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Journal of Functional Analysis 272 (2017) 631–660

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Journal of Functional Analysis

www.elsevier.com/locate/jfa

Sobolev-BMO and fractional integrals on

super-critical ranges of Lebesgue spaces

Lucas Chaffee a, Jarod Hart b,∗,1, Lucas Oliveira c

a Department of Mathematics, Western Washington University, Bellingham, WA 98225, United Statesb Department of Mathematics, University of Kansas, Lawrence, KS 66045, United Statesc Department of Mathematics, Universidade Federal do Rio Grande do Sul, Porto Alegre, Rio Grande do Sul, Brazil

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 February 2016Accepted 14 October 2016Available online 19 October 2016Communicated by P. Auscher

Keywords:Fractional integral operatorSobolev-BMOSobolev inequalityBilinear operator

In this article, we explore the mapping and boundedness properties of linear and bilinear fractional integral operators acting on Lebesgue spaces with large indices. The prototype ν-order fractional integral operator is the Riesz potential Iν , and the standard estimates for Iν are from Lp into Lq when 1 < p < n

νand 1

p= 1

q+ ν

n. We show that a ν-order linear

fractional integral operator can be continuously extended to a bounded operator from Lp into the Sobolev-BMO space Is(BMO) when n

ν≤ p < ∞ and 0 ≤ s < ν satisfy 1

p= ν−s

n.

Likewise, we prove estimates for ν-order bilinear fractional integral operators from Lp1 × Lp2 into Is(BMO) for various ranges of the indices p1, p2, and s satisfying 1

p1+ 1

p2= ν−s

n.

© 2016 Elsevier Inc. All rights reserved.

* Corresponding author.E-mail addresses: [email protected] (L. Chaffee), [email protected] (J. Hart),

[email protected] (L. Oliveira).1 Hart was partially supported by an AMS-Simons Travel Grant.

http://dx.doi.org/10.1016/j.jfa.2016.10.0150022-1236/© 2016 Elsevier Inc. All rights reserved.

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632 L. Chaffee et al. / Journal of Functional Analysis 272 (2017) 631–660

1. Introduction

The purpose of this article is to explore the behavior of linear and bilinear fractional integral operators acting on Lebesgue spaces beyond their critical index. Consider the operator T defined via

Tf(x) =∫Rn

K(x, y)f(y)dy

for appropriate functions f : Rn → C. Such an operator T is a ν-order fractional integral operator with kernel regularity M + γ for some integer M ≥ 0 and 0 < γ ≤ 1, denoted T ∈ SIOν(M + γ), if the kernel K satisfies

|DαxK(x, y)| � 1

|x− y|n+|α|−νfor α ∈ Nn

0 with |α| ≤ M and

|DαxK(x, y) −Dα

xK(x + h, y)| � |h|γ|x− y|n+|α|+γ−ν

for α ∈ Nn0 with |α| = M

for all x, y, h ∈ Rn with x �= y and |h| < |x −y|/2. For a multi-index α = (α1, ..., αn) ∈ Nn0 ,

we denote |α| = α1 + · · · + αn. We call T ∈ SIOν(M + γ) a fractional integral operator to agree with typical convention, but do not use the seemingly natural acronym FIO

to avoid confusion with the common notation for Fourier integral operators. The Riesz potential Iν , defined in this way with K(x, y) = |x −y|−(n−ν), is the prototypical ν-order linear fractional integral operator, and it is typically understood as a smoothing operator. One way to realize Iν as a smoothing operator is that Iν is bounded from Lp into Lq

for any p and q satisfying 1p = 1

q + νn , with 1 < p < n

ν and 0 < ν < n. The action of Iν is smoothing in the sense that Iνf behaves better locally than the input function f since Iνf ∈ Lq whenever f ∈ Lp with q > p according to the scaling 1

p = 1q + ν

n . These estimates are very well known for the Riesz potential operator Iν , and go back to the work of Littlewood, Paley, Sobolev, and Riesz [15,16,29,28] in the first half of the 20th century. Since |Tf(x)| ≤ CIν(|f |)(x) for ν-order fractional integral operators, it trivially follows that any operator T ∈ SIOν(M + γ) for any M + γ > 0 satisfies the same Lp-smoothing properties.

In some senses, the range 1 < p < nν makes up a sub-critical regime for a ν-order

fractional integral operator T acting on Lp spaces. This can be observed, for example, in that Tf ∈ Lq for some 1 < q < ∞ in this range. The same cannot be said when p ≥ n/ν. Estimates for the critical index p = n

ν are also known; any T ∈ SIOν(M + γ) can be extended to a bounded linear operator from Ln/ν into BMO, as long as 0 < ν < n. Once again, this demonstrates the smoothing properties of T in terms of local behavior since Tf ∈ BMO allows only logarithmic blowup when f ∈ Ln/ν ; this is better local behavior for Tf since Tf ∈ BMO implies Tf ∈ Lq

loc for any 1 < q < ∞. The precise origins of this Ln/ν to BMO result are not completely clear, but early results along the

L. Chaffee et al. / Journal of Functional Analysis 272 (2017) 631–660 633

lines were given by Stein and Zygmund [30] and Muckenhoupt and Wheeden [26]. Some more recent extensions of this estimate can be found in the work of Harboure, Salinas, and Viviani [12,13] and Gatto and Vagi [8].

Based on the discussion in the last two paragraphs, it is apparent that larger values of p assure better local behavior for Tf when f ∈ Lp, at least when 1 < p ≤ n/ν. However, there is less known about the behavior of fractional integral operators when n/ν < p < ∞, particularly for non-convolution type operators. Harboure, Salinas, and Viviani extended the definition of the Riesz potential Iν to Lp for a restricted range of values for p ≥ n/ν in [12,13], and they proved some estimates for Iν mapping into Campanato type spaces.

We prove estimates for such operators mapping Lp into Sobolev-BMO spaces Is(BMO) for n/ν < p < ∞ and appropriate s > 0. These Is(BMO) spaces were initially introduced by Neri [27] and further developed by Strichartz [31,32]. See also the article [2] by El Baraka for some work on related spaces. We will return to discuss Sobolev-BMO spaces more in depth later, but for now we simply note that Is(BMO) is the collection of tempered distributions f modulo polynomials such that |∇|sf ∈ BMO

where |∇|s is the Fourier multiplier |∇|sf(ξ) = |ξ|sf(ξ).

Theorem 1.1. Let T ∈ SIOν(M + γ) for some ν > 0, integer M ≥ 0, and 0 < γ ≤ 1. Then T is bounded from Lp into Is(BMO) for all s ≥ 0 and n/ν ≤ p < ∞ such that ν − n < s < min(ν, M + γ) and 1

p = ν−sn .

Note that it is a trivial matter to say that Iν maps Lp into Is(BMO) for 1p = ν−s

n

since Iν = Is ◦ Iν−s and Iν−s is bounded from Lp into BMO for 1p = ν−s

n . However, Theorem 1.1 does not easily follow from these mapping properties of Iν . In particular, the property |Tf(x)| � Iν(|f |)(x) was useful in extending the Lp-smoothing properties to T , but this estimate is no longer directly applicable when considering mapping properties into Is(BMO).

Non-convolution type operators related to SIOν(M+γ) were studied, for example, by Folland and Stein in [4], by Torres in [33], and by Youssfi in [34]. The fractional integral operators from [33] and [34], which are denoted OPKm([α] + δ) and SIO(s, δ) respec-tively in those works, are most closely related to the ones in this article. Torres proved that if a ν-order fractional integral operator T satisfies certain cancelation conditions, then T is bounded from F s,q

p into F s+ν,qp (which coincide with the homogeneous Sobolev

mapping W s,p to W s+ν,p when q = 2) for appropriate ranges of indices. Youssfi proved similar boundedness results, from F s,q

p into F s+ν,qp for these operators under relaxed can-

celation conditions from those in [33], though stronger kernel regularity and cancelationconditions for T1 are assumed in [34] than are used in this article. These boundedness results demonstrate a smoothing property of these operators along the Triebel–Lizorkin smoothness scale, much the same as the smoothing properties of Iν . The estimates in Theorem 1.1, written in the notation of Triebel–Lizorkin spaces, say that any ν-order fractional integral operator T with sufficient kernel regularity is bounded from F 0,2

p into


F s,2∞ where 1

p = ν−sn . Our estimates differ from those in [33,34] in that the mapping of

T moves along both the regularity scale from 0 to s and along the “size” scale from pto ∞. As we will see later, this allows us to conclude pointwise regularity estimates for Tf when f ∈ Lp and n/ν < p < ∞.

In the second part of this work, we turn our attention to bilinear versions of Theo-rem 1.1. Consider the bilinear operator T defined via

T (f, g)(x) =∫

R2n

K(x, y, z)f(y)g(z)dy dz

for appropriate functions f, g : Rn → C. Such an operator T is a ν-order bilinear frac-tional integral operator with kernel regularity M + γ for some integer M ≥ 0 and 0 < γ ≤ 1, denoted T ∈ BSIOν(M + γ), if the kernel K satisfies

|DαxK(x, y, z)| � 1

(|x− y| + |x− z|)2n+|α|−νfor α ∈ Nn

0 with |α| ≤ M and

|DαxK(x, y, z) −Dα

xK(x + h, y, z)| � |h|γ(|x− y| + |x− z|)2n+|α|+γ−ν

for α ∈ Nn0 with |α| = M

for all x, y, z, h ∈ Rn with |x − y| + |x − z| �= 0 and |h| < (|x − y| + |x − z|)/4. The typical example of a ν-order Bilinear Fractional Operator Iν ∈ BSIO(M + γ) is the bilinear Riesz potential defined

Iν(f, g)(x) =∫

R2n

1(|x− y| + |x− z|)2n−ν

f(y)g(z)dy dz.

