The effect of the smoothness of fractional type operators over their commutators with Lipschitz symbols on weighted spaces

TL;DR

This paper establishes boundedness results for fractional integral operators and their commutators with Lipschitz symbols on weighted spaces, including new cases with less regular kernels and characterizations of symbol classes.

Contribution

It provides new boundedness results for fractional operators and their commutators with Lipschitz symbols, including cases with less regular kernels and characterizations involving symbols.

Findings

Boundedness of fractional operators on weighted spaces.

New results for commutators with less regular kernels.

Characterization of symbols related to commutator boundedness.

Abstract

We prove boundedness results for integral operators of fractional type and their higher order commutators between weighted spaces, including $L^p$-$L^q$, $L^p$-$BMO$ and $L^p$-Lipschitz estimates. The kernels of such operators satisfy certain size condition and a Lipschitz type regularity, and the symbol of the commutator belongs to a Lipschitz class. We also deal with commutators of fractional type operators with less regular kernels satisfying a H\"ormander's type inequality. As far as we know, these last results are new even in the unweighted case. Moreover, we give a characterization result involving symbols of the commutators and continuity results for extreme values of $p$.

Equations

$$I_{\alpha }f(x)={\int }_{{\mathbb{R}}^n}\frac{f(y)}{|x-y{|}^{n-\alpha }}dy,$$

$$[b,I_{\alpha }](f)=b{I}_{\alpha }f-I_{\alpha }(bf).$$

$$T_{\alpha }f(x)={\int }_{{\mathbb{R}}^n}{K}_{\alpha }(x-y)f(y)dy,$$

$$|b(x)-b(y)|\le C|x-y{|}^{\delta }.$$

$${\int }_{s<|x|\le 2s}|K_{\alpha }(x)|dx\le C{s}^{\alpha },$$

$$\left|K_{\alpha }(x-y)-K_{\alpha }(x^{\prime }-y)\right|+\left|K_{\alpha }(y-x)-K_{\alpha }(y-x^{\prime })\right|\le C\frac{{\left|x-x^{\prime }\right|}^{\eta }}{{\left|x-y\right|}^{n-\alpha +\eta }},$$

$$T_{\alpha ,b}^{m}f(x)={\int }_{{\mathbb{R}}^n}(b(x)-b(y){)}^m{K}_{\alpha }(x-y)f(y)dy,$$

$$\underset{B}{\sup }\frac{\Vert w{\chi }_B{\Vert }_{\infty }}{|B{|}^{1+\frac{\delta }{n}}}{\int }_B|f(x)-f_B|dx<\infty ,$$

$$[w{]}_{A_{p,q}}:=\underset{B}{\sup }{\left(\frac{1}{|B|}{\int }_{B}w(x{)}^{q}dx\right)}^{1/q}{\left(\frac{1}{|B|}{\int }_{B}w(x{)}^{-p^{\prime }}dx\right)}^{1/p^{\prime }}<\infty .$$

$${\left({\int }_{{\mathbb{R}}^n}|T_{\alpha ,b}^{m}f(x){|}^{q}w(x{)}^{q}dx\right)}^{1/q}\le C{\Vert b\Vert }_{\Lambda (\delta )}^m{\left({\int }_{{\mathbb{R}}^n}|f(x){|}^{p}w(x{)}^{p}dx\right)}^{1/p}$$

$${\Vert T_{\alpha ,b}^{m}f\Vert }_{{\mathbb{L}}_w(\tilde{\delta })}\le C{\Vert b\Vert }_{\Lambda (\delta )}^m{\Vert fw\Vert }_{L^r}$$

$$\frac{{\Vert w{\chi }_B\Vert }_{\infty }}{|B{|}^{1+\frac{\delta }{n}}}{\int }_B\left|\sum _{k=0}^m{c}_k\left[S(x,u,k)-(S(\cdot ,u,k){)}_B\right]\right|dx\le C{\Vert fw\Vert }_r,$$

$$\frac{1}{|B{|}^{1+\frac{\delta }{n}}}{\int }_B\left|\sum _{k=0}^1{c}_k\left[S(x,u,k)-(S(\cdot ,u,k){)}_B\right]\right|dx\le C{\Vert f\Vert }_{n/\alpha }.$$

$$\frac{1}{|B{|}^{1+\frac{\delta }{n}}}{\int }_B\left|\sum _{k=0}^1{c}_k\left[S(x,u,k)-(S(\cdot ,u,k){)}_B\right]\right|dx=\frac{1}{{\left|B\right|}^{1+\frac{\delta }{n}}}{\int }_B\left|b(x)-b_B\right|dx\left|{\int }_{(2B{)}^c}\frac{f(y)}{{\left|u-y\right|}^{n-\alpha }}dy\right|,$$

$$\frac{1}{{\left|B\right|}^{1+\frac{\delta }{n}}}{\int }_B\left|b(x)-b_B\right|dx\left|{\int }_{(2B{)}^c}\frac{f(y)}{{\left|u-y\right|}^{n-\alpha }}dy\right|\le C{\Vert f\Vert }_{n/\alpha }.$$

$$\frac{1}{{\left|B\right|}^{1+\frac{\delta }{n}}}{\int }_B\left|b(x)-b_B\right|dx{\left|{\int }_{(2B{)}^c\cup \{|u-y|<N\}}\frac{dy}{{\left|u-y\right|}^n}\right|}^{1-\alpha /n}\le C.$$

$$\Vert f{\Vert }_{\Phi ,B}=\inf \left\{\lambda >0:\frac{1}{|B|}{\int }_B\Phi \left(\frac{|f(x)|}{\lambda }\right)dx\le 1\right\}.$$

$$M_{\alpha ,\Phi }f(x)=\underset{B\ni x}{\sup }|B{|}^{\alpha /n}\Vert f{\Vert }_{\Phi ,B},$$

$$\frac{1}{|B|}{\int }_B|f(x)g(x)|dx\le 2\Vert f{\Vert }_{\Phi ,B}\Vert g{\Vert }_{\tilde{\Phi },B},$$

$$\tilde{\Phi }(t)=\underset{s>0}{\sup }\{st-\varphi (s)\}.$$

$$\Vert fg{\Vert }_{\Theta ,B}\lesssim \Vert f{\Vert }_{\Phi ,B}\Vert g{\Vert }_{\Psi ,B}.$$

