Boundedness of some singular integrals operators in weighted generalized Grand Lebesgue spaces

Abstract

If $I\subset {\mathbb {R}}$ is a bounded interval, we prove the boundedness of Calderón singular operator and of Hardy-Littlewood Maximal operator in the generalized weighted Grand Lebesgue spaces $L_p^{p),{\delta }}(I)$, $1<p<\infty$.

Introduction

Let $\Omega \subset {\mathbb {R}}^n$, $n\ge 2$, be a measurable, bounded domain, and $f : \Omega \rightarrow {\mathbb {R}}^n , f=(f^1,\ldots ,f^n)$ be a mapping defined on $\Omega$.

In 1992 Iwaniec and Sbordone introduced the Grand Lebesgue space $L^{n)}(\Omega )$, [16], they studied the integrability of the Jacobian determinant, by proving that, under the assumption $|Df| \in L^{n)}(\Omega )$, the Jacobian of f is locally integrable. Such assumption, weaker than $|Df| \in L^{n}(\Omega )$, is shown to be optimal.

Since than, a very rich literature arose on the generalization of these spaces to the more general context of the grand Orlicz-Sobolev space , hence the Grand Lebesgue spaces played an important role in PDEs theory and in Function Spaces theory (see e.g. [4, 8, 10,11,12, 14, 15, 22, 23] and references therein).

Later, in [9], Fiorenza introduced the Small Lebesgue spaces $L^{(p}(\Omega )$, characterizing them as the associate space of the Grand Lebesgue spaces $L^{p)}(\Omega )$, in [1] has been proved that the spaces $L^{p)'}(\Omega )$ and $L^{(p'}(\Omega )$ coincide, and a generalization of the Grand and the Small Lebesgue spaces, denoted, respectively, by $L^{p), \Theta }(\Omega )$ and $L^{(p, \Theta }(\Omega )$, $\Theta >0$, was introduced.

In [3], a new generalization of such spaces is given. More precisely

Definition 1.1

Let $1<p<\infty$. A function $\delta$, left continuous on $(0,p-1)$, is said to be in the class ${\mathcal {B}}_p$ if

$$\begin{aligned}&(j)&\delta (0+)=0\\&(jj)&0<\delta \le 1\\&(jjj)&\delta (\varepsilon )^\frac{1}{p-\varepsilon }\, \mathrm{is} \, \mathrm{increasing}\, \mathrm{in} \, \varepsilon \end{aligned}$$

Definition 1.2

Let $\Omega \subset {\mathbb {R}}^n$, $n\ge 1$, be a measurable set of Lebesgue measure $|\Omega |<+\infty$, let $1<p<\infty$ and let $\delta \in {\mathcal {B}}_p$. The Grand Lebesgue space $L^{p),\delta }(\Omega )$ with respect to $\delta$ is the Banach Function Space defined on the set of all measurable functions $f\in \mathcal{M}_o$, such that

$$\begin{aligned} \Vert f\Vert _{p),\delta }==\rho _{p),\delta }(|f|)= \!\!\!\!\sup _{0<\varepsilon<p-1} \delta (\varepsilon )^\frac{1}{p-\varepsilon }\left( \frac{1}{|\Omega |}\int _{\Omega }|f(y)|^{p-\epsilon }dy\right) ^{\frac{1}{p-\epsilon }}<+ \infty \, . \end{aligned}$$

The goal of this paper is to prove the boundedness of some singular integral operators in the generalized weighted Grand Lebesgue spaces $L_{w}^{p),\delta }(I)$, $1<p<\infty$, equipped with the norm

$$\begin{aligned} \Vert f\Vert _{L_w^{p),\delta }} = \!\!\!\!\sup _{0<\varepsilon <p-1} \delta (\varepsilon )^\frac{1}{p-\varepsilon }\left( \frac{1}{|I|}\int _{I}|f(y)|^{p-\epsilon }w(y)dy\right) ^{\frac{1}{p-\epsilon }} \end{aligned}$$