It is well-known that Iν satisfies Lp smoothing type conditions, much like those of Iν the linear fractional integral operators. In particular, the operator Iν is bounded from Lp1 × Lp2 into Lq for any 1 < p1, p2 < ∞ and 0 < ν < 2n satisfying 1

p =1p1

+ 1p2

= 1q + ν

n , and any T ∈ BSIOν(M + γ) satisfies the same estimates since |T (f, g)(x)| � Iν(|f |, |g|)(x). A number of Lp smoothing type estimates for bilinear fractional integrals are proved in the work of Grafakos [9], Kenig and Stein [21], Kalton and Grafakos [10], Moen [24,25], Lacey, Moen, Pérez, and Torres [22], Maldonado, Moen, and Naibo [23], and Bernicot, Maldonado, Moen, and Naibo [3]. We consider the action of T on Lp1 × Lp2 in the super-critical range where 1

p1+ 1

p2< ν

n , where we should expect better local behavior for T (f, g), much like linear situation. Aimar, Hartzstein, Iaffei, and Viviani [1] prove some estimates from products of Lebesgue spaces with large exponents into Campanato type spaces for a slight modification of Iν. In the next result, we prove estimates for the class of non-convolution operators BSIOν(M + γ) mapping into Is(BMO) spaces.


Theorem 1.2. Let T ∈ BSIOν(M + γ) for some integer M ≥ 0, 0 < γ ≤ 1, ν > 0, and 0 ≤ s < min(M + γ, ν).

• If 0 < ν < n and 1 < p1, p2 ≤ ∞ such that 1p1

+ 1p2

= ν−sn , then

‖T (f, g)‖Is(BMO) � ‖f‖Lp1‖g‖Lp2

for all f ∈ Lp1 and g ∈ Lp2 .• If ν = n, s �= 0, and 1 < p1, p2 < ∞ such that 1

p1+ 1

p2= ν−s

n , then


for all f ∈ Lp1 and g ∈ Lp2 .• If n < ν < 2n, s > ν − n, and 1 < p1, p2 < ∞ such that min(p1, p2) < n

ν−n and 1p1

+ 1p2

= ν−sn , then


for all f ∈ Lp1 and g ∈ Lp2 .

Initially, it may appear that Theorem 1.2 obtains better results for T ∈ BSIOν(M+γ)for smaller values of ν. In fact, these estimates are better for smaller values of ν in terms of asymptotic estimates, but it is better for larger values of ν in terms of local estimates. The reason for this is the restrictions that are implicitly imposed on p1, p2 through the size of ν. For all of the estimates in Theorem 1.2, the index relation 1

p1+ 1

p2= ν−s

n is required. In the first estimate, where 0 < ν < n, this forces p1, p2 ≥ n

ν . Hence, when 0 < ν < n, the input functions f and g must behave well locally in the sense that they belong to Lp1 and Lp2 classes for relatively larger exponents p1, p2. On the other hand, if we consider the situation n < ν < 2n, the input functions f and g can belong to Lp1

and Lp2 classes with smaller indices given by the inequality 1 < min(p1, p2) < nν−n . So

we conclude that Theorem 1.2 does in fact exhibit the appropriate behavior in terms of smoothing action according to the size of ν. Furthermore, it seems that ν = n is a threshold value. The operators in BSIOν(M +γ) behave significantly different on either side of the ν = n mark. For 0 < ν < n, the indices p1 and p2 must belong to an interval of the form [nν , ∞), whereas for larger values n < ν < 2n the indices are allowed to range all the way down to 1 and at least one of p1 or p2 must belong to (1, n

ν−n ). In the next result, we extend boundedness results for BSIOν(M + γ) to situations where ν is large.

Theorem 1.3. Let T ∈ BSIOν(ν) for some ν > 0.

• If n ≥ 2 and ν > n, then T is bounded from Lp1 × Lp2 into Is(BMO) for all s ∈ Rsatisfying ν + 1 − n < �s� ≤ s < ν and 1 < p1, p2 ≤ ∞ satisfying 1 + 1 = ν−s .
p1 p2 n


• If ν > 1 is not an integer, then T is bounded from Lp1 × Lp2 into Is(BMO) for all �ν� ≤ s < ν and 1 < p1, p2 ≤ ∞ satisfying 1

p1+ 1

p2= ν−s

n .• If ν > n is an integer, then T is bounded from Lp1 × Lp2 into Is(BMO) for all

ν − n < s < ν and 1 < p1, p2 < ∞ satisfying 1p1

+ 1p2

= ν−sn .

There is an article forthcoming from the second author, Torres, and Wu [20] concern-ing the smoothing properties of bilinear fractional integral operators. In that work, the authors extend the notion of bilinear smoothing in a different direction than we do in this article. Here we prove estimates for T mapping Lp1 × Lp2 into Is(BMO) where p1

and p2 are large; hence this can be interpreted as extending the Lp smoothing properties of T to a super-critical range where the outputs Tf have high regularity when f ∈ Lp

and p is large. In [20], estimates of the form Lp1 × Lp2 into W ν,p are proved for appro-priate ν-order fractional integral operators, which allows one to view such operators as smoothing in a stronger sense than the Lp smoothing already known for these operators; that is, in the sense that a ν-order bilinear fractional integral operator T (f, g) yields differentiable functions (in the Sobolev space sense) even when f and g only belong to Lebesgue spaces. The two types of results are related in the sense that they both exhibit ways in which ν-order bilinear fractional integral operators can be viewed as smoothing operators, but beyond that the two works are very different.

Given Theorems 1.1–1.3 and the discussion of the smoothing behavior of operators in SIOν(M + γ) and BSIOν(M + γ), we consider the following natural question: How smooth is Tf when f ∈ Lp or T (f, g) when f ∈ Lp1 and g ∈ Lp2 for large values of p, p1, and p2? Given the boundedness results in Theorems 1.1–1.3, this question can be re-duced to measuring the smoothness properties of Is(BMO) functions for s > 0. A fairly comprehensive answer to this question was provided by Strichartz in [31], where he devel-oped many interesting properties of Sobolev-BMO spaces. He proved that any element of Is(BMO) for s strictly larger than some integer k ≥ 0 is a k-times continuously differentiable function. Furthermore, he proved that if k < s < k + 1 for some integer k ≥ 0 and f ∈ Is(BMO), then the kth order partial derivatives of f are (s −k)-Lipschitz continuous. Hence Tf is s-Lipschitz continuous whenever T ∈ SIOν(M + γ), f ∈ Lp, and p is large enough, and Tf is k-times differentiable with (s − k)-Lipschitz continuous kth order derivatives for an appropriate integer k as long as ν and p are large enough. Similar conclusions can be drawn for T (f, g) when f ∈ Lp1 and g ∈ Lp2 . The precise details of when Tf and T (f, g) are s-Lipschitz functions can be obtained by combining the boundedness in Theorems 1.1–1.3 with the Lipschitz estimates for Sobolev-BMO

functions proved by Strichartz in [31], which are stated in Theorem 4.1.We take a moment to mention a few interesting properties relating Sobolev-BMO

spaces and Lipschitz spaces. For the purposes of this discussion, we only consider these spaces with index 0 < s ≤ 1 in dimension n = 1. Define Λs to be the space of s-Lipschitz functions, taken modulo constants, with norm


‖f‖Λs= sup

x�=y

|f(x) − f(y)||x− y|s .

Radermacher’s theorem says that f ∈ Λ1 if and only if f is almost everywhere differen-tiable with f ′ ∈ L∞, and Strichartz proved in [31] that f ∈ I1(BMO) if and only if f is almost everywhere differentiable with f ′ ∈ BMO. Hence the inclusion Λ1 � I1(BMO)immediately follows. For instance f(x) = x(log(|x|) − 1) is in I1(BMO) but not in Λ1. On the other hand, this inclusion is reversed when 0 < s < 1; that is, Is(BMO) � Λs

for 0 < s < 1. We discuss this situation further in Remark 4.2.This article is organized in the following way. Sections 2 and 3 are dedicated to de-

veloping the theory of fractional integral operators on Lp spaces for large values of p in the linear and bilinear settings, respectively, and to prove Theorems 1.1, 1.2, and 1.3. We also discuss some applications and a calculus for SIOν(M + γ) and BSIOν(M + γ)in sections 2 and 3. In section 4, we give a new proof of some of the regularity proper-ties of Sobolev-BMO functions proved by Strichartz. We will also discuss some subtle differences in the regularity of Is(BMO) functions depending on whether s is an integer or non-integer.

2. Linear fractional integral operators

The main goal of this section is to prove boundedness properties of ν-order fractional integral operators from Lp into Is(BMO) for 1

p = ν−sn . Prior to doing this, we set some

notation and provide a definition for T on functions in Lp (first via a dense subset of Lp). This definition of T on Lp is not, strictly speaking, necessary in order to prove Theorem 1.1, since that result only asserts that T can be extended to a bounded operator from Lp into Is(BMO) for appropriate indices; since Tf is a priori defined when f is a Schwartz function, a collection that is dense in Lp for 1 < p < ∞, there is no need to define T on Lp directly. We include the computations in Proposition 2.1 to verify this since we think it is helpful in understanding the behavior of fractional integrals on Lp

functions for large p.Let S be the Schwartz class of smooth, rapidly decreasing functions with the typical

semi-norm topology, S ′ be its dual, and S∞ be the subspace of S made up of all f ∈ S

such that ∫Rn

f(x)xαdx = 0

for all α ∈ Nn0 . One can realize Iν for ν > 0 as a Fourier multiplier Iνf(ξ) = cn|ξ|−ν f(ξ)

for f ∈ S∞ and some dimensional constant cn. Furthermore, it follows that Iν maps S∞into S∞ continuously, and hence can be extended to a continuous operator from S ′

∞into S ′

∞. Also define |∇|sf for s > 0 to be the Fourier multiplier |∇|sf(ξ) = |ξ|sf(ξ), which again is continuous from S∞ into S∞ and can be extended to a continuous


operator from S ′∞ into S ′

∞. For s > 0, define Is(BMO) to be the collection of all distributions f modulo polynomials such that |∇|sf belongs to BMO. Taking this class modulo polynomials with the norm ‖f‖Is(BMO) = ‖ |∇|sf‖BMO makes Is(BMO) a Banach space. In fact, using the identification F 0,2

∞ = BMO with comparable norms, it follows that F s,2

∞ = Is(BMO) with comparable norms. Here F s,q∞ are the typical p = ∞

Triebel–Lizorkin norms; see for example [6] for more on these spaces.

