Multilinear theory of strongly singular Calderón–Zygmund operators and applications

Nonlinear Analysis 192 (2020) 111699

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Nonlinear Analysis www.elsevier.com/locate/na

Multilinear theory of strongly singular Calderón–Zygmund operators and applications✩ Yan Lin School of Science, China University of Mining and Technology, Beijing 100083, China

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Article history: Received 11 June 2019 Accepted 7 November 2019 Communicated by Vicentiu D Radulescu MSC: 42B20 42B35 Keywords: Multilinear strongly singular Calderón–Zygmund operator Sharp maximal function BM O function LM O function

abstract The main purpose of this paper is to study the multilinear strongly singular Calderón–Zygmund operator whose kernel is more singular near the diagonal than that of the standard multilinear Calderón–Zygmund operator. The sharp maximal estimate for this class of multilinear singular integrals is established, and as applications, its boundedness on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces is obtained, respectively. Moreover, the endpoint estimates of the types L∞ × . . . × L∞ → BM O, BM O × . . . × BM O → BM O, and LM O × . . . × LM O → LM O are established for the multilinear strongly singular Calderón–Zygmund operator. These results improve the corresponding known ones for the standard multilinear Calderón–Zygmund operator. Furthermore, we remove the size condition assumption for the kernel of the multilinear strongly singular Calderón–Zygmund operator which has been imposed in the literature for the case of the standard multilinear Calderón–Zygmund operator to get the sharp maximal estimates (see Remarks 1.1-1.4). Extra care is needed to deal with the mean oscillation over balls with small radius to overcome the stronger singularity in establishing such sharp maximal estimates and endpoint estimates. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction The model for the strongly singular integral operator is the multiplier operator defined by α

(Tα,β f )∧ (ξ) = θ(ξ)

ei|ξ| ˆ f (ξ), β |ξ|

where 0 < α < 1, 0 < β ≤ nα/2 and θ(ξ) is a standard cutoff function. This operator was studied by Hirschman [23], Stein [42], Wainger [45] and C. Fefferman [15] and was named weakly–strongly singular Calder´ on–Zygmund operator by C. Fefferman in [15]. ✩ This work was supported by the National Natural Science Foundation of China (No. 11671397), the Fundamental Research Funds for the Central Universities, China (No. 2009QS16), and the Yue Qi Young Scholar Project of China University of Mining and Technology, Beijing, China. E-mail address: [email protected].

https://doi.org/10.1016/j.na.2019.111699 0362-546X/© 2019 Elsevier Ltd. All rights reserved.

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The convolution form of Tα,β can be roughly written as ∫ Tα,β (f )(x) = p.v. where λ =

nα/2−β 1−α

and α′ =

−α′

ei|x−y|

n+λ

|x − y|

α 1−α [ n. It ]was shown β 2 +λ and Tα,β n β+λ

χ(|x − y|)f (y)dy,

that Tα,β is bounded on Lp (Rn ) by Hirschman in [23]

is not bounded on Lp (Rn ) by Wainger [45] if | 21 − p1 | > and Stein [42] when − < [n ] [n ] β β 1 1 2 +λ 2 +λ . At the critical exponent p satisfying − = 0 n β+λ 2 p0 n β+λ , C. Fefferman proved that Tα,β is bounded | 21

1 p|

′

from Lp0 (Rn ) to Lorentz space Lp0 ,p0 (Rn ), where p′0 is the dual exponent to p0 . When β = nα/2, the sharp endpoint estimate for strongly singular operators were established by C. Fefferman and Stein in [16] using the duality of Hardy space H 1 and BM O space. The weighted norm estimates were established by Chanillo in [6]. Alvarez and Milman in [2] introduced the strongly singular non-convolution operator which is called strongly singular Calder´ on–Zygmund operator, whose properties are similar to those of classical Calder´onZygmund operators, but the kernel is more singular near the diagonal than that of the standard case. Definition 1.1. Let T : S → S ′ be a bounded linear operator. T is called a strongly singular Calder´ onZygmund operator if the following conditions are satisfied. (1) T can be extended into a continuous operator from L2 (Rn ) into itself. (2) There exists a function K(x, y) continuous away from the diagonal {(x, y) : x = y} such that δ

|y − z|

|K(x, y) − K(x, z)| + |K(y, x) − K(z, x)| ≤ C

|x − z|

δ n+ α

,

α

if 2|y − z| ≤ |x − z| for some 0 < δ ≤ 1 and 0 < α < 1. And ∫ ∫ ⟨T f, g⟩ = K(x, y)f (y)g(x)dydx, for f, g ∈ S with disjoint supports. (3) For some n(1−α)/2 ≤ β < n/2, both T and its conjugate operator T ∗ can be extended into continuous operators from Lq to L2 , where 1/q = 1/2 + β/n. Following a suggestion of Stein, Alvarez and Milman showed that the pseudo-differential operator with −β symbol in the H¨ ormander’s class Sα, δ , where 0 < δ ≤ α < 1 and n(1 − α)/2 ≤ β < n/2, is included in the strongly singular Calder´ on-Zygmund operator. This fact shows that the research of strongly singular Calder´ on-Zygmund operators has important value both on the theory of singular integrals in harmonic analysis and related subjects in PDE. Alvarez and Milman [2,3] discussed the boundedness of the strongly singular Calder´on-Zygmund operator on Lebesgue spaces. Lin [29] and Lin and Lu [33] established the sharp maximal estimates and endpoint estimates of the strongly singular Calder´ on-Zygmund operator. For the boundedness of strongly singular Calder´ on-Zygmund operators and related topics, one can refer to [4,29–34,40,46] and so on. In this paper, we will focus on the multilinear form of the strongly singular Calder´on-Zygmund operator. The multilinear Calder´ on-Zygmund theory was first studied by Coifman and Meyer in [7–9]. The study of multilinear singular integrals was motivated not only as generalizations of the theory of linear ones but also its natural appearance in harmonic analysis. In recent years, this topic has received increasing attentions and well development, such as the systemic treatment of multilinear Calder´on–Zygmund operators by Grafakos– Torres in [19,20] and multilinear fractional integrals by Kenig and Stein in [25]. Recently, a lot of research

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work involves these operators from various points of view. One can refer to [5,21,22,24,27,28,36–39] and their references for details. We now review briefly the definition of the multilinear Calder´on-Zygmund operator. Let m ∈ N+ and K(y0 , y1 , . . . , ym ) be a function defined away from the diagonal y0 = y1 = · · · = ym in (Rn )m+1 . T is an m-linear operator defined on product of test functions such that for K, the integral representation below is valid ∫ ∫ m ∏ T (f1 , . . . , fm )(x) = ··· K(x, y1 , . . . , ym ) fj (yj )dy1 · · · dym , (1.1) Rn

Rn

j=1

where fj (j = 1, . . . , m) are smooth functions with compact support and x ∈ / ∩m j=1 suppfj . Especially, we call K a standard m-linear Calder´ on-Zygmund kernel if it satisfies the following size and smoothness estimates. C , (1.2) |K(y0 , y1 , . . . , ym )| ≤ ∑m ( k, l=0 |yk − yl |)mn for some C > 0 and all (y0 , y1 , . . . , ym ) ∈ (Rn )m+1 away from the diagonal. ε

C|yj − yj′ | |K(y0 , . . . , yj , . . . , ym ) − K(y0 , . . . , yj′ , . . . , ym )| ≤ ∑m , ( k, l=0 |yk − yl |)mn+ε

(1.3)

for some ε > 0, whenever 0 ≤ j ≤ m and |yj − yj′ | ≤ 21 max0≤k≤m |yj − yk |. According to [20], if an m-linear operator T defined by (1.1) associated with a standard m-linear Calder´ on-Zygmund kernel K, and satisfies either of the following two conditions for given numbers 1 ≤ t1 , t2 , . . . , tm , t < ∞ with 1/t = 1/t1 + 1/t2 + · · · + 1/tm , (C1) T maps Lt1 , 1 × · · · × Ltm , 1 into Lt, ∞ if t > 1, (C2) T maps Lt1 , 1 × · · · × Ltm , 1 into L1 if t = 1, where Lt1 , 1 , . . . , Ltm , 1 and Lt, ∞ are Lorentz spaces, then we say that T is a standard m-linear Calder´ onZygmund operator. In this paper, we are interested in the multilinear strongly singular Calder´on-Zygmund operator defined as follows. Definition 1.2. Let T be an m-linear operator defined by (1.1). T is called an m-linear strongly singular Calder´ on-Zygmund operator if the following conditions are satisfied. (1) For some ε > 0 and 0 < α ≤ 1, ε

|K(x, y1 , . . . , ym ) − K(x′ , y1 , . . . , ym )| ≤

C|x − x′ | , (|x − y1 | + · · · + |x − ym |)mn+ε/α

(1.4)

