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

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.

1. Introduction and Results

Let 0<𝛼<𝑛, Ω∈𝐿^𝑠(𝑆^𝑛−1)(𝑠>𝑛/(𝑛−𝛼)) is homogeneous of degree zero on 𝑅^𝑛, 𝑆^𝑛−1 denotes the unit sphere in 𝑅^𝑛, the fractional multilinear singular integral operator with rough kernel 𝑇_Ω,𝛼,𝐴 is defined by 𝑇_Ω,𝛼,𝐴𝑓(𝑥)=_𝑅𝑛Ω𝑥−𝑦𝑥−𝑦^{𝑛−𝛼+𝑚−1}𝑅_𝑚𝐴;𝑥,𝑦𝑓𝑦𝑑𝑦, (1) where 𝑅_𝑚(𝐴;𝑥,𝑦) denotes the 𝑚th remainder of the Taylor series of a function 𝐴 defined on 𝑅^𝑛 at 𝑥 about 𝑦. More precisely, 𝑅_𝑚𝐴;𝑥,𝑦=𝐴(𝑥)−_|𝛾|<𝑚1𝐷𝛾!^𝛾𝐴𝑦𝑥−𝑦^𝛾, (2) and the corresponding fractional multilinear maximal operator is defined by 𝑀_Ω,𝛼,𝐴𝑓(𝑥)=sup_𝑟>01𝑟^{𝑛−𝛼+𝑚−1}×_{|𝑥−𝑦|<𝑟}Ω𝑥−𝑦𝑅_𝑚𝐴;𝑥,𝑦𝑓𝑦𝑑𝑦. (3)

Multilinear operator was first introduced by Calderón 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 𝑇_Ω,𝛼,𝐴 is obviously the commutator operator 𝐴,𝑇_Ω,𝛼𝑓(𝑥)=𝐴(𝑥)𝑇_Ω,𝛼𝑓(𝑥)−𝑇_Ω,𝛼𝐴𝑓(𝑥), (4) 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 of 𝑇_Ω,𝛼,𝐴 and 𝑀_Ω,𝛼,𝐴 had been obtained on Lebesgue spaces in [4–7]. However, the corresponding results have not been obtained on 𝐿^𝑝(⋅)(𝑅^𝑛). Nowadays, there is an evident increase of investigations related to both the theory of the spaces 𝐿^𝑝(⋅) 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 𝑇_Ω,𝛼,𝐴 and 𝑀_Ω,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 function 𝑝(⋅)∶𝑅^𝑛→[1,∞), for some 𝜆>0, then, the variable exponent Lebesgue space 𝐿^𝑝(⋅)(𝑅^𝑛) is defined by 𝐿^𝑝(⋅)𝑅^𝑛=𝑓ismeasurable∶_𝑅𝑛𝑓(𝑥)𝜆^𝑝(𝑥)𝑑𝑥<∞, (5) with norm 𝑓_{𝐿𝑝(⋅)(𝑅𝑛)}=inf𝜆>0∶_𝑅𝑛𝑓(𝑥)𝜆^𝑝(𝑥)𝑑𝑥≤1. (6) We denote 𝑝₋=essinf𝑝(𝑥)∶𝑥∈𝑅^𝑛,𝑝₊=esssup𝑝(𝑥)∶𝑥∈𝑅^𝑛. (7) Using this notation we define a class of variable exponent as follows: Φ𝑅^𝑛=𝑝(⋅)∶𝑅^𝑛⟶[1,∞),𝑝₋>1,𝑝₊<∞. (8) The exponent 𝑝^′(⋅) means the conjugate of 𝑝(⋅), namely, 1/𝑝(𝑥)+1/𝑝^′(𝑥)=1 holds.

Definition 2. For 𝛽>0, the homogeneous Lipschitz space Λ̇_𝛽 is the space of functions 𝑓, such that 𝑓_̇Λ𝛽=sup_{𝑥,ℎ∈𝑅𝑛,ℎ≠0}Δ_ℎ^[𝛽]+1𝑓(𝑥)|ℎ|^𝛽<∞, (9) where Δ¹_ℎ𝑓(𝑥)=𝑓(𝑥+ℎ)−𝑓(𝑥), Δ_ℎ^𝑘+1𝑓(𝑥)=Δ^𝑘_ℎ𝑓(𝑥+ℎ)−Δ^𝑘_ℎ𝑓(𝑥), 𝑘≥1.

