# Lipschitz Estimates for Fractional Multilinear Singular Integral on Variable Exponent Lebesgue Spaces

Authors: Hui-Ling Wu; Jia-Cheng Lan
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We obtain the Lipschitz boundedness for a class of fractional multilinear operators with rough kernels on variable exponent Lebesgue spaces. Our results generalize the related conclusions on Lebesgue spaces with constant exponent.


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## Body

## 1. Introduction and Results

Let0<α<n, Ω∈Ls(Sn-1)(s>n/(n-α)) is homogeneous of degree zero on Rn, Sn-1 denotes the unit sphere in Rn, the fractional multilinear singular integral operator with rough kernel TΩ,α,A is defined by
(1)TΩ,α,Af(x)=∫RnΩ(x-y)|x-y|n-α+m-1Rm(A;x,y)f(y)dy,
where Rm(A;x,y) denotes the mth remainder of the Taylor series of a function A defined on Rn at x about y. More precisely,
(2)Rm(A;x,y)=A(x)-∑|γ|<m1γ!DγA(y)(x-y)γ,
and the corresponding fractional multilinear maximal operator is defined by
(3)MΩ,α,Af(x)=supr>01rn-α+m-1×∫|x-y|<r|Ω(x-y)||Rm(A;x,y)||f(y)|dy.Multilinear operator was first introduced by Caldero´n in [1], and then Meyer [2] studied it in depth and extended such type of operators. Multilinear singular integral operator was later introduced by Professor Lu during 1999 [3]. Especially as m=1, the fractional multilinear singular integral operator TΩ,α,A is obviously the commutator operator
(4)[A,TΩ,α]f(x)=A(x)TΩ,αf(x)-TΩ,α(Af)(x),
the commutator is a typical non-convolution singular operator. Since the commutator has a close relation with partial differential equations and pseudo-differential operator, multilinear operator has been receiving more widely attention.It is well known that the boundedness ofTΩ,α,A and MΩ,α,A had been obtained on Lebesgue spaces in [4–7]. However, the corresponding results have not been obtained on Lp(·)(Rn). Nowadays, there is an evident increase of investigations related to both the theory of the spaces Lp(·) themselves and the operator theory in these spaces [8–11]. This is caused by possible applications to models with nonstandard local growth in elasticity theory, fluid mechanics, and differential equations [12–14]. The purpose of this paper is to study the behaviour of TΩ,α,A and MΩ,α,A on variable Lebesgue spaces.To state the main results of this paper, we need to recall some notions.Definition 1.
Suppose a measurable functionp(·):Rn→[1,∞), for some λ>0, then, the variable exponent Lebesgue space Lp(·)(Rn) is defined by
(5)Lp(·)(Rn)={fismeasurable:∫Rn(|f(x)|λ)p(x)dx<∞},
with norm
(6)∥f∥Lp(·)(Rn)=inf{λ>0:∫Rn(|f(x)|λ)p(x)dx≤1}.
We denote
(7)p-=essinf{p(x):x∈Rn},p+=esssup{p(x):x∈Rn}.
Using this notation we define a class of variable exponent as follows:
(8)Φ(Rn)={p(·):Rn⟶[1,∞),p->1,p+<∞}.
The exponent p′(·) means the conjugate of p(·), namely, 1/p(x)+1/p′(x)=1 holds.Definition 2.
Forβ>0, the homogeneous Lipschitz space Λ˙β is the space of functions f, such that
(9)∥f∥Λ˙β=supx,h∈Rn,h≠0|Δh[β]+1f(x)||h|β<∞,
where Δh1f(x)=f(x+h)-f(x), Δhk+1f(x)=Δhkf(x+h)-Δhkf(x), k≥1.Definition 3.
For0<α<n, the fractional integral operator with rough kernel is defined by
(10)TΩ,αf(x)=∫RnΩ(x-y)|x-y|n-αf(y)dy,T¯Ω,αf(x)=∫Rn|Ω(x-y)||x-y|n-α|f(y)|dy.