α

whenever |x − x′ | ≤ 21 max1≤j≤m |x − yj |. (2) For some given numbers 1 ≤ r1 , . . . , rm < ∞ with 1/r = 1/r1 + · · · + 1/rm , T maps Lr1 × · · · × Lrm into Lr, ∞ . (3) For some given numbers 1 ≤ l1 , . . . , lm < ∞ with 1/l = 1/l1 + · · · + 1/lm , T maps Ll1 × · · · × Llm into q, ∞ L , where 0 < l/q ≤ α. The following remarks are in order to illustrate common points and the differences between the standard multilinear Calder´ on-Zygmund operator and the multilinear strongly singular Calder´on-Zygmund operator. Remark 1.1. It is easy to see that the condition (1.3) implies the condition (1.4) when α = 1. In this special case, we can take lj = rj , j = 1, . . . , m, and q = l = r in (3) of Definition 1.2. Then the condition (3) of Definition 1.2 is completely in agreement with the condition (2), thus we can remove it in this situation. From

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this point of view, we can say that the multilinear strongly singular Calder´on-Zygmund operator generalizes the standard multilinear Calder´ on-Zygmund operator. Remark 1.2. However, for the case 0 < α < 1, the kernel of the multilinear strongly singular Calder´onZygmund operator defined by Definition 1.2 is more singular than that of the standard one near the diagonal. This is the reason why we call it “strongly singular”. And the stronger singularities will bring up new difficulties and new techniques are needed to handle such operators. Remark 1.3. It should be pointed out that we do not need any size condition like (1.2) for the kernel of the multilinear strongly singular Calder´ on-Zygmund operator to obtain our main results in this paper. Remark 1.4. On the other hand, by comparing Definition 1.1 with Definition 1.2, one can find out that the linear strongly singular Calder´ on-Zygmund operator satisfies the conditions of Definition 1.2 for the situations m = 1 and 0 < α < 1. Thus, the multilinear strongly singular Calder´on-Zygmund operators also generalize the linear ones. In what follows, for 1 ≤ p ≤ ∞, p′ is the conjugate index of p, that is, 1/p + 1/p′ = 1. E c = Rn \ E is the complementary set of E. C’s will be constants which are independent of the main parameter and may vary from line to line. We will always denote by B(x, R) the ball centered at x with radius R > 0, CB(x, R) = ∫ B(x, CR) for C > 0, |B(x, R)| the Lebesgue measure of B(x, R) and fB(x, R) = |B(x,1 R)| B(x, R) f (y)dy. This paper will be organized as follows. The sharp maximal pointwise estimate for the multilinear strongly singular Calder´ on-Zygmund operator will be established in Section 2. And as applications, we can obtain the boundedness of the multilinear strongly singular Calder´on-Zygmund operator on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces, respectively. In Section 3, three kinds of endpoint estimates for the multilinear strongly singular Calder´on-Zygmund operator will be discussed. We will establish the boundedness of L∞ × · · · × L∞ → BM O, BM O × · · · × BM O → BM O, and LM O × · · · × LM O → LM O types, respectively. 2. Sharp maximal pointwise estimates and applications 2.1. Sharp maximal pointwise estimates The definition and properties of BM O functions inspire us naturally to study the sharp maximal function f , associated to any locally integrable function f . It is defined by ∫ ∫ 1 1 M ♯ (f )(x) = sup |f (y) − fB |dy ∼ sup inf |f (y) − a|dx, B∋x |B| B B∋x a∈C |B| B ♯

where the supremum is taken over all balls B containing x. By M we will denote the Hardy–Littlewood maximal operator defined by ∫ 1 M f (x) = sup |f (y)|dy. B∋x |B| B s

s

For 0 < s < ∞, we use Ms and Ms♯ to denote the operators Ms (f ) = [M (|f | )]1/s and Ms♯ (f ) = [M ♯ (|f | )]1/s , respectively. In this section, the pointwise estimate for the sharp maximal function of the multilinear strongly singular Calder´ on-Zygmund operator is established by means of other classes of maximal functions. The following is the main result of this section.

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Theorem 2.1. Let T be an m-linear strongly singular Calder´ on-Zygmund operator and s = max{r1 , . . . , rm , l1 , . . . , lm }, where rj and lj are given as in Definition 1.2, j = 1, . . . , m. If 0 < δ < 1/m, then Mδ♯ (T (f⃗))(x) ≤ C

m ∏

Ms (fj )(x),

j=1

for all m-tuples f⃗ = (f1 , . . . , fm ) of bounded measurable functions with compact support. For the special case α = 1, as we have discussed in Remarks 1.1–1.4, the condition (3) of Definition 1.2 can be removed. Then the standard multilinear Calder´ on-Zygmund operator satisfies the conditions of Definition 1.2 by taking r1 = · · · = rm = 1 since the L1 × · · · × L1 → L1/m,∞ boundedness obtained in [20]. Thus in this special case, s = max{r1 , . . . , rm , l1 , . . . , lm } = 1. Then we can obtain the corresponding sharp maximal estimate for the standard multilinear Calder´ on-Zygmund operator as a corollary of Theorem 2.1. Corollary 2.1. Let T be a standard m-linear Calder´ on-Zygmund operator. If 0 < δ < 1/m, then Mδ♯ (T (f⃗))(x) ≤ C

m ∏

M (fj )(x),

j=1

for all m-tuples f⃗ = (f1 , . . . , fm ) of bounded measurable functions with compact support. Remark 2.1. It should be pointed out that the sharp maximal estimate for the standard multilinear Calder´ on-Zygmund operator obtained in Corollary 2.1 was established earlier in [38]. However, differently from the method in [38], here we do not need any size condition assumption for the kernel. In order to prove Theorem 2.1, we need the following two lemmas. Lemma 2.1 (See [17,27]). Let 0 < p < r < ∞, then there is a positive constant C = Cp,r such that for any measurable function f there has −1/p

|Q|

−1/r

∥f ∥Lp (Q) ≤ C|Q|

∥f ∥Lr,∞ (Q) .

Lemma 2.2. Let δ > 0, x ∈ Rn and f be a locally integrable function. Then for any ball B = B(x0 , r) containing x with r > 0, there is ∫ |f (y)| dy ≤ Cr−δ M (f )(x), n+δ B c |x0 − y| where C is a positive constant independent of f , x, x0 and r. We omit the proof of Lemma 2.2 since it is conventional. Now, we can return to prove Theorem 2.1. Proof of Theorem 2.1. In order to simplify the proof, we only consider the situation when m = 2. Actually, the similar procedure works for all other situations. Let f1 , f2 be bounded measurable functions with compact support, then for any ball B = B(x0 , rB ) containing x with rB > 0, we will consider two cases, respectively. Case 1: rB ≥ 1. Write f1 = f1 χ2B + f1 χ(2B)c := f11 + f12 , f2 = f2 χ2B + f2 χ(2B)c := f21 + f22 . (2.1)

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Take a c0 = T (f12 , f21 )(x0 ) + T (f11 , f22 )(x0 ) + T (f12 , f22 )(x0 ), then (

1 |B| (

∫

⏐ ⏐ ⏐|T (f1 , f2 )(z)|δ − |c0 |δ ⏐dz

)1/δ

B

)1/δ ∫ 1 δ |T (f1 , f2 )(z) − c0 | dz |B| B ( )1/δ ∫ 1 δ ≤C |T (f11 , f21 )(z)| dz |B| B )1/δ ( ∫ 1 δ |T (f12 , f21 )(z) − T (f12 , f21 )(x0 )| dz +C |B| B ( )1/δ ∫ 1 δ +C |T (f11 , f22 )(z) − T (f11 , f22 )(x0 )| dz |B| B ( )1/δ ∫ 4 ∑ 1 δ +C |T (f12 , f22 )(z) − T (f12 , f22 )(x0 )| dz := Ij . |B| B j=1

≤

Notice that 0 < δ < r < ∞, where r is given as in Definition 1.2. It follows from Lemma 2.1 and (2) of Definition 1.2 that I1 ≤ C|B|

−1/δ

∥T (f11 , f21 )∥Lδ (B)

−1/r

≤ C|B| ∥T (f11 , f21 )∥Lr,∞ (B) ( ) r1 ( ) r1 ∫ ∫ 1 2 1 1 r r ≤C |f1 (y1 )| 1 dy1 |f2 (y2 )| 2 dy2 |2B| 2B |2B| 2B ≤ CMr1 (f1 )(x)Mr2 (f2 )(x) ≤ CMs (f1 )(x)Ms (f2 )(x). α

α ≤ rB ≤ 12 |y1 − x0 |. By H¨older’s inequality, the For z ∈ B and y1 ∈ (2B)c , there is |z − x0 | ≤ rB condition of the kernel in (1) of Definition 1.2 and Lemma 2.2, we have ∫ 1 |T (f12 , f21 )(z) − T (f12 , f21 )(x0 )|dz I2 ≤ C |B| B ∫ ∫ ∫ 1 ≤C |K(z, y1 , y2 ) − K(x0 , y1 , y2 )∥f1 (y1 )∥f2 (y2 )|dy2 dy1 dz |B| B (2B)c 2B ∫ ∫ ∫ ε 1 |z − x0 | ≤C |f1 (y1 )||f2 (y2 )|dy2 dy1 dz |B| B (2B)c 2B (|x0 − y1 | + |x0 − y2 |)2n+ε/α (∫ )(∫ ) |f1 (y1 )| ε ≤ CrB |f2 (y2 )|dy2 dy1 2n+ε/α 2B (2B)c |x0 − y1 | −(n+ε/α)

ε ≤ CrB |B|M (f2 )(x)rB

≤

M (f1 )(x)

ε−ε/α CM (f1 )(x)M (f2 )(x)rB

≤ CMs (f1 )(x)Ms (f2 )(x). Similarly we can get that I3 ≤ CMs (f1 )(x)Ms (f2 )(x).

