Remarks on localized sharp functions on certain sets in ${\mathbb {R}}^n$

Abstract

The aim of this note is to define localized sharp functions on certain domains in ${\mathbb {R}}^n$ and prove $L^p$ estimates analogue to that of Fefferman–Stein. The proofs go by modifications of the good lambda inequality.

1 Introduction

On ${\mathbb {R}}^n$ let $f^{\#}_{\varDelta }(x)$ and $M^{\varDelta }f(x)$ denote the classical dyadic sharp function and dyadic maximal function respectively, that is,

$$\begin{aligned}&f^{\#}_{\varDelta }(x)=\sup _{x \in Q\in \varDelta }\frac{1}{|Q|}\int _{Q}|f(y)-f_Q|\,dy,\\&\quad M^{\varDelta }f(x)=\sup \limits _{x\in Q\in \varDelta } \frac{1}{|Q|}\int _Q |f(y)|\, dy, \end{aligned}$$

where here and subsequently, $\varDelta$ denotes the collection of all dyadic cubes in ${\mathbb {R}}^n$ and

$$\begin{aligned} f_Q=\frac{1}{|Q|} \int _Q f(x)\,dx. \end{aligned}$$

Suppose that $f\in L^{p_0}({\mathbb {R}}^n)$ for some $p_0$. The well-known Fefferman–Stein inequality asserts that if $1<p<\infty$, $1\le p_0\le p$, and $f^{\#}_{\varDelta }\in L^p({\mathbb {R}}^n)$, then $M^{\varDelta }f\in L^p({\mathbb {R}}^n)$ and

$$\begin{aligned} \Vert M^{\varDelta }f\Vert _{L^p({\mathbb {R}}^n)}\le C_n(p) \Vert f^{\#}_{\varDelta }\Vert _{L^p({\mathbb {R}}^n)} \end{aligned}$$

(1.1)

(see [2, Sect. 3], [4, Chapter 4]). The inequality (1.1) implies that for every $1<p<\infty$ one has

$$\begin{aligned} \Vert f\Vert _{L^p({\mathbb {R}}^n)}\le C_n(p)\Vert f^{\#}_{\varDelta }\Vert _{L^p({\mathbb {R}}^n)}. \end{aligned}$$

The estimate (1.1) is a consequence of the following good lambda distributional inequality

$$\begin{aligned} \left| \{x\in {\mathbb {R}}^n: M^{\varDelta }f(x)> \lambda , \, f^{\#}_{\varDelta }(x) \le c\lambda \}\right| \le a\left| \{x\in {\mathbb {R}}^n: M^{\varDelta }f(x)> b\lambda \}\right| , \end{aligned}$$

(1.2)

where $\lambda >0$, $c>0$, $0<b<1$, $a=2^nc/(1-b)$, and $f\in L^1_{\mathrm{loc}}({\mathbb {R}}^n)$ (see [4]).

Let $\varOmega$ be a domain in ${\mathbb {R}}^n$. Our goal is to define for $f\in L^1_{\mathrm{loc}} (\varOmega )$ a localized version $f^{\#}_{\mathrm{loc}}$ of the sharp function which will satisfy

$$\begin{aligned} \Vert f\Vert _{L^p(\varOmega )} \le C_p\Vert f^{\#}_{\mathrm{loc}}\Vert _{L^p(\varOmega )} \ \ \text {for}\ \ f\in L^p(\varOmega ). \end{aligned}$$

(1.3)