The corresponding fractional maximal operator with rough kernel is defined by(11)MΩ,αf(x)=supr>01rn-α∫|x-y|<r|Ω(x-y)||f(y)|dy.
Whenα=0, TΩ,α is much more closely related to the elliptic partial equations of second order with variable coefficients. In 1955, Calderón and Zygmund [15] proved the Lp boundedness. In 1971, Muckenhoupt and Wheeden [16] proved the (Lp,Lq) boundedness of TΩ,α with power weights.
In this paper, we state some properties of variable exponents belonging to classB(Rn).Proposition 4.
Ifp(·)∈Φ(Rn) satisfies
(12)|p(x)-p(y)|≤-Clog(|x-y|),|x-y|≤12,|p(x)-p(y)|≤Clog(e+|x|),|y|≥|x|,
Then, one has p(·)∈B(Rn).Recently, Mitsuo Izuki has proved the condition as below.Theorem A (see [17]).
Suppose thatp(·)∈Φ(Rn) satisfies conditions (12) in Proposition 4. Let 0<α<n/p+, and define the variable exponent q(·) by
(13)1p(x)-1q(x)=αn.
Then, one has that for all f∈Lp(·)(Rn),
(14)∥[b,Iα]f∥Lq(·)(Rn)≤C∥b∥BMO(Rn)∥f∥Lp(·)(Rn)
for all f∈Lp(·)(Rn) and b∈BMO(Rn).Next, we will discuss the boundedness ofTΩ,α,A and MΩ,α,A on variable Lebesgue spaces. We can get TΩ,α,A and MΩ,α,A are bounded from Lp(·)(Rn) to Lq(·)(Rn). In fact, the results generalize Theorem A in [17] from classical Lebesgue spaces to variable exponent Lebesgue spaces. Now, let us formulate our results as follows.Theorem 5.
Suppose thatp(·)∈Φ(Rn) satisfies conditions (12) in Proposition 4. Let 0<α<n/p+, 0<β<1, 0<α+β<n/p+, and 1<p+<n/(α+β), and define the variable exponent q(·) by
(15)1q(x)-1p(x)=α+βn.
If DγA∈Λ˙β(|γ|=m-1), then, there is a C>0, independent of f and A, such that
(16)∥TΩ,α,Af∥Lq(·)(Rn)≤C(∑|γ|=m-1∥DγA∥Λ˙β)∥f∥Lp(·)(Rn).Theorem 6.
Suppose thatp(·)∈Φ(Rn) satisfies conditions (12) in Proposition 4. Let 0<α<n/p+, 0<β<1, 0<α+β<n/p+, and 1<p+<n/(α+β), and define the variable exponent q(·) by
(17)1q(x)-1p(x)=α+βn.
If DγA∈Λ˙β(|γ|=m-1), then, there is a C>0, independent of f and A, such that
(18)∥MΩ,α,Af∥Lq(·)(Rn)≤C(∑|γ|=m-1∥DγA∥Λ˙β)∥f∥Lp(·)(Rn).Remark 7.
We point out thatC will denote positive constants whose values may change at different places.


## 2. Lemmas and Proof of Theorems

Lemma 8 (see [15]).
LetA(x) be a function on Rn with mth order derivatives in Llocl(Rn) for some l>n. Then,
(19)|Rm(A;x,y)|≤C|x-y|m∑|r|=m(1|Qxy|∫Qxy|DγA(z)|ldz)1/l,
where Qxy is the cube centered at x and having diameter 5n|x-y|.Lemma 9 (see [18]).
For0<β<1, 1≤q<∞, we have
(20)∥f∥Λ˙β=supQ1|Q|1+β/n∫Q|f(x)-mQ(f)|dx≈supQ1|Q|β/n(1|Q|∫Q|f(x)-mQ(f)|qdx)1/q.Lemma 10 (see [18]).
LetQ*⊂Q, g∈Λ˙β(0<β<1), then,
(21)|mQ*(g)-mQ(g)|≤C|Q|β/n∥g∥Λ˙β.We state the following important lemma.Lemma 11.