Definition 3. For 0<𝛼<𝑛, the fractional integral operator with rough kernel is defined by 𝑇_Ω,𝛼𝑓(𝑥)=_𝑅𝑛Ω𝑥−𝑦𝑥−𝑦^𝑛−𝛼𝑓𝑦𝑑𝑦,𝑇_Ω,𝛼𝑓(𝑥)=_𝑅𝑛Ω𝑥−𝑦𝑥−𝑦^𝑛−𝛼𝑓𝑦𝑑𝑦. (10)
The corresponding fractional maximal operator with rough kernel is defined by 𝑀_Ω,𝛼𝑓(𝑥)=sup_𝑟>01𝑟^𝑛−𝛼_{|𝑥−𝑦|<𝑟}Ω𝑥−𝑦𝑓𝑦𝑑𝑦. (11)
When 𝛼=0, 𝑇_Ω,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 𝐿^𝑝 boundedness. In 1971, Muckenhoupt and Wheeden [16] proved the (𝐿^𝑝,𝐿^𝑞) boundedness of 𝑇_Ω,𝛼 with power weights.
In this paper, we state some properties of variable exponents belonging to class 𝐵(𝑅^𝑛).

Proposition 4. If 𝑝(⋅)∈Φ(𝑅^𝑛) satisfies 𝑝(𝑥)−𝑝𝑦≤−𝐶,≤1log𝑥−𝑦𝑥−𝑦2,𝑝(𝑥)𝑦≤𝐶−𝑝(|𝑥,𝑦≥|𝑥|,log𝑒+|) (12) Then, one has 𝑝(⋅)∈𝐵(𝑅^𝑛).

Recently, Mitsuo Izuki has proved the condition as below.

Theorem A (see [17]). Suppose that 𝑝(⋅)∈Φ(𝑅^𝑛) satisfies conditions (12) in Proposition 4. Let 0<𝛼<𝑛/𝑝₊, and define the variable exponent 𝑞(⋅) by 1𝑝(𝑥)−1𝑞(𝑥)=𝛼𝑛. (13) Then, one has that for all 𝑓∈𝐿^𝑝(⋅)(𝑅^𝑛), 𝑏,𝐼^𝛼𝑓_{𝐿𝑞(⋅)(𝑅𝑛)}≤𝐶‖𝑏‖_{𝐵𝑀𝑂(𝑅}^𝑛₎𝑓_{𝐿𝑝(⋅)(𝑅𝑛)} (14) for all 𝑓∈𝐿^𝑝(⋅)(𝑅^𝑛) and 𝑏∈𝐵𝑀𝑂(𝑅^𝑛).

Next, we will discuss the boundedness of 𝑇_Ω,𝛼,𝐴 and 𝑀_Ω,𝛼,𝐴 on variable Lebesgue spaces. We can get 𝑇_Ω,𝛼,𝐴 and 𝑀_Ω,𝛼,𝐴 are bounded from 𝐿^𝑝(⋅)(𝑅^𝑛) to 𝐿^𝑞(⋅)(𝑅^𝑛). 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 that 𝑝(⋅)∈Φ(𝑅^𝑛) 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 1𝑞(𝑥)−1𝑝(𝑥)=𝛼+𝛽𝑛. (15) If 𝐷^𝛾𝐴∈Λ̇_𝛽(|𝛾|=𝑚−1), then, there is a 𝐶>0, independent of 𝑓 and 𝐴, such that 𝑇_Ω,𝛼,𝐴𝑓_{𝐿𝑞(⋅)(𝑅𝑛)}≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽𝑓_{𝐿𝑝(⋅)(𝑅𝑛)}. (16)

Theorem 6. Suppose that 𝑝(⋅)∈Φ(𝑅^𝑛) 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 1𝑞(𝑥)−1𝑝(𝑥)=𝛼+𝛽𝑛. (17) If 𝐷^𝛾𝐴∈Λ̇_𝛽(|𝛾|=𝑚−1), then, there is a 𝐶>0, independent of 𝑓 and 𝐴, such that 𝑀_Ω,𝛼,𝐴𝑓_{𝐿𝑞(⋅)(𝑅𝑛)}≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽𝑓_{𝐿𝑝(⋅)(𝑅𝑛)}. (18)

Remark 7. We point out that 𝐶 will denote positive constants whose values may change at different places.