By localized we mean that the cubes which are taken in the definition of $f^{\#}_{\mathrm{loc}}(x)$ are contained in a bounded set ${\mathcal {B}}_x\subset \varOmega$. So one possible definition can be taken as follows. Let $\tau :\varOmega \rightarrow (0,\infty )$. For $f\in L^1_{\mathrm{loc}}(\varOmega )$ we set

$$\begin{aligned} f^{\#}_{{\mathrm{loc}}, \, \tau }(x)=\sup _{x\in Q\subset \varOmega , \ \ell (Q)<\tau (x)} \frac{1}{|Q|}\int _{Q}|f(y)-f_Q|\,dy, \end{aligned}$$

where Q is any cube (not necessarily dyadic) and $\ell (Q)$ denotes its side-length. Note that $\tau$ cannot be taken arbitrarily. For example, if $\varOmega =(0,\infty )$ and $\tau$ is such that $\lim _{x\rightarrow \infty } \tau (x)/x=0$, then taking $f(x)=\chi _{(0,R)}(x)$ we have $\Vert f\Vert _{L^p(\varOmega )}=R^{1\slash p}$ while $\lim _{R\rightarrow \infty } R^{-1\slash p} \Vert f^{\#}_{{\mathrm{loc}},\, \tau }\Vert _{L^p(\varOmega )}= 0$. On the other hand, we shall show that for certain sets $\varOmega$ in ${\mathbb {R}}^n$ if $\tau (x)$ behaves like $\frac{1}{2}\text { dist}(x,\partial \varOmega )$, then $f^{\#}_{{\mathrm{loc}},\, \tau }$ satisfies (1.3). Moreover, the inequality (1.3) holds for $p=1$, provided f is supported by a bounded set and $|f|\log (2+|f|)$ is integrable. These will be obtained by proving modifications of the good lambda inequality (see Propositions 2.4, 3.7, and 4.5).

2 Localized sharp function on ${\mathbb {R}}^n{\setminus } \{0\}$

Let $\varOmega ={\mathbb {R}}^n{\setminus } \{0\}$. We define the localized sharp function on $\varOmega$ as

$$\begin{aligned} f^{\#}_{\mathrm{loc},\,\varOmega }(x)=\sup _{x \in K \subset \left\{ y \in \varOmega :\,\frac{|x|}{4\sqrt{n}} \le |y| \le 4\sqrt{n}|x|\right\} }\frac{1}{|K|}\int _{K}|f(y)-f_{K}|\,dy, \end{aligned}$$

where the supremum is taken over all cubes K (not necessarily dyadic) contained in the set

$$\begin{aligned} \left\{ y \in \varOmega :\frac{|x|}{4\sqrt{n}} \le |y| \le 4\sqrt{n}|x|\right\} . \end{aligned}$$

We now turn to define the local dyadic maximal function associated with a Whitney decomposition of $\varOmega$. For this purpose, put

$$\begin{aligned}&\rho (x,y)=\max (|x_1-y_1|,|x_2-y_2|,\ldots ,|x_n-y_n|). \end{aligned}$$

Set

$$\begin{aligned} {\mathcal {L}}= \left\{ Q\in \varDelta : \ell (Q)=\rho (0,Q)\right\} . \end{aligned}$$

The set ${\mathcal {L}}$ forms a Whitney covering of $\varOmega$. For every integer k we define the k -th layer $L_k$ of ${\mathcal {L}}$ as

$$\begin{aligned} L_k=\left\{ Q\in {\mathcal {L}}: \ell (Q)=2^{-k}\right\} . \end{aligned}$$

Clearly, $Q\in L_k$ if and only if $2^{-m}Q\in L_{k+m}$. Figure 1 shows three k-layers for $n = 2$. Here and subsequently, $\alpha Q=\{\alpha x: x\in Q\}$, $\alpha >0$.

Fig. 1

Cubes from three k-layers of $\mathcal {L}$ for $n=2$ which are contained in $(0,\infty )^n$

For every positive integer m the partition ${\mathcal {L}}_m$ of $\varOmega$ is obtained by dividing each cube Q from ${\mathcal {L}}$ into $2^{nm}$ dyadic cubes each of side-length $2^{-m}\ell (Q)$. Let

$$\begin{aligned} {\mathcal {D}}=\mathcal {L} \cup \bigcup _{m=1}^{\infty }\mathcal {L}_{m}. \end{aligned}$$

The local dyadic maximal function associated with the Whitney covering ${\mathcal {L}}$ of $\varOmega$ is defined by

$$\begin{aligned} M_{\mathcal {D}}f(x)=\sup _{x \in K \in {\mathcal {D}}}\frac{1}{|K|}\int _{K}|f(y)|\,dy. \end{aligned}$$

Our goal of this section is to prove the following theorem.