$$\sum _{j=1}^{\infty }(2^{j}R{)}^{n-\alpha }\Vert \left(K_{\alpha }(\cdot -y)-K_{\alpha }(\cdot )\right){\chi }_{|\cdot |\sim 2^{j}R}{\Vert }_{\Phi ,B(0,2^{j+1}R)}\le C,$$

$$\sum _{j=1}^{\infty }(2^{j}R{)}^{n-\alpha }{\left(\frac{1}{(2^{j}R{)}^n}{\int }_{|x|\sim 2^{j}R}|K_{\alpha }(x-y)-K_{\alpha }(x){|}^{q}dx\right)}^{1/q}\le C.$$

$${\int }_0^1{\omega }_{\Phi }(t)\frac{dt}{t}<\infty ,$$

$${\omega }_{\Phi }(t)=\underset{|y|\le t}{\sup }||\Omega (\cdot +y)-\Omega (\cdot )|{|}_{\Phi ,S^{n-1}}<\infty ,$$

$$\sum _{j=1}^{\infty }(2^j{)}^{m\delta }(2^{j}R{)}^{n-\alpha }\Vert \left(K_{\alpha }(\cdot -y)-K_{\alpha }(\cdot )\right){\chi }_{|\cdot |\sim 2^{j}R}{\Vert }_{\Phi ,B(0,2^{j+1}R)}\le C.$$

$${\int }_c^{\infty }\frac{{\Psi }^{1+\frac{\rho \alpha }{n}}(t)}{t^{\rho }}\frac{dt}{t}<\infty$$

$${\left({\int }_{{\mathbb{R}}^n}|T_{\alpha ,b}^{m}f(x){|}^{q}w(x{)}^{q}dx\right)}^{1/q}\le C\Vert b{\Vert }_{\Lambda (\delta )}^m{\left({\int }_{{\mathbb{R}}^n}|f(x){|}^{p}w(x{)}^{p}dx\right)}^{1/p}$$

$${\Vert T_{\alpha ,b}^{m}f\Vert }_{{\mathbb{L}}_w(\tilde{\delta })}\le C{\Vert b\Vert }_{\Lambda (\delta )}^m{\Vert fw\Vert }_{L^r}$$

$$\frac{\Vert w{\chi }_B{\Vert }_{\infty }}{|B{|}^{1+\frac{\delta }{n}}}{\int }_B\left|\sum _{k=0}^m{c}_k\left[S(x,u,k)-(S(\cdot ,u,k){)}_B\right]\right|dx\le C{\Vert fw\Vert }_r,$$

Full text

1 Introduction

There is a close relationship between the theory of Partial Differential Equations and Harmonic Analysis. This is evidenced, for instance, by the existence of a mechanism that provides us with regular solutions of PDE’s when we equip this machinery with continuity properties of certain related operators (see, for example, [4], [5], [8], [9], [12], [34]). Therefore, it seems appropriate to explore the boundedness properties of the mentioned operators and, particularly, we shall be concerned with commutators of integral operators of fractional type.

It is well known that the fractional integral operator of order $\alpha$, $0<\alpha<n$, is defined by

[TABLE]

whenever this integral is finite. There is a vast amount of information about the behavior of the operator above (see for example [18], [20], [26], [30] and [36]). In [6], Chanillo introduced the first order commutator of $I_{\alpha}$ with a symbol $b\in L_{\rm loc}^{1}(\mathbb{R}^{n})$, formally defined by

[TABLE]

Particularly, if $b\in BMO$, the space of bounded mean oscillation, the author proved that, for $1<p<n/\alpha$ and $1/q=1/p-\alpha/n$, the operator $[b,I_{\alpha}]$ is bounded from $L^{p}(\mathbb{R}^{n})$ into $L^{q}(\mathbb{R}^{n})$. Some topics related to properties of boundedness for commutators of fractional integral operators for extreme values of $p$ can be found in [19]. (For more information about $BMO$ spaces see [21]).

The continuity properties of the commutator of the fractional integral operator acting on different spaces were studied by several authors contributing, in this way, to the development of PDE’s. Some related articles are given by [2], [10], [14], [17], [23], [27], [28], [29], [33], [35]. In [25] the authors consider the commutators of certain fractional type operators with Lipschitz symbols and prove the boundedness between Lebesgue spaces, including the boundedness from Lebesgue spaces into $BMO$ and Lipschitz spaces on non-homogeneous spaces. (See also [32] in the context of variable Lebesgue spaces).

Nevertheless, there is not enough information about the behavior of the commutators acting between weighted Lebesgue spaces, even less for extreme values of $p$, that is, the weighted $L^{p}$-$BMO$ or $L^{p}$-Lipschitz boundedness. Hence, one of our main aims is, precisely, to give sufficient conditions on the weights in order to obtain these continuity properties. Some previous results in this direction were given in [1] where the authors study the boundedness between Lebesgue spaces with variable exponent for commutators of fractional type operators with $BMO$ symbols, (see also [11] in the framework of Orlicz spaces).

We shall first consider fractional type operators, and their commutators, which kernels satisfy certain size condition and a Lipschitz type regularity. For this type of operators we prove boundedness results of the type described above. Particularly we prove a characterization result involving symbols of the commutators and continuity results for extreme values of $p$.

Later, we study commutators of fractional type operators with less regular kernels. These type of operators include a great variety of operators and were introduced in [3]. See section 2.2 for examples and more information.

The paper is organized as follows. In section §2 we give the preliminaries definitions in order to state the main results of the article, which are also included in this section. Then, in §3 we give some auxiliary results which allow us to prove the main results in §4.

2 Preliminaries and main results

In this section we give the definitions of the operators we shall be dealing with and the functional class of the symbols in order to define the commutators.

We shall consider fractional operators of convolution type $T_{\alpha}$, $0<\alpha<n$, defined by

[TABLE]

where the kernel $K_{\alpha}$ is not identically zero and verifies certain size and smoothness conditions.

Let $0<\delta<1$. We say that a function $b$ belongs to the space $\Lambda(\delta)$ if there exists a positive constant $C$ such that, for every $x$, $y\in\mathbb{R}^{n}$

[TABLE]

The smallest of such constants will be denoted by $\|b\|_{\Lambda(\delta)}$. We will be dealing with commutators with symbols belonging to this class of functions.

Given a weight $w$, that is, a non-negative and locally integrable function, we say that a measurable function $f$ belongs to $L^{p}_{w}(\mathbb{R}^{n})$ for some $1<p<\infty$ if $fw\in L^{p}(\mathbb{R}^{n})$

We classify the operators defined in (2.1) into two different types, according to the conditions satisfied by $K_{\alpha}$.