Proposition 2.1. Let ν > 0 and T ∈ SIOν(M + γ) for M + γ > ν. Define Lp0 to be the

collection of Lp functions supported away from a neighborhood of the origin. For any max(n/ν, 1) < p < ∞ and f ∈ Lp

0, define

T f(x) =∫Rn

⎛⎝K(x, y) −

∑|α|≤�ν�

Dα1 K(0, y)

α! xα

⎞⎠ f(y)dy, (2.1)

which is an absolutely convergent integral. Furthermore for any f ∈ S ∩Lp0, there exists

a polynomial p(x) of degree at most �ν� such that Tf(x) = T f(x) + p(x). In fact, the precise polynomial p(x) can be identified in (2.1).

Proof. Fix ν > 0 and max(n/ν, 1) < p < ∞. Let f ∈ Lp0, for which there exists δ > 0

such that supp(f) ⊂ Rn\B(0, δ). Now we show that the integral used to define T f in (2.1) is absolutely convergent for all x ∈ Rn. First note that if x = 0, then for all y �= 0

K(x, y) −∑

|α|≤�ν�

Dα1 K(0, y)

α! xα = K(0, y) −K(0, y) = 0.

Hence the integrand in (2.1) is zero almost everywhere and so T f(0) = 0, according to the definition of T f in (2.1). Now for any x, y ∈ Rn with x �= 0 and |y| > 2|x|, using the mean value theorem we have

∣∣∣∣∣∣K(x, y) −∑

|α|≤�ν�

Dα1 K(0, y)

α! xα

∣∣∣∣∣∣ �|x|�ν�+1

|y|n+�ν�−ν+1 .

Noting that �ν� − ν + 1 > 0, it follows that

∫|y|>2|x|

∣∣∣∣∣∣K(x, y) −∑

|α|≤�ν�

Dα1 K(0, y)

α! xα

∣∣∣∣∣∣ |f(y)|dy

≤∫ |x|�ν�+1

|y|n+�ν�−ν+1 |f(y)|dy
|y|≥2|x|


≤ |x|�ν�+1‖f‖Lp

⎛⎜⎝ ∫

|y|≥2|x|

1|y|(n+�ν�−ν+1)p′ dy

⎞⎟⎠

1/p′

� ‖f‖Lp |x|ν−n/p.

Using the support assumption for f ∈ Lp0 it follows that

∫|y|≤2|x|

∣∣∣∣∣∣K(x, y) −∑

|α|≤�ν�

Dα1 K(0, y)

α! xα

∣∣∣∣∣∣ |f(y)|dy

�∫

|y|≤2|x|

1|x− y|n−ν

|f(y)|dy +∑

|α|≤�ν�

∫|y|≤2|x|

|x||α||y|n+|α|−ν

|f(y)|dy

�∫

|x−y|≤3|x|

1|x− y|n−ν

|f(y)|dy + δ−�ν�|x|�ν�∫

|y|≤2|x|

1|y|n−ν

|f(y)|dy

≤ ‖f‖Lp

⎛⎜⎝ ∫

|x−y|≤3|x|

1|x− y|(n−ν)p′ dy

⎞⎟⎠

1/p′

+ ‖f‖Lpδ−�ν�|x|�ν�

⎛⎜⎝ ∫

|y|≤2|x|

1|y|(n−ν)p′ dy

⎞⎟⎠

1/p′

≤ ‖f‖Lp |x|ν−n/p + ‖f‖Lpδ−�ν�|x|�ν�+ν−n/p

� ‖f‖Lp |x|ν−n/p(1 + δ−�ν�|x|�ν�).

Note that n/ν < p < ∞ implies that (n −ν)p′ < n, and so the integrals in y above are con-vergent. Therefore, we have shown that the integral in (2.1) is absolutely convergent, and furthermore that T f as defined in (2.1) satisfies |T f(x)| � ‖f‖Lp |x|ν−n/p(1 + δ−1|x|)�ν�for f ∈ Lp

0.Now we show that T f as defined in (2.1) coincides with the given definition of Tf as

a distribution modulo polynomials for all f ∈ S supported away from the origin. Let f ∈ S with supp(f) ⊂ Rn\B(0, δ) for some δ > 0 and ψ ∈ S with vanishing moments of all orders up to order �ν�. Then it follows that∫

Rn

Tf(x)ψ(x)dx =∫

R2n

K(x, y)ψ(x)f(y)dx dy

=∫

R2n

⎡⎣K(x, y) −

∑|α|≤�ν�

D1K(0, y)α! xα

⎤⎦ψ(x)f(y)dx dy

=∫

T f(x)ψ(x)dx.
Rn


We use the vanishing moment properties of ψ to subtract the appropriate polynomial, and we can switch order of integration here since the integrand is absolutely integrable in R2n, which follows from the computations above. On the left hand side is the a priori definition of Tf paired with ψ, and on the right side is the definition of T f given in (2.1)paired with ψ. Therefore these two agree modulo a polynomial of degree up to �ν�. �Remark 2.2. We make a quick remark that we can extend the definition of T directly for f with a slight modification Proposition 2.1. For f ∈ Lp when n/ν < p < ∞, consider

T f(x) = T (f · χB(0,1))(x) + T (f · χB(0,1)c)(x).

The second term here exists by Proposition 2.1 since f · χB(0,1)c ∈ Lp0, and it easily

follows that for f ∈ S this definition of T f also agrees with the a priori definition of T modulo polynomials. When identifying Tf as elements in spaces that are defined modulo polynomials, both of these definitions are the same; in fact, one can generate many other definitions for T f that agree with T modulo polynomials by splitting f =f · χB(0,δ) + f · χB(0,δ)c . So these are all acceptable ways to extend the definition of Twhen working with Sobolev-BMO spaces, which are defined modulo polynomials.

Now we start to work towards proving Theorem 1.1. Our approach here involves de-veloping a sort of calculus for SIOν(M +γ), which involves taking both weak derivatives and fractional derivatives of operators in SIOν(M + γ). Roughly speaking, for a deriva-tive operator Ds of order 0 < s < min(ν, M + γ), we show DsT ∈ SIOν−s(M + γ − s). In this way estimates for T ∈ SIOν(M + γ) mapping Ln/(ν−s) into Is(BMO) can be reduced to the standard estimate for DsT ∈ SIOν−s(M + γ − s) from Ln/(ν−s) into BMO. Roughly speaking, this reduction is

‖Tf‖Is(BMO) ≈ ‖DsTf‖BMO � ‖f‖Lp ,

which holds since DsT ∈ SIOν−s(M + γ − s). In practice, we do not take Ds = |∇|sfor technical reasons. Instead, we take Ds to be the composition of weak derivatives and |∇|s where 0 < s < 1. There is another technical reduction that must be made for our argument involving |∇|sT , which does not allow us to conclude that |∇|sT itself is in SIOν−s(M + γ − s), but it is sufficient for our purposes working with Is(BMO).

Proposition 2.3. If T ∈ SIOν(M + γ) where ν > 1 and 0 < γ ≤ 1, then DαT ∈SIOν−|α|(M − |α| + γ) for all α ∈ Nn

0 with |α| < min(ν, M + 1).

Proof. Let ν > 1, T ∈ SIOν , and α ∈ Nn0 be such that |α| < min(ν, M+1). Let f, g ∈ S ,

and it follows that

〈DαTf, g〉 = (−1)|α|∫

Tf(x)Dαg(x)dx = (−1)|α|∫ ⎛

⎝ ∫K(x, y)Dαg(x)dx

⎞⎠ f(y)dy.

Rn Rn Rn


It is easy to show that Tf ∈ S ′ for any f ∈ S and ν > 1 (even when ν ≥ n), and Tf(x) is a locally integrable function with at most polynomial growth. So everything in the last computation is valid. Let η ∈ C∞(Rn) be a smooth function such that η(x) = 0for 0 ≤ |x| ≤ 1/2 and η(x) = 1 for |x| ≥ 1. Also define ηN (x) = η(Nx), and it follows that ηN (x) → 1 as N → ∞ for x �= 0. Then

(−1)|α|∫Rn

K(x, y)Dαxg(x)dx

= limN→∞

(−1)|α|∫Rn

K(x, y)ηN (x− y)Dαx g(x)dx

=∫Rn

DαxK(x, y)g(x)dx +

∑β+μ=α:μ�=0

cβ,μ limN→∞

∫Rn

DβxK(x, y)Dμ [ηN (x− y)] g(x)dx.

The sum here is over all multi-indices β, μ ∈ Nn0 such that β + μ = α and μ �= 0. Note

that |DαxK(x, y)| � |x −y|n+|α|−ν , and hence is locally integrable, which we use to apply

dominated convergence to the first term above. Now if β+μ = α with μ �= 0 and |α| < ν, it follows that

∣∣∣∣∣∣∫Rn

DβxK(x, y)Dμ [ηN (x− y)] g(x)dx

∣∣∣∣∣∣� ‖g‖L∞

∫(2N)−1≤|x−y|≤N−1

N |μ||(Dμη)(N(x− y))||x− y|n+|β|−ν

dx

� ‖g‖L∞N |μ|∫

(2N)−1≤|x−y|≤N−1

(N |x− y|)n+|β|−ν

|x− y|n+|β|−νdx � ‖g‖L∞N |μ|+|β|−ν .