Theorem 2.1

For every $1 \le p<\infty$ there is a constant $C>0$ such that for every locally integrable function f on $\varOmega$ for which there exists $0<p_0\le p$ such that $M_{\mathcal {D}} f\in ~L^{p_0}(\varOmega )$ one has

$$\begin{aligned} \Vert M_{\mathcal {D}}f\Vert _{L^p(\varOmega )} \le C \Vert f^{\#}_{\mathrm{loc},\,\varOmega }\Vert _{L^p(\varOmega )}. \end{aligned}$$

Corollary 2.2

For every $1<p<\infty$ there is a constant $C_p>0$ such that

$$\begin{aligned} \Vert f\Vert _{L^{p}(\varOmega )} \le C_p \Vert f^{\#}_{\mathrm{loc},\,\varOmega }\Vert _{L^p(\varOmega )} \ \ \ \text {for} \ \ f\in L^p(\varOmega ). \end{aligned}$$

Corollary 2.3

There is a constant $C>0$ such that if f is supported by a bounded set and $|f|\log (2+|f|)$ is integrable, then

$$\begin{aligned} \Vert f\Vert _{L^{1}(\varOmega )} \le C \Vert f^{\#}_{\mathrm{loc},\,\varOmega }\Vert _{L^1(\varOmega )}. \end{aligned}$$

There is no loss of generality if we assume that all the functions under consideration take values in ${\mathbb {R}}$. Clearly, for almost every $x\in \varOmega$ there is a unique cube $Q\in {\mathcal {L}}$ such that $x\in Q$. For such an x let

$$\begin{aligned} Sf(x)=|f|_Q, \ \ S'f(x) = f_Q. \end{aligned}$$

(2.1)

The proof of Theorem 2.1 is a consequence of the following modified version of the good lambda inequality, which is stated in the proposition below.

Proposition 2.4

(modified good lambda inequality) For every constant $0<b<1$ there is a constant $C>0$ such that for all $c,\alpha >0$, and every locally integrable function f which satisfies $\lim _{|x|\rightarrow \infty } Sf(x)=0$ we have

$$\begin{aligned}&\left| \{x \in \varOmega :M_{\mathcal {D}}f(x)>\alpha , \, f^{\#}_{\mathrm{loc},\,\varOmega }(x)<c\alpha \}\right| \\&\quad \le Cc\left| \{x \in \varOmega :M_{\mathcal {D}}f(x)>b\alpha \}\right| +\left| \{x \in \varOmega : {\mathbb {S}}f(x)>b\alpha \}\right| , \end{aligned}$$

where here and subsequently,

$$\begin{aligned} {\mathbb {S}}f(x)=2^{1-n}\cdot 3^{2n}\sum _{j=1}^\infty f^{\#}_{\mathrm{loc},\,\varOmega }(2^jx). \end{aligned}$$

Proof of Theorem 2.1

If we assume Proposition 2.4, the proof of the theorem is a slight modification of that in the classical case (see [2,3,4]). For the convenience of the reader, we provide details. We may assume that $\Vert f^{\#}_{\mathrm{loc},\,\varOmega }\Vert _{L^p(\varOmega )}$ is finite. Then, by the Minkowski inequality,

$$\begin{aligned} \Vert {\mathbb {S}}f \Vert _{L^p(\varOmega )}\le 2^{1-n}\cdot 3^{2n}\sum _{j=1}^\infty 2^{-jn\slash p} \Vert f^{\#}_{\mathrm{loc},\,\varOmega }\Vert _{L^p(\varOmega )} = C_p\Vert f^{\#}_{\mathrm{loc},\,\varOmega }\Vert _{L^p(\varOmega )}. \end{aligned}$$