(1)

The Calder$\acute{\mathrm{o}}$n-Zygmund theory of singular integral operators plays a basic and extremely important role in harmonic analysis. In many applications, a crucial step has been to show that one of the classical operators of harmonic analysis, e.g. maximal operators, singular integrals, fractional integrals, is bounded on the standard Lebesgue spaces. A lot of inequalities involving such kind of classical singular operators of harmonic analysis have been extensively investigated, by obtaining a very intensive literature regarding their boundedness in various function spaces (see e.g. [5,6,7, 18, 19, 21]), with weights basically in the $A_p$ class of Muckenoupt, see Sect. 2.

In particular, in what follows, we will refer to Calderón singular operator and to Hardy-Littlewood Maximal operator, defined, respectively as

$$\begin{aligned} C_af(x)=\int _I \frac{a(x)-a(t)}{(x-t)^2}f(t)dt, \qquad x \in I \end{aligned}$$

and

$$\begin{aligned} Mf(x)= \sup _{J\subset I}\frac{1}{|J|}\int _J |f(y)|dy, \qquad x\in I \end{aligned}$$

where I is a bounded interval in ${\mathbb {R}}$, ${\overline{I}}$ is its closure, $a: {\overline{I}} \rightarrow {\mathbb {R}}$ is a Lip1 function on ${\overline{I}}$ and $J \subset I$ is a subinterval.

We will prove the following results

Theorem 1.1

Let I be a finite interval, $1<p<\infty$ and $\delta \in \mathcal{B}_p$. Then

(i) If $w \in A_p(I)$ then $\mathcal{C}_a$ is bounded in $L_{w}^{p),\delta }(\Omega )$.

(ii) If $\mathcal{C}_a$ is bounded in $L_{w}^{p),\delta }(\Omega )$ and there exists a constant k such that

$0< k \le |a'(x)|$ for all $x\in I$, then $w \in A_p(I)$.

Theorem 1.2

Let I be a bounded interval, $1<p<\infty$ and $\delta \in \mathcal{B}_p$. Then the Hardy-Littlewood maximal operator is bounded in $L_{w}^{p),\delta }(\Omega )$ iff $w \in A_p$.

Preliminaries

Let us recall some definitions and results which will be useful in the sequel.

Lemma 2.1

Let $1<p<\infty$ and $0<\sigma <p-1$. If $\delta \in {\mathcal {B}}_p$, there exists a constant $c=c(p,\delta , \sigma )$ such that

$$\begin{aligned}&\sup _{0<\varepsilon<\sigma } \delta (\varepsilon )^\frac{1}{p-\varepsilon }\left( \frac{1}{|\Omega |} \int _{\Omega } f^{p-\varepsilon }dx\right) ^\frac{1}{p-\varepsilon }\le \rho _{p),\delta }(f)\le c\,\\&\quad \sup _{0<\varepsilon <\sigma } \delta (\varepsilon )^\frac{1}{p-\varepsilon }\left( \frac{1}{|\Omega |} \int _{\Omega } f^{p-\varepsilon }dx\right) ^\frac{1}{p-\varepsilon } \, . \end{aligned}$$

Definition 2.1

Let $1<p<\infty$, a function w(x) satisfies the Muckenhoupt condition $A_p$, on an interval I, if w(x) is nonnegative and, for all subinterval J the following condition

$$\begin{aligned} A_p(w,I):= \sup _J\left( \frac{1}{|J|}\int _Jw(x)dx\right) \left( \frac{1}{|J|}\int _Jw(x)^{-\frac{1}{p-1}}dx\right) ^{p-1}<\infty \end{aligned}$$

(2)

holds. The function w(x) is said to be a weight of the Muckenhoupt class, $w(x) \in A_p(I)$.