Since |β| + |μ| = |α| < ν, it follows that this term tends to zero as N → ∞. Then we have

〈DαTf, g〉 =∫Rn

⎛⎝ ∫

Rn

DαxK(x, y)g(x)dx

⎞⎠ f(y)dy = 〈Tαf, g〉

for all f, g ∈ S , where Tα is given by integration against the kernel DαxK(x, y). The

kernel properties for DαxK(x, y) easily follow, and hence Tα = DαT ∈ SIOν−|α|(M −

|α| + γ) with kernel DαxK(x, y). �

The next lemma is a fairly standard one in Fourier analysis and Littlewood–Paley–Stein theory.


Lemma 2.4. Let ψ ∈ S∞ such that supp(ψ) ⊂ B(0, 4)\B(0, 1/2) and∑k∈Z

ψ(2−kξ) = 1

for all ξ �= 0. Then for any h ∈ S∞∑k∈Z

ψk ∗ h = limN→∞

∑|k|<N

ψk ∗ h = h

in the Schwartz semi-norm topology, where ψk(x) = 2knψ(2kx).

Theorem 2.5. Assume T ∈ SIOν(γ) for some 0 < γ ≤ 1 and 0 < ν < n. For any s, γ > 0such that 0 < s + 2γ < min(γ, ν), there exists T ∈ SIOν−s(γ) such that |∇|sT = T in S ′ modulo polynomials. That is, for all f ∈ S and g ∈ S∞ we have

〈|∇|sTf, g〉 =∫Rn

T f(x)g(x)dx.

Proof. Let T ∈ SIOν(γ) for some 0 < γ ≤ 1 and 0 < ν < n, and also let s, γ > 0 such that s + 2γ < min(γ, ν). Note that |∇|sTf ∈ S ′

∞ for any f ∈ S . Let ψ ∈ S with ψsupported in the annulus 1/2 < |ξ| < 4, and such that∑

k∈Z

ψ(2−kξ) = 1

for ξ �= 0. Define Qkf = ψk ∗ f . For g ∈ S∞, it follows that |∇|sg ∈ S∞ and that

〈Tf, |∇|sg〉 =∑k∈Z

〈Tf,Qk|∇|sg〉 =∑k∈Z

2sk⟨Tf, Qkg

⟩=

∑k∈Z

2sk∫Rn

QkTf(x)g(x)dx,

where ψ = |∇|sψ ∈ S∞ and Qkf = ψk ∗ f . Motivated by this computation, we define

T f(x) =∑k∈Z

2skQkTf(x).

Now it remains to show that T ∈ SIOν−s(γ), and that

∑k∈Z

2sk∫Rn

QkTf(x)g(x)dx =∫Rn

T f(x)g(x)dx

whenever f ∈ S and g ∈ S∞. We start by estimating the kernel of T , which is given by

K(x, y) =∑k∈Z

2skT ∗ψxk(y).

We split this into two situations, where |x − y| > 22−k and where |x − y| ≤ 22−k.


Size estimate for K, case 1: Assume that |x − y| > 22−k. Here we prove that

|T ∗ψxk(y)| � 2−kγ

|x− y|n+γ−ν. (2.2)

Using the mean zero property of ψ, we have

|T ∗ψxk(y)| =

∣∣∣∣∣∣∫Rn

(K(u, y) −K(x, y))ψxk(u)du

∣∣∣∣∣∣≤

∫|x−u|<|x−y|/2

|K(u, y) −K(x, y)| |ψxk(u)|du

+∫

|x−y|≤2|x−u|

|K(u, y)| |ψxk(u)|du

+∫

|x−y|≤2|x−u|

|K(x, y)| |ψxk(u)|du = Ak(x, y) + Bk(x, y) + Ck(x, y).

We estimate the first term Ak in the following way

Ak(x, y) �∫

|x−u|<|x−y|/2

|x− u|γ|x− y|n+γ−ν

2kn

(1 + 2k|x− u|)n+1+γdu

� 2−kγ

|x− y|n+γ−ν

∫Rn

2kn

(1 + 2k|x− u|)n+1 du � 2−kγ

|x− y|n+γ−ν.

For the second term, we consider the following

Bk(x, y) �∫

|x−y|≤2|x−u|

1|u− y|n−ν

|ψxk(u)|du

�∫

|x−y|≤2|x−u||u−y|≤|x−y|

1|u− y|n−ν

2kn

(1 + 2k|x− u|)n+γdu

+ 1|x− y|n−ν

∫|x−y|≤2|x−u||u−y|>|x−y|

2kn

(1 + 2k|x− u|)n+γdu

� 2−kγ

|x− y|n+γ

∫|u−y|≤|x−y|

du

|u− y|n−ν+ 2−kγ

|x− y|n−ν

∫|x−y|≤2|x−u|

du

|x− u|n+γ

� 2−kγ

n+γ−ν.
|x− y|


Finally, the third term is bounded in a similar fashion

Ck(x, y) �1

|x− y|n−ν

∫|x−y|≤2|x−u|

2kn

(1 + 2k|x− u|)n+γdu

� 2−kγ

|x− y|n−ν

∫|x−y|≤2|x−u|

1|x− u|n+γ

du � 2−kγ

|x− y|n+γ−ν.

This completes the proof of the estimate in (2.2).

Size estimate for K, case 2: Assume that |x − y| ≤ 22−k. In this situation, we estimate

|T ∗ψxk(y)| directly in the following way

|T ∗ψxk(y)| �

∫Rn

1|u− y|n−ν

2kn

(1 + 2k|x− u|)n+1 du

� 2k(n−ν)∫

|u−y|>2−k

2kn

(1 + 2k|u− y|)n+1 du

+ 2kn∫

|u−y|≤2−k

1|u− y|n−ν

du � 2k(n−ν). (2.3)

Using (2.2) and (2.3), the kernel of T satisfies

|K(x, y)| ≤∑k∈Z

2sk|T ∗ψxk(y)| �

∑k:2−k<|x−y|

2−k(γ−s)

|x− y|n+γ−ν+

∑k:2−k≥|x−y|

2k(n+s−ν)

� 1|x− y|n+s−ν

.

Now we verify the γ regularity estimate for K(x, y), for which we fix |h| < |x − y|/2. Note that if |x − y|/4 ≤ |h| < |x − y|/2, then

|K(x, y) − K(x + h, y)| � 1|x− y|n+s−ν

+ 1|x + h− y|n+s−ν

� 1|x− y|n+s−ν

� |h|γ|x− y|n+s+γ−ν

.

So assume that |h| < |x −y|/4, and we consider |K(x, y) −K(x +h, y)| in a few situations.

Regularity estimate for K, case 1: Assume that |x − y| > 22−k. Here we prove that

|T ∗ψxk(y) − T ∗ψx+h

k (y)| � 2−k(γ−γ)|h|γn+γ−ν

+ 2−k(γ−2γ)|h|γn+γ−γ−ν

. (2.4)
|x− y| |x− y|


Similar to above

|T ∗ψxk(y) − T ∗ψx+h

k (y)| =

∣∣∣∣∣∣∫Rn

(K(u, y) −K(x, y))(ψxk(u)du− ψx+h

k (u)du)

∣∣∣∣∣∣≤

∫|x−u|<|x−y|/2

|K(u, y) −K(x, y)| |ψxk(u) − ψx+h

k (u)|du

+∫

|x−y|≤2|x−u|

|K(u, y)| |ψxk(u) − ψx+h

k (u)|du

+∫

|x−y|≤2|x−u|

|K(x, y)| |ψxk(u) − ψx+h

k (u)|du

= Ak(x, y) + Bk(x, y) + Ck(x, y).

We have

Ak(x, y) �∫

|x−u|<|x−y|/2

|x− u|γ|x− y|n+γ−ν

2kn(2k|h|)γ(1 + 2k min(|x− u|, |x + h− u|))n+1+γ

du

�∫

|x−u|<|x−y|/2

min(|x− u|, |x + h− u|)γ + |h|γ|x− y|n+γ−ν

× 2kn(2k|h|)γ(1 + 2k min(|x− u|, |x + h− u|))n+1+γ

du

� |h|γ + 2kγ |h|2γ|x− y|n+γ−ν

∫|x−u|<|x−y|/2

2kn

(1 + 2k min(|x− u|, |x + h− u|))n+1 du

� |h|γ + 2kγ |h|2γ|x− y|n+γ−ν

= |h|γ|x− y|n+γ−ν

+ 2kγ |h|2γ|x− y|n+γ−ν

.

If |h| ≤ 2−k, then

Ak(x, y) �2−k(γ−γ)|h|γ|x− y|n+γ−ν

,

and the estimate in (2.4) is complete. If |h| > 2−k, then we note that

Ak(x, y) ≤ |T ∗ψxk(y)| + |T ∗ψx+h

k (y)| � 2−kγ

|x− y|n+γ−ν+ 2−kγ

|x + h− y|n+γ−ν

� 2−kγ

n+γ−ν
|x− y|


and

Ak(x, y) �|h|γ

|x− y|n+γ−ν+ 2kγ |h|2γ

|x− y|n+γ−ν� 2kγ |h|2γ

|x− y|n+γ−ν.

Choose λ = γ2γ ∈ (0, 1), and it follows that

Ak(x, y) �(

2kγ |h|2γ|x− y|n+γ−ν

)λ ( 2−kγ

|x− y|n+γ−ν

)1−λ

= 2k(2γλ−γ)|h|2λγ|x− y|n+γ−ν

= 2−k(γ−γ)|h|γ|x− y|n+γ−ν

,

which completes the estimate in (2.4) for Ak(x, y). For the second and third term, we argue in almost exactly the same way as when proving (2.2),

Bk(x, y) �∫

|x−y|≤2|x−u|

1|u− y|n−ν

|ψxk(u) − ψx+h

k (u)|du

�∫

|x−y|≤2|x−u||u−y|≤|x−y|/4

1|u− y|n−ν

2kn(2k|h|)γ(1 + 2k min(|x− u|, |x + h− u|))n+γ

du

+ 1|x− y|n−ν

∫|x−y|≤2|x−u||u−y|>|x−y|/4


du

� 2−k(γ−γ)|h|γ|x− y|n+γ

∫|x−y|≤2|x−u||u−y|≤|x−y|/4

1|u− y|n−ν

du

+ 2kγ |h|γ|x− y|n−ν

∫|x−y|≤2|x−u||u−y|>|x−y|/4

2kn

(1 + 2k min(|x− u|, |x + h− u|))n+γdu

� 2−k(γ−γ)|h|γ|x− y|n+γ−ν

+ 2kγ |h|γ|x− y|n−ν

.