2. Lemmas and Proof of Theorems

Lemma 8 (see [15]). Let 𝐴(𝑥) be a function on 𝑅^𝑛 with 𝑚th order derivatives in 𝐿^𝑙_𝑙𝑜𝑐(𝑅^𝑛) for some 𝑙>𝑛. Then, 𝑅_𝑚𝐴;𝑥,𝑦≤𝐶𝑥−𝑦^𝑚_|𝑟|=𝑚1𝑄^𝑦_𝑥^𝑄𝑦_𝑥𝐷^𝛾𝐴(𝑧)^𝑙𝑑𝑧^1/𝑙, (19) where 𝑄^𝑦_𝑥 is the cube centered at 𝑥 and having diameter 5√𝑛|𝑥−𝑦|.

Lemma 9 (see [18]). For 0<𝛽<1, 1≤𝑞<∞, we have 𝑓_̇Λ𝛽=sup_𝑄1|𝑄|^{1+𝛽/𝑛𝑄}𝑓(𝑥)−𝑚_𝑄𝑓𝑑𝑥≈sup_𝑄1|𝑄|^𝛽/𝑛1|𝑄|_𝑄𝑓(𝑥)−𝑚_𝑄𝑓^𝑞𝑑𝑥^1/𝑞. (20)

Lemma 10 (see [18]). Let 𝑄^∗⊂𝑄, 𝑔∈Λ̇_𝛽(0<𝛽<1), then, 𝑚_𝑄^∗𝑔−𝑚_𝑄𝑔≤𝐶|𝑄|^𝛽/𝑛𝑔_̇Λ𝛽. (21)

We state the following important lemma.

Lemma 11. Suppose 0<𝛼<𝑛, 0<𝛽<1, with 0<𝛼+𝛽<𝑛, Ω∈𝐿^𝑠(𝑆^𝑛−1)(𝑠>𝑛/(𝑛−(𝛼+𝛽))), 𝐷^𝛾𝐴∈Λ̇_𝛽. Then, there exists a constant 𝐶 only depends on 𝑚,𝑛,𝛼, and 𝛽, such that 𝑇_Ω,𝛼,𝐴𝑓(𝑥)≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽𝑇_Ω,𝛼+𝛽𝑓(𝑥). (22)