(2.2)

Since $M_{\mathcal {D}} f\in L^{p_0}(\varOmega )$, $\lim _{|x|\rightarrow \infty } Sf(x)=0$. Let

$$\begin{aligned} I_R&=p\int _{0}^{R}\alpha ^{p-1}\left| \{x \in \varOmega :M_{\mathcal {D}}f(x)>\alpha \}\right| \,d\alpha \\&= p\int _{0}^{R}\alpha ^{p-1}\left| \{x \in \varOmega :M_{\mathcal {D}}f(x)>\alpha , \, f^{\#}_{\mathrm{loc},\,\varOmega }(x)<c\alpha \}\right| \,d\alpha \\&\quad +p\int _{0}^{R}\alpha ^{p-1}\left| \{x \in \varOmega :M_{\mathcal {D}}f(x)>\alpha , \, f^{\#}_{\mathrm{loc},\,\varOmega }(x)\ge c\alpha \}\right| \,d\alpha . \end{aligned}$$

Applying Proposition 2.4, we obtain

$$\begin{aligned} I_R&\le Ccp\int _{0}^{R}\alpha ^{p-1}|\{x \in \varOmega :M_{\mathcal {D}}f(x)>b\alpha \}|\,d\alpha \\&\quad + p\int _{0}^{R}\alpha ^{p-1}|\{x \in \varOmega :{\mathbb {S}}f(x)>b\alpha \}|\,d\alpha \\&\quad +p\int _{0}^{R}\alpha ^{p-1}|\{x \in \varOmega :f^{\#}_{\mathrm{loc},\,\varOmega }(x) \ge c\alpha \}|\,d\alpha \\&\le Ccb^{-p} I_{bR}+p\int _{0}^{R}\alpha ^{p-1}|\{x \in \varOmega :{\mathbb {S}}f(x)>b\alpha \}|\,d\alpha \\&\quad +p\int _{0}^{R}\alpha ^{p-1}|\{x \in \varOmega :f^{\#}_{\mathrm{loc},\,\varOmega }(x)\ge c\alpha \}|\,d\alpha . \end{aligned}$$

Clearly, $I_R<\infty$, since, by assumption, $M_{\mathcal {D}}f\in L^{p_0}(\varOmega )$ and $0<p_0\le p$. Moreover, $I_{bR}\le I_b$ because $0<b<1$. Taking c small enough such that $Ccb^{-p}<1$ we obtain

$$\begin{aligned} I_R&\le C'p\int _{0}^{R}\alpha ^{p-1}|\{x \in \varOmega :{\mathbb {S}}f(x)>b\alpha \}|\,d\alpha \\&\quad +C'p\int _{0}^{R}\alpha ^{p-1}|\{x \in \varOmega :f^{\#}_{\mathrm{loc},\,\varOmega }(x) \ge c\alpha \}|\,d\alpha . \end{aligned}$$

Letting $R\rightarrow \infty$, we conclude

$$\begin{aligned} \Vert M_{\mathcal {D}}f\Vert _{L^p(\varOmega )} \le C_1\Vert {\mathbb {S}}f\Vert _{L^p(\varOmega )}+C_2\Vert f^{\#}_{\mathrm{loc},\,\varOmega }\Vert _{L^p(\varOmega )}\le C_3 \Vert f^{\#}_{\mathrm{loc},\,\varOmega }\Vert _{L^p(\varOmega )}, \end{aligned}$$

where in the last inequality we have used (2.2). $\square$

The remaining part of the section is devoted to proving Proposition 2.4. The following two lemmas will play a crucial role in the proof.