Suppose0<α<n, 0<β<1, with 0<α+β<n, Ω∈Ls(Sn-1)(s>n/(n-(α+β))), DγA∈Λ˙β. Then, there exists a constant C only depends on m,n,α, and β, such that
(22)|TΩ,α,Af(x)|≤C(∑|γ|=m-1∥DγA∥Λ˙β)T¯Ω,α+βf(x).Proof.
For anyx∈Rn, let the cube be centered at x and having the diameter be l, where l>0, we have
(23)TΩ,α,Af(x)=(∫Q+∫Qc)Ω(x-y)|x-y|n-α+m-1Rm(A;x,y)f(y)dy:=H1+H2.
Below, we give estimates of H1. Let
(24)|H1|≤∑j=0∞∫2-jQ∖2-j-1Q|Ω(x-y)||Rm(A;x,y)||x-y|n-α+m-1|f(y)|dy≤∑j=0∞∫2-jQ∖2-j-1Q|Ω(x-y)||Rm(A2-jQ;x,y)||x-y|n-α+m-1|f(y)|dy.
Note thatA2-jQ(y)=A(y)-∑|γ|=m-1(1/γ!)m2-jQ(DγA)yr. When y∈2-jQ∖2-j-1Q, by Lemmas 8, 9, and 10, we have
(25)|Rm(A2-jQ;x,y)|≤C(2-jl)β|x-y|m-1∑|γ|=m-1∥DγA∥Λ˙β.
Note that |x-y|≥2-j-1, we have |x-y|β≥2-β(2-jl)β, such that
(26)|H1|≤C(∑|γ|=m-1∥DγAk∥Λ˙β)×∑j=0∞(2-jl)β∫2-jQ∖2-j-1Q|Ω(x-y)||f(y)||x-y|n-αdy≤C(∑|γ|=m-1∥DγA∥Λ˙β)×∑j=0∞∫2-jQ∖2-j-1Q(2-jl)β|Ω(x-y)||f(y)||x-y|n-αdy≤C(∑|γ|=m-1∥DγA∥Λ˙β)×∑j=0∞∫2-jQ∖2-j-1Q2β|x-y|β|Ω(x-y)||x-y|n-α|f(y)|dy≤C(∑|γ|=m-1∥DγA∥Λ˙β)∫Q|Ω(x-y)||f(y)||x-y|n-(α+β)dy≤C(∑|γ|=m-1∥DγA∥Λ˙β)∫Rn|Ω(x-y)||f(y)||x-y|n-(α+β)dy=C(∑|γ|=m-1∥DγA∥Λ˙β)T¯Ω,α+βf(x).
Below, we give the estimates ofH2. For 0<α+β<n, we get
(27)|H2|≤∑j=0∞∫2j+1Q∖2jQ|Ω(x-y)||Rm(A;x,y)||x-y|n-α+m-1|f(y)|dy≤∑j=0∞∫2j+1Q∖2jQ|Ω(x-y)||Rm(A2j+1Q;x,y)||x-y|n-α+m-1|f(y)|dy.
For any y∈2j+1Q∖2jQ,
(28)A2j+1Q(y)=A(y)-∑|γ|m-11γ!m2j+1Q(DγA).
Thus, by Lemmas 8 and 9, we obtain
(29)|Rm(A2j+1Q;x,y)|≤C(2jl)β|x-y|m-1∑|γ|=m-1∥DγA∥Λ˙β.
And for |x-y|≥2jl, we have |x-y|β≥(2jl)β. Hence,
(30)|H2|≤(∑|γ|=m-1∥DγA∥Λ˙β)×∑j=0∞(2jl)β∫2j+1Q∖2jQ|Ω(x-y)||f(y)||x-y|n-αdy≤C(∑|γ|=m-1∥DγA∥Λ˙β)×∑j=0∞∫2j+1Q∖2jQ|x-y|β|Ω(x-y)||f(y)||x-y|n-αdy≤C(∑|γ|=m-1∥DγA∥Λ˙β)×∑j=0∞∫2j+1Q∖2jQ|Ω(x-y)||f(y)||x-y|n-(α+β)dy≤C(∑|γ|=m-1∥DγA∥Λ˙β)∫Rn|Ω(x-y)||f(y)||x-y|n-(α+β)dy≤C(∑|γ|=m-1∥DγA∥Λ˙β)T¯Ω,α+βf(x).From the proof above, we obtain(31)|TΩ,α,Af(x)|≤|H1|+|H2|≤C(∑|γ|=m-1∥DγA∥Λ˙β)T¯Ω,α+βf(x).Lemma 12 (see [19]).
