−y|mK(x,y)f(y)dy=∫RnRm(˜A;x,y)|x−y|mK(x,y)f1(y)dy−∑|β|=m1β!∫RnK(x,y)(x−y)β|x−y|mDβ˜A(y)f1(y)dy+∫RnRm+1(˜A;x,y)|x−y|mK(x,y)f2(y)dy, |
then
1|Q|∫Q|TAδ(f)(x)−T˜Aδ(f2)(0)|dx≤1|Q|∫Q|Tδ(Rm(˜A;x,⋅)|x−⋅|mf1)(x)|dx+∑|β|=m1β!1|Q|∫Q|Tδ((x−⋅)β|x−⋅|mDβ˜Af1)(x)|dx+|T˜Aδ(f2)(x)−T˜Aδ(f2)(0)|dx:=I+II+III. |
For I, note that for x∈Q and y∈˜Q, using Lemma 2, we get
Rm(˜A;x,y)≤C|x−y|m∑|β|=m||DβA||BMO, |
thus, by the Lp(Rn) to Lq(Rn)-boundedness of TAδ for 1<p,q<∞ with 1/q=1/p−δ/n, we get
I≤C|Q|∫Q|Tδ(∑|β|=m||DβA||BMOf1)(x)|dx≤C∑|β|=m||DβA||BMO(1|Q|∫Q|Tδ(f1)(x)|qdx)1/q≤C∑|β|=m||DβA||BMO|Q|−1/q||f1||Lp≤C∑|β|=m||DβA||BMOr−n(1/p−δ/n)||fχ˜Q||Lp≤C∑|β|=m||DβA||BMO||f||Bδp. |
For II, taking 1<s<p such that 1/r=1/s−δ/n, by the (Ls,Lr)-boundedness of Tδ and Holder's inequality, we gain
II≤C|Q|∫Q|Tδ(∑|β|=m(DβA−(DβA)˜Q)f1)(x)|dx≤C∑|β|=m(1|Q|∫Q|Tδ((DβA−(DβA)˜Q)f1)(x)|rdx)1/r≤C|Q|−1/r∑|β|=m||(DβA−(DβA)˜Q)f1||Ls≤C|Q|−1/r||f1||Lp∑|β|=m(1|Q|∫˜Q|DβA(y)−(DβA)˜Q|ps/(p−s)dy)(p−s)/(ps)|Q|(p−s)/(ps)≤C∑|β|=m||DβA||BMOr−n/q||fχ˜Q||Lp≤C∑|β|=m||DβA||BMO||f||Bδp. |
To estimate III, we write
T˜Aδ(f2)(x)−T˜Aδ(f2)(0)=∫Rn[K(x,y)|x−y|m−K(0,y)|y|m]Rm(˜A;x,y)f2(y)dy+∫RnK(0,y)f2(y)|y|m[Rm(˜A;x,y)−Rm(˜A;0,y)]dy−∑|β|=m1β!∫Rn(K(x,y)(x−y)β|x−y|m−K(0,y)(−y)β|y|m)Dβ˜A(y)f2(y)dy:=III1+III2+III3. |
By Lemma 2 and the following inequality (see [15])
|bQ1−bQ2|≤Clog(|Q2|/|Q1|)||b||BMO for Q1⊂Q2, |
we know that, for x∈Q and y∈2k+1˜Q∖2k˜Q,
|Rm(˜A;x,y)|≤C|x−y|m∑|β|=m(||DβA||BMO+|(DβA)˜Q(x,y)−(DβA)˜Q|)≤Ck|x−y|m∑|β|=m||DβA||BMO. |
Note that |x−y|∼|y| for x∈Q and y∈Rn∖˜Q, we obtain, by the condition of K,
|III1|≤C∫Rn(|x||y|m+n+1−δ+|x|ε|y|m+n+ε−δ)|Rm(˜A;x,y)||f2(y)|dy≤C∑|β|=m||DβA||BMO∞∑k=0∫2k+1˜Q∖2k˜Qk(|x||y|n+1−δ+|x|ε|y|n+ε−δ)|f(y)|dy≤C∑|β|=m||DβA||BMO∞∑k=1k(2−k+2−εk)(2kr)−n(1/p−δ/n)||fχ2k˜Q||Lp≤C∑|β|=m||DβA||BMO∞∑k=1k(2−k+2−εk)||f||Bδp≤C∑|β|=m||DβA||BMO||f||Bδp. |
For III2, by the formula (see [5]):
Rm(˜A;x,y)−Rm(˜A;x0,y)=∑|γ|<m1γ!Rm−|γ|(Dγ˜A;x,x0)(x−y)γ |
and Lemma 2, we have
|Rm(˜A;x,y)−Rm(˜A;x0,y)|≤C∑|γ|<m∑|β|=m|x−x0|m−|γ||x−y||γ|||DβA||BMO, |
thus, similar to the estimates of III1, we get
|III2|≤C∑|β|=m||DβA||BMO∞∑k=0∫2k+1˜Q∖2k˜Q|x||y|n+1−δ|f(y)|dy≤C∑|β|=m||DβA||BMO||f||Bδp. |
For III3, by Holder's inequality, similar to the estimates of III1, we get
|III3|≤C∑|β|=m∞∑k=0∫2k+1˜Q∖2k˜Q(|x||y|n+1−δ+|x|ε|y|n+ε−δ)|Dβ˜A(y)||f(y)|dy≤C∑|β|=m∞∑k=1(2−k+2−εk)(2kr)−n(1/p−δ/n)(|2k˜Q|−1∫2k˜Q|DβA(y)−(DβA)˜Q|p′dy)1/p′||fχ2k˜Q||Lp≤C∑|β|=m||DβA||BMO∞∑k=1(2−k+2−εk)(2kr)−n(1/p−δ/n)||fχ2k˜Q||Lp≤C∑|β|=m||DβA||BMO||f||Bδp. |
Thus
III≤C∑|β|=m||DβA||BMO||f||Bδp, |
which together with the estimates for I and II yields the desired result. This finishes the proof of Theorem 1.