Lemma 2.5

For every locally integrable function f and almost every $x \in \varOmega$ one has

$$\begin{aligned} S'f(x) \le S'f(2x)+ 2^{-n}\cdot 3^{2n}f^{\#}_{\mathrm{loc},\,\varOmega }(2x). \end{aligned}$$

Proof

It suffices to prove the lemma for $x=(x_1,x_2,\ldots ,x_n)$ such that $x_j>0$ for every $j=1,2,\ldots ,n$. Let $Q_1$ be the unique dyadic cube from ${\mathcal {L}}$ which contains x. Let k be such that $2^{-(k+1)}=\ell ( Q_1)$. Set $Q_2=2Q_1$. Let

$$\begin{aligned} Q_0=\{x=(x_1,\ldots ,x_n)\in {\mathbb {R}}^n: \, 0\le x_j\le 1 \ \ \text {for all } j=1,2,\ldots ,n\}. \end{aligned}$$

(2.3)

Then there is the unique vector $p=(p_1,p_2,\ldots ,p_n)\ne \mathbf 0$, $p_j\in \{0,1\}$, such that $Q_1=2^{-k-1}Q_0+2^{-k-1} p$. Set $K=3\cdot 2^{-k-1}Q_0+2^{-k-1} p$. Then

$$\begin{aligned} Q_1\cup Q_2\subset K\subset \left\{ y\in \varOmega : \frac{|2x|}{4\sqrt{n}}\le |y|\le 4\sqrt{n}|2x|\right\} . \end{aligned}$$

We shall call the set $M=K{\setminus } (Q_1\cup Q_2)$ the complementary neighborhood of the pair of cubes $(Q_1,Q_2)$.

Let us remark that for $n=1$ the complementary neighborhood of two intervals is the empty set. For $n=2$ the complementary neighborhoods are presented in Fig. 2.

Fig. 2

Complementary neighborhoods for $n=2$

We have

$$\begin{aligned}&\frac{|Q_1|}{|K|}=\frac{1}{3^n},\ \ \frac{|Q_2|}{|K|}=\frac{2^n}{3^n}, \ \ \frac{|M|}{|K|}=\frac{3^n-2^n-1}{3^n},\nonumber \\&\quad f_{K}=\frac{1}{3^n}f_{Q_1}+\frac{2^n}{3^n}f_{Q_2}+\frac{3^n-2^n-1}{3^n}f_{M}. \end{aligned}$$

(2.4)

We consider two cases.

Case 1 $S'f(x)\ge f_M$. Then

$$\begin{aligned} f^{\#}_{\mathrm{loc},\,\varOmega }(2x)&\ge \frac{1}{|K|}\int _{K}|f(y)-f_{K}|\,dy \\&= \frac{1}{|K|}\int _{K}|f(y)-\left( \frac{1}{3^n}f_{Q_1} +\frac{2^n}{3^n}f_{Q_2}+\frac{3^n-2^n-1}{3^n}f_{M}\right) |\,dy \\&\ge \frac{1}{|K|}\int _{Q_1}\left| f(y)-\left( \frac{1}{3^n}f_{Q_1} +\frac{2^n}{3^n}f_{Q_2}+\frac{3^n-2^n-1}{3^n}f_{M}\right) \right| \,dy \\&\ge \frac{1}{|K|}\left| \int _{Q_1}f(y)-\left( \frac{1}{3^n}f_{Q_1} +\frac{2^n}{3^n}f_{Q_2}+\frac{3^n-2^n-1}{3^n}f_{M}\right) \,dy \right| \\&= \left| f_{Q_1}\frac{1}{3^n}-f_{Q_1}\frac{1}{3^{2n}}-f_{Q_2}\frac{2^n}{3^{2n}}-f_{M}\frac{3^n-1-2^n}{3^{2n}}\right| \\&\ge f_{Q_1}\frac{1}{3^n}-f_{Q_1}\frac{1}{3^{2n}}-f_{Q_2}\frac{2^n}{3^{2n}}-f_{M}\frac{3^n-1-2^n}{3^{2n}} \\ {}&= f_{Q_1}\frac{3^n-1}{3^{2n}}-f_{Q_2}\frac{2^n}{3^{2n}}-f_{M}\frac{3^n-1-2^n}{3^{2n}}. \end{aligned}$$