Proof. For any 𝑥∈𝑅^𝑛, let the cube be centered at 𝑥 and having the diameter be 𝑙, where 𝑙>0, we have 𝑇_Ω,𝛼,𝐴𝑓(𝑥)=_𝑄+_𝑄𝑐Ω𝑥−𝑦𝑥−𝑦^{𝑛−𝛼+𝑚−1}𝑅_𝑚𝐴;𝑥,𝑦𝑓𝑦𝑑𝑦∶=𝐻₁+𝐻₂. (23) Below, we give estimates of 𝐻₁. Let 𝐻₁≤^∞_{𝑗=02−𝑗𝑄⧵2−𝑗−1𝑄}Ω𝑥−𝑦𝑅_𝑚𝐴;𝑥,𝑦𝑥−𝑦^{𝑛−𝛼+𝑚−1}𝑓𝑦≤𝑑𝑦_{∞𝑗=02−𝑗𝑄⧵2−𝑗−1𝑄}Ω𝑥−𝑦𝑅_𝑚𝐴₂^−𝑗_𝑄;𝑥,𝑦𝑥−𝑦^{𝑛−𝛼+𝑚−1}𝑓𝑦𝑑𝑦. (24)
Note that 𝐴₂^−𝑗_𝑄(𝑦)=𝐴(𝑦)−∑_{|𝛾|=𝑚−1}(1/𝛾!)𝑚₂^−𝑗_𝑄(𝐷^𝛾𝐴)𝑦^𝑟. When 𝑦∈2^−𝑗𝑄⧵2^−𝑗−1𝑄, by Lemmas 8, 9, and 10, we have 𝑅_𝑚𝐴₂^−𝑗_𝑄;𝑥,𝑦≤𝐶2^−𝑗𝑙^𝛽𝑥−𝑦^𝑚−1_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽. (25) Note that |𝑥−𝑦|≥2^−𝑗−1, we have |𝑥−𝑦|^𝛽≥2^−𝛽(2^−𝑗𝑙)^𝛽, such that 𝐻₁≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_𝑘̇Λ𝛽×^∞_𝑗=02^−𝑗𝑙^𝛽_{2−𝑗𝑄⧵2−𝑗−1𝑄}Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^𝑛−𝛼𝑑𝑦≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽×^∞_{𝑗=02−𝑗𝑄⧵2−𝑗−1𝑄}2^−𝑗𝑙^𝛽Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^𝑛−𝛼𝑑𝑦≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽×^∞_{𝑗=02−𝑗𝑄⧵2−𝑗−1𝑄}2^𝛽𝑥−𝑦^𝛽Ω𝑥−𝑦𝑥−𝑦^𝑛−𝛼𝑓𝑦𝑑𝑦≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽𝑄Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^{𝑛−(𝛼+𝛽)}𝑑𝑦≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_{̇Λ𝛽𝑅𝑛}Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^{𝑛−(𝛼+𝛽)}𝑑𝑦=𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽𝑇_Ω,𝛼+𝛽𝑓(𝑥). (26)
Below, we give the estimates of 𝐻₂. For 0<𝛼+𝛽<𝑛, we get 𝐻₂≤^∞_{𝑗=02𝑗+1𝑄⧵2𝑗𝑄}Ω𝑥−𝑦𝑅_𝑚𝐴;𝑥,𝑦𝑥−𝑦^{𝑛−𝛼+𝑚−1}𝑓𝑦≤𝑑𝑦_{∞𝑗=02𝑗+1𝑄⧵2𝑗𝑄}Ω𝑥−𝑦𝑅_𝑚𝐴₂^𝑗+1_𝑄;𝑥,𝑦𝑥−𝑦^{𝑛−𝛼+𝑚−1}𝑓𝑦𝑑𝑦. (27) For any 𝑦∈2^𝑗+1𝑄⧵2^𝑗𝑄, 𝐴₂^𝑗+1_𝑄𝑦=𝐴𝑦−_{|𝛾|𝑚−1}1𝑚𝛾!^2𝑗+1𝑄𝐷^𝛾𝐴. (28) Thus, by Lemmas 8 and 9, we obtain 𝑅_𝑚𝐴₂^𝑗+1_𝑄2;𝑥,𝑦≤𝐶^𝑗𝑙^𝛽𝑥−𝑦^𝑚−1_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽. (29) And for |𝑥−𝑦|≥2^𝑗𝑙, we have |𝑥−𝑦|^𝛽≥(2^𝑗𝑙)^𝛽. Hence, 𝐻₂≤_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽×^∞_𝑗=02^𝑗𝑙^𝛽_{2𝑗+1𝑄⧵2𝑗𝑄}Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^𝑛−𝛼𝑑𝑦≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽×^∞_{𝑗=02𝑗+1𝑄⧵2𝑗𝑄}𝑥−𝑦^𝛽Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^𝑛−𝛼𝑑𝑦≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽×^∞_{𝑗=02𝑗+1𝑄⧵2𝑗𝑄}Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^{𝑛−(𝛼+𝛽)}𝑑𝑦≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_{̇Λ𝛽𝑅𝑛}Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^{𝑛−(𝛼+𝛽)}𝑑𝑦≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽𝑇_Ω,𝛼+𝛽𝑓(𝑥). (30)

From the proof above, we obtain 𝑇_Ω,𝛼,𝐴𝑓(𝑥)≤𝐻₁+𝐻₂≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽𝑇_Ω,𝛼+𝛽𝑓(𝑥). (31)