Note that when |h| ≤ |x − y|/2 and |u − y| ≤ |x − y|/4, we have

|x + h− u| ≥ |x− u| − |h| ≥ |x− y| − |u− y| − |h| ≥ |x− y|/4,

which we used in the computations above. Now we use λ = γγ , and it follows that

(2kγ |h|γ

|x− y|n−ν

)λ ( 2−kγ

|x− y|n+γ−ν

)1−λ

= 2λkγ |h|λγ|x− y|λ(n−ν)

2−k(1−λ)γ

|x− y|(1−λ)(n+γ−ν)

= 2k(2λγ−γ)

n−ν+(1−λ)γ |h|λγ = 2−k(γ−2γ)|h|γ

n−ν+γ−γ.
|x− y| |x− y|


Finally, the third term is bounded in a similar fashion

Ck(x, y) �1

|x− y|n−ν

∫|x−y|≤2|x−u|


du

� 2−k(γ−γ)|h|γ|x− y|n−ν

∫|x−y|≤2|x−u|

1min(|x− u|, |x + h− u|)n+γ

du � 2−k(γ−γ)|h|γ|x− y|n+γ−ν

.

This completes the proof of the estimate in (2.4).

Regularity estimate for K, case 2: Assume that |x − y| ≤ 22−k. In this situation, we

estimate |T ∗ψx+hk (y) − T ∗ψx

k(y)| directly in the following way

|T ∗ψx+hk (y) − T ∗ψx

k(y)| �∫Rn

1|u− y|n−ν

2kn(2k|h|)γ(1 + 2k|x− u|)n+1 du

� 2k(n+γ−ν)|h|γ∫

|u−y|>23−k

2kn

(1 + 2k|x− u|)n+1 du

+ 2k(n+γ)|h|γ∫

|u−y|≤23−k

1|x− u|n−ν

du

� 2k(n+γ−ν)|h|γ∫

|u−y|>23−k

2kn

(1 + 2k|u− y|)n+1 du

+ 2k(n+γ)|h|γ∫

|x−u|≤24−k

1|x− u|n−ν

du

� 2k(n+γ−ν)|h|γ . (2.5)

Using (2.4) and (2.5), the kernel of T satisfies

|K(x, y) − K(x + h, y)| ≤∑k∈Z

2sk|T ∗ψxk(y) − T ∗ψx+h

k (y)|

�∑

k:2−k<|x−y|

2−k(γ−s−γ)|h|γ|x− y|n+γ−ν

+ 2−k(γ−s−2γ)|h|γ|x− y|n+γ−γ−ν

+∑

k:2−k≥|x−y|2k(n+γ+s−ν)|h|γ � |h|γ

|x− y|n+s−ν+γ.

Note that we do not care about the regularity in the y variable of K(x, y) for T to belong to SIOν−s(γ). Hence we have shown that T ∈ SIOν−s(γ) whenever s, γ > 0


and s + 2γ < γ. Finally, we note that for f ∈ S and g ∈ S∞, it follows by dominated convergence that

〈|∇|sTf, g〉 =∑k∈Z

2sk∫Rn

QkTf(x)g(x)dx =∫Rn

∑k∈Z

2skQkTf(x)g(x)dx

=∫Rn

T f(x)g(x)dx,

which completes the proof. �The next lemma is well known, for instance it can be found in [30,26,13,8], among

other places.

Lemma 2.6. If T ∈ SIOν(γ) for any 0 < γ ≤ 1 and 0 < ν < n, then T can be extended to a bounded linear operator from Ln/ν into BMO.

Proposition 2.7. Assume that T ∈ SIOν(γ) for some 0 < γ ≤ 1 and 0 < ν < n. Then T can be extended to a bounded operator from L

nν−s into Is(BMO) for all 0 ≤ s <

min(γ, ν).

Proof. Assume that T ∈ SIOν(γ) for some 0 < γ ≤ 1 and 0 < ν < n. Fix 0 < s <min(γ, ν) and choose γ > 0 small enough so that 0 < s + 2γ < min(γ, ν). Then there exists T ∈ SIOν−s(γ) such that T = |∇|sT in S ′

∞. Then for any f ∈ S , it follows that for any cube Q

1|Q|

∫Q

∑k:2−k≤�(Q)

22sk|QkTf(x)|2 = 1|Q|

∫Q

∑k:2−k≤�(Q)

|Qk(|∇|sTf)(x)|2

= 1|Q|

∫Q

∑k:2−k≤�(Q)

|Qk(T f)(x)|2

� ‖T f‖BMO � ‖f‖L

nν−s

.

Note that T can be extended to a bounded operator from Ln

ν−s since T ∈ SIOν−s(γ). Therefore by density of S in L

nν−s , it follows that T can be extended to a bounded

operator from Ln

ν−s into Is(BMO). �Proof of Theorem 1.1. Suppose ν > 0, M ≥ 0 is an integer, and 0 < γ ≤ 1. If in addition ν ≤ 1, then this was proved in Proposition 2.7. Otherwise, assume that ν > 1 and fix s ≥ 0 such that ν − n < s < min(ν, M + γ). If s is an integer, then by Proposition 2.3 it follows that DαT ∈ SIOν−s(M − s + γ) for all |α| = s, and hence using the derivative characterization of Is(BMO) proved by Strichartz in [31] and Lemma 2.6 we have


‖Tf‖Is(BMO) ≈∑|α|=s

‖DαTf‖BMO � ‖f‖L

nν−s

.

If s is not an integer, then we choose an integer L ≥ 0 and 0 < δ < 1 such that s = L +δ. It follows that DαT ∈ SIOν−L(M − L + γ) for all |α| = L, and furthermore there exist Tα ∈ SIOν−L−δ(M + γ − L − δ) = SIOν−s(M + γ − s) such that Tα = |∇|δ(DαT ) in S ′

∞ for all |α| = L. Therefore

‖Tf‖Is(BMO) ≈∑

|α|=L

‖DαTf‖Iδ(BMO) ≈∑

|α|=L

‖Tαf‖BMO � ‖f‖L

nν−s

.

Note that since Proposition 2.3 allows for ν ≥ n, we are always able to reduce the estimates in this proof to the situation where 0 < ν ≤ 1, in which case Theorem 2.5, Lemma 2.6, and Proposition 2.7 are all applicable. �Remark 2.8. We make a few quick remarks about how Theorem 1.1 can be applied in various situations. One can define a kernel K(x, y) = g(x −y), where g : Rn → C satisfies

|Dαg(x)| � |x|μ−|α| for |α| ≤ M and x ∈ Rn

|Dαg(x) −Dαg(y)| � |x− y|γ |x|μ−|α|−γ for |α| = M and |x− y| < |x|/2

for some μ ≥ 0, integer M ≥ 0, and 0 < γ ≤ 1. It follows that Tg ∈ SIOn+μ(M + γ), where Tg = g ∗ f . Hence Theorem 1.1 can be applied to such operators to conclude that g ∗ f ∈ Is(BMO) for appropriate f and s. In light of Theorem 4.1, this provides a way to conclude precise regularity results for these convolutions. Some typical choices for gare g(x) = |x|μ for some μ > 0 or g(x) = xα for some α ∈ Nn

0 .Consider for a moment a function g that satisfies the estimates above with M = 0,

i.e.

|g(x)| � |x|μ for x ∈ Rn

|g(x) − g(y)| � |x− y|γ |x|μ−γ for |x− y| < |x|/2,

for some μ ≥ 0, integer M ≥ 0, and 0 < γ ≤ 1. Also let T be defined in terms of a kernel K(x, y) satisfying |DαK(x, y)| � |x − y|−(n+|α|−ρ) for appropriate α ∈ Nn

0 and for some ρ ∈ R. In this case, we consider the kernel Kg(x, y) = (g(x) − g(y))K(x, y), which can be used to define the commutator [T, Mg]f = g · Tf − T (f · g). As long as ρ + μ > 0and K is sufficiently smooth, it follows that [T, Mg] ∈ SIOμ+ρ(γ). So again, we can apply Theorem 1.1, Theorem 4.1 to arrive at precise regularity estimates for [T, Mg]ffor appropriate f ∈ Lp.

In addition, operators defined via the kernel K(x, y)(x − y)α, where K is a standard singular integral kernel were studied in [5,7,17,19], where they played a significant role in proving boundedness properties for singular integral operators.


A primary take-away from this remark is that we can arrive at some new regularity estimates by considering the fractional integral operators in SIOν(M + γ) when ν ≥ n, which is outside the realm of many for the estimates currently known for fractional integrals.

3. Bilinear fractional integral operators

In this section, we apply the linear fractional integral operator results from the last sec-tion to bilinear operators. The approach is motivated primarily by the work of Grafakos and Torres in [11]. In that article, the authors reduce estimates for a bilinear Calderón–Zygmund operator T (f, g) to a related linear Calderón–Zygmund operator Tgf = T (f, g), which satisfies all pertinent estimates with constants bounded by a constant times ‖g‖L∞ . In fact, our estimates in the fractional integral setting go through much more simply than in the singular integral operator setting because there are fewer issues with defining Tg

caused by local integrability of the kernel of T .