Proof of Theorem 2. Let f∈H˙Kp(Rn), by Lemma 1, f=∑∞j=−∞λjaj, where a′js are the central (n(1−1/p),p)-atom with suppaj⊂Bj=B(0,2j) and ||f||H˙Kp≈∑j|λj|. We write
||˜TAδ(f)||˙Kαq=∞∑k=−∞2kn(1−1/p)||χk˜TAδ(f)||Lq≤∞∑k=−∞2kn(1−1/p)k−1∑j=−∞|λj|||χk˜TAδ(aj)||Lq+∞∑k=−∞2kn(1−1/p)∞∑j=k|λj|||χk˜TAδ(aj)||Lq=J+JJ. |
For JJ, by the (Lp,Lq)-boundedness of ˜TAδ for 1/q=1/p−δ/n, we get
JJ≤C∞∑k=−∞2kn(1−1/p)∞∑j=k|λj|||aj||Lp≤C∞∑k=−∞2kn(1−1/p)∞∑j=k|λj|2jn(1/p−1)≤C∞∑j=−∞|λj|j∑k=−∞2(k−j)n(1−1/p)≤C∞∑j=−∞|λj|≤C||f||H˙Kp. |
To obtain the estimate of J, we denote that ˜A(x)=A(x)−∑|β|=m1β!(DβA)2Bjxβ. Then Qm(A;x,y)=Qm(˜A;x,y) and Qm+1(A;x,y)=Rm(A;x,y)−∑|β|=m1β!(x−y)βDβA(x). We write, by the vanishing moment of a and for x∈Ck with k≥j+1,
˜TAδ(aj)(x)=∫RnK(x,y)Rm(A;x,y)|x−y|maj(y)dy−∑|β|=m1β!∫RnK(x,y)Dβ˜A(x)(x−y)β|x−y|maj(y)dy=∫Rn[K(x,y)|x−y|m−K(x,0)|x|m]Rm(˜A;x,y)aj(y)dy+∫RnK(x,0)|x|m[Rm(˜A;x,y)−Rm(˜A;x,0)]aj(y)dy−∑|β|=m1β!∫Rn[K(x,y)(x−y)β|x−y|m−K(x,0)xβ|x|m]Dβ˜A(x)aj(y)dy. |
Similar to the proof of Theorem 1, we obtain
|˜TAδ(aj)(x)|≤C∫Rn[|y||x|m+n+1−δ+|y|ε|x|m+n+ε−δ]|Rm(˜A;x,y)||aj(y)|dy+C∑|β|=m∫Rn[|y||x|n+1−δ+|y|ε|x|n+ε−δ]|Dβ˜A(x)||aj(y)|dy≤C∑|β|=m||DβA||BMO[2j2k(n+1−δ)+2jε2k(n+ε−δ)]+C∑|β|=m[2j2k(n+1−δ)+2jε2k(n+ε−δ)]|Dβ˜A(x)|, |
thus
J≤C∑|β|=m||DβA||BMO∞∑k=−∞2kn(1−1/p)k−1∑j=−∞|λj|[2j2k(n+1−δ)+2jε2k(n+ε−δ)]2kn/q+C∑|β|=m∞∑k=−∞2kn(1−1/p)k−1∑j=−∞|λj|[2j2k(n+1−δ)+2jε2k(n+ε−δ)](∫Bk|Dβ˜A(x)|qdx)1/q≤C∑|β|=m||DβA||BMO∞∑k=−∞2kn(1−δ/n)k−1∑j=−∞|λj|[2j2k(n+1−δ)+2jε2k(n+ε−δ)]≤C∑|β|=m||DβA||BMO∞∑j=−∞|λj|∞∑k=j+1[2j−k+2(j−k)ε]≤C∑|β|=m||DβA||BMO∞∑j=−∞|λj|≤C∑|β|=m||DβA||BMO||f||H˙Kp. |
This completes the proof of Theorem 2.