Ifp(·)∈Φ(Rn), for all f∈Lp(·)(Rn), then, the norm ∥f∥Lp(·)(Rn) has the following equivalence:
(32)∥f∥Lp(·)(Rn)≤sup{∫Rn|f(x)g(x)|dx:∥g∥Lp′(·)(Rn)≤1}≤rp∥f∥Lp′(·)(Rn),
where rp:=1+1/p--1/p+.Lemma 13 (see [19], the generalized Ho¨lder inequality).
Ifp(·)∈Φ(Rn), then, for all f∈Lp(·)(Rn) and for all g∈Lp′(·)(Rn), we have
(33)∫Rn|f(x)g(x)|dx≤C∥f∥Lp(·)(Rn)∥g∥Lp′(·)(Rn).By a similar method of Ding and Lu [20], it is easy to verify the following result.Lemma 14.
For anyɛ>0 with 0<α+β-ɛ<α+β+ɛ<n, we have
(34)|T¯Ω,α+βf(x)|≤C[MΩ,α+β+ɛf(x)]1/2[MΩ,α+β-ɛf(x)]1/2,
where C depends only on α,β,ɛ, and n.Lemma 15 (see [19]).
Given thatp(·):Rn→[1,∞), such that p+<∞, then, ∥f∥Lp(·)(Rn)<C1 if and only if |f|Lp(·)(Rn)<C2. In particular, if either constant equals 1, one can make the other equals 1 as well.Remark 16.
We denote|f|Lp(·)(Rn)=∫Rn|f(y)|p(y)dy.Lemma 17 (see [21]).
Suppose thatp(·)∈Φ(Rn) satisfies conditions (12) in Proposition 4. Let 0<α+β<n/p+, and define the variable exponent q(·) by
(35)1p(x)-1q(x)=α+βn.
Then, one has that for all f∈Lp(·)(Rn),
(36)∥MΩ,α+βf∥Lq(·)(Rn)≤C∥f∥Lp(·)(Rn).Lemma 18.
Let0<α<n, Ω∈Ls(Sn-1), then, for x∈Rn,
(37)T¯Ω,α,Af(x)≥MΩ,α,Af(x),
where
(38)T¯Ω,α,Af(x)=∫Rn|Ω(x-y)||x-y|n-α+m-1|Rm(A;x,y)||f(y)|dy.Proof.
Since(39)T¯Ω,α,Af(x)=∫Rn|Ω(x-y)||x-y|n-α+m-1|Rm(A;x,y)||f(y)|dy≥∫|x-y|<r|Ω(x-y)||x-y|n-α+m-1|Rm(A;x,y)||f(y)|dy≥1rn-α+m-1∫|x-y|<r|Ω(x-y)||Rm(A;x,y)||f(y)|dy,
then,
(40)T¯Ω,α,Af(x)≥MΩ,α,Af(x).Proof of Theorem5.
Since(41)|TΩ,α,Af(x)|≤C(∑|γ|=m-1∥DγA∥Λ˙β)T¯Ω,α+βf(x),
by Lemma 12, then, we have
(42)∥TΩ,α,Af(x)∥Lq(·)(Rn)≤C(∑|γ|=m-1∥DγA∥Λ˙β)×sup{∫RnT¯Ω,α+βf(x)|g(x)|dx:∥g∥Lq′(·)(Rn)≤1}.