Proposition 3.1. Let T ∈ BSIOν(M + γ) for some ν > 0, M ≥ 0, and 0 < γ ≤ 1. For appropriate functions g, define

Tgf(x) = limN→∞

∫|x−y|>1/N

Kg(x, y)f(y)dy, where Kg(x, y) =∫Rn

K(x, y, z)g(z)dz.

Then the following hold.

• If 0 < ν < n and g ∈ L∞, then Tgf = T (f, g) for f ∈ S and Tg ∈ SIOν(M + γ).• If 0 < ν ≤ n and g ∈ Lp for some n/ν < p < ∞, then Tgf = T (f, g) for f ∈ S and

Tg ∈ SIOν−n/p(M + γ).• If n < ν < 2n and g ∈ Lp for some 1 < p < n

ν−n , then Tgf = T (f, g) for f ∈ S and Tg ∈ SIOν−n/p(M + γ).

Proof. Let 0 < ν < n, T ∈ BSIOν(M + γ), and g ∈ L∞. Then for x �= y

|DαxKg(x, y)| � ‖g‖L∞

∫Rn

1(|x− y| + |x− z|)2n+|α|−ν

dz � ‖g‖L∞1

|x− y|n+|α|−ν

for all α ∈ Nn0 with |α| ≤ M and x �= y. Note that this integral is absolutely convergent

since 0 < ν < n and x �= y. Also

|DαxKg(x, y) −Dα

xKg(x + h, y)| � ‖g‖L∞

∫Rn

|h|γ(|x− y| + |x− z|)2n+|α|+γ−ν

dz

� ‖g‖L∞|h|γn+|α|+γ−ν
|x− y|


for all α ∈ Nn0 with |α| = M and |h| < |x − y|/2.

The other two cases follow with similar computations. Let 0 < ν ≤ n, T ∈BSIOν(M + γ), and g ∈ Lp for some n/ν < p < ∞. For x �= y and α ∈ Nn

0 with |α| ≤ M

|DαxKg(x, y)| � ‖g‖Lp

⎛⎝ ∫

Rn

1(|x− y| + |x− z|)(2n+|α|−ν)p′ dz

⎞⎠

1/p′

� ‖g‖Lp

1|x− y|n(1+1/p)+|α|−ν

,

and similarly


xKg(x + h, y)| � ‖g‖Lp

|h|γ|x− y|n(1+1/p)+|α|+γ−ν

for all α ∈ Nn0 with |α| = M and |h| < |x − y|/2. Note that the integrals above converge

because (2n − ν)p′ > n whenever 0 < ν < n.Let n < ν < 2n, T ∈ BSIOν(M + γ), and g ∈ Lp for some 1 < p < n/ν. For x �= y

and α ∈ Nn0 with |α| ≤ M

|DαxKg(x, y)| � ‖g‖Lp

1|x− y|n(1+1/p)+|α|−ν

,

and similarly


xKg(x + h, y)| � ‖g‖Lp

|h|γ|x− y|n(1+1/p)+|α|+γ−ν

for all α ∈ Nn0 with |α| = M and |h| < |x − y|/2. Again, the integrals above converge

because (2n − ν)p′ > n whenever n < ν < 2n and p < n/ν.Note that a priori, it may not be clear that Tgf = T (f, g) for f, g ∈ S since K(x, y, z)

is not necessarily integrable in z when x = y. However, we know that Kg(x, y) is locally integrable for appropriate g, which allows for the following argument. For f ∈ S and g ∈ Lp for the appropriate p depending on how ν relates to n, we have

T (f, g)(x) =∫

R2n

K(x, y, z)f(y)g(z)dy dz

= limN→∞

∫Rn

⎛⎝ ∫

Rn

K(x, y, z)g(z)dz

⎞⎠ f(y)χ|x−y|>1/Ndy

= limN→∞

∫Kg(x, y)f(y)χ|x−y|>1/Ndy =

∫Kg(x, y)f(y)dy,

Rn Rn


where we note that |Kg(x, y)f(y)|χ|x−y|<1/N ≤ |Kg(x, y)f(y)| uniformly in N and Kg(x, y)f(y) is an L1 function in y since Kg(x, y) is uniformly bounded by a constant times |x −y|−n(1+1/p)−ν , |x −y|−n(1+1/p)−ν is locally integrable in y, and f ∈ S . So we can apply dominated convergence in the above computation to verify that Tgf = T (f, g). �Proposition 3.2. Let ν > 0 and T ∈ BSIOν(M + γ) for some integer M ≥ 0 and 0 < γ ≤ 1. For any multi-index α ∈ Nn

0 with |α| < min(ν, M + γ), it follows that DαT ∈ BSIOν−|α|(M − |α| + γ).

Proof. Let ν > 0, T ∈ BSIOν(M + γ), and α ∈ Nn0 be such that |α| < min(ν, M + γ).

Also let η ∈ C∞(R2n) be a smooth function such that η(x, y) = 0 for 0 ≤ |x| + |y| ≤ 1/2and η(x, y) = 1 for |x| + |y| ≥ 1. Define ηN (x, y) = η(Nx, Ny), and it follows that ηN (x, y) → 1 as N → ∞ for x, y ∈ R such that |x| + |y| �= 0. For f, g, h ∈ S ,

〈DαT (f, g), h〉

= (−1)|α|∫Rn

T (f, g)(x)Dαh(x)dx

= (−1)|α|∫Rn

⎛⎝ lim

N→∞

∫R2n

K(x, y, z)f(y)g(z)ηN (x− y, x− z)dy dz

⎞⎠Dαh(x)dx

= (−1)|α| limN→∞

∫R2n

⎛⎝ ∫

Rn

K(x, y, z)ηN (x− y, x− z)Dαh(x)dx

⎞⎠ f(y)g(z)dy dz.

Now we consider∫Rn


= (−1)|α|∫Rn

Dα [K(x, y, z)ηN (x− y, x− z)]h(x)dx

= (−1)|α|∫Rn

DαK(x, y, z)ηN (x− y, x− z)h(x)dx

+ (−1)|α|∑

β+μ=α:μ�=0

cβ,μ

∫Rn

DβK(x, y, z)Dμ [ηN (x− y, x− z)]h(x)dx

= (−1)|α|∫Rn

DαK(x, y, z)h(x)ηN (x− y, x− z)dx

+ (−1)|α|∑

β+μ=α:μ�=0

cβ,μ∫

DβK(x, y, z)Dμ [ηN (x− y, x− z)]h(x)dx.
Rn


Fix 0 < ε < min(1, ν − |α|), which is possible since |α| < ν. Then

∫R2n

⎛⎝ ∫

Rn

|DβK(x, y, z)Dμ [ηN (x− y, x− z)]h(x)|dx

⎞⎠ |f(y)g(z)|dy dz

� N |μ|+|β|+ε−ν

∫(2N)−1≤|x−y|+|x−z|≤N−1

1(|x− y| + |x− z|)2n−ε

× |h(x)f(y)g(z)|dx dy dz

� N |α|+ε−ν

∫Rn

Iε(|f |, |g|)(x)|h(x)|dx � N |α|+ε−ν‖f‖Lp1‖g‖Lp2‖h‖Lq′ ,

where 1 < p1, p2, q < ∞ are some fixed set of indices satisfying 1p1

+ 1p2

= 1q + ε

n . By our selection of ε, it follows that |α| + ε − ν < 0, and hence for β + μ = α with μ �= 0, we have

limN→∞

∣∣∣∣∣∣∫

R2n

⎛⎝ ∫

Rn


⎞⎠ f(y)g(z)dy dz

∣∣∣∣∣∣� lim

N→∞N |α|+ε−ν‖f‖Lp1‖g‖Lp2‖h‖Lq′ = 0.

Finally, we have that

〈DαT (f, g), h〉

= (−1)|α| limN→∞

∫R2n

⎛⎝ ∫

Rn



= limN→∞

∫R2n

⎛⎝ ∫

Rn

DαK(x, y, z)h(x)ηN (x− y, x− z)dx


+∫

R2n

⎛⎝ ∑

β+μ=α:μ�=0

cβ,μ

∫Rn



=∫Rn

⎛⎝ ∫

R2n

DαK(x, y, z)f(y)g(z)dy dz

⎞⎠h(x)dx =

∫Rn

Tα(f, g)(x)h(x)dx.

Therefore DαT (f, g) = Tα(f, g) for all f, g ∈ S , where Tα ∈ BSIOν−|α|(M − |α| + γ)is given by integration against Dα

xK(x, y, z). Note that the kernel estimates for Tα to belong to BSIOν−|α|(M−|α| +γ) follow trivially from those of T ∈ BSIOν(M +γ). �


Remark 3.3. We note that Proposition 3.2 provides a sort of calculus for BSIOν(M +γ). Indeed, for T ∈ BSIOν(M+γ) and α ∈ Nn

0 with |α| < min(M+γ, ν), we can make sense of the weak derivative Dα moving down the scale of BSIOν−s(M + γ − s) collections. Furthermore, with the reduction to SIOν(M + γ) provided by Proposition 3.1 and the calculus of SIOν(M + γ), one can extend the calculus for BSIOν(M + γ) to include fractional derivatives.