2.1 Fractional integral operators with Lipschitz regularity

We say that $K_{\alpha}$ satisfies the size condition $S_{\alpha}$ if it verifies the following inequality

[TABLE]

for every $s>0$ and some positive constant $C$.

We shall also assume that $K_{\alpha}$ satisfies the smoothness condition $H_{\alpha,\infty}^{*}$, that is, there exist a positive constant $C$ and $0<\eta\leq 1$ such that

[TABLE]

whenever $\left|x-y\right|\geq 2\left|x-x^{\prime}\right|$.

A typical example is the fractional integral operator $I_{\alpha}$, whose kernel $K_{\alpha}(x)=|x|^{\alpha-n}$ satisfies conditions $S_{\alpha}$ and $H_{\alpha,\infty}^{*}$, as it can be easily checked.

Related with the fractional integral operators $T_{\alpha}$, we can formally define the higher order commutators with symbol $b\in L^{1}_{{\rm loc}}(\mathbb{R}^{n})$, by

[TABLE]

where $m\in\mathbb{N}_{0}$ is the order of the commutator. Clearly, $T_{\alpha,b}^{0}=T_{\alpha}$.

As we have said, we are interested in studying the boundedness properties of the commutators $T_{\alpha,b}^{m}$, with symbol $b\in\Lambda(\delta)$, on weighted spaces. We shall first consider their continuity on weighted Lebesgue spaces of the type defined previously. We shall also analyze the boundedness of $T_{\alpha,b}^{m}$ from weighted Lebesgue spaces into certain weighted version of Lipschitz spaces. For $0<\delta<1$ and $w$ a weight, these spaces are denoted by $\mathbb{L}_{w}(\delta)$ and collect the functions $f\in L^{1}_{{\rm loc}}(\mathbb{R}^{n})$ that satisfy

[TABLE]

where $\|g\|_{\infty}$ denotes the essential supremum of a measurable function $g$. The case $\delta=0$ of the space above was introduced in [26] as a weighted version of the space of functions with bounded mean oscillation.

The classes of weights we will be dealing with are the well-known $A_{p,q}$ classes of Muckenhoupt and Wheeden ([26]). For $1\leq p,q<\infty$ these classes are defined as the weights $w$ such that

[TABLE]

When $q=\infty$, we understand that $w\in A_{p,\infty}$ as $w^{-p^{\prime}}\in A_{1}$.

In this subsection we shall assume that the operator $T_{\alpha}$ has a convolution kernel $K_{\alpha}$ that verifies conditions $S_{\alpha}$ and $H_{\alpha,\infty}^{*}$ with $0<\eta\leq 1$. In order to simplify the hypothesis we shall supposse that $m\in\mathbb{N}\cup\{0\}$ with the convention that $\beta/0=\infty$ if $\beta>0$.

We now give the boundedness result between weighted Lebesgue spaces for the higher order commutators of $T_{\alpha}$ with Lipschitz symbols.

Theorem 2.1**.**

Let $0<\alpha<n$ and $0<\delta<\min\{\eta,(n-\alpha)/m\}$. Let $1<p<n/(m\delta+\alpha)$, $1/q=1/p-(m\delta+\alpha)/n$ and $b\in\Lambda(\delta)$. If $w\in A_{p,q}$, then there exists a positive constant $C$ such that

[TABLE]

for every $f\in L^{p}_{w}(\mathbb{R}^{n})$.

Remark 2.2*.*

When $m=0$ and $T_{\alpha}=I_{\alpha}$, the result above was proved in [26]. Notice that there are no symbols or parameters $\delta$ in the hypothesis in this case.

The next result gives the continuity properties of $T_{\alpha,b}^{m}$ between weighted Lebesgue spaces and $\mathbb{L}_{w}(\widetilde{\delta})$ spaces. By $\beta^{+}$ we understand $\beta$ if $\beta>0$ and [math] is $\beta\leq 0$.

Theorem 2.3**.**

Let $0<\alpha<n$, and $0<\delta<\min\{\eta,(n-\alpha)/m\}$. Let $n/(m\delta+\alpha)\leq r<n/(\alpha+(m-1)\delta)$, if $m\geq 1$ or $n/\alpha\leq r<n/(\alpha-\eta)^{+}$, if $m=0$. Let $\widetilde{\delta}=m\delta+\alpha-\frac{n}{r}$ and $b\in\Lambda(\delta)$. If $w\in A_{r,\infty}$, then there exists a positive constant $C$ such that

[TABLE]

for every $f\in L^{r}_{w}(\mathbb{R}^{n})$.

Remark 2.4*.*

When $m=1$, $w=1$ and $T_{\alpha}=I_{\alpha}$, the result above was proved in [25] in the general context of non-doubling measures.

Remark 2.5*.*

If $r=n/(m\delta+\alpha)$, then $\widetilde{\delta}=0$ and the space $\mathbb{L}_{w}(\widetilde{\delta})$ is the weighted version of the $BMO$ spaces introduced in [26]. By taking into account the range of $p$ in Theorem 2.1, it is clear that this is the endpoint value from which the Lebesgue spaces change into $BMO$ and Lipschitz spaces when the commutator acts. Particularly, if $m=0$ and $T_{\alpha}=I_{\alpha}$, this is the well-known result proved in [26]. Notice again that there are no parameters $\delta$ or symbols in this case.

On the other hand, if $m=0$, $T_{\alpha}=I_{\alpha}$ and $n/\alpha\leq r<n/(\alpha-1)^{+}$, that is, $\eta=1$ in the definition of the class $H_{\alpha,\infty}^{*}$, the result above was proved in [31].

For the extreme value $r=n/((m-1)\delta+\alpha)$, $m\in\mathbb{N}$ and $0<\delta<\eta\leq 1$, we obtain the following endpoint result in order to characterize the symbol $b$ in $\Lambda(\delta)$ in terms of the boundedness of $T_{\alpha,b}^{m}$ in the sense of Theorem 2.3. In order to give this result we introduce some previous notation. For $k=0,1,...,m$ we denote $c_{k}={m!}/{(k!(m-k)!)}$. If also $x$, $u\in\mathbb{R}^{n}$, we denote $S(x,u,k)=(b(x)-b_{B})^{m-k}T_{\alpha}((b-b_{B})^{k}f_{2})(u)$, where $f_{2}=f\chi_{\mathbb{R}^{n}\setminus B}$ for a given ball $B$ and a locally integrable function $f$.