Lemma 12 (see [19]). If 𝑝(⋅)∈Φ(𝑅^𝑛), for all 𝑓∈𝐿^𝑝(⋅)(𝑅^𝑛), then, the norm ‖𝑓‖_{𝐿𝑝(⋅)(𝑅𝑛)} has the following equivalence: 𝑓_{𝐿𝑝(⋅)(𝑅𝑛)}≤sup_𝑅𝑛𝑓(𝑥)𝑔(𝑥)𝑑𝑥∶𝑔_{𝐿𝑝′(⋅)(𝑅𝑛)}≤1≤𝑟_𝑝𝑓_{𝐿𝑝′(⋅)(𝑅𝑛)}, (32) where 𝑟_𝑝∶=1+1/𝑝₋−1/𝑝₊.

Lemma 13 (see [19], the generalized Hölder inequality). If 𝑝(⋅)∈Φ(𝑅^𝑛), then, for all 𝑓∈𝐿^𝑝(⋅)(𝑅^𝑛) and for all 𝑔∈𝐿^𝑝′(⋅)(𝑅^𝑛), we have _𝑅𝑛𝑓(𝑥)𝑔(𝑥)𝑑𝑥≤𝐶𝑓_{𝐿𝑝(⋅)(𝑅𝑛)}𝑔_{𝐿𝑝′(⋅)(𝑅𝑛)}. (33)

By a similar method of Ding and Lu [20], it is easy to verify the following result.

Lemma 14. For any 𝜀>0 with 0<𝛼+𝛽−𝜀<𝛼+𝛽+𝜀<𝑛, we have 𝑇_Ω,𝛼+𝛽𝑓(𝑥)≤𝐶𝑀_{Ω,𝛼+𝛽+𝜀}𝑓(𝑥)^1/2𝑀_{Ω,𝛼+𝛽−𝜀}𝑓(𝑥)^1/2, (34) where 𝐶 depends only on 𝛼,𝛽,𝜀, and 𝑛.

Lemma 15 (see [19]). Given that 𝑝(⋅)∶𝑅^𝑛→[1,∞), such that 𝑝₊<∞, then, ‖𝑓‖_{𝐿𝑝(⋅)(𝑅𝑛)}<𝐶₁ if and only if |𝑓|_𝐿^𝑝(⋅)_(𝑅^𝑛₎<𝐶₂. In particular, if either constant equals 1, one can make the other equals 1 as well.

Remark 16. We denote |𝑓|_𝐿^𝑝(⋅)_(𝑅^𝑛₎=∫_𝑅𝑛|𝑓(𝑦)|^𝑝(𝑦)𝑑𝑦.

Lemma 17 (see [21]). Suppose that 𝑝(⋅)∈Φ(𝑅^𝑛) satisfies conditions (12) in Proposition 4. Let 0<𝛼+𝛽<𝑛/𝑝₊, and define the variable exponent 𝑞(⋅) by 1𝑝(𝑥)−1𝑞(𝑥)=𝛼+𝛽𝑛. (35) Then, one has that for all 𝑓∈𝐿^𝑝(⋅)(𝑅^𝑛), 𝑀_Ω,𝛼+𝛽𝑓_{𝐿𝑞(⋅)(𝑅𝑛)}≤𝐶𝑓_{𝐿𝑝(⋅)(𝑅𝑛)}. (36)

Lemma 18. Let 0<𝛼<𝑛, Ω∈𝐿^𝑠(𝑆^𝑛−1), then, for 𝑥∈𝑅^𝑛, 𝑇_Ω,𝛼,𝐴𝑓(𝑥)≥𝑀_Ω,𝛼,𝐴𝑓(𝑥), (37) where 𝑇_Ω,𝛼,𝐴𝑓(𝑥)=_𝑅𝑛Ω𝑥−𝑦𝑥−𝑦^{𝑛−𝛼+𝑚−1}𝑅_𝑚𝐴;𝑥,𝑦𝑓𝑦𝑑𝑦. (38)