Y. Lin / Nonlinear Analysis 192 (2020) 111699

α

7

α

For z ∈ B and y1 , y2 ∈ (2B)c , there are |z − x0 | ≤ 21 |y1 − x0 | and |z − x0 | ≤ 21 |y2 − x0 |. It follows from H¨ older’s inequality, (1) of Definition 1.2 and Lemma 2.2 that I4 ≤ C

∫

1 |B|

|T (f12 , f22 )(z) − T (f12 , f22 )(x0 )|dz

B

∫ ∫

1 ≤C |B|

∫ |K(z, y1 , y2 ) − K(x0 , y1 , y2 )∥f1 (y1 )∥f2 (y2 )|dy2 dy1 dz

B

(2B)c

(2B)c

∫ ∫ ∫ ε 1 |z − x0 | |f1 (y1 )||f2 (y2 )|dy2 dy1 dz |B| B (2B)c (2B)c (|x0 − y1 | + |x0 − y2 |)2n+ε/α (∫ )(∫ ) |f1 (y1 )| |f2 (y2 )| ε ≤ CrB dy dy 1 2 n+ε/(2α) n+ε/(2α) (2B)c |x0 − y1 | (2B)c |x0 − y2 | ≤C

−ε/(2α)

ε ≤ CrB rB

−ε/(2α)

M (f1 )(x)rB

M (f2 )(x)

≤ CMs (f1 )(x)Ms (f2 )(x). Case 2: 0 < rB < 1. ˜ = B(x0 , rα ). Write Denote by B B ˜1 ˜2 f1 = f1 χ2B˜ + f1 χ(2B) ˜ c := f1 + f1 ,

˜1 ˜2 f2 = f2 χ2B˜ + f2 χ(2B) ˜ c := f2 + f2 .

(2.2)

Take a c˜0 = T (f˜12 , f˜21 )(x0 ) + T (f˜11 , f˜22 )(x0 ) + T (f˜12 , f˜22 )(x0 ), then )1/δ ∫ ⏐ 1 δ⏐ ⏐|T (f1 , f2 )(z)|δ − |˜ c0 | ⏐dz |B| B ( )1/δ ∫ 1 δ ≤C |T (f˜11 , f˜21 )(z)| dz |B| B )1/δ ( ∫ 1 δ 2 ˜1 2 ˜1 ˜ ˜ |T (f1 , f2 )(z) − T (f1 , f2 )(x0 )| dz +C |B| B ( )1/δ ∫ 1 δ 1 ˜2 1 ˜2 ˜ ˜ +C |T (f1 , f2 )(z) − T (f1 , f2 )(x0 )| dz |B| B ( )1/δ ∫ 4 ∑ 1 δ +C |T (f˜12 , f˜22 )(z) − T (f˜12 , f˜22 )(x0 )| dz := I˜j . |B| B j=1 (

Notice that 0 < δ < q < ∞ and 0 < l/q ≤ α, where l and q are given as in Definition 1.2. It follows from Lemma 2.1 and (3) of Definition 1.2 that −1/δ I˜1 ≤ C|B| ∥T (f˜11 , f˜21 )∥Lδ (B) −1/q

∥T (f˜11 , f˜21 )∥Lq,∞ (B) )1( )1 ( ∫ ∫ l1 l2 1 1 l l −1/q ˜ 1/l |f1 (y1 )| 1 dy1 |f2 (y2 )| 2 dy2 ≤ C|B| |B| ˜ 2B˜ ˜ 2B˜ |2B| |2B| ≤ C|B|

n(α/l−1/q)

≤ CrB

Ml1 (f1 )(x)Ml2 (f2 )(x)

≤ CMs (f1 )(x)Ms (f2 )(x).

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α

˜ c , there is |z − x0 | ≤ rα ≤ 1 |y1 − x0 |. By H¨older’s inequality, (1) of For z ∈ B and y1 ∈ (2B) B 2 Definition 1.2 and Lemma 2.2, we have ∫ 1 ˜ I2 ≤ C |T (f˜12 , f˜21 )(z) − T (f˜12 , f˜21 )(x0 )|dz |B| B ∫ ∫ ∫ ε |z − x0 | 1 |f1 (y1 )||f2 (y2 )|dy2 dy1 dz ≤C |B| B (2B) ˜ c 2B ˜ (|x0 − y1 | + |x0 − y2 |)2n+ε/α (∫ )(∫ ) |f1 (y1 )| ε ≤ CrB |f2 (y2 )|dy2 dy 1 ˜ c |x0 − y1 |2n+ε/α ˜ 2B (2B) ε ˜ α −(n+ε/α) ≤ CrB |B|M (f2 )(x)(rB ) M (f1 )(x) ≤ CMs (f1 )(x)Ms (f2 )(x). Similarly we can get that I˜3 ≤ CMs (f1 )(x)Ms (f2 )(x). ˜ c , there are |z − x0 |α ≤ 1 |y1 − x0 | and |z − x0 |α ≤ 1 |y2 − x0 |. It follows from For z ∈ B and y1 , y2 ∈ (2B) 2 2 H¨ older’s inequality, (1) of Definition 1.2 and Lemma 2.2 that ∫ 1 I˜4 ≤ C |T (f˜12 , f˜22 )(z) − T (f˜12 , f˜22 )(x0 )|dz |B| B ∫ ∫ ∫ ε |z − x0 | 1 |f1 (y1 )||f2 (y2 )|dy2 dy1 dz ≤C |B| B (2B) ˜ c (2B) ˜ c (|x0 − y1 | + |x0 − y2 |)2n+ε/α (∫ )(∫ ) |f1 (y1 )| |f2 (y2 )| ε ≤ CrB dy dy 1 2 ˜ c |x0 − y1 |n+ε/(2α) ˜ c |x0 − y2 |n+ε/(2α) (2B) (2B) ε α −ε/(2α) α −ε/(2α) ≤ CrB (rB ) M (f1 )(x)(rB ) M (f2 )(x) ≤ CMs (f1 )(x)Ms (f2 )(x). Thus, combining the estimates in both cases, there is Mδ♯ (T (f1 , f2 ))(x) ∼ sup inf

B∋x a∈C

(

1 |B|

∫

⏐ ⏐ ⏐|T (f1 , f2 )(z)|δ − a⏐dz

)1/δ

B

≤ CMs (f1 )(x)Ms (f2 )(x), which completes the proof of the theorem. 2.2. Boundedness on product of weighted Lebesgue spaces A non-negative measurable function w is said to be in the Muckenhoupt class Ap with 1 < p < ∞ if for every cube Q in Rn , there exists a positive constant C independent of Q such that (

1 |Q|

)( )p−1 ∫ 1 1−p′ ω(x)dx w(x) dx ≤ C, |Q| Q Q

∫

where Q denotes a cube in Rn with the side parallel to the coordinate axes and 1/p + 1/p′ = 1. When p = 1, a non-negative measurable function w is said to belong to A1 , if there exists a constant C > 0 such that for any cube Q, ∫ 1 w(y)dy ≤ Cw(x), a.e. x ∈ Q. |Q| Q

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⋃ Denote by A∞ = p≥1 Ap . The well known property of the Muckenhoupt class is that if w ∈ Ap with 1 < p < ∞, then w ∈ Ar for all r > p, and w ∈ Aq for some 1 < q < p. By means of the pointwise estimate for the sharp maximal function, we can establish the boundedness of the multilinear strongly singular Calder´ on-Zygmund operator on product of weighted Lebesgue spaces as follows. Theorem 2.2. Let T be an m-linear strongly singular Calder´ on-Zygmund operator and s = max{r1 , . . . , rm , l1 , . . . , lm }, where rj and lj are given as in Definition 1.2, j = 1, . . . , m. Then for any s < p1 , . . . , pm < ∞ with 1/p = 1/p1 + · · · + 1/pm , T can be extended into a bounded operator from Lp1 (w1 ) × · · · × Lpm (wm ) into p ∏m p Lp (w), where (w1 , . . . , wm ) ∈ (Ap1 /s , . . . , Apm /s ) and w = j=1 wj j . For the special case α = 1, as we have discussed in Section 2.1, the standard multilinear Calder´onZygmund operator satisfies the conditions of Definition 1.2 by taking r1 = · · · = rm = 1. Thus we can obtain the corresponding weighted estimate for the standard multilinear Calder´on-Zygmund operator as a corollary of Theorem 2.2. Corollary 2.2. Let T be a standard m-linear Calder´ on-Zygmund operator. Then for any 1 < p1 , . . . , pm < ∞ with 1/p = 1/p1 + · · · + 1/pm , T is bounded from Lp1 (w1 ) × · · · × Lpm (wm ) into Lp (w), where (w1 , . . . , wm ) ∈ p ∏m pj (Ap1 , . . . , Apm ) and w = j=1 wj . Remark 2.2. It should be pointed out that in the special case α = 1, the result we obtain in Corollary 2.2 is in agreement with the one obtained in [18]. From this point of view, we can conclude that the range of indexes in Theorem 2.2 is reasonable. Now, we come back to the proof of Theorem 2.2. Firstly, we need the following two necessary lemmas. Lemma 2.3 (See [27]). Let 0 < p, δ < ∞ and w ∈ A∞ . Then there exists a constant C > 0 depending only on the A∞ constant of w such that ∫ ∫ p [Mδ (f )(x)] w(x)dx ≤ C [Mδ♯ (f )(x)]p w(x)dx, Rn