Theorem 2.6**.**

Let $m\in\mathbb{N}$, $0<\delta<\min\{\eta,(n-\alpha)/m\}$ and $r=n/((m-1)\delta+\alpha)$. If $w\in A_{n/(m\delta+\alpha),\infty}$ and $b\in\Lambda(\delta)$, the following statements are equivalent.

1. (i)

$T_{\alpha,b}^{m}:L^{r}_{w}(\mathbb{R}^{n})(\mathbb{R}^{n})\hookrightarrow\mathbb{L}_{w}(\delta)$; 2. (ii)

There exists a positive constant $C$ such that

[TABLE]

for every ball $B\subset\mathbb{R}^{n}$, $x,u\in B$ and $f\in L^{1}_{{\rm loc}}(\mathbb{R}^{n})$.

Remark 2.7*.*

When $m=1$, $w=1$ and $T_{\alpha}=I_{\alpha}$ the result above was proved in [25] in a more general context of non-homogeneous spaces. Certainly, their result was inspired in the article of [19], where the same result is proved for $m=1$, $w=1$, $T_{\alpha}=I_{\alpha}$ and $b\in BMO$.

Remark 2.8*.*

In [19] the authors also have obtained that, in the case of $T_{\alpha}=I_{\alpha}$, $b\in BMO$, $m=1$ and $w=1$, the boundedness of the commutator $I_{\alpha,b}$ from $L^{n/\alpha}$ into $BMO$ can only occur if $b$ is constant. In our case, if $b\in\Lambda(\delta)$, when $T_{\alpha}=I_{\alpha}$, $m=1$ and $w=1$, we can deduce, by (2.3), that, if $I_{\alpha,b}$ is bounded from $L^{n/\alpha}$ into $\mathbb{L}(\delta)$, then

[TABLE]

Since it is easy to see that

[TABLE]

we have that

[TABLE]

Following the same guidelines as in [19] with $f_{N}(y)=\left|u-y\right|^{-\alpha}\chi_{B(0,N)}(u-y)\chi_{(2B)^{c}}(y)$ for $N\in\mathbb{N}$, we obtain that

[TABLE]

Due tu the fact that $\left|\int_{(2B)^{c}\cup\{\left|u-y\right|<N\}}\frac{dy}{\left|u-y\right|^{n}}\right|^{1-\alpha/n}\to\infty$ when $N\to\infty$, we have $b(x)=b_{B}$ almost everywhere, for every ball $B$, which yields that $b$ is essentially constant.

2.2 Fractional integral operators with Hörmander type

regularity

We now introduce the conditions on the kernel that will be considered in this section. First, we must give some notation.

It is well-known that the commutators of fractional integral operators can be controlled, in some sense, by maximal type operators associated to Young functions. By a Young function we mean a function $\Phi:[0,\infty)\rightarrow[0,\infty)$ that is increasing, convex and verifies $\Phi(0)=0$ and $\Phi(t)\rightarrow\infty$ when $t\rightarrow\infty$. The $\Phi$-Luxemburg average over a ball $B$ is defined, for a locally integrable function $f$, by

[TABLE]

The maximal type operators that control the commutators involve these averages. More precisely, if $f\in L^{1}_{{\rm loc}}(\mathbb{R}^{n})$ and $0<\alpha<n$, we define the fractional type maximal operator associated to a Young function $\Phi$, by

[TABLE]

where the supremum is taken over every ball $B$ that contains $x\in\mathbb{R}^{n}$.

Given a Young function $\Phi$, the following Hölder’s type inequality holds for every pair of measurable functions $f,g$

[TABLE]

where $\widetilde{\Phi}$ is the complementary Young function of $\Phi$, defined by

[TABLE]

It is easy to see that $t\leq\Phi^{-1}(t)\widetilde{\Phi}^{-1}(t)\leq 2t$ for every $t>0$.

Moreover, given $\Phi,\Psi$ and $\Theta$ Young functions verifying that $\Phi^{-1}(t)\Psi^{-1}(t)\lesssim\Theta^{-1}(t)$ for every $t>0$, the following generalization holds

[TABLE]

We are now in position to define the smoothness condition on $K_{\alpha}$.

We say that $K_{\alpha}\in H_{\alpha,\Phi}$ if there exist $c\geq 1$ and $C>0$ such that for every $y\in\mathbb{R}^{n}$ and $R>c|y|$

[TABLE]

where $|\cdot|\sim s$ means the set $\{x\in\mathbb{R}^{n}:s<|x|\leq 2s\}$.

When $\Phi(t)=t^{q}$, $1\leq q<\infty$, we denote this class by $H_{\alpha,q}$ and it can be written as

[TABLE]

The kernels given above are, a priori, less regular than the kernel of the fractional integral operator $I_{\alpha}$ and they have been studied by several authors. For example, in [22], the author studied fractional integrals given by a multiplier. If $m:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is a function, the multiplier operator $T_{m}$ is defined, through the Fourier transform, as $\widehat{T_{m}f}(\zeta)=m(\zeta)\widehat{f}(\zeta)$ for $f$ in the Schwartz class. Under certain conditions on the derivatives of $m$, the multiplier operator $T_{m}$ can be seen as the limit of convolution operators $T_{m}^{N}$, having a simpler form. Their corresponding kernels $K_{\alpha}^{N}$ belong to the class $S_{\alpha}\cap H_{\alpha,r}$ with constant independent of $N$, for certain values of $r>1$ given by the regularity of the function $m$ (see [22]).

Other examples of this type of operators are fractional integrals with rough kernels, that is, with kernel $K_{\alpha}(x)=\Omega(x)|x|^{\alpha-n}$ where $\Omega$ is a function defined on the unit sphere $S^{n-1}$ of $\mathbb{R}^{n}$, extended to $\mathbb{R}^{n}\setminus\{0\}$ radially. The function $\Omega$ is an homogeneous function of degree [math]. In [3, Proposition 4.2], the authors showed that $K_{\alpha}\in S_{\alpha}\cap H_{\alpha,\Phi}$, for certain Young function $\Phi$, provided that $\Omega\in L^{\Phi}(S^{n-1})$ with

[TABLE]

where $\omega_{\Phi}$ is the $L^{\Phi}$-modulus of continuity given by

[TABLE]

for every $t\geq 0$. This type of operators where also studied in [7] and [13].