In 1972, B. Muckenhoupt [20] characterized the weights $w:{\mathbb {R}}^n \rightarrow [0,\infty )$ such that the Hardy-Littlewood Maximal operator, M, is bounded on $L^p({\mathbb {R}}^n, wdx)$, $(p>1)$ as those verifying the $A_p$ condition (2), where the supremum is taken over all cubes in ${\mathbb {R}}^n$ containing x. A generalization to weighted Orlicz spaces was provided by Kerman–Torchinsky [17] and a different proof was given in [2] with a precise dependence of the constants in the resulting inequalities.

Let us conclude the Section by the following results:

Lemma 2.2

[18] Let $1<p<\infty$ and $w \in A_p(I)$. Then there exist positive constants $\sigma$ and L such that $w \in A_{p-\sigma }(I)$ and

$$\begin{aligned} ||\mathcal{C}_a||_{L_w^{p-\epsilon }\rightarrow L_w^{p-\epsilon }}\le L \end{aligned}$$

(3)

for all $0<\epsilon <\sigma$.

Lemma 2.3

Let $1<p<\infty$, I a bounded interval and $\delta \in \mathcal{B}_p$ . Then there exists a positive constant c such that

$$\begin{aligned} ||f\chi _J||_{L_{w}^{p),\delta }(I)}\le c w(J)^{-1/p}||f\chi _J||_{L_{w}^{p}(I)}||\chi _J||_{L_{w}^{p), \delta }(I)} \end{aligned}$$

(4)

for all $f \in L_w^{p}(I)$ and for all subinterval $J\subset I$.

Proof

Let $f\ge 0$, then, by Hölder inequality

$$\begin{aligned} ||f\chi _J||_{L_{w}^{p),\delta }(I)}= & {} \sup _{0<\epsilon<p-1}\left( \delta (\epsilon )\frac{1}{|J|}\int _{J}|f(y)|^{p-\epsilon }w(y)dy\right) ^{\frac{1}{p-\epsilon }} \\= & {} \sup _{0<\epsilon<p-1}\left( \delta (\epsilon )\frac{1}{|J|}\int _{J}|f(y)|^{p-\epsilon }w(y)^{\frac{p-\epsilon }{p}}w(y)^{\frac{\epsilon }{p}}dy\right) ^{\frac{1}{p-\epsilon }} \\\le & {} \sup _{0<\epsilon<p-1}\delta (\epsilon )^{\frac{1}{p-\epsilon }}\left( \frac{1}{|J|}\int _{J}|f(y)|^{p}w(y)dy\right) ^{\frac{1}{p}}\cdot \\&\quad \left( \frac{1}{|J|}\int _Jw(y)dy\right) ^{\frac{\epsilon }{p(p-\epsilon )}} \\= & {} \sup _{0<\epsilon<p-1}\delta (\epsilon )^{\frac{1}{p-\epsilon }} \left( \int _{J}|f(y)|^{p}w(y)dy\right) ^{\frac{1}{p}}\cdot \\&\quad \left( w(J)\right) ^ {-\frac{1}{p}}\left( \frac{1}{|J|}\right) ^{\frac{1}{p-\epsilon }} \left( w(J)\right) ^{\frac{1}{p-\epsilon }} \\= & {} \left( w(J)\right) ^{-\frac{1}{p}}||f\chi _J||_{L_{w}^{p}(I)}\sup _{0<\epsilon <p-1}\delta (\epsilon )^{\frac{1}{p-\epsilon }}\left( \int _J\chi _J(y)w(y)dy\right) ^{\frac{1}{p-\epsilon }} \\= & {} \left( w(J)\right) ^{-\frac{1}{p}}||f\chi _J||_{L_{w}^{p}(I)}||\chi _J||_{L_{w}^{p), \delta }(I)} \end{aligned}$$

$\square$

As a consequence of lemma 2.3. we have the following result, for its proof we refer to [18, lemma 2.3], which can be, easily, applied to our generalized spaces.