Rn

for every function f such that the left-hand side is finite. Lemma 2.4 (See [18]). For (w1 , . . . , wm ) ∈ (Ap1 , . . . , Apm ) with 1 ≤ p1 , . . . , pm < ∞, and for 0 < θm θ1 , . . . , θm < 1 such that θ1 + · · · + θm = 1, we have w1θ1 · · · wm ∈ Amax{p1 ,...,pm } . Then, we can prove Theorem 2.2 as follows. Proof of Theorem 2.2. It follows from Lemma 2.4 that w ∈ Amax{p1 /s,...,pm /s} ⊂ A∞ . For every j = 1, . . . , m, wj ∈ Apj /s and pj > s imply that the Hardy–Littlewood maximal operator M is bounded on Lpj /s (wj ). Take a δ such that 0 < δ < 1/m, then by Lemma 2.3 and Theorem 2.1, we have ∥T (f⃗)∥Lp (w) ≤ ∥Mδ (T (f⃗))∥Lp (w) ≤ C∥Mδ♯ (T (f⃗))∥Lp (w) ∏  m ∏   m  ≤ C Ms (fj ) ≤ C ∥Ms (fj )∥Lpj (wj )  p j=1

L (w)

j=1

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10

=C =C

m ∏ j=1 m ∏

s

1/s

∥M (|fj | )∥

p /s L j (wj )

≤C

m ∏  s 1/s |fj |  p /s j j=1

L

(wj )

∥fj ∥Lpj (wj ) ,

j=1

which gives the desired result. 2.3. Boundedness on product of variable exponent Lebesgue spaces The theory of variable exponent function spaces has been intensely investigated recently since some elementary results were established by Kov´ aˇcik and R´akosn´ık in [26]. For the properties of Calder´onZygmund operators and many classical operators on the variable exponent Lebesgue space, one can refer to [10,11,13,14,26] and their references. Definition 2.1. Let p(·) : Rn → [1, ∞) be a measurable function. The variable exponent Lebesgue space, Lp(·) (Rn ), is defined by )p(x) } { ∫ ( |f (x)| dx < ∞ for some constant λ > 0 . Lp(·) (Rn ) = f is measurable : λ Rn It is well known that the set Lp(·) (Rn ) becomes a Banach space with respect to the norm )p(x) } { ∫ ( |f (x)| dx ≤ 1 . ∥f ∥Lp(·) (Rn ) = inf λ > 0 : λ Rn Denote by P(Rn ) the set of all measurable functions p(·) : Rn → [1, ∞) such that 1 < p− := essinf x∈Rn p(x) and p+ := esssupx∈Rn p(x) < ∞, and by B(Rn ) the set of all p(·) ∈ P(Rn ) such that the Hardy–Littlewood maximal operator M is bounded on Lp(·) (Rn ). By means of the pointwise estimate for the sharp maximal function, we can establish the boundedness of the multilinear strongly singular Calder´ on-Zygmund operator on product of variable exponent Lebesgue spaces as follows. Theorem 2.3. Let p(·), p1 (·), . . . , pm (·) ∈ B(Rn ) with 1/p(·) = 1/p1 (·) + · · · + 1/pm (·). Let q0j be given as in Lemma 2.5 for pj (·), j = 1, . . . , m. Let T be an m-linear strongly singular Calder´ on-Zygmund operator and s = max{r1 , . . . , rm , l1 , . . . , lm }, where rj and lj are given as in Definition 1.2, j = 1, . . . , m. If s ≤ min1≤j≤m q0j , then T can be extended into a bounded operator from Lp1 (·) (Rn ) × · · · × Lpm (·) (Rn ) into Lp(·) (Rn ). For the special case α = 1, as we have discussed in Section 2.1, the standard multilinear Calder´onZygmund operator satisfies the conditions of Definition 1.2 by taking r1 = · · · = rm = 1. Thus we can obtain the corresponding boundedness for the standard multilinear Calder´on-Zygmund operator on product of variable exponent Lebesgue spaces as a corollary of Theorem 2.3. Corollary 2.3. Let p(·), p1 (·), . . . , pm (·) ∈ B(Rn ) with 1/p(·) = 1/p1 (·) + · · · + 1/pm (·) and T be a standard m-linear Calder´ on-Zygmund operator. Then T is bounded from Lp1 (·) (Rn ) × · · · × Lpm (·) (Rn ) into Lp(·) (Rn ). Now, we come back to the proof of Theorem 2.3. Firstly, we need the following necessary lemmas.

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11

Lemma 2.5 (See [12]). Let p(·) ∈ P(Rn ). Then M is bounded on Lp(·) (Rn ) if and only if Mq0 is bounded on Lp(·) (Rn ) for some 1 < q0 < ∞. Lemma 2.6 (See [35]). Let p(·), p1 (·), . . . , pm (·) ∈ P(Rn ) so that 1/p(x) = 1/p1 (x) + · · · + 1/pm (x). Then for any fj ∈ Lpj (·) (Rn ), j = 1, . . . , m, there has  ∏ m ∏  m  m−1   ≤2 ∥fj ∥Lpj (·) (Rn ) . fj   j=1

Lp(·) (Rn )

j=1

Lemma 2.7 (See [13]). Given a family F of ordered pairs of measurable functions, suppose for some fixed 0 < p0 < ∞, every (f, g) ∈ F and every w ∈ A1 , ∫ ∫ p p |f (x)| 0 w(x)dx ≤ C0 |g(x)| 0 w(x)dx. Rn

Rn

( p(·) )′

Let p(·) ∈ P(Rn ) with p0 ≤ p− . If p0 (f, g) ∈ F, ∥f ∥Lp(·) (Rn ) ≤ C∥g∥Lp(·) (Rn ) .

∈ B(Rn ), then there exists a constant C > 0 such that for all

Lemma 2.8 (See [13]). If p(·) ∈ P(Rn ), then C0∞ (Rn ) is dense in Lp(·) (Rn ). Lemma 2.9 (See [12]). Let p(·) ∈ P(Rn ). Then the following conditions are equivalent. (i) p(·) ∈ B(Rn ). (ii) p′ (·) ∈ B(Rn ). (iii) p(·) ∈ B(Rn ) for some 1 < p0 < p− . 0 ) (pp(·) ′ (iv) p0 ∈ B(Rn ) for some 1 < p0 < p− . Then, we are able to prove Theorem 2.3. Proof of Theorem 2.3. Since p(·) ∈ B(Rn ), then by Lemma 2.9, there exists a p0 such that 1 < p0 < p− ( )′ and p(·) ∈ B(Rn ). Take a δ such that 0 < δ < 1/m. For any w ∈ A1 , it follows from Lemma 2.3 and p0 Theorem 2.1 that ∫ p0 |T (f⃗)(x)| w(x)dx n R∫ ≤ [Mδ (T (f⃗))(x)]p0 w(x)dx n R∫ ≤C [Mδ♯ (T (f⃗))(x)]p0 w(x)dx n R ]p0 ∫ [∏ m ≤C Ms (fj )(x) w(x)dx Rn j=1 [∏ m

∫ ≤C

Rn j=1

]p0 Mqj (fj )(x) w(x)dx 0

holds for all m-tuples f⃗ = (f1 , . . . , fm ) of bounded measurable functions with compact support. ( ) ∏m Applying Lemma 2.7 to the pair T (f⃗), j=1 Mqj (fj ) , we have 0

∏   m  ⃗  ∥T (f )∥Lp(·) (Rn ) ≤ C  Mqj (fj )  j=1

0

Lp(·) (Rn )

.