Proof of Theorem 3. For any cube Q=Q(0,r) with r>1, let f∈Bδp and ˜A(x)=A(x)−∑|β|=m1β!(DβA)˜Qxβ. We write, for f=fχ4Q+fχ(4Q)c=f1+f2 and z∈3Q∖2Q,
˜TAδ(f)(x)=˜TAδ(f1)(x)+∫RnRm(˜A;x,y)|x−y|mK(x,y)f2(y)dy−∑|β|=m1β!(DβA(x)−(DβA)Q)(Tδ,β(f2)(x)−Tδ,β(f2)(z))−∑|β|=m1β!(DβA(x)−(DβA)Q)Tδ,β(f2)(z)=I1(x)+I2(x)+I3(x,z)+I4(x,z), |
where Tδ,β is the singular integral operator with the kernel (x−y)β|x−y|mK(x,y) for |β|=m. Note that (I4(⋅,z))Q=0, we have
˜TAδ(f)(x)−(˜TAδ(f))Q=I1(x)−(I1(⋅))Q+I2(x)−I2(z)−[I2(⋅)−I2(z)]Q−I3(x,z)+(I3(x,z))Q−I4(x,z). |
By the (Lp,Lq)-bounded of ˜TAδ, we get
1|Q|∫Q|I1(x)|dx≤(1|Q|∫Q|˜TAδ(f1)(x)|qdx)1/q≤C|Q|−1/q||f1||Lp≤C||f||Bδp. |
Similar to the proof of Theorem 1, we obtain
|I2(x)−I2(z)|≤C||f||Bδp |
and
1|Q|∫Q|I3(x,z)|dx≤C||f||Bδp. |
Then integrating in x on Q and using the above estimates, we obtain the equivalence of the estimate
1|Q|∫Q|˜TAδ(f)(x)−(˜TAδ(f))Q|dx≤C||f||Bδp |
and the estimate
1|Q|∫Q|I4(x,z)|dx≤C||f||Bδp. |
This completes the proof of Theorem 3.
In this section we shall apply the theorems of the paper to some particular operators such as the Calderón-Zygmund singular integral operator and fractional integral operator.
Application 1. Calderón-Zygmund singular integral operator.
Let T be the Calderón-Zygmund operator defined by (see [10,11,15])
T(f)(x)=∫RnK(x,y)f(y)dy, |
the multilinear operator related to T is defined by
TA(f)(x)=∫RnRm+1(A;x,y)|x−y|mK(x,y)f(y)dy. |
Then it is easily to see that T satisfies the conditions in Theorems 1–3, thus the conclusions of Theorems 1–3 hold for TA.
Application 2. Fractional integral operator with rough kernel.
For 0<δ<n, let Tδ be the fractional integral operator with rough kernel defined by (see [2,7])
Tδf(x)=∫RnΩ(x−y)|x−y|n−δf(y)dy, |
the multilinear operator related to Tδ is defined by
TAδf(x)=∫RnRm+1(A;x,y)|x−y|m+n−δΩ(x−y)f(y)dy, |
where Ω is homogeneous of degree zero on Rn, ∫Sn−1Ω(x′)dσ(x′)=0 and Ω∈Lipε(Sn−1) for some 0<ε≤1, that is there exists a constant M>0 such that for any x,y∈Sn−1, |Ω(x)−Ω(y)|≤M|x−y|ε. Then Tδ satisfies the conditions in Theorem 1. In fact, for suppf⊂(2Q)c and x∈Q=Q(x0,d), by the condition of Ω, we have (see [16])
|Ω(x−y)|x−y|n−δ−Ω(x0−y)|x0−y|n−δ|≤C(|x−x0|ε|x0−y|n+ε−δ+|x−x0||x0−y|n+1−δ), |
thus, the conclusions of Theorems 1–3 hold for TAδ.
The author would like to express his deep gratitude to the referee for his/her valuable comments and suggestions. This research was supported by the National Natural Science Foundation of China (Grant No. 11901126), the Scientific Research Funds of Hunan Provincial Education Department. (Grant No. 19B509).
The authors declare that they have no competing interests.
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