Lemma 2.4

Let $1<p<\infty$, I a bounded interval and $\delta \in \mathcal{B}_p$ . If $\mathcal{C}_a$ is bounded in $L_{w}^{p),\delta }(I)$ and there exists a constant m such that $0<m\le |a'(t)|$. Then

$$\begin{aligned} w^{-\frac{1}{p-1}} < \infty \end{aligned}$$

(5)

Proof of Theorem 1.1

Let us assume $I:=[0,1]$. Since $w \in A_p$, then there exists $\epsilon >0$ such that $w \in A_{p-\epsilon }(I)$. If $f\in L_{w}^{p),\delta }(I)$, by definition it follows that $f\in L_{w}^{p-\epsilon }(I)$, which garantees the existence of $\mathcal{C}_a f(x)$ a.e. in [0, 1].

By lemma 2.2, there exists a constant $\sigma \in (0,1)$ such that

$$\begin{aligned} ||\mathcal{C}_af||_{L_{w}^{p-\epsilon }(I)}\le c ||f||_{L_{w}^{p-\epsilon }(I)} \end{aligned}$$

for all $\epsilon \in (0,\sigma ]$ and with c a constant independent on f and on $\epsilon$.

Let us fix $\epsilon \in (0,p-1)$, then, by Hölder inequality we get

$$\begin{aligned}&||\mathcal{C}_af||_{L_{w}^{p-\epsilon }(I)}=\left( \int _I |\mathcal{C}_af|^{p-\epsilon }w(x)dx\right) ^{\frac{1}{p-\epsilon }}\le \left( \int _I |\mathcal{C}_af|^{p-\sigma }w(x)dx\right) ^{\frac{1}{p-\sigma }}\cdot \\&\quad \left( w(I)\right) ^{\frac{\epsilon -\sigma }{(p-\sigma )(p-\epsilon )}} \end{aligned}$$

By using this estimate we deduce

$$\begin{aligned} ||\mathcal{C}_af||_{L_{w}^{p),\delta }(I)}= & {} \max \left\{ \sup _{0<\epsilon<\sigma }\delta (\epsilon )^{\frac{1}{p-\epsilon }}||\mathcal{C}_af||_{L_{w}^{p-\epsilon }(I)}\,, \, \sup _{\sigma \le \epsilon<p-1}\delta (\epsilon )^{\frac{1}{p-\epsilon }}||\mathcal{C}_af||_{L_{w}^{p-\epsilon }(I)} \right\} \\\le & {} \max \biggm \{\sup _{0<\epsilon<\sigma }\delta (\epsilon )^{\frac{1}{p-\epsilon }}||\mathcal{C}_af||_{L_{w}^{p-\epsilon }(I)}\, ,\\&\sup _{\sigma \le \epsilon<p-1}\delta (\epsilon )^{\frac{1}{p-\epsilon }}||\mathcal{C}_af||_{L_{w}^{p-\sigma }(I)} \cdot (w(I))^{\frac{\epsilon -\sigma }{(p-\sigma )(p-\epsilon )}} \biggm \}\\= & {} \max \biggm \{\sup _{0<\epsilon<\sigma }\delta (\epsilon )^{\frac{1}{p-\epsilon }}||\mathcal{C}_af||_{L_{w}^{p-\epsilon }(I)}\, ,\\&\delta (\sigma )^{\frac{1}{p-\sigma }}||\mathcal{C}_af||_{L_{w}^{p-\sigma }(I)}\sup _{\sigma \le \epsilon<p-1}\delta (\epsilon )^{\frac{1}{p-\epsilon }}\delta (\sigma )^{-\frac{1}{p-\sigma }}\cdot (w(I))^{\frac{\epsilon -\sigma }{(p-\sigma )(p-\epsilon )}} \biggm \}\\ \quad\le & {} \max \biggm \{\sup _{0<\epsilon<\sigma }\delta (\epsilon )^{\frac{1}{p-\epsilon }}||\mathcal{C}_af||_{L_{w}^{p-\epsilon }(I)}\,, \, \left( \sup _{0<\epsilon<\sigma }\delta (\epsilon )^{\frac{1}{p-\epsilon }}||\mathcal{C}_af||_{L_{w}^{p-\epsilon }(I)}\right) \\&\left( \sup _{\sigma \le \epsilon <p-1}\delta (\epsilon )^{\frac{1}{p-\epsilon }}\delta (\sigma )^{-\frac{1}{p-\sigma }}\cdot (w(I))^{\frac{\epsilon -\sigma }{(p-\sigma )(p-\epsilon )}}\right) \biggm \} \end{aligned}$$