Using the generalized Ho¨lder inequality, then,
(43)∥TΩ,α,Af∥Lq(·)(Rn)≤C(∑|γ|=m-1∥DγA∥Λ˙β)∥T¯Ω,α+βf∥Lq(·)(Rn)∥g∥Lq′(·)(Rn)≤C(∑|γ|=m-1∥DγA∥Λ˙β)∥T¯Ω,α+βf∥Lq(·)(Rn).
Next, we will prove∥T¯Ω,α+βf∥Lq(·)(Rn)≤∥f∥Lp(·)(Rn). Fix f∈Lp(·)(Rn), without loss of generality we may assume that ∥f∥Lp(·)(Rn)=1. Since q+<∞, by Lemma 15 it will suffice to prove that |TΩ,α+βf|Lq(·)(Rn)≤C.
Fixɛ, 0<ɛ<min(α+β,n-(α+β)), such that
(44)2(ɛq+/n)+1>1,
define r(·):Rn→[1,+∞) by
(45)r(x)=2(ɛq(x)/n)+1.
Then, by (44), we have r->1. Moreover, by elementary algebra, for all x∈Rn,
(46)1p(x)-1r(x)q(x)/2=α+β-ɛn,(47)1p(x)-1r′(x)q(x)/2=α+β+ɛn.
So that by Lemma 14, we have
(48)∫Rn|T¯Ω,α+βf(x)|q(x)dx≤C∫Rn[MΩ,α+β-ɛf(x)]q(x)/2[MΩ,α+β+ɛf(x)]q(x)/2dx.
By Lemma 13, then,
(49)∫Rn|T¯Ω,α+βf(x)|q(x)dx≤C∥[MΩ,α+β-ɛf(x)]q(x)/2∥Lr(·)(Rn)×∥[MΩ,α+β+ɛf(x)]q(x)/2∥Lr′(·)(Rn).
Without loss of generality, we may assume that each is greater than 1, since, otherwise, there is nothing to prove. In this case, in the definition of each norm we may assume that the infimum is taken over by values of λ which are greater than 1. But then, since for all x∈Rn and λ>1, λ2/q(x)≥λ2/q+, we have
(50)∫Rn([MΩ,α+β-ɛf(x)]q(x)/2λ)r(x)dx=∫Rn(MΩ,α+β-ɛf(x)λ2/q(x))r(x)q(x)/2dx≤∫Rn(MΩ,α+β-ɛf(x)λ2/q+(x))r(x)q(x)/2dx.
Therefore, by (46) and Lemma 17, we have
(51)∥[MΩ,α+β-ɛf(x)]q(x)/2∥Lr(x)(Rn)≤∥[MΩ,α+β-ɛf(x)]∥Lr(x)q(x)/2(Rn)q+/2≤C∥f∥Lp(x)(Rn)q+/2≤C.
In the same way, we have(52)∫Rn([MΩ,α+β+ɛf(x)]q(x)/2λ)r′(x)dx=∫Rn(MΩ,α+β+ɛf(x)λ2/q(x))r′(x)q(x)/2dx≤∫Rn(MΩ,α+β+ɛf(x)λ2/(q)+(x))r′(x)q(x)/2dx.
Therefore, by (47) and Lemma 17, then,
(53)∥[MΩ,α+β+ɛf(x)]q(x)/2∥Lr′(x)(Rn)≤∥[MΩ,α+β+ɛf(x)]∥Lr′(x)q(x)/2(Rn)q+/2≤C∥f∥Lp(x)(Rn)q+/2≤C.
Hence,(54)|T¯Ω,α+βf|Lq(·)(Rn)=∫Rn|T¯Ω,α+βf(x)|q(x)dx≤C.
So, we have
(55)∥T¯Ω,α+βf∥Lq(·)(Rn)≤∥f∥Lp(·)(Rn),∥TΩ,α,Af∥Lq(·)(Rn)≤C(∑|γ|=m-1∥DγA∥Λ˙β)∥T¯Ω,α+βf∥Lq(·)(Rn)≤C(∑|γ|=m-1∥DγA∥Λ˙β)∥f∥Lp(·)(Rn).
This completes the proof of Theorem 5.By Lemmas15 and 18 and Theorem 5, the proof of Theorem 6 is directly deduced.



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