By the assumption $-f_M\ge - S'f(x)= - f_{Q_1}$, hence

$$\begin{aligned}&f_{Q_1}\frac{3^n-1}{3^{2n}}-f_{Q_2}\frac{2^n}{3^{2n}}-f_{M}\frac{3^n-1-2^n}{3^{2n}} \ge f_{Q_1}\frac{3^n-1}{3^{2n}}-f_{Q_2}\frac{2^n}{3^{2n}}-f_{Q_1}\frac{3^n-1-2^n}{3^{2n}} \\&\quad =f_{Q_1}\frac{2^n}{3^{2n}}-f_{Q_2}\frac{2^n}{3^{2n}}=S'f(x)\frac{2^n}{3^{2n}}-S'f(2x)\frac{2^n}{3^{2n}}, \end{aligned}$$

which gives the lemma.

Case 2 $S'f(x)\le f_M$. The proof in this case is similar to that in Case 1. The only difference is that we diminish the area of integration to $Q_2$ instead of $Q_1$. We omit the details. If $n=1$, then $M=\emptyset$. In this case we set $f_M=0$ and proceed as in Case 1. $\square$

Remark 2.6

If we apply the lemma to the function $-f$, we obtain the inequality

$$\begin{aligned} S'f(2x)\le S'f(x)+ 2^{-n}\cdot 3^{2n} f^{\#}_{\mathrm{loc},\,\varOmega }(2x). \end{aligned}$$

Lemma 2.7

For every locally integrable function f and almost every $x \in \varOmega$ one has

$$\begin{aligned} Sf(x) \le S f(2x)+ 2^{1-n}\cdot 3^{2n} f^{\#}_{\mathrm{loc},\,\varOmega }(2x). \end{aligned}$$

(2.5)

Proof

Using Lemma 2.5 to |f| we get $Sf(x) \le Sf(2x)+ 2^{-n}\cdot 3^{2n}|f|^{\#}_{\mathrm{loc},\,\varOmega }(2x)$. The inequality (2.5) holds because $|f|^{\#}_{\mathrm{loc},\,\varOmega }(2x)\le 2 f^{\#}_{\mathrm{loc},\,\varOmega }(2x)$. $\square$

Iterating the inequality (2.5) we obtain the following corollary.

Corollary 2.8

Assume that a locally integrable function f on $\varOmega$ satisfies

$$\begin{aligned} \lim _{|x|\rightarrow \infty }Sf(x)~=~0. \end{aligned}$$

Then

$$\begin{aligned} Sf(x)\le {\mathbb {S}}f(x) \ \ \ \text {for every } x\in \varOmega . \end{aligned}$$

Proof of Proposition 2.4

The proof of the proposition is a modification of that of the classical good lambda inequality (cf. [1, 4]). Let $\{Q_j\}$ be the partition of the set $\{x\in \varOmega : M_{\mathcal {D}}f(x)>b\alpha \}$ which consists of maximal dyadic cubes $Q_j$ from ${\mathcal {D}}$ which satisfy $|f|_{Q_j} >b\alpha$. Obviously, the cubes $Q_j$ have disjoint interiors. Further,

$$\begin{aligned}&\left\{ x\in \varOmega :M_{\mathcal {D}}f(x)>\alpha ,\,f^{\#}_{\mathrm{loc},\,\varOmega }(x)<c\alpha \right\} \subset \left\{ x \in \varOmega :M_{\mathcal {D}}f(x)>b\alpha \right\} =\bigcup _j Q_j,\\&\left\{ x\in \varOmega :M_{\mathcal {D}}f(x)>\alpha ,\,f^{\#}_{\mathrm{loc},\,\varOmega }(x)<c\alpha \right\} = \bigcup _{j}\left\{ x \in Q_j:M_{\mathcal {D}}f(x)>\alpha , \, f^{\#}_{\mathrm{loc},\,\varOmega }(x)<c\alpha \right\} . \end{aligned}$$