As we said previously, we are interested in studying the higher order commutators of $T_{\alpha}$. Since we are dealing with symbols of Lipschitz type, the smoothness condition associated to these commutators is defined as follows.

Definition 2.9**.**

Let $m\in\mathbb{N}_{0}$, $0<\alpha<n$, $0\leq\delta<\min\{1,(n-\alpha)/m\}$ and let $\Phi$ be a Young function. We say that $K_{\alpha}\in H_{\alpha,\Phi,m}(\delta)$ if

[TABLE]

for some constants $c\geq 1$ and $C>0$ and for every $y\in\mathbb{R}^{n}$ with $R>c|y|$.

Clearly, when $\delta=0$ or $m=0$, $H_{\alpha,\Phi,m}(\delta)=H_{\alpha,\Phi}$.

Remark 2.10*.*

It is easy to see that $H_{\alpha,\Phi,m}(\delta_{2})\subset H_{\alpha,\Phi,m}(\delta_{1})\subset H_{\alpha,\Phi}$ whenever $0\leq\delta_{1}<\delta_{2}<\min\{1,(n-\alpha)/m\}$.

Recall that Fourier multipliers and fractional integrals with rough kernels are examples of fractional integral operators with $K_{\alpha}\in H_{\alpha,\Phi}$ for certain Young function. By assuming adequate conditions depending on $\delta$ on the multiplier $m$, or on the $L^{\Phi}$-modulus of continuity $\omega_{\Phi}$, one can obtain kernels $K_{\alpha}\in H_{\alpha,\Phi,m}(\delta)$. This fact can be proved by adapting Proposition 4.2 and Corollary 4.3 given in [3] (see also [24]).

We shall also deal with a class of Young functions that arises in connection with the boundedness of the fractional maximal operator $M_{\alpha,\Psi}$ on weighted Lebesgue spaces (see §3). Given $0<\alpha<n$, $1\leq\beta<p<n/\alpha$ and a Young function $\Psi$, we shall say that $\Psi\in\mathcal{B}_{\alpha,\beta}$ if $t^{-\alpha/n}\Psi^{-1}(t)$ is the inverse of a Young function and $\Psi^{1+\frac{\rho\alpha}{n}}\in B_{\rho}$ for every $\rho>n\beta/(n-\alpha\beta)$, that is, there exists a positive constant $c$ such that

[TABLE]

for each of those values of $\rho$.

We now state the following generalizations of Theorems 2.1 and 2.3. We shall consider again $m\in\mathbb{N}_{0}$.

Theorem 2.11**.**

Let $0<\alpha<n$ and $0<\delta<\min\{1,(n-\alpha)/m\}$. Let $1<p<n/(m\delta+\alpha)$, $1/q=1/p-(m\delta+\alpha)/n$ and $b\in\Lambda(\delta)$. Assume that $T_{\alpha}$ has a kernel $K_{\alpha}\in S_{\alpha}\cap H_{\alpha,\Phi}$ for a Young function $\Phi$ such that its complementary function $\widetilde{\Phi}\in\mathcal{B}_{m\delta+\alpha,\beta}$ for some $1\leq\beta<p$. Then, if $w$ is a weight verifying $w^{\beta}\in A_{\frac{p}{\beta},\frac{q}{\beta}}$, there exists a positive constant $C$ such that

[TABLE]

for every $f\in L^{p}_{w}(\mathbb{R}^{n})$.

Remark 2.12*.*

If we consider, for example, $\Phi(t)=e^{t^{1/\gamma}}-e$ with $\gamma>0$, then $\widetilde{\Phi}(t)\thickapprox t(1+\log^{+}t)^{\gamma}$ and this function verifies condition $\mathcal{B}_{m\delta+\alpha,1}$. Thus, $\Phi$ satisfies the hypothesis of the theorem above and, in this case, we can take $w\in A_{p,q}$. As we have mentioned before, this condition $\mathcal{B}_{m\delta+\alpha,\beta}$ is related with the boundedness of the corresponding fractional maximal operator $M_{m\delta+\alpha,\widetilde{\Phi}}$ between $L^{p}_{w}$ and $L^{q}_{w}$ when $w^{\beta}\in A_{\frac{p}{\beta},\frac{q}{\beta}}$ (see Theorem 3.6 below). When $\beta>1$, a typical example is $\widetilde{\Phi}(t)=t^{\beta}(1+\log^{+}t)^{\gamma}$ for $\gamma\geq 0$. In this case, the Young function $\Phi$ related with the smoothness condition on the kernel $K_{\alpha}$ given in the theorem above is $\Phi(t)=t^{\beta^{\prime}}(1+\log^{+}t)^{-\gamma/(\beta-1)}$, where $\beta^{\prime}=\beta/(\beta-1)$.

Theorem 2.13**.**

Let $0<\alpha<n$, $0<\delta<\min\{1,(n-\alpha)/m\}$ and $n/(m\delta+\alpha)\leq r<n/((m-1)\delta+\alpha)$ such that $\widetilde{\delta}=m\delta+\alpha-\frac{n}{r}$. Let $w$ be a weight such that $w^{\beta}\in A_{r/\beta,\infty}$ for some $1<\beta<r$. Assume that $T_{\alpha}$ has a kernel $K_{\alpha}\in S_{\alpha}\cap H_{\alpha,\Phi,m}(\delta)$ for a Young function $\Phi$ such that $\Phi^{-1}(t)\lesssim t^{\frac{\beta-1}{r}}$ for every $t>0$. If $b\in\Lambda(\delta)$, then there exists a positive constant $C$ such that

[TABLE]

for every $f\in L^{r}_{w}(\mathbb{R}^{n})$.

Theorem 2.14**.**

Let $m\in\mathbb{N}$, $0<\delta<\min\{1,(n-\alpha)/m\}$ and $r=n/((m-1)\delta+\alpha)$. Let $w$ be a weight such that $w^{\beta}\in A_{r/\beta,\infty}$ for some $1<\beta<r$. Let $T_{\alpha}$ be a fractional integral operator with kernel $K_{\alpha}\in S_{\alpha}\cap H_{\alpha,\Phi,m}(\delta)$ where $\Phi$ is a Young function verifying $\Phi^{-1}(t)\lesssim t^{\frac{\beta-1}{r}}$ for every $t>0$, and $\widetilde{\Phi}\in\mathcal{B}_{m\delta+\alpha,\beta}$. If $b\in\Lambda(\delta)$, the following statements are equivalent,

1. (i)

$T_{\alpha,b}^{m}:L^{r}_{w}(\mathbb{R}^{n})\hookrightarrow\mathbb{L}_{w}(\delta)$; 2. (ii)

There exists a positive constant $C$ such that

[TABLE]

for every ball $B\subset\mathbb{R}^{n}$, $x,u\in B$ and $f\in L^{1}_{{\rm loc}}(\mathbb{R}^{n})$.