Let us set

$$\begin{aligned} S=\sup _{0<\epsilon <\sigma }\delta (\epsilon )^{\frac{1}{p-\epsilon }}||\mathcal{C}_af||_{L_{w}^{p-\epsilon }(I)} \end{aligned}$$

and

$$\begin{aligned} T=\sup _{\sigma \le \epsilon <p-1}\delta (\epsilon )^{\frac{1}{p-\epsilon }}\delta (\sigma )^{-\frac{1}{p-\sigma }}(w(I))^{\frac{\epsilon - \sigma }{(p-\sigma )(p-\epsilon )}} \end{aligned}$$

previous inequality becomes

$$\begin{aligned} ||\mathcal{C}_af||_{L_{w}^{p)\delta }(I)}=\max \left\{ 1\cdot S, T\cdot S\right\} \le \max \left\{ 1, T\right\} S \end{aligned}$$

Hence

$$\begin{aligned} ||\mathcal{C}_af||_{L_{w}^{p),\delta }(I)}\le & {} \max \biggm \{1, \sup _{\sigma \le \epsilon<p-1}\delta (\epsilon )^{\frac{1}{p-\epsilon }}\delta (\sigma )^{-\frac{1}{p-\sigma }}(w(I))^{\frac{\epsilon -\sigma }{(p-\sigma )(p-\epsilon )}}\biggm \}\cdot S \\\le & {} \max \biggm \{1, \sup _{\sigma \le \epsilon <p-1}\delta (\epsilon )^{\frac{1}{p-\epsilon }}\delta (\sigma )^{-\frac{1}{p-\sigma }}(1+w(I))^{\frac{\epsilon -\sigma }{(p-\sigma )(p-\epsilon )}}\biggm \}\cdot S \end{aligned}$$

Let us set

$$\begin{aligned} h(\epsilon )=\delta (\epsilon )^{\frac{1}{p-\epsilon }} \end{aligned}$$

and

$$\begin{aligned} g(\epsilon )=(1+w(I))^{\frac{\epsilon -\sigma }{(p-\sigma )(p-\epsilon )}}\, , \end{aligned}$$

both of them are increasing functions in $(\sigma , p-1)$.

Hence, by using the boundedness of $C_a$ in $L_w^{p-\epsilon }$ we get

$$\begin{aligned} ||\mathcal{C}_af||_{L_{w}^{p),\delta }(I)}\le & {} \max \biggm \{1, \delta (\sigma )^{-\frac{1}{p-\sigma }}\delta (p-1)(1+w(I))^ {\frac{p-\sigma -1}{p-\sigma }}\biggm \}\cdot \\&\quad \sup _{0<\epsilon <\sigma }\delta (\epsilon )^{\frac{1}{p-\epsilon }}||f||_{L_{w}^{p-\epsilon }(I)}\\\le & {} c\left[ \delta (p-1)\delta (\sigma )^{-\frac{1}{p-\sigma }}(1+w(I))^{\frac{p-\sigma -1}{p-\sigma }}\right] ||f||_{L_{w}^{p)\delta }(I)} \end{aligned}$$

Let us prove the necessity.