Proof. Since 𝑇_Ω,𝛼,𝐴𝑓(𝑥)=_𝑅𝑛Ω𝑥−𝑦𝑥−𝑦^{𝑛−𝛼+𝑚−1}𝑅_𝑚𝐴;𝑥,𝑦𝑓𝑦𝑑𝑦≥_{|𝑥−𝑦|<𝑟}Ω𝑥−𝑦𝑥−𝑦^{𝑛−𝛼+𝑚−1}𝑅_𝑚𝑓𝑦≥1𝐴;𝑥,𝑦𝑑𝑦𝑟^{𝑛−𝛼+𝑚−1}_{|𝑥−𝑦|<𝑟}Ω𝑥−𝑦𝑅_𝑚𝐴;𝑥,𝑦𝑓𝑦𝑑𝑦, (39) then, 𝑇_Ω,𝛼,𝐴𝑓(𝑥)≥𝑀_Ω,𝛼,𝐴𝑓(𝑥). (40)

Proof of Theorem 5. Since 𝑇_Ω,𝛼,𝐴𝑓(𝑥)≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽𝑇_Ω,𝛼+𝛽𝑓(𝑥), (41) by Lemma 12, then, we have 𝑇_Ω,𝛼,𝐴𝑓(𝑥)_{𝐿𝑞(⋅)(𝑅𝑛)}≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽×sup_𝑅𝑛𝑇_Ω,𝛼+𝛽𝑓(𝑥)𝑔(𝑥)𝑑𝑥∶𝑔_{𝐿𝑞′(⋅)(𝑅𝑛)}≤1. (42) Using the generalized Hölder inequality, then, 𝑇_Ω,𝛼,𝐴𝑓_{𝐿𝑞(⋅)(𝑅𝑛)}≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽𝑇_Ω,𝛼+𝛽𝑓_{𝐿𝑞(⋅)(𝑅𝑛)}𝑔_{𝐿𝑞′(⋅)(𝑅𝑛)}≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽𝑇_Ω,𝛼+𝛽𝑓_{𝐿𝑞(⋅)(𝑅𝑛)}. (43)
Next, we will prove ‖𝑇_Ω,𝛼+𝛽𝑓‖_{𝐿𝑞(⋅)(𝑅𝑛)}≤‖𝑓‖_{𝐿𝑝(⋅)(𝑅𝑛)}. Fix 𝑓∈𝐿^𝑝(⋅)(𝑅^𝑛), without loss of generality we may assume that ‖𝑓‖_{𝐿𝑝(⋅)(𝑅𝑛)}=1. Since 𝑞₊<∞, by Lemma 15 it will suffice to prove that |𝑇_Ω,𝛼+𝛽𝑓|_𝐿^𝑞(⋅)_(𝑅^𝑛₎≤𝐶.
Fix 𝜀, 0<𝜀<min(𝛼+𝛽,𝑛−(𝛼+𝛽)), such that 2𝜀𝑞₊/𝑛+1>1, (44) define 𝑟(⋅)∶𝑅^𝑛→[1,+∞) by 𝑟(𝑥)=2(𝑥).𝜀𝑞/𝑛+1 (45)
Then, by (44), we have 𝑟₋>1. Moreover, by elementary algebra, for all 𝑥∈𝑅^𝑛, 1𝑝(𝑥)−1𝑟(𝑥)𝑞(𝑥)=/2𝛼+𝛽−𝜀𝑛,1𝑝(𝑥)−1𝑟^′(𝑥)𝑞(𝑥)=/2𝛼+𝛽+𝜀𝑛. (46)(47) So that by Lemma 14, we have _𝑅𝑛𝑇_Ω,𝛼+𝛽𝑓(𝑥)^𝑞(𝑥)𝑑𝑥≤𝐶_𝑅𝑛𝑀_{Ω,𝛼+𝛽−𝜀}𝑓(𝑥)^𝑞(𝑥)/2𝑀_{Ω,𝛼+𝛽+𝜀}𝑓(𝑥)^𝑞(𝑥)/2𝑑𝑥. (48) By Lemma 