Proof of Theorem 1.2. Let T ∈ BSIOν(M + γ) for some integer M ≥ 0, 0 < γ ≤ 1, and 0 < ν < n. Also let 0 ≤ s < min(M +γ, ν), and 1 < p1, p2 ≤ ∞ such that 1

p1+ 1

p2= ν−s

n . Since 0 < ν−s

n = 1p1

+ 1p2

, it follows that at least one of p1, p2 must be finite. Without loss of generality, assume that p1 < ∞. Fix g ∈ Lp2 , and by Proposition 3.1 it follows that Tg ∈ SIOν−n/p2(M + γ). By Theorem 1.1, Tg is bounded from Lp1 into Is(BMO)with a norm that depends linearly on ‖g‖Lp2 . Note that the equation 1

p1+ 1

p2= ν−s

n

forces 0 ≤ s < ν − n/p2. Therefore T is bounded from Lp1 × Lp2 into Is(BMO).Now suppose ν = n, and let 0 < s < min(M + γ, ν) and 1 < p1, p2 < ∞ such

that 1p1

+ 1p2

= ν−sn . Again fix g ∈ Lp2 , and it follows from Proposition 3.1 that Tg ∈

SIOν−n/p2(M+γ). Since 1p1

+ 1p2

= ν−sn , it follows that s < ν−n/p2 and 1

p1= (ν−n/p2)−s

n . Therefore Tg is bounded from Lp1 into Is(BMO), and hence T is bounded from Lp1×Lp2

into Is(BMO).If n < ν < 2n, ν − n < s < min(M + γ, ν), and 1 < min(p1, p2) < n

ν−n such that 1p1

+ 1p2

= ν−sn . It follows that either p1 or p2 is strictly smaller than n

ν−n ; without loss of generality assume that p2 < n

ν−n . Then by Proposition 3.1, it follows that Tg ∈SIOν−n/p2(M + γ) when g ∈ Lp2 . Note that by assumption p1 < ∞. Therefore Tg is bounded from Lp1 into Is(BMO), and hence T is bounded from Lp1×Lp2 into Is(BMO)by the third part of Proposition 3.1 and Theorem 1.1. �Proof of Theorem 1.3. Let T ∈ BSIOν(ν) for some ν > n, and assume that n ≥ 2. Let ν−n ≤ s < ν and 1 < p1, p2 ≤ ∞ satisfying 1

p1+ 1

p2= ν−s

n . Let L ≥ 0 be the integer and 0 < δ ≤ 1 such that s = L +δ. Note if s is an integer, then L = s −1 and δ = 1. When s is not an integer L = �s� and δ = s −�s�. Then condition �s� > ν+1 −n implies ν−L < n. It also follows that L < s < ν, and so, by Proposition 3.2, DαT ∈ BSIOν−L(ν − L) for all α ∈ Nn

0 with |α| = L. Hence we can apply the first estimate of Theorem 1.2 to DαT

for |α| = L. Then it follows that

‖T (f, g)‖Is(BMO) ≈∑

|α|=L

‖DαT (f, g)‖Iδ(BMO) � ‖f‖Lp1‖g‖Lp2

for all f ∈ Lp1 and g ∈ Lp2 satisfying 1 < p1, p2 ≤ ∞ and 1p1

+ 1p2

= ν−L−δn = ν−s

n .Now assume that ν > 1 is not an integer and n ≥ 1. This time fix an integer

L ≥ 0 and 0 < δ < 1 such that ν = L + δ. By Proposition 2.3 it follows that DαT ∈ BSIOν−L(ν − L) = BSIOδ(δ) for all α ∈ Nn

0 with |α| = L. By the first estimate in Theorem 1.2, for any 0 ≤ t < δ it follows that DαT is bounded from Lp1 × Lp2 into


It(BMO) as long as 1 < p1, p2 ≤ ∞ and 1p1

+ 1p2

= δ−tn . Then for any L = �ν� ≤ s < ν

and 1 < p1, p2 < ∞ with 1p1

+ 1p2

= ν−sn , it follows that


|α|=L

‖DαT (f, g)‖Is−L(BMO) � ‖f‖Lp1‖g‖Lp2 ,

where we have applied the estimate for DαT with t = s −L ∈ [0, δ) since L = �ν� ≤ s < ν.Assume that ν > n is an integer. Then DαT ∈ BSIOn(n) for all α ∈ Nn

0 with |α| = ν − n. By the second part of Theorem 1.2 it follows that


|α|=ν−n

‖DαT (f, g)‖Is−(ν−n)(BMO) � ‖f‖Lp1‖g‖Lp2

for all f ∈ Lp1 and g ∈ Lp2 , as long as ν − n < s < ν, and 1 < p1, p2 < ∞ such that 1p1

+ 1p2

= ν−sn . �

Remark 3.4. Let μ > 0, and consider

Tμ(f, g)(x) =∫

R2n

(|x− y| + |x− z|)μf(y)g(z)dy dz

for μ > 0. It follows that Tμ ∈ BSIO2n+μ(M) for all M > 0. Therefore we can apply Theorems 1.2 and 1.3 to conclude Tμ is bounded from Lp1 × Lp2 into Is(BMO) for various ranges of parameters p1, p2, and s satisfying 1

p1+ 1

p2= 2n+μ−s

n = 2 − s−μn . Then

we can conclude precise regularity estimates for Tμ(f, g) using Theorem 4.1.One can also consider operators defined via the kernels K(x, y, z)(x − y)α(x − z)β

where K is a standard bilinear singular integral operator kernel and α, β ∈ Nn0 . Such

an operator is a factional integral operator of order |α| + |β|, and falls in the scope of analysis in this paper. This type of bilinear operator was used in [18] in connection with the Hardy space mapping properties of bilinear singular integral operators.

In fact, this construction can be extended to bilinear convolution type operators where the kernel is defined K(x, y, z) = G(x −y, x −z), K(x, y, z) = G(x −y, y−z), K(x, y, z) =G(x − z, y − z) for an appropriate function G : R2n → C. We can also consider the commutators defined via the kernels K(x, y, z) = G(x, z) −G(y, z), K(x, y, z) = G(x, y) −G(z, y), or K(x, y, z) = G(x, y) − G(x, z). These formulations can be carried out with techniques similar to the ones discussed in Remark 2.8.

Finally, consider the operator Bμ defined via the kernel |x − y − z|μ. Then it follows that

Bμ(f, g)(x) = 2−(2n+μ)∫

|2x− y − z|μf(y/2)g(z/2)dy dz = 2−(2n+μ)Bμ(f , g)(x).
R2n


The kernel of Bμ is given by bμ(x, y, z) = |2x − y − z|μ, which satisfies

Dαx bμ(x, y, z) � (|x− y| + |x− z|)μ−|α|

as long as |α| < μ. Hence we can apply our estimates to Bμ(f, g), and consequently to Bμ(f, g) as well.

4. Regularity of Sobolev-BMO functions

In this final section, we present a new proof for a result due to Strichartz in [31].

Theorem 4.1. If f ∈ BMO and 0 < ν < 1, then Iνf , defined initially as a distribution modulo polynomials, agrees with a continuous function Iνf(x) that satisfies

|Iνf(x) − Iνf(y)| � |x− y|ν‖f‖BMO (4.1)

for all x, y ∈ Rn. Consequently, if f ∈ Iν(BMO) for some 0 < ν < 1, then f is ν-Lipschitz continuous and |f(x) − f(y)| � |x − y|ν‖f‖Iν(BMO) for all x, y ∈ Rn.

In Strichartz’s proof of this estimate, he passed through a duality argument and proves estimates for the L1 norm of |x −y|−(n−ν)−|y|−(n−ν) as well as its Riesz transforms. We provide a proof here that used BMO oscillation estimates directly to verify Theorem 4.1.

Proof of Theorem 4.1. Let f ∈ BMO, 0 < ν < 1, and ψ ∈ S with integral zero. Then

〈Iνf, ψ〉 =∫Rn

f(u)

⎛⎝ ∫

Rn

1|u− x|n−ν

ψ(x)dx

⎞⎠ du =

∫Rn

F (x)ψ(x)dx,

where

F (x) =∫Rn

[1

|u− x|n−ν− 1

|u|n−ν

]f(u)du.

It is not hard to see that F is well-defined for f ∈ BMO. Furthermore, from the com-putations below, it follows that

∫R2n

∣∣∣∣ 1|x− u|n−ν

− 1|u|n−ν

∣∣∣∣ |f(u)ψ(x)|du dx < ∞.

We use this to apply Fubini and Toneli’s theorem to interchange order of the order of integration above to realize 〈Iνf, ψ〉 as the integral of F (x)ψ(x). It follows that Iνf(x) =F (x) + c for some constant c. Also note that


∫Rn

(1

|u− x|n−ν− 1

|u|n−ν

)du =

∫Rn

(1

|u− x/2|n−ν− 1

|u + x/2|n−ν

)du

= limR→∞

∫|u|≤R

du

|u− x/2|n−ν−

∫|u|≤R

du

|u + x/2|n−ν= 0.

Then

|Iνf(x) − Iνf(y)| =

∣∣∣∣∣∣∫Rn

[1

|x− u|n−ν− 1

|y − u|n−ν

](f(u) − fB(x,|x−y|))du

∣∣∣∣∣∣�

∫|x−u|>2|x−y|

|x− y||x− u|n+1−ν

|f(u) − fB(x,|x−y|)|du

+∫

|x−u|≤2|x−y|

1|x− u|n−ν


+∫

|x−u|≤2|x−y|

1|y − u|n−ν


= A(x, y) + B(x, y) + C(x, y).

We estimate the first term A(x, y) with the following computation;

A(x, y) ≤∞∑k=1

|x− y|(2k|x− y|)n+1−ν

∫2k|x−y|<|x−u|≤2k+1|x−y|


� |x− y|ν∞∑k=1

2−(1−ν)k 1(2k+1|x− y|)n

∫|x−u|≤2k+1|x−y|

|f(u) − fB(x,2k+1|x−y|)|du

+ |x− y|ν∞∑k=1

2−(1−ν)kk∑

j=0|fB(x,2j+1|x−y|) − fB(x,2j |x−y|)|

� |x− y|ν‖f‖BMO

∞∑k=1

2−(1−ν)k + |x− y|ν‖f‖BMO

∞∑k=1

(k + 1) 2−(1−ν)k

� |x− y|ν‖f‖BMO.