Since w is integrable on I we have $||1||_{L_{w}^{p),\delta }(I)}<+\infty$. If $J:=(a,b) \subset I$ is such that $b-a \le 1/4$, let us set

$$\begin{aligned} J':= {\left\{ \begin{array}{ll} (b,2b-a) &{} \text{ if } \qquad b<2b-a \\ (2a-b,a)&{}\text{ otherwise } \end{array}\right. } \end{aligned}$$

We want to prove that

$$\begin{aligned} ||\chi _J||_{L_{w}^{p),\delta }(I)}\le c ||\chi _J'||_{L_{w}^{p),\delta }(I)} \end{aligned}$$

(6)

Without loss of generality, we can assume $J'=(b,2b-a)$. Hence $0 \le t-x \le 2|J|$, for all $t\in J'$ and $x\in J$.

Let $f=\chi _{J'}$ then

$$\begin{aligned} |\mathcal{C}_af(x)| =\left| \int _{J'}\frac{a(x)-a(t)}{(x-t)^2}dt\right| =\left| \frac{a(x)-a(t)}{x-\xi }\right| \int _{J'}\frac{dt}{t-x} \ge |a'(\xi )|\int _{J'}\frac{dt}{t-x} \ge \frac{1}{2}m \end{aligned}$$

for arbitrary $x \in J$ and some $\xi \in (b,2b-a)$.

Thus

$$\begin{aligned} ||\mathcal{C}_af||_{L_{w}^{p),\delta }(I)} \ge ||\mathcal{C}_af||_{L_{w}^{p),\delta }(J)}\ge m \left| \left| \int _{J'}\frac{dt}{t-x}\right| \right| \ge \frac{1}{2}m ||{\chi _J}||_{L_{w}^{p),\delta }(I)} \end{aligned}$$

Hence, by using the boundedness of $\mathcal{C}_a$, previous inequality implies

$$\begin{aligned} ||{\chi _{J'}}||_{L_{w}^{p),\delta }(I)}= ||f||_{L_{w}^{p),\delta }(I)}\ge c ||\mathcal{C}_af||_{L_{w}^{p),\delta }(I)}\ge \frac{1}{2}m ||{\chi _J}||_{L_{w}^{p),\delta }(I)} \end{aligned}$$

hence

$$\begin{aligned} ||{\chi _J}||_{L_{w}^{p),\delta }(I)} \le c ||{\chi _{J'}}||_{L_{w}^{p),\delta }(I)} \end{aligned}$$

i.e. (6).

Let us consider the test function $\displaystyle {f=w^{1-p'}\chi _J}$, if $x \in J'$ we have

$$\begin{aligned} |\mathcal{C}_af(x)|= \left| \int _{J'}\frac{a(x)-a(t)}{(x-t)^2}w^{1-p^{'}}(t)dt\right| \ge \frac{m}{2|J|}\int _{J}w^{1-p^{'}}(t)dt \end{aligned}$$

hence

$$\begin{aligned} ||\mathcal{C}_af||_{L_{w}^{p),\delta }(I)} \ge c \left( \frac{1}{|J|}\int _Jw^{1-p^{'}}(t)dt\right) \cdot ||{\chi _{J'}}||_{L_{w}^{p),\delta }(0,1)} \end{aligned}$$

so that

$$\begin{aligned} \frac{1}{|J|}w^{1-p^{'}}(J)||{\chi _{J'}}||_{L_{w}^{p),\delta }(0,1)}\le & {} c ||\mathcal{C}_af||_{L_{w}^{p),\delta }(I)}\le c ||f||_{L_{w}^{p),\delta }(I)}= c||f\chi _J||_{L_{w}^{p),\delta }(J)} \\\le & {} c w(J)^{-\frac{1}{p}}||f\chi _J||_{L_{w}^{p}(I)}||\chi _J||_{L_{w}^{p),\delta }(I)} \\\le & {} c w(J)^{-\frac{1}{p}}||f\chi _J||_{L_{w}^{p}(I)}||\chi _{J'}||_{L_{w}^{p),\delta }(I)} \\= & {} c w(J)^{-\frac{1}{p}}\left( \int _Jw^{1-p^{'}}(t)dt\right) ^{\frac{1}{p}}||\chi _{J'}||_{L_{w}^{p),\delta }(I)} \end{aligned}$$

Thus

$$\begin{aligned} \frac{1}{|J|}w^{1-p^{'}}(J) \le c w(J)^{-\frac{1}{p}}(w^{1-p^{'}}(J))^{\frac{1}{p}} \end{aligned}$$

that is $w \in A_p$.