3 Auxiliary results

In this section we give some previous results. We begin with some inequalities involving functions in $\Lambda(\delta)$.

Lemma 3.1**.**

Let $0<\delta<1$ and $B\subset\mathbb{R}^{n}$ a ball. If $b\in\Lambda(\delta)$, then

1. (i)

for every $y\in\lambda B$, $\lambda\geq 1$,

[TABLE] 2. (ii)

for every $j\in\mathbb{N}$

[TABLE]

The following lemma is an easy consequence of condition $S_{\alpha}$.

Lemma 3.2**.**

Let $K_{\alpha}$ be a kernel verifying condition $S_{\alpha}$ with $0<\alpha<n$. Then, for any ball $B=B(x_{B},r_{B})$, we have

[TABLE]

Proof.

By changing variables first, we then split the integral into dyadic sets and use $S_{\alpha}$ condition in each set as it follows

[TABLE]

since $\alpha>0$. ∎

In order to obtain the boundedness result between Lebesgue spaces, we prove the following key estimate, which shows how can we control the higher order commutators of $T_{\alpha}$ by a fractional maximal function via the sharp maximal operator $M^{\sharp}_{0,\gamma}$, $0<\gamma<1$, given by

[TABLE]

where $M^{\sharp}_{0,\gamma}f:=M^{\sharp}_{0}(|f|^{\gamma})^{1/\gamma}$.

Lemma 3.3**.**

Let $m\in\mathbb{N}$, $0<\gamma<1/m$, $0<\alpha<n$ and $0<\delta<\min\{1,(n-\alpha)/m\}$. Let $b\in\Lambda(\delta)$ and $T_{\alpha}$ a fractional integral operator with kernel $K_{\alpha}\in S_{\alpha}$. Then, there exists a positive constant $C$ such that

1. (i)

if $K_{\alpha}\in H_{\alpha,\infty}^{*}$,

[TABLE]

where $\theta_{j}=\delta(m-j)$, $j=0,\dots,m$. 2. (ii)

if $K_{\alpha}\in H_{\alpha,\Phi}$ for some Young function $\Phi$,

[TABLE]

where $\theta_{j}=\delta(m-j)$, $j=0,\dots,m$, and $\widetilde{\Phi}$ is the complementary function of $\Phi$.

Remark 3.4*.*

For $0<\delta<1$, $m=1$ and $K_{\alpha}\in H_{\alpha,\infty}^{*}$ and homogeneous of degree $\alpha-n$, the proof of (i) can be found in [32] for a larger class of Lipschitz spaces with variable parameter.

Proof of Lemma 3.3:.

Fix $B$ a ball containing $x$, and decompose the commutator in the following way (see, for instance, [16] or [27])

[TABLE]

If we split $f=f_{1}+f_{2}$ where $f_{2}=f\chi_{2B}$, it is sufficient to estimate, for $0<\gamma<1/m$, the average

[TABLE]

where $x_{B}$ denotes the center of $B$, and

[TABLE]

For simplicity, we will assume $\|b\|_{\Lambda(\delta)}=1$. We shall first estimate $I$. From Lemma 3.1 (i) we have

[TABLE]

where $\theta_{j}=(m-j)\delta$. Note that the last maximal operator is of fractional-type since $0<\theta_{j}<(m-j)(n-\alpha)/m\leq n$ for every $0\leq j\leq m-1$.

We will now estimate $II$. If $y\in B$ and $z\in 2B$, then $|y-z|<3R$ and we have, by Lemma 3.2, that

[TABLE]

From Lemma 3.1 (i), we can estimate $|b(z)-b_{2B}|^{m}$ by $\|b\|_{\Lambda(\delta)}^{m}|2B|^{\frac{m\delta}{n}}=\|b\|_{\Lambda(\delta)}^{m}|2B|^{\frac{\theta_{0}}{n}}$ to obtain

[TABLE]

Since $0<\delta<(n-\alpha)/m$, it is clear that $0<\theta_{0}+\alpha<n$, so $M_{\theta_{0}+\alpha}$ is a fractional-type maximal operator.

In order to estimate $III$, we first observe that, by Jensen’s inequality

[TABLE]

and, setting $B_{j}=2^{j}B$, the integrand can be estimated, using Lemma 3.1 (i), as follows

[TABLE]

Here, we must distinguish the cases $K_{\alpha}\in H_{\alpha,\infty}^{*}$ and $K_{\alpha}\in H_{\alpha,\Phi}$.

If $K_{\alpha}\in H_{\alpha,\infty}^{*}$,

[TABLE]

since $\eta>0$. Therefore

[TABLE]

Let us now consider the case $K_{\alpha}\in H_{\alpha,\Phi}$. Applying Hölder’s inequality with $\Phi$ and $\widetilde{\Phi}$ in (3.3), we obtain

[TABLE]

Therefore,

[TABLE]

Combining all these estimates, we obtain the desired pointwise inequalities. ∎

The following result is a variant of the well-known Fefferman-Stein’s inequality (see [15]) and it will be a key estimate to prove Theorem 2.1.

Lemma 3.5** ([28]).**

Let $0<p<\infty$ and $0<\gamma<1$. Let $w$ be a weight in the $A_{\infty}$ class. Then, there exists a positive constant $C$ such that

[TABLE]

for every measurable function $f$.

We shall also need two results involving the boundedness of fractional maximal operators associated with Young functions, that can be found in [1].

Theorem 3.6** ([1]).**

Let $0<\alpha<n$, $1\leq\beta<p<n/\alpha$ and $1/q=1/p-\alpha/n$. Let $w$ be a weight such that $w^{\beta}\in A_{p/\beta,q/\beta}$. Let $\Psi$ be a Young function that satisfies $\Psi\in\mathcal{B}_{\alpha,\beta}$. Then, $M_{\alpha,\Psi}$ is bounded from $L^{p}(w^{p},\mathbb{R}^{n})$ into $L^{q}(w^{q},\mathbb{R}^{n})$.