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12

Noticing the choice of q0j , we can get that Mqj is bounded on Lpj (·) (Rn ) by Lemma 2.5, j = 1, . . . , m. 0 Then it follows from Lemma 2.6 that m ∏

∥T (f⃗)∥Lp(·) (Rn ) ≤ C

∥Mqj (fj )∥Lpj (·) (Rn ) ≤ C 0

j=1

m ∏

∥fj ∥Lpj (·) (Rn ) .

j=1

This completes the proof of the theorem. 3. Endpoint estimate 3.1. Boundedness of L∞ × · · · × L∞ → BM O type In this section, we will focus on the behaviors when p1 = · · · = pm = ∞ and establish the endpoint estimate for the multilinear strongly singular Calder´on-Zygmund operator from L∞ × · · · × L∞ into BM O. Theorem 3.1. Let T be an m-linear strongly singular Calder´ on-Zygmund operator, q be given as in (3) of Definition 1.2 and q > 1. Then T can be extended into a bounded operator from L∞ × · · · × L∞ into BM O. Proof . We only give the proof when m = 2 and omit other situations since their similarities. Let f1 , f2 ∈ L∞ , then for any ball B = B(x0 , rB ) with rB > 0, we will consider two cases, respectively. Case 1: rB ≥ 1. Use the same decomposition as (2.1) and choose the same c0 , then ∫ 1 |T (f1 , f2 )(x) − c0 |dx |B| B ∫ ∫ 1 1 1 1 ≤ |T (f1 , f2 )(x)|dx + |T (f12 , f21 )(x) − T (f12 , f21 )(x0 )|dx |B| B |B| B ∫ ∫ 1 1 |T (f11 , f22 )(x) − T (f11 , f22 )(x0 )|dx + |T (f12 , f22 )(x) − T (f12 , f22 )(x0 )|dx + |B| B |B| B 4 ∑ := Jj . j=1

Denote by s = max{r1 , . . . , rm , l1 , . . . , lm }, where rj and lj are given as in Definition 1.2, j = 1, . . . , m. Take p1 , . . . , pm such that max{s, m} < p1 , . . . , pm < ∞. Let 1/p = 1/p1 + · · · + 1/pm , then 1 < p < ∞. It follows from Theorem 2.2 that T is bounded from Lp1 × · · · × Lpm into Lp . By H¨ older’s inequality and the Lp1 × Lp2 → Lp boundedness of T , we have ( J1 ≤

1 |B|

∫

p

|T (f11 , f21 )(x)| dx

B −1/p

≤ C|B|

) p1

∥f11 ∥Lp1 ∥f21 ∥Lp2

≤ C∥f1 ∥∞ ∥f2 ∥∞ . α

For x ∈ B and y1 ∈ (2B)c , there is |x − x0 | ≤ 12 |y1 − x0 |. By the condition of the kernel in (1) of Definition 1.2, we have ∫ ∫ ∫ 1 J2 ≤ |K(x, y1 , y2 ) − K(x0 , y1 , y2 )∥f1 (y1 )∥f2 (y2 )|dy2 dy1 dx |B| B (2B)c 2B ∫ ∫ ∫ ε 1 |x − x0 | ≤ C∥f1 ∥∞ ∥f2 ∥∞ dy2 dy1 dx |B| B (2B)c 2B (|x0 − y1 | + |x0 − y2 |)2n+ε/α

Y. Lin / Nonlinear Analysis 192 (2020) 111699

ε ≤ C∥f1 ∥∞ ∥f2 ∥∞ rB |B|

∫

1

(2B)c

|x0 − y1 |

2n+ε/α

13

dy1

≤ C∥f1 ∥∞ ∥f2 ∥∞ . Similarly we can get that J3 ≤ C∥f1 ∥∞ ∥f2 ∥∞ . α

α

For x ∈ B and y1 , y2 ∈ (2B)c , there are |x − x0 | ≤ 21 |y1 − x0 | and |x − x0 | ≤ 12 |y2 − x0 |. It follows from (1) of Definition 1.2 that ∫ ∫ ∫ 1 J4 ≤ |K(x, y1 , y2 ) − K(x0 , y1 , y2 )∥f1 (y1 )∥f2 (y2 )|dy2 dy1 dx |B| B (2B)c (2B)c ∫ ∫ ∫ ε 1 |x − x0 | ≤ C∥f1 ∥∞ ∥f2 ∥∞ dy2 dy1 dx |B| B (2B)c (2B)c (|x0 − y1 | + |x0 − y2 |)2n+ε/α (∫ )(∫ ) 1 1 ε ≤ C∥f1 ∥∞ ∥f2 ∥∞ rB dy1 dy2 n+ε/(2α) n+ε/(2α) (2B)c |x0 − y1 | (2B)c |x0 − y2 | −ε/(2α) −ε/(2α) rB

ε ≤ C∥f1 ∥∞ ∥f2 ∥∞ rB rB

≤ C∥f1 ∥∞ ∥f2 ∥∞ . Case 2: 0 < rB < 1. ˜ = B(x0 , rα ). Use the same decomposition as (2.2) and choose the same c˜0 , then Denote by B B ∫ 1 |T (f1 , f2 )(x) − c˜0 |dx |B| B ∫ ∫ 1 1 1 ˜1 ˜ |T (f1 , f2 )(x)|dx + |T (f˜12 , f˜21 )(x) − T (f˜12 , f˜21 )(x0 )|dx ≤ |B| B |B| B ∫ ∫ 1 1 |T (f˜11 , f˜22 )(x) − T (f˜11 , f˜22 )(x0 )|dx + |T (f˜12 , f˜22 )(x) − T (f˜12 , f˜22 )(x0 )|dx + |B| B |B| B 4 ∑ := J˜j . j=1

Notice that 1 < q < ∞ and 0 < l/q ≤ α, where l is given as in Definition 1.2. It follows from Lemma 2.1 and (3) of Definition 1.2 that −1 J˜1 ≤ C|B| ∥T (f˜11 , f˜21 )∥L1 (B) −1/q ≤ C|B| ∥T (f˜1 , f˜1 )∥Lq,∞ (B) 1

−1/q

≤ C|B|

2

∥f˜11 ∥Ll1 ∥f˜21 ∥Ll2 n(α/l−1/q)

≤ C∥f1 ∥∞ ∥f2 ∥∞ rB ≤ C∥f1 ∥∞ ∥f2 ∥∞ . α

˜ c , there is |x − x0 | ≤ 1 |y1 − x0 |. By the condition of the kernel in (1) of For x ∈ B and y1 ∈ (2B) 2 Definition 1.2, we have ∫ ∫ ∫ ε 1 |x − x0 | J˜2 ≤ C∥f1 ∥∞ ∥f2 ∥∞ dy2 dy1 dx |B| B (2B) ˜ c 2B ˜ (|x0 − y1 | + |x0 − y2 |)2n+ε/α ∫ 1 ε ˜ ≤ C∥f1 ∥∞ ∥f2 ∥∞ rB |B| dy1 ˜ c |x0 − y1 |2n+ε/α (2B) ≤ C∥f1 ∥∞ ∥f2 ∥∞ .

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14

Similarly we can get that J˜3 ≤ C∥f1 ∥∞ ∥f2 ∥∞ . α

α

˜ c , there are |x − x0 | ≤ 1 |y1 − x0 | and |x − x0 | ≤ 1 |y2 − x0 |. It follows from For x ∈ B and y1 , y2 ∈ (2B) 2 2 (1) of Definition 1.2 that ∫ ∫ ∫ ε 1 |x − x0 | ˜ dy2 dy1 dx J4 ≤ C∥f1 ∥∞ ∥f2 ∥∞ |B| B (2B) ˜ c (2B) ˜ c (|x0 − y1 | + |x0 − y2 |)2n+ε/α (∫ )(∫ ) 1 1 ε ≤ C∥f1 ∥∞ ∥f2 ∥∞ rB dy1 dy2 ˜ c |x0 − y1 |n+ε/(2α) ˜ c |x0 − y2 |n+ε/(2α) (2B) (2B) ≤ C∥f1 ∥∞ ∥f2 ∥∞ . Thus, combining the estimates in both cases, there is ∫ 1 |T (f1 , f2 )(x) − a|dx ≤ C∥f1 ∥∞ ∥f2 ∥∞ , ∥T (f1 , f2 )∥BM O ∼ sup inf B a∈C |B| B which completes the proof of the theorem. 3.2. Boundedness of BM O × · · · × BM O → BM O type The definition of the BM O space is well known. The most useful property of the BM O function is the classical John–Nirenberg inequality, which shows that functions in the BM O space are locally exponentially integrable. A well known result in [43] showed that the classical Calder´on-Zygmund operator is bounded on the BM O space. Lin and Lu also gave the BM O boundedness of the linear strongly singular Calder´on-Zygmund operator in [33]. These conclusions essentially depend on the cancellation condition of the kernel, which can be expressed as T 1 = 0. A natural question is: if we give some kinds of cancellation conditions to the multilinear strongly singular Calder´ on-Zygmund operator, whether it can also be bounded on the product of BM O spaces? Actually, the answer is affirmative. Theorem 3.2. Let T be an m-linear strongly singular Calder´ on–Zygmund operator, q be given as in (3) of Definition 1.2 and q > 1. If T (f1 , . . . , fj−1 , 1, fj+1 , . . . , fm ) = 0, j = 1, . . . , m, then T can be extended into a bounded operator from BM O × · · · × BM O into BM O. We need the following two lemmas to develop the proof of Theorem 3.2. Lemma 3.1 (See [33]).. Let f be a function in BM O. Suppose 1 ≤ p < ∞, x ∈ Rn , and r1 , r2 > 0. Then (

1 |B(x, r1 )|

∫ B(x, r1 )

p

|f (y) − fB(x, r2 ) | dy

)1/p

( ) ⏐ r1 ⏐ ≤ C 1 + ⏐ ln ⏐ ∥f ∥BM O , r2

where C > 0 is independent of f , x, r1 and r2 . Lemma 3.2. Let δ > 0 and f ∈ BM O. Then for any ball B = B(x, r) with r > 0 and x ∈ Rn , there is ∫ |f (y) − fB | dy ≤ Cr−δ ∥f ∥BM O , n+δ B c |x − y| where C is a positive constant independent of f , x and r.