Proof of Theorem 1.2

Let us consider a sub interval $J\subset I$. By definition of $\mathcal{M}$ we get

$$\begin{aligned} \frac{1}{|J|}\int _J|f(t)|dt \le M(f\chi _J)(x), \quad x\in J \end{aligned}$$

The boundedness of $\mathcal{M}$ implies

$$\begin{aligned} \left| \left| \left( \frac{1}{|J|}\int _J|f(y)|dy\right) \chi _J\right| \right| _{L_{w}^{p),\delta }(I)}= & {} \sup _{0<\epsilon<p-1}\left( \delta (\epsilon ) \left( \frac{1}{|I|}\int _I\left( \frac{1}{|J|}\int _J|f(y)|dy\right) \right. \right. \\&\quad \left. \left. \chi _J(x)\right) ^{p-\epsilon }w(x)dx\right) ^{\frac{1}{p-\epsilon }} \\= & {} \sup _{0<\epsilon<p-1}\left( \delta (\epsilon )\left( \frac{1}{|J|}\int _J|f(y)|dy\right) ^{p-\epsilon }\right. \\&\quad \left. \frac{1}{|I|}\int _I\chi _J(x)w(x)dx\right) ^{\frac{1}{p-\epsilon }} \\= & {} \left( \frac{1}{|J|}\int _J|f(y)|dy\right) \sup _{0<\epsilon <p-1}\\&\quad \left( \delta (\epsilon )\frac{1}{|I|}\int _J\chi _J(x)w(x)dx\right) ^{\frac{1}{p-\epsilon }} \\= & {} \left( \frac{1}{|J|}\int _J|f(y)dy \right) ||\chi _J||_{L_{w}^{p),\delta }(I)} \end{aligned}$$

hence, by Hölder’s inequality, with exponents $\displaystyle {\frac{p}{p-\epsilon }}$ and $\displaystyle {\frac{p}{\epsilon }}$, we get

$$\begin{aligned}&\left( \frac{1}{|J|}\int _J|f(y)|dy \right) ||\chi _J||_{L_{w}^{p),\delta }(I)} \le ||\mathcal{M}(f \cdot \chi _J)||_{L_{w}^{p),\delta }(I)} \le c||f \cdot \chi _J||_{L_{w}^{p),\delta }(I)} \\&\quad =c \sup _{0<\epsilon<p-1}\left( \delta (\epsilon )\frac{1}{|I|}\int _J|f(y)|^{p-\epsilon }w(y)dy\right) ^{\frac{1}{p-\epsilon }} \\&\quad \le c \sup _{0<\epsilon<p-1}\delta (\epsilon )^{\frac{1}{p-\epsilon }}\left( \frac{1}{|I|}\int _J|f(y)|^p w(y)dy\right) ^{\frac{1}{p}}\cdot \left( \frac{1}{|I|}\int _Jw(y)dy\right) ^{\frac{\epsilon }{p(p-\epsilon )}} \\&\quad =c w(J)^{-\frac{1}{p}}\left( \int _J|f(y)|^pw(y)dy\right) ^{\frac{1}{p}}\sup _{0<\epsilon <p-1}\left( \delta (\epsilon )\frac{1}{|I|}w(J)\right) ^{\frac{1}{p-\epsilon }} \\&\quad = c w(J)^{-\frac{1}{p}}\left( \int _J|f(y)|^pw(y)dy\right) ^{\frac{1}{p}}||\chi _J||_{L_{w}^{p),\delta }(I)} \end{aligned}$$

Consequently

$$\begin{aligned} \frac{1}{|J|}\int _J|f(y)|dy \le c w(J)^{-\frac{1}{p}}\left( \int _J|f(y)|^pw(y)dy\right) ^{\frac{1}{p}} \end{aligned}$$

which, for $f=w^{-\frac{1}{p-1}}\chi _J$,   implies $w\in A_p(I)$.