Note that if $\Psi=t^{\beta}(1+\log^{+}t)^{\gamma}$ for any $\gamma\geq 0$, then $\Psi\in\mathcal{B}_{\alpha,\beta}$ and the following result holds.

Theorem 3.7** ([1]).**

Let $0<\alpha<n$, $1<p<n/\alpha$ and $1/q=1/p-\alpha/n$. Let $w$ be a weight and $\Psi(t)=t^{\beta}(1+\log^{+}t)^{\gamma}$ where $1\leq\beta<p$ and $\gamma\geq 0$. Then, $M_{\alpha,\Psi}$ is bounded from $L^{p}(w^{p},\mathbb{R}^{n})$ into $L^{q}(w^{q},\mathbb{R}^{n})$ if and only if $w^{\beta}\in A_{p/\beta,q/\beta}$.

In order to prove Theorem 2.6, we shall need the following estimate.

Lemma 3.8**.**

Let $0<\delta<\min\{\eta,(n-\alpha)/(m-1)\}$, for $0<\eta\leq 1$. Let $r=n/((m-1)\delta+\alpha)$, $w\in A_{r,\infty}$, $b\in\Lambda(\delta)$ and $f\in L^{r}_{w}(\mathbb{R}^{n})$. Let $B\subset\mathbb{R}^{n}$ be a ball and $f_{2}=f\chi_{\mathbb{R}^{n}\setminus 2B}$. If $T_{\alpha}$ is a fractional integral operator with kernel $K_{\alpha}\in S_{\alpha}\cap H_{\alpha,\infty}^{*}$, then, for every $x,u\in B$,

[TABLE]

for each $k=0,...,m$.

Proof of Lemma 3.8.

If $K_{\alpha}\in S_{\alpha}\cap H_{\alpha,\infty}^{*}$, by taking $x,u\in B$, and $0\leq k\leq m$, and setting $B_{j}=2^{j}B$, we have from Lemma 3.1 (i) that

[TABLE]

Now by Hölder’s inequality and the fact that $w\in A_{r,\infty}$ with $r=n/(\alpha+(m-1)\delta)$, we get

[TABLE]

where the series is summable since $0\leq k\leq m$ and $\delta<\eta$. ∎

Lemma 3.9**.**

Let $m\in\mathbb{N}$, $0<\alpha<n$, $0<\delta<\min\{1,(n-\alpha)/(m-1)\}$, $r=n/((m-1)\delta+\alpha)$, $b\in\Lambda(\delta)$ and $f\in L^{r}_{w}(\mathbb{R}^{n})$ where $w$ is a weight such that $w^{\beta}\in A_{r/\beta,\infty}$ for some $1<\beta<r$. Let $B\subset\mathbb{R}^{n}$ be a ball and $f_{2}=f\chi_{\mathbb{R}^{n}\setminus 2B}$. If $T_{\alpha}$ is a fractional integral operator with kernel $K_{\alpha}\in H_{\alpha,\Phi,m}(\delta)$, where $\Phi$ is a Young function verifying $\Phi^{-1}(t)\lesssim t^{\frac{\beta-1}{r}}$ for every $t>0$, then, for every $x,u\in B$,

[TABLE]

for each $k=0,...,m$.

Proof.

Fix $x,u\in B$ and $0\leq k\leq m$. Setting $B_{u}=B(u,R)$ which satisfies $B\subset 2B_{u}\subset 4B$, and using Lemma 3.1 (i) we have

[TABLE]

Since $1/r+1/(r/\beta)^{\prime}=1-(\beta-1)/r$, we can use Hölder’s inequality with $\Phi^{-1}(t)t^{1/r}t^{1/(r/\beta)^{\prime}}\lesssim t$ and the fact that $w^{\beta}\in A_{r/\beta,\infty}$, to get

[TABLE]

where we have used that $\delta k+\alpha-n/r\leq m\delta$ for $m\in\mathbb{N}$, and that $K_{\alpha}\in H_{\alpha,\Phi,m}(\delta)$. ∎

4 Proofs of main results

Proof of Theorem 2.1:.

The proof will be done by induction and, without loss of generality, we shall assume $\|b\|_{\Lambda(\delta)}=1$. Notice that when $m=0$, $1/q=1/p-\alpha/n$ and the boundedness result is known to be true for $A_{p,q}$ weights (see [1] in the more general setting of variable Lebesgue spaces).

Fix $m\in\mathbb{N}$ and define the following auxiliary exponents

[TABLE]

Clearly, $p_{m}=q$ and, if $\theta_{j}=(m-j)\delta$, we have that

[TABLE]

Notice also that $p\leq p_{j}\leq p_{l}\leq q$ for every $0\leq j\leq l\leq m$.

It is easy to see that $w\in A_{p,q}$ yields $w^{\gamma}\in A_{\frac{p}{\gamma},\frac{q}{\gamma}}$ for every $0<\gamma<1$. Moreover, from the properties of these classes, we have that $w^{\gamma}\in A_{\frac{p_{j}}{\gamma},\frac{p_{l}}{\gamma}}$ for every $0\leq j\leq l\leq m$.

By applying Fefferman-Stein’s inequality (3.4) with $w^{q}\in A_{1+q/p^{\prime}}\subset A_{\infty}$, we get

[TABLE]

Now, since $K_{\alpha}\in S_{\alpha}\cap H_{\alpha,\infty}^{*}$, from Lemma 3.3 we have that

[TABLE]

Since $w\in A_{p,q}$ and $1/q=1/p-(\theta_{0}+\alpha)/n$, we have that

[TABLE]

On the other hand, since $w^{\gamma}\in A_{\frac{p_{j}}{\gamma},\frac{q}{\gamma}}$ for every $j=1,\dots,m-1$, then the fractional maximal operator $M_{\theta_{j}\gamma}$ is bounded from $L^{\frac{p_{j}}{\gamma}}(\mathbb{R}^{n})$ to $L^{\frac{q}{\gamma}}(\mathbb{R}^{n})$. Thus, we have that

[TABLE]

Since $1/p_{j}=1/p-(j\delta+\alpha)/n$ and $w\in A_{p,p_{j}}$, and recalling that $\|b\|_{\Lambda(\delta)}=1$, we apply the inductive hypothesis to get

[TABLE]

Proof of Theorem 2.3:.