13, then, _𝑅𝑛𝑇_Ω,𝛼+𝛽𝑓(𝑥)^𝑞(𝑥)𝑀𝑑𝑥≤𝐶_{Ω,𝛼+𝛽−𝜀}𝑓(𝑥)^𝑞(𝑥)/2_{𝐿𝑟(⋅)(𝑅𝑛)}×𝑀_{Ω,𝛼+𝛽+𝜀}𝑓(𝑥)^𝑞(𝑥)/2_{𝐿𝑟′(⋅)(𝑅𝑛)}. (49) 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∈𝑅^𝑛 and 𝜆>1, 𝜆^2/𝑞(𝑥)≥𝜆^2/𝑞+, we have _𝑅𝑛𝑀_{Ω,𝛼+𝛽−𝜀}𝑓(𝑥)^𝑞(𝑥)/2𝜆^𝑟(𝑥)=𝑑𝑥_𝑅𝑛𝑀_{Ω,𝛼+𝛽−𝜀}𝑓(𝑥)𝜆^{2/𝑞(𝑥)𝑟(𝑥)𝑞(𝑥)/2}≤𝑑𝑥_𝑅𝑛𝑀_{Ω,𝛼+𝛽−𝜀}𝑓(𝑥)𝜆^{2/𝑞+(𝑥)𝑟(𝑥)𝑞(𝑥)/2}𝑑𝑥. (50) Therefore, by (46) and Lemma 17, we have 𝑀_{Ω,𝛼+𝛽−𝜀}𝑓(𝑥)^𝑞(𝑥)/2_{𝐿𝑟(𝑥)(𝑅𝑛)}≤𝑀_{Ω,𝛼+𝛽−𝜀}𝑓(𝑥)^𝑞+_𝐿^/2_{𝑟(𝑥)𝑞(𝑥)/2(𝑅𝑛)}≤𝐶𝑓^𝑞+_𝐿^/2_{𝑝(𝑥)(𝑅𝑛)}≤𝐶. (51)
In the same way, we have _𝑅𝑛𝑀_{Ω,𝛼+𝛽+𝜀}𝑓(𝑥)^𝑞(𝑥)/2𝜆^{𝑟′(𝑥)}=𝑑𝑥_𝑅𝑛𝑀_{Ω,𝛼+𝛽+𝜀}𝑓(𝑥)𝜆^{2/𝑞(𝑥)𝑟′(𝑥)𝑞(𝑥)/2}≤𝑑𝑥_𝑅𝑛𝑀_{Ω,𝛼+𝛽+𝜀}𝑓(𝑥)𝜆^{2/(𝑞)+(𝑥)𝑟′(𝑥)𝑞(𝑥)/2}𝑑𝑥. (52) Therefore, by (47) and Lemma 17, then, 𝑀_{Ω,𝛼+𝛽+𝜀}𝑓(𝑥)^𝑞(𝑥)/2_{𝐿𝑟′(𝑥)(𝑅𝑛)}≤𝑀_{Ω,𝛼+𝛽+𝜀}𝑓(𝑥)^𝑞+_𝐿^/2_{𝑟′(𝑥)𝑞(𝑥)/2(𝑅𝑛)}≤𝐶𝑓^𝑞+_𝐿^/2_{𝑝(𝑥)(𝑅𝑛)}≤𝐶. (53)
Hence, 𝑇_Ω,𝛼+𝛽𝑓_{𝐿𝑞(⋅)(𝑅𝑛)}=_𝑅𝑛𝑇_Ω,𝛼+𝛽𝑓(𝑥)^𝑞(𝑥)𝑑𝑥≤𝐶. (54) So, we have 𝑇_Ω,𝛼+𝛽𝑓_{𝐿𝑞(⋅)(𝑅𝑛)}≤𝑓_{𝐿𝑝(⋅)(𝑅𝑛)},𝑇_Ω,𝛼,𝐴𝑓_{𝐿𝑞(⋅)(𝑅𝑛)}≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽𝑇_Ω,𝛼+𝛽𝑓_{𝐿𝑞(⋅)(𝑅𝑛)}≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_̇Λ𝛽𝑓_{𝐿𝑝(⋅)(𝑅𝑛)}. (55) This completes the proof of Theorem 5.

By Lemmas 15 and 18 and Theorem 5, the proof of Theorem 6 is directly deduced.

Acknowledgments

This paper is supported by the Natural Science Foundation of Zhejiang Province (M6090681) and supported by the Education Deptartment of Zhejiang Province (Y201120509).