On the other hand, let $w \in A_p(I)$, then, it is well known [20] that the Hardy-Littlewood Maximal Operator is bounded on $L^p({\mathbb {R}}^n, wdx), (p>1)$, where the supremum is taken over all cubes in ${\mathbb {R}}^n$ containing x. Since the class $A_p(I)$ is open, then there exists a small constant $\sigma$ such that $w \in L^{p-\sigma }({\mathbb {R}}^n, wdx)$, then, there exist positive constants $M_1\, M_2$, such that

$$\begin{aligned} ||\mathcal{M}f||_{L_{w}^{p-\sigma }(I)}\le M_1||f||_{L_{w}^{p-\sigma }(I)}, \qquad ||\mathcal{M}f||_{L_{w}^{p}(I)}\le M_1||f||_{L_{w}^{p}(I)} . \end{aligned}$$

For arbitrary $0<\epsilon <\sigma$, there exists $0<t_{\epsilon }<1$, such that

$$\begin{aligned} \frac{1}{p-\epsilon }=\frac{1-t_{\epsilon }}{p-\sigma }+\frac{t_{\epsilon }}{p} \end{aligned}$$

By Riesz-Thorin interpolation theorem, we get

$$\begin{aligned} ||\mathcal{M}f||_{L_{w}^{p-\epsilon }(I)}\le M_1^{1-t_{\epsilon }} M_2^{t_{\epsilon }}||f||_{L_{w}^{p-\epsilon }(I)} \end{aligned}$$

$\forall f \in L_{w}^{p}(I)\bigcap L_{w}^{p-\epsilon }(I)$, i.e. there exists a positive constant L, such that

$$\begin{aligned} ||\mathcal{M}f||_{L_{w}^{p-\epsilon }(I)}\le L ||f||_{L_{w}^{p-\epsilon }(I)}. \end{aligned}$$

(7)

From here on, in order to prove the boundedness of $\mathcal{M}$, we can proceed by using, exactly the same arguments as for the operator $\mathcal{C}_a$, or, alternatively, we can use the following

Alternative proof

Let $w \in A_p(I)$, let us prove the boundedness of the Hardy-Littlewood Maximal operator. Indeed, by setting , we get

Therefore, since $\displaystyle \max _{\sigma \le \epsilon <p-1}\delta (\epsilon )^{\frac{1}{p-\epsilon }}\,\delta (\sigma )^{-\frac{1}{p-\sigma }}\ge 1$, then

(8)

Let us take $0<\epsilon \le \sigma$, so that $p-\epsilon >1$, hence we can apply the weighted Hardy inequality with the exponent p replaced by $p-\epsilon$, to get

By taking the supremum over $0<\epsilon < \sigma$ on both sides, the previous inequality implies

and therefore, by substituting in (8), we get

$$\begin{aligned} \le \max _{\sigma \le \epsilon<p-1}\delta (\epsilon )^{\frac{1}{p-\epsilon }}\,\delta (\sigma )^{-\frac{1}{p-\sigma }}\frac{p-\sigma }{p-\sigma -1} \sup _{0<\epsilon <p-1}\left( \delta (\epsilon ) \frac{1}{|I|}\int _{I} f^{p-\epsilon }w dx \right) ^{\frac{1}{p-\epsilon }} \end{aligned}$$

(9)

By setting $c(p,\delta )=\displaystyle {\inf _{0<\sigma<p-1}\max _{\sigma \le \epsilon <p-1}\delta (\epsilon )^{\frac{1}{p-\epsilon }}\,\delta (\sigma )^{-\frac{1}{p-\sigma }}\frac{p-\sigma }{p-\sigma -1}}$, and taking the supremum over all cubes $J\subset I$ containing x we get the boundedness of $\mathcal{M}$, as we desired.