B(x, y) is bounded in a similar fashion,

B(x, y) ≤∫ 1

|x− u|n−ν|f(u) − fB(x,|x−y|)|du

|x−u|≤2|x−y|


≤∞∑k=0

1(2−k|x− y|)n−ν

∫2−k|x−y|<|x−u|≤21−k|x−y|


≤ |x− y|ν∞∑k=0

2−νk 1(2−k|x− y|)n

∫|x−u|≤21−k|x−y|

|f(u) − fB(x,21−k|x−y|)|du

+ |x− y|ν∞∑k=0

2−νkk∑

j=0|fB(x,21−j |x−y|) − fB(x,2−j |x−y|)|

� |x− y|ν‖f‖BMO

∞∑k=0

2−νk + |x− y|ν‖f‖BMO

∞∑k=0

(k + 1)2−νk


The final term is bounded as well,

C(x, y) ≤∫

|y−u|≤3|x−y|

1|y − u|n−ν


≤∫

|y−u|≤3|x−y|

1|y − u|n−ν

|f(u) − fB(y,|x−y|)|du

+ |fB(y,|x−y|) − fB(x,|x−y|)|∫

|y−u|≤3|x−y|

du

|y − u|n−ν

� |x− y|ν‖f‖BMO + |x− y|ν 1|B(y, |x− y|)|

∫|y−u|≤|x−y|


� |x− y|ν‖f‖BMO + |x− y|ν 1|B(x, |x− y|)|

∫|x−u|≤2|x−y|



Note that the first term in the estimate of C(x, y) can be estimated in exactly the same way as the B(x, y) term with the roles of x and y switched and with the domain of integration |y − u| ≤ 3|x − y| in place of |x − u| ≤ 2|x − y|. This completes the proof of estimate (4.1). The second estimate follows immediately. For f ∈ Iν(BMO) with 0 < ν < 1, there exists f ∈ BMO such that f = Iν(f) +c for some constant c. Then by the first estimate that was just proved, it follows that |f(x) −f(y)| = |Iν(f)(x) −Iν(f)(y)| �‖f‖Iν(BMO)|x − y|ν for all x, y ∈ Rn. �Remark 4.2. We return to discuss the interesting regularity behavior of Lipschitz spaces and Sobolev-BMO spaces for integers versus non-integers. For an integer s > 0, define Λs


to be the collection of all (s −1)-times differentiable functions, taken modulo polynomials, such that

‖f‖Λs= sup

x�=y

∑|α|=s−1

|Dαf(x) −Dαf(y)||x− y| < ∞.

For a non-integer s > 0, define Λs to be the collection of all L-times differentiable functions, taken modulo polynomials, such that

‖f‖Λs= sup

x�=y

∑|α|=L

|Dαf(x) −Dαf(y)||x− y|s−L

< ∞,

where L = �s� is the integer part of s. It follows that Λs � Is(BMO) whenever sis an integer and Is(BMO) � Λs whenever s is not an integer (see [31] for more on this). It was also observed by Strichartz that a slightly different version of the Lipschitz space Λs for integers gives the inclusion we might expect. Define the Zygmund class of functions for s > 0, borrowing notation from [31], Λ0

s(∞, ∞) to be the collection of L-times differentiable functions f such that

‖f‖Λ0s(∞,∞) = sup

x�=y

|ΔLy f(x)|

|x− y|s < ∞,

where L = �s� + 1 and ΔLy is the Lth order difference operator. With this definition,

it follows that Is(BMO) � Λs = Λ0s(∞, ∞) whenever s > 0 is not an integer and

Λs � Is(BMO) � Λ0s(∞, ∞) when s is an integer. The ideas behind these inclusions

can be traced all the way back to some results of Hardy that extended the construction of Weierstrass’s continuous, nowhere differentiable function, as well as to some of Zyg-mund’s work on the rate of convergence of Fourier coefficients for functions satisfying relaxed smoothness conditions. For more information on this, see [14,35] and [31]. These inclusions provide a rather precise understanding on the regularity of Sobolev-BMO

functions, and through Theorems 1.1–1.3, provide a good understanding for the regu-larity properties of linear and bilinear fractional integral operators acting on Lebesgue spaces with super-critical indices.

References

[1] H. Aimar, S. Hartzstein, B. Iaffei, B. Viviani, The Riesz potential as a multilinear operator into general BMOβ spaces, Problems in mathematical analysis. No. 55, J. Math. Sci. (N. Y.) 173 (6) (2011) 643–655.

[2] A. El Baraka, Littlewood–Paley characterization for Campanato spaces, J. Funct. Spaces Appl. 4 (2) (2006) 193–220.

[3] F. Bernicot, D. Maldonado, K. Moen, V. Naibo, Bilinear Sobolev–Poincaré inequalities and Leibniz-type rules, J. Geom. Anal. 24 (2) (2014) 1144–1180.

[4] G.B. Folland, E.M. Stein, Estimates for the ∂b complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974) 429–522.

http://refhub.elsevier.com/S0022-1236(16)30319-6/bib41484956s1





http://refhub.elsevier.com/S0022-1236(16)30319-6/bib424D4D4Es1





[5] M. Frazier, Y.S. Han, B. Jawerth, G. Weiss, The T1 Theorem for Triebel–Lizorkin Spaces, Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989.

[6] M. Frazier, B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1) (1990) 34–170.

[7] M. Frazier, R.H. Torres, G. Weiss, The boundedness of Calderón–Zygmund operators on the spaces Fα,qp , Rev. Mat. Iberoam. 4 (1) (1988) 41–72.

[8] A.E. Gatto, S. Vági, Boundedness of fractional integrals on spaces of homogeneous type of finite measure, X latin American school of mathematics, Rev. Un. Mat. Argentina 37 (1–2) (1991) 67–72.

[9] L. Grafakos, On multilinear fractional integrals, Studia Math. 102 (1) (1992) 49–56.[10] L. Grafakos, N. Kalton, Multilinear estimates and fractional integration, Math. Ann. 319 (1) (2001)

151–180.[11] L. Grafakos, R.H. Torres, Multilinear Calderón–Zygmund theory, Adv. Math. 165 (1) (2002)

124–164.[12] E. Harboure, O. Salinas, B. Viviani, Boundedness of the fractional integral in Orlicz and φ-bounded

mean oscillation spaces, in: Proceedings of the Second “Dr. António A. R. Monteiro” Congress on Mathematics, Univ. Nac. del Sur, Bahía Blanca, 1993, pp. 41–50.

[13] E. Harboure, O. Salinas, B. Viviani, Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces, Trans. Amer. Math. Soc. 349 (1) (1997) 235–255.

[14] G.H. Hardy, Weierstrass’s non-differentiable function, Trans. Amer. Math. Soc. 17 (3) (1916) 301–325.

[15] G.H. Hardy, J.E. Littlewood, Some properties of fractional integrals I, Math. Z. 27 (1927) 565–606.[16] G.H. Hardy, J.E. Littlewood, Some properties of fractional integrals II, Math. Z. 27 (1932) 403–439.[17] J. Hart, G. Lu, Hardy space estimates for Littlewood–Paley–Stein square functions and Calderón–

Zygmund operators, J. Fourier Anal. Appl. 22 (1) (2016) 159–186.[18] J. Hart, G. Lu, Hardy space estimates for bilinear Littlewood–Paley square functions and

singular integral operators, Indiana Univ. Math. J. 65 (5) (2016), http://dx.doi.org/10.1512/iumj.2016.65.5898.

[19] J. Hart, L. Oliveira, Hardy spaces estimates for limited ranges of Muckenhoupt Ap weights, sub-mitted for publication.

[20] J. Hart, R.H. Torres, X. Wu, Smoothing properties bilinear operators, in preparation.[21] C. Kenig, E. Stein, Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1) (1999)

1–15.[22] M. Lacey, K. Moen, C. Pérez, R. Torres, Sharp weighted bounds for fractional integral operators,

J. Funct. Anal. 259 (5) (2010) 1073–1097.[23] D. Maldonado, K. Moen, V. Naibo, Weighted multilinear Poincaré inequalities for vector fields of

Hörmander type, Indiana Univ. Math. J. 60 (2) (2011) 473–506.[24] K. Moen, Weighted inequalities for multilinear fractional integral operators, Collect. Math. 60 (2)

(2009) 213–238.[25] K. Moen, Linear and multilinear fractional operators: Weighted inequalities, sharp bounds, and

other properties, Thesis (Ph.D.)–University of Kansas, 2009, 143 pp.[26] B. Muckenhoupt, R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer.

Math. Soc. 192 (1974) 261–274.[27] U. Neri, Fractional integration on the space H1 and its dual, Studia Math. 53 (1975) 175–189.[28] M. Riesz, L’intégrale de Riemann–Liouville et le problème de Cauchy, Acta Math. 81 (1949) 1–223.[29] S.L. Sobolev, On a theorem in functional analysis, Mat. Sob. 46 (1938) 471–497.[30] E.M. Stein, A. Zygmund, Boundedness of translation invariant operators on Hölder spaces and

Lp-spaces, Ann. of Math. (2) 85 (1967) 337–349.[31] R. Strichartz, Bounded mean oscillation and Sobolev spaces, Indiana Univ. Math. J. 29 (4) (1980)

539–558.[32] R. Strichartz, Traces of BMO-Sobolev spaces, Proc. Amer. Math. Soc. 83 (3) (1981) 509–513.[33] R.H. Torres, Boundedness results for operators with singular kernels on distribution spaces, Mem.

Amer. Math. Soc. 90 (442) (1991).[34] A. Youssfi, Regularity properties of singular integral operators, Studia Math. 119 (3) (1996) 199–217.[35] A. Zygmund, Smooth functions, Duke Math. J. 12 (1945) 47–76.

http://refhub.elsevier.com/S0022-1236(16)30319-6/bib46484A57s1


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http://refhub.elsevier.com/S0022-1236(16)30319-6/bib4B47s1











http://refhub.elsevier.com/S0022-1236(16)30319-6/bib4C5031s1




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http://refhub.elsevier.com/S0022-1236(16)30319-6/bib4C4D5054s1

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http://refhub.elsevier.com/S0022-1236(16)30319-6/bib4D6F31s1

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http://refhub.elsevier.com/S0022-1236(16)30319-6/bib4D57s1

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