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We omit the proof of Lemma 3.2 since it is conventional by using the definition of BM O functions and Lemma 3.1. Now, we are able to return to prove the main result in this section. Proof of Theorem 3.2. We only give the proof when m = 2 and omit other situations since their similarities. Let f1 , f2 ∈ BM O, then for any ball B = B(x0 , rB ) with rB > 0, we will consider two cases, respectively. Case 1: rB ≥ 1. Write f1 = (f1 )2B + (f1 − (f1 )2B )χ2B + (f1 − (f1 )2B )χ(2B)c := f11 + f12 + f13 , f2 = (f2 )2B + (f2 − (f2 )2B )χ2B + (f2 − (f2 )2B )χ(2B)c := f21 + f22 + f23 . It follows from the hypothesis of the theorem that T (f1 , f2 ) = T (f12 , f22 ) + T (f12 , f23 ) + T (f13 , f22 ) + T (f13 , f23 ). Take a d0 = T (f12 , f23 )(x0 ) + T (f13 , f22 )(x0 ) + T (f13 , f23 )(x0 ), then ∫ 1 |T (f1 , f2 )(x) − d0 |dx |B| B ∫ ∫ 1 1 |T (f12 , f22 )(x)|dx + |T (f12 , f23 )(x) − T (f12 , f23 )(x0 )|dx ≤ |B| B |B| B ∫ ∫ 1 1 3 2 3 2 |T (f1 , f2 )(x) − T (f1 , f2 )(x0 )|dx + |T (f13 , f23 )(x) − T (f13 , f23 )(x0 )|dx + |B| B |B| B 4 ∑ Lj . := j=1

Choose p1 , . . . , pm , p the same as in the proof of Theorem 3.1. Then T is bounded from Lp1 × · · · × Lpm into Lp and 1 < p < ∞. It follows from H¨ older’s inequality that ( L1 ≤

1 |B| (

∫

) p1 p |T (f12 , f22 )(x)| dx

B

) p1 ( ) p1 ∫ ∫ 1 2 1 1 p1 p2 ≤C |f1 (y1 ) − (f1 )2B | dy1 |f2 (y2 ) − (f2 )2B | dy2 |2B| 2B |2B| 2B ≤ C∥f1 ∥BM O ∥f2 ∥BM O . α

For x ∈ B and y2 ∈ (2B)c , there is |x − x0 | ≤ 12 |y2 − x0 |. By the condition of the kernel in Definition 1.2 and Lemma 3.2, we have ∫ ∫ ∫ 1 L2 ≤ |K(x, y1 , y2 ) − K(x0 , y1 , y2 )| |B| B (2B)c 2B ×|f1 (y1 ) − (f1 )2B ||f2 (y2 ) − (f2 )2B |dy1 dy2 dx ∫ ∫ ∫ ε 1 |x − x0 | ≤C |B| B (2B)c 2B (|x0 − y1 | + |x0 − y2 |)2n+ε/α ×|f1 (y1 ) − (f1 )2B ||f2 (y2 ) − (f2 )2B |dy1 dy2 dx (∫ )(∫ ) |f2 (y2 ) − (f2 )2B | ε ≤ CrB |f1 (y1 ) − (f1 )2B |dy1 dy2 2n+ε/α 2B (2B)c |x0 − y2 | −(n+ε/α)

ε ≤ CrB ∥f1 ∥BM O |B|rB

≤ C∥f1 ∥BM O ∥f2 ∥BM O .

∥f2 ∥BM O

Y. Lin / Nonlinear Analysis 192 (2020) 111699

16

Similarly we can get that L3 ≤ C∥f1 ∥BM O ∥f2 ∥BM O . α

α

For x ∈ B and y1 , y2 ∈ (2B) , there are |x − x0 | ≤ 21 |y1 − x0 | and |x − x0 | ≤ 12 |y2 − x0 |. It follows from (1) of Definition 1.2 and Lemma 3.2 that ∫ ∫ ∫ 1 L4 ≤ |K(x, y1 , y2 ) − K(x0 , y1 , y2 )| |B| B (2B)c (2B)c c

×|f1 (y1 ) − (f1 )2B ||f2 (y2 ) − (f2 )2B |dy1 dy2 dx ∫ ∫ ∫ ε |x − x0 | 1 ≤C |B| B (2B)c (2B)c (|x0 − y1 | + |x0 − y2 |)2n+ε/α ×|f1 (y1 ) − (f1 )2B ||f2 (y2 ) − (f2 )2B |dy1 dy2 dx (∫ )(∫ ) |f1 (y1 ) − (f1 )2B | |f2 (y2 ) − (f2 )2B | ε ≤ CrB dy dy 1 2 n+ε/(2α) n+ε/(2α) (2B)c |x0 − y1 | (2B)c |x0 − y2 | −ε/(2α)

ε ≤ CrB rB

−ε/(2α)

∥f1 ∥BM O rB

∥f2 ∥BM O

≤ C∥f1 ∥BM O ∥f2 ∥BM O . Case 2: 0 < rB < 1. ˜ = B(x0 , rα ). Write Denote by B B ˜1 ˜2 ˜3 f1 = (f1 )2B˜ + (f1 − (f1 )2B˜ )χ2B˜ + (f1 − (f1 )2B˜ )χ(2B) ˜ c := f1 + f1 + f1 ,

(3.1)

˜1 ˜2 ˜3 f2 = (f2 )2B˜ + (f2 − (f2 )2B˜ )χ2B˜ + (f2 − (f2 )2B˜ )χ(2B) ˜ c := f2 + f2 + f2 .

(3.2)

It follows from the hypothesis of the theorem that T (f1 , f2 ) = T (f˜12 , f˜22 ) + T (f˜12 , f˜23 ) + T (f˜13 , f˜22 ) + T (f˜13 , f˜23 ). Take a d˜0 = T (f˜12 , f˜23 )(x0 ) + T (f˜13 , f˜22 )(x0 ) + T (f˜13 , f˜23 )(x0 ), then ∫ 1 |T (f1 , f2 )(x) − d˜0 |dx |B| B ∫ ∫ 1 1 2 ˜2 ˜ ≤ |T (f1 , f2 )(x)|dx + |T (f˜12 , f˜23 )(x) − T (f˜12 , f˜23 )(x0 )|dx |B| B |B| B ∫ ∫ 1 1 3 ˜2 3 ˜2 ˜ ˜ |T (f1 , f2 )(x) − T (f1 , f2 )(x0 )|dx + |T (f˜13 , f˜23 )(x) − T (f˜13 , f˜23 )(x0 )|dx + |B| B |B| B :=

4 ∑

˜j . L

j=1

Notice that 1 < q < ∞ and 0 < l/q ≤ α, where l is given as in Definition 1.2. It follows from Lemma 2.1 and (3) of Definition 1.2 that ˜ 1 ≤ C|B|−1 ∥T (f˜12 , f˜22 )∥L1 (B) L −1/q

∥T (f˜12 , f˜22 )∥Lq,∞ (B) ( )1 ∫ l1 1 l −1/q ˜ 1/l |f1 (y1 ) − (f1 )2B˜ | 1 dy1 ≤ C|B| |B| ˜ 2B˜ |2B| ≤ C|B|

(

1 × ˜ |2B|

∫ ˜ 2B

l2

|f2 (y2 ) − (f2 )2B˜ | dy2

)1

l2

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17

n(α/l−1/q)