Fix $f\in L^{r}_{w}(\mathbb{R}^{n})$. For a ball $B\subset\mathbb{R}^{n}$, set $f_{1}=f\chi_{2B}$, $f_{2}=f-f_{1}$ and $a_{B}=\frac{1}{|B|}\int_{B}T_{\alpha,b}^{m}f_{2}$. Then,

[TABLE]

Let us first notice that, since $w\in A_{r,\infty}$, there exists $1<s^{\prime}<r$ such that $w\in A_{s^{\prime},\infty}$ and, we can choose $1<q<\infty$ such that $\frac{1}{q}+\frac{1}{r}+\frac{1}{s}=1$.

For $I_{1}$ we write

[TABLE]

By using Tonelli’s theorem and the fact that $b\in\Lambda(\delta)$, we obtain

[TABLE]

We notice that for $x,y\in 2B$, if $R$ is the radius of $B$, then $x\in B(y,4R)$ so we can use Lemma 3.2 and Hölder’s inequality to have

[TABLE]

For $I_{2}$, we first estimate the difference $\left|T^{m}_{\alpha,b}f_{2}(x)-(T^{m}_{\alpha,b}f_{2})_{B}\right|$ for every $x\in B$. Since

[TABLE]

we analyze $A=\left|T^{m}_{\alpha,b}f_{2}(x)-T^{m}_{\alpha,b}f_{2}(y)\right|$. If $x,y\in B$

[TABLE]

By the definition of $\Lambda(\delta)$, we get that

[TABLE]

Then, by Hölder’s inequality, the definition of $\tilde{\delta}$ and the fact that $w\in A_{r,\infty}$, we deduce that

[TABLE]

In order to estimate $I_{4}$, we use that $b\in\Lambda(\delta)$ and the smoothness condition $S_{\alpha}$ on the kernel to get that

[TABLE]

Then, by Tonelli’s Theorem and the smoothness condition $S_{\alpha}$

[TABLE]

since $\tilde{\delta}<\delta$. Therefore, we deduce the inequality

[TABLE]

and, thus,

[TABLE]

so it remains to take supremum over all the balls $B$ to get the desired result. ∎

Proof of Theorem 2.6 .

Let $B\subset\mathbb{R}^{n}$ be a ball and $x\in B$. Let $f=f_{1}+f_{2}$ with $f_{1}=f\chi_{2B}$. Then,

[TABLE]

We can rewrite the above identity in the following form

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

For $\sigma_{1}$, since $w\in A_{\frac{n}{m\delta+\alpha},\infty}$, there exists $1<p<\frac{n}{m\delta+\alpha}$ such that $w\in A_{p,\infty}$. We take $\frac{1}{q}=\frac{1}{p}-\frac{m\delta+\alpha}{n}$, so $q>p$ and $w\in A_{q,\infty}$ and, moreover, $w\in A_{p,q}$. By applying Hölder’s inequality with $q$ and $q^{\prime}$ and the boundedness of $T^{m}_{\alpha,b}$ from $L^{p}_{w}(\mathbb{R}^{n})$ to $L^{q}_{w}(\mathbb{R}^{n})$ (Theorem 2.1) we obtain that

[TABLE]

Since $\frac{1}{p}=\frac{\alpha+(m-1)\delta}{n}+\frac{1}{q}+\frac{\delta}{n}$, we can apply again Hölder’s inequality and the fact that $w\in A_{q,\infty}$ to get

[TABLE]

In order to estimate $\sigma_{2}$ we use the inequality

[TABLE]

and Lemma 3.8 to obtain

[TABLE]

Consequently, since

[TABLE]

by first assuming that $T^{m}_{\alpha,b}f:L^{\frac{n}{\alpha+(m-1)\delta}}_{w}\hookrightarrow\mathbb{L}_{w}(\delta)$, then

[TABLE]

On the other hand, if we suppose that (2.3) holds, it is easy to see that $T^{m}_{\alpha,b}f:L^{\frac{n}{\alpha+(m-1)\delta}}_{w}(\mathbb{R}^{n})\hookrightarrow\mathbb{L}_{w}(\delta)$. ∎

Proof of Theorem 2.11:.

We proceed by induction. We must point out that the case $m=0$ was already proved in [1]. As in the proof of Theorem 2.1 we have that

[TABLE]

We shall now use the second part of Lemma 3.3, since we have that $K_{\alpha}\in S_{\alpha}\cap H_{\alpha,\Phi}$. Thus, we obtain that

[TABLE]

where we have assumed, without loss of generality, that $\|b\|_{\Lambda(\delta)}=1$.

From the hypothesis on the weight $w$ and the Young function $\widetilde{\Phi}$, by Theorem 3.6 we know that $\|wM_{m\delta+\alpha,\widetilde{\Phi}}f\|_{q}\lesssim\|fw\|_{p}$.

The proof now follows in the same way as in the proof of Theorem 2.1. ∎

Proof of Theorem 2.13:.

Take $f,f_{1},f_{2}$ and $a_{B}$ as in the proof of Theorem 2.3, and define $I_{1}$ and $I_{2}$ likewise.

Since in $I_{1}$ we have only used the size condition $S_{\alpha}$, the estimation is the same, by taking into account that $w^{\beta}\in A_{r/\beta,\infty}$ yields $w\in A_{r,\infty}$ for any $\beta\geq 1$.

For $I_{2}$ we proceed similarly but we have to use now that $K_{\alpha}\in H_{\alpha,\Phi,m}(\delta)$ with $\Phi^{-1}(t)\lesssim t^{\frac{\beta-1}{r}}$ for some $1<\beta<r$ and all $t>0$. We split the average into $I_{3}$ and $I_{4}$ as in the proof of Theorem 2.3. The last one can be controlled in the same form. The difference will be in $I_{3}$. Recall that

[TABLE]

for $x\in B$.

By the definition of $\Lambda(\delta)$, we get that

[TABLE]

Now, since $K_{\alpha}\in H_{\alpha,\Phi,m}(\delta)$, $w^{\beta}\in A_{r/\beta,\infty}$ and $\Phi^{-1}(t)t^{1/r}t^{1/(r/\beta)^{\prime}}\lesssim t$, we can prodeed as in (3.6) with $k=m$ to obtain

[TABLE]

Proof of Theorem 2.14:.

We proceed as in the proof of Theorem 2.6. We must only use the corresponding hypothesis on the kernel, that guarantees the validity of Theorem 2.11 and Lemma 2.14, which are immediate from the fact that $S_{\alpha}\cap H_{\alpha,\Phi,m}(\delta)\subset S_{\alpha}\cap H_{\alpha,\Phi}$ (see Remark 2.10). ∎