≤ C∥f1 ∥BM O ∥f2 ∥BM O rB ≤ C∥f1 ∥BM O ∥f2 ∥BM O . α

˜ c , there is |x − x0 | ≤ 1 |y2 − x0 |. By the condition of the kernel in Definition 1.2 For x ∈ B and y2 ∈ (2B) 2 and Lemma 3.2, we have ∫ ∫ ∫ ε |x − x0 | ˜2 ≤ C 1 L |B| B (2B) ˜ c 2B ˜ (|x0 − y1 | + |x0 − y2 |)2n+ε/α ×|f1 (y1 ) − (f1 )2B˜ ||f2 (y2 ) − (f2 )2B˜ |dy1 dy2 dx ) (∫ )(∫ |f2 (y2 ) − (f2 )2B˜ | ε dy2 ≤ CrB |f1 (y1 ) − (f1 )2B˜ |dy1 ˜ ˜ c |x0 − y2 |2n+ε/α 2B (2B) ε ˜ α )−(n+ε/α) ∥f2 ∥BM O ≤ CrB ∥f1 ∥BM O |B|(r B ≤ C∥f1 ∥BM O ∥f2 ∥BM O . Similarly we can get that ˜ 3 ≤ C∥f1 ∥BM O ∥f2 ∥BM O . L ˜ c , there are |x − x0 |α ≤ 1 |y1 − x0 | and |x − x0 |α ≤ 1 |y2 − x0 |. It follows from For x ∈ B and y1 , y2 ∈ (2B) 2 2 (1) of Definition 1.2 and Lemma 3.2 that ∫ ∫ ∫ ε |x − x0 | ˜4 ≤ C 1 L |B| B (2B) ˜ c (2B) ˜ c (|x0 − y1 | + |x0 − y2 |)2n+ε/α ×|f1 (y1 ) − (f1 )2B˜ ||f2 (y2 ) − (f2 )2B˜ |dy1 dy2 dx )(∫ ) (∫ |f2 (y2 ) − (f2 )2B˜ | |f1 (y1 ) − (f1 )2B˜ | ε dy1 dy2 ≤ CrB ˜ c |x0 − y1 |n+ε/(2α) ˜ c |x0 − y2 |n+ε/(2α) (2B) (2B) ε α −ε/(2α) α −ε/(2α) ≤ CrB (rB ) ∥f1 ∥BM O (rB ) ∥f2 ∥BM O ≤ C∥f1 ∥BM O ∥f2 ∥BM O . Thus, combining the estimates in both cases, there is ∫ 1 ∥T (f1 , f2 )∥BM O ∼ sup inf |T (f1 , f2 )(x) − a|dx ≤ C∥f1 ∥BM O ∥f2 ∥BM O , B a∈C |B| B which completes the proof of the theorem. 3.3. Boundedness of LM O × · · · × LM O → LM O type The LM O space is essentially a special case of the function space introduced by Spanne in [41]. Some properties of the LM O function are similar to those of the BM O function. But there are also some new interesting phenomena for the LM O function itself. One can refer to [1] and the references there for details. Definition 3.1. LM O is a subspace of BM O, equipped with the semi-norm ∫ ∫ 1 + | ln r| 1 [f ]LM O = sup |f (x) − fBr |dx + sup |f (x) − fBr |dx, |Br | 0
1 |Br |

∫

p

|f (x) − fBr | dx Br

)1/p .

18

Y. Lin / Nonlinear Analysis 192 (2020) 111699

The authors in [33,41,44] obtained the LM O-boundedness of classical Calder´on-Zygmund operators and linear strongly singular Calder´ on-Zygmund operators, respectively. In this section, we will establish the boundedness of the multilinear strongly singular Calder´on-Zygmund operator on product of LM O spaces. Theorem 3.3. Let T be an m-linear strongly singular Calder´ on-Zygmund operator, q be given as in (3) of Definition 1.2 and q > 1. If T (f1 , . . . , fj−1 , 1, fj+1 , . . . , fm ) = 0, j = 1, . . . , m, then T can be extended into a bounded operator from LM O × · · · × LM O into LM O. Before developing the proof of Theorem 3.3, we need the following necessary lemmas. Lemma 3.3 (See [1]). If f ∈ LM O, then for any 1 ≤ p < ∞, there exists a constant C > 0 depending only on n and p such that [f ]LM Op ≤ C[f ]LM O . Lemma 3.4 (See [33]). Let δ > 0 and f ∈ LM O. Then for any ball B = B(x, r) with 0 < r < 21 , ∫ |f (y) − fB | dy ≤ Cr−δ (1 + | ln r|)−1 [f ]LMO , n+δ B c |x − y| where C > 0 is independent of f , x and r. Lemma 3.5. For any a, b ∈ R, there is 1 + |a + b| ≥ (1 + |a|)−1 (1 + |b|).

We omit the details of the proof of Lemma 3.5 since it is elementary. Now, we can start the proof of Theorem 3.3. Proof of Theorem 3.3. We only consider the situation when m = 2. Actually, the similar procedure works for all other situations. For f1 , f2 ∈ LM O and any ball B = B(x0 , rB ) with rB ≥ 1, it follows from Theorem 3.2 and Definition 3.1 that ∫ 1 |T (f1 , f2 )(x) − (T (f1 , f2 ))B |dx |B| B ≤ ∥T (f1 , f2 )∥BM O ≤ C∥f1 ∥BM O ∥f2 ∥BM O ≤ C[f1 ]LM O [f2 ]LM O . Thus it is sufficient to prove that, for any ball B = B(x0 , rB ) with 0 < rB < 1, the following inequality holds. ∫ 1 + | ln rB | |T (f1 , f2 )(x) − (T (f1 , f2 ))B |dx ≤ C[f1 ]LM O [f2 ]LM O . (3.3) |B| B We consider two cases, respectively. Case 1: 4−1/α ≤ rB < 1. Theorem 3.2 also implies that ∫ 1 + | ln rB | |T (f1 , f2 )(x) − (T (f1 , f2 ))B |dx |B| ∫ B 1 ≤C |T (f1 , f2 )(x) − (T (f1 , f2 ))B |dx |B| B ≤ C∥T (f1 , f2 )∥BM O ≤ C[f1 ]LM O [f2 ]LM O .

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Case 2: 0 < rB < 4−1/α . ˜ = B(x0 , rα ). Use the same decompositions as (3.1)–(3.2) and choose the same d˜0 , then Denote by B B ∫ 1 + | ln rB | |T (f1 , f2 )(x) − (T (f1 , f2 ))B |dx |B| B ∫ 1 + | ln rB | ≤C |T (f1 , f2 )(x) − d˜0 |dx |B| B ∫ 1 + | ln rB | ≤C |T (f˜12 , f˜22 )(x)|dx |B| B ∫ 1 + | ln rB | +C |T (f˜12 , f˜23 )(x) − T (f˜12 , f˜23 )(x0 )|dx |B| B ∫ 1 + | ln rB | |T (f˜13 , f˜22 )(x) − T (f˜13 , f˜22 )(x0 )|dx +C |B| B ∫ 1 + | ln rB | +C |T (f˜13 , f˜23 )(x) − T (f˜13 , f˜23 )(x0 )|dx |B| B 4 ∑ Hj . := j=1 α < 1/2, 1 < q < ∞ and 0 < l/q ≤ α, where l is given as in Definition 1.2. It follows Notice that 0 < 2rB from Lemma 2.1, (3) of Definition 1.2, Lemmas 3.3 and 3.5 that −1

∥T (f˜12 , f˜22 )∥L1 (B) −1/q ≤ C(1 + | ln rB |)|B| ∥T (f˜12 , f˜22 )∥Lq,∞ (B) ( )1 ∫ l1 1 l −1/q ˜ 1/l |f1 (y1 ) − (f1 )2B˜ | 1 dy1 ≤ C(1 + | ln rB |)|B| |B| ˜ |2B| 2B˜ ( )1 ∫ l2 1 l2 |f2 (y2 ) − (f2 )2B˜ | dy2 × ˜ |2B| 2B˜ −1/q ˜ 1/l α −1 α −1 ≤ C(1 + | ln rB |)|B| |B| [f1 ]LM Ol1 (1 + | ln 2rB |) [f2 ]LM Ol2 (1 + | ln 2rB |)

H1 ≤ C(1 + | ln rB |)|B|

≤ C[f1 ]LM O [f2 ]LM O (1 + | ln rB |)|B|

−1/q

˜ |B|

1/l

n(α/l−1/q)

≤ C[f1 ]LM O [f2 ]LM O (1 + | ln rB |)rB

(1 + | ln 2 + α ln rB |)−1

(1 + α| ln rB |)−1 (1 + ln 2)

≤ C[f1 ]LM O [f2 ]LM O . α

˜ c , there is |x − x0 | ≤ 1 |y2 − x0 |. By the condition of the kernel in Definition 1.2, For x ∈ B and y2 ∈ (2B) 2 α the fact 0 < 2rB < 1/2, Lemmas 3.4 and 3.5, we have ∫ ∫ ∫ 1 + | ln rB | H2 ≤ C |K(x, y1 , y2 ) − K(x0 , y1 , y2 )| |B| ˜ c 2B ˜ B (2B) ×|f1 (y1 ) − (f1 )2B˜ ||f2 (y2 ) − (f2 )2B˜ |dy1 dy2 dx ∫ ∫ ∫ ε 1 + | ln rB | |x − x0 | ≤C |B| ˜ c 2B ˜ (|x0 − y1 | + |x0 − y2 |)2n+ε/α B (2B) ×|f1 (y1 ) − (f1 )2B˜ ||f2 (y2 ) − (f2 )2B˜ |dy1 dy2 dx (∫ )(∫ ) |f2 (y2 ) − (f2 )2B˜ | ε ≤ CrB (1 + | ln rB |) |f1 (y1 ) − (f1 )2B˜ |dy1 dy 2 ˜ ˜ c |x0 − y2 |2n+ε/α 2B (2B) ε α −(n+ε/α) α −1 ˜ B ≤ CrB (1 + | ln rB |)∥f1 ∥BM O |B|(r ) [f2 ]LM O (1 + | ln 2rB |) ≤ C[f1 ]LM O [f2 ]LM O (1 + | ln rB |)(1 + | ln 2 + α ln rB |)−1 ≤ C[f1 ]LM O [f2 ]LM O .

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Similarly we can get that H3 ≤ C[f1 ]LM O [f2 ]LM O . ˜ c , there are |x − x0 |α ≤ 1 |y1 − x0 | and |x − x0 |α ≤ 1 |y2 − x0 |. It follows from For x ∈ B and y1 , y2 ∈ (2B) 2 2 (1) of Definition 1.2, Lemmas 3.4 and 3.5 that ∫ ∫ ∫ 1 + | ln rB | H4 ≤ C |K(x, y1 , y2 ) − K(x0 , y1 , y2 )| |B| ˜ c (2B) ˜ c B (2B) ×|f1 (y1 ) − (f1 )2B˜ ||f2 (y2 ) − (f2 )2B˜ |dy1 dy2 dx ∫ ∫ ∫ ε |x − x0 | 1 + | ln rB | ≤C |B| ˜ c (2B) ˜ c (|x0 − y1 | + |x0 − y2 |)2n+ε/α B (2B) ×|f1 (y1 ) − (f1 )2B˜ ||f2 (y2 ) − (f2 )2B˜ |dy1 dy2 dx (∫ )(∫ ) |f1 (y1 ) − (f1 )2B˜ | |f2 (y2 ) − (f2 )2B˜ | ε ≤ CrB (1 + | ln rB |) dy1 dy2 ˜ c |x0 − y1 |n+ε/(2α) ˜ c |x0 − y2 |n+ε/(2α) (2B) (2B) ε α −ε/(2α) α −1 ≤ CrB (1 + | ln rB |)(rB ) [f1 ]LM O (1 + | ln 2rB |) α −ε/(2α) α −1 ×(rB ) [f2 ]LM O (1 + | ln 2rB |)

≤ C[f1 ]LM O [f2 ]LM O (1 + | ln rB |)(1 + | ln 2 + α ln rB |)−1 ≤ C[f1 ]LM O [f2 ]LM O . Thus, combining the estimates in both cases, (3.3) holds, which completes the proof of the theorem. References [1] P. Acquistapace, On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl. 161 (1992) 231–269. [2] J. Alvarez, M. Milman, H p Continuous properties of Calder´ on-Zygmund-type operators, J. Math. Anal. Appl. 118 (1986) 63–79. [3] J. Alvarez, M. Milman, Vector valued inequalities for strongly singular Calder´ on-Zygmund operators, Rev. Mat. Iberoamericana 2 (1986) 405–426. [4] T.A. Bui, Global W 1,p(·) estimate for renormalized solutions of quasilinear equations with measure data on Reifenberg domains, Adv. Nonlinear Anal. 7 (2018) 517–533. [5] T.A. Bui, X.T. Duong, Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers, Bull. Sci. Math. 137 (2013) 63–75. [6] S. Chanillo, Weighted norm inequalities for strongly singular convolution operators, Trans. Amer. Math. Soc. 281 (1984) 77–107. [7] R.R. Coifman, Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975) 315–331. [8] R.R. Coifman, Y. Meyer, Au del des oprateurs pseudo-diffrentiels, Ast` erisque 57 (1978). [9] R.R. Coifman, Y. Meyer, Commutateurs dintgrales singuli` eres et oprateurs multilinaires, Ann. Inst. Fourier (Grenoble) 28 (3) (1978) 177–202. [10] D. Cruz-Uribe, A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Birkh¨ auser, Springer, Basel, 2013. [11] D. Cruz-Uribe, A. Fiorenza, J.M. Martell, C. P´ erez, The boundedness of classical operators on variable Lp spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006) 239–264. [12] L. Diening, Maximal function on Musielak-Orlicz spaces and generlaized Lebesgue spaces, Bull. Sci. Math. 129 (2005) 657–700. [13] L. Diening, P. Harjulehto, P. H¨ ast¨ o, M. Ruˇ ziˇ cka, Lebesgue and Sobolev Spaces with Variable Exponents, in: Lecture Notes in Math, vol. 2017, Springer-Verlag, Berlin, 2011. [14] L. Diening, M. Ruˇ ziˇ cka, Calder´ on-Zygmund operators on generalized Lebesgue spaces Lp(·) and problems related to fluid dynamics, J. Reine Angew. Math. 563 (2003) 197–220. [15] C. Fefferman, Inequalities for strongly singular convolution operators, Acta Mater. 124 (1970) 9–36. [16] C. Fefferman, E.M. Stein, H p Spaces of several variables, Acta Mater. 129 (1972) 137–191. [17] J. Garcia-Cuerva, J.L. Rubio de Francia, Weighted norm inequalities and related topics, in: North-Holland Math Studies, vol. 116, North-Holland Publishing Co, Amsterdam, 1985. [18] L. Grafakos, J.M. Martell, Extrapolation of weighted norm inequalities for multivariable operators and applications, J. Geom. Anal. 14 (1) (2004) 19–46. [19] L. Grafakos, R. Torres, Maximal operator and weighted norm inequalities for multilinear singular integrals, Indiana Univ. Math. J. 51 (2002) 1261–1276.

Y. Lin / Nonlinear Analysis 192 (2020) 111699

21

[20] L. Grafakos, R. Torres, Multilinear Calder´ on-Zygmund theory, Adv. Math. 165 (2002) 124–164. [21] J. Hart, Bilinear square functions and vector-valued Calder´ on-Zygmund operators, J. Fourier Anal. Appl. 18 (2012) 1291–1313. [22] J. Hart, A new proof of the bilinear T (1) theorem, Proc. Amer. Math. Soc. 142 (2014) 3169–3181. [23] I.I. Hirschmann, Multiplier transformations, Duke Mat. J. 26 (1959) 222–242. [24] T. Iida, Y. Komori-Furuya, E. Sato, A note on multilinear fractional integrals (English summary), Anal. Theory Appl. 26 (2010) 301–307. [25] C. Kenig, E.M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1999) 1–5. [26] O. Kov´ aˇ cik, J. R´ akosn´ık, On spaces Lp(x) and W k,p(x) , Czechoslovak Math. J. 41 (4) (1991) 592–618. [27] A.K. Lerner, S. Ombrosi, C. P´ erez, R.H. Torres, R. Trujillo-Gonz´ alez, New maximal functions and multiple weights for the multilinear Calder´ on-Zygmund theory, Adv. Math. 220 (4) (2009) 1222–1264. [28] K. Li, W. Sun, Weak and strong type weighted estimates for multilinear Calder´ on-Zygmund operators, Adv. Math. 254 (2014) 736–771. [29] Y. Lin, Strongly singular Calder´ on-Zygmund operator and commutator on Morrey type spaces, Acta Math. Sin. 23 (2007) 2097–2110. [30] Y. Lin, Z.G. Liu, W.L. Cong, Weighted Lipschitz estimates for commutators on weighted Morrey spaces, J. Inequal. Appl. 2015 (2015) 338. [31] Y. Lin, S.Z. Lu, Toeplitz operators related to strongly singular Calder´ on-Zygmund operators, Sci. China Ser. A 49 (2006) 1048–1064. [32] Y. Lin, S.Z. Lu, Boundedness of commutators on Hardy-type spaces, Integral Equations Operator Theory 57 (2007) 381–396. [33] Y. Lin, S.Z. Lu, Strongly singular Calder´ on-Zygmund operators and their commutators, Jordan J. Math. Stat. 1 (2008) 31–49. [34] Y. Lin, G.F. Sun, Strongly singular Calder´ on-Zygmund operators and commutators on weighted Morrey spaces, J. Inequal. Appl. 2014 (2014) 519. [35] G. Lu, P. Zhang, Multilinear Calder´ on-Zygmund operator with kernels of Dini’s type and applications, Nonlinear Anal. TMA 107 (2014) 92–117. [36] D. Maldonado, V. Naibo, Weighted norm inequalities for paraproducts and bilinear pseudodifferential operators with mild regularity, J. Fourier Anal. Appl. 15 (2009) 218–261. [37] K. Moen, Weighted inequalities for multilinear fractional integral operators, Collect Math. 60 (2009) 213–238. [38] C. P´ erez, R.H. Torres, Sharp maximal function estimates for multilinear singular integrals, Contemp. Math. 320 (2003) 323–331. [39] C. P´ erez, R.H. Torres, Minimal regularity conditions for the end-point estimate of bilinear Calder´ on-Zygmund operators, Proc. Amer. Math. Soc. Ser. B 1 (2014) 1–13. [40] V.D. Radulescu, D.D. Repovs, Partial Differential Equations with Variable Exponents, Variational Methods and Qualitative Analysis, in: Monographs and Research Notes in Math, CRC Press, Boca Raton, FL, 2015. [41] S. Spanne, Some function spaces defined using the mean oscillation over cubes, Ann. Sc. Norm. Super. Pisa 19 (1965) 593–608. [42] E.M. Stein, Singular integrals, harmonic functions and differentiability properties of functions of several variables, Proc. Symposia Pure Appl. Math. 10 (1967) 316–335. [43] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ, Princeton, New Jersey, USA, 1993. [44] Y.Z. Sun, W.Y. Su, An endpoint estimate for the commutator of singular integrals, Acta Math. Sin. 21 (2005) 1249–1258. [45] S. Wainger, Special trigonometric series in k-dimensions, Mem. Amer. Math. Soc. 59 (1965). [46] Y. Wang, J. Xiao, Well/ill-posedness for the dissipative Navier–Stokes system in generalized Carleson measure spaces, Adv. Nonlinear Anal. 8 (2019) 203–224.