Thus, it suffices to show that either

$$\begin{aligned} \left| \left\{ x \in Q_j:M_{\mathcal {D}}f(x)>\alpha ,\, f^{\#}_{\mathrm{loc},\,\varOmega }(x)<c\alpha \right\} \right| \le cC|Q_j| \end{aligned}$$

(2.6)

or

$$\begin{aligned} \left\{ x \in Q_j:M_{\mathcal {D}}f(x)>\alpha ,\, f^{\#}_{\mathrm{loc},\,\varOmega }(x)<c\alpha \right\} \subset \left\{ x \in \varOmega :{\mathbb {S}}f(x)>b\alpha \right\} . \end{aligned}$$

(2.7)

Assume that the set $A_j=\{x\in Q_j :M_{\mathcal {D}}f(x)>\alpha ,\, f^{\#}_{\mathrm{loc},\,\varOmega }(x)<c\alpha \}$ is not empty, otherwise there is nothing to prove. Fix $x_0\in A_j$. We consider two cases.

Case 1 $Q_j\in {\mathcal {L}}$. Then $b\alpha < |f|_{Q_j}=Sf(y)\le \mathbb Sf(y)$ for every $y\in Q_j$, where in the last inequality we have used Corollary 2.8. Thus $Q_j\subset \{x \in \varOmega :{\mathbb {S}}f(x)>b\alpha \}$ and (2.7) holds in this case.

Case 2 $Q_j\in {\mathcal {L}}_m$ for $m\ge 1$. In this case the proof follows the pattern from [4]. Indeed, first observe that for every $Q\in \mathcal D$ such that $Q_j\varsubsetneq Q$ one has $|f|_Q\le b\alpha$. Thus, for every $x\in Q_j$ such that $M_{\mathcal {D}}f(x)>\alpha$ one has $M_{\mathcal {D}} \{(f-f_{Q})\chi _{Q_j}\}(x)>(1-b)\alpha$. Let ${\widetilde{Q}}_j$ be the parent of $Q_j$. Clearly, ${\widetilde{Q}}_j\in {\mathcal {D}}$ and

$$\begin{aligned} A_{j}= \{x \in Q_j:M_{\mathcal {D}}(f-f_{\widetilde{Q_j}})\chi _{Q_j}(x)>(1-b)\alpha , \, M_{\mathcal {D}}f(x)>\alpha , \, f^{\#}_{\mathrm{loc},\,\varOmega }(x)<c\alpha \}. \end{aligned}$$

Since $M_{\mathcal {D}}$ satisfies the weak type (1,1) inequality with the constant $C'=1$, we have

$$\begin{aligned} \begin{aligned} | A_j |&\le \frac{1}{(1-b)\alpha }\int _{Q_j}|f-f_{\widetilde{Q_j}}|\,dx \le \frac{1}{(1-b)\alpha }\int _{\widetilde{Q_j}}|f-f_{\widetilde{Q_j}}|\,dx \le \frac{|\widetilde{Q_j}|}{(1-b)\alpha }f^{\#}_{\mathrm{loc},\,\varOmega }(x_0) \\&\le \frac{|\widetilde{Q_j}|}{(1-b)\alpha }c\alpha = \frac{2^nc}{1-b}|Q_j|, \end{aligned} \end{aligned}$$

so (2.6) holds in this case with $C=2^n(1-b)^{-1}$. $\square$

3 Localized sharp function for $(0,\infty )^n$

In this section ${\widetilde{\varOmega }} =\{x=(x_1,\ldots ,x_n)\in {\mathbb {R}}^n: x_j>0, \, j=1,2,\ldots ,n\}$ denotes the generalized first quoter in ${\mathbb {R}}^n$. The distance of $x\in {\widetilde{\varOmega }}$ from the boundary is given by ${\widetilde{\rho }}(x,\partial {\widetilde{\varOmega }})=\min \{x_j: j=1,2,\ldots , n\}$. We define the partition ${\widetilde{{\mathcal {L}}}}$ of ${\widetilde{\varOmega }}$:

$$\begin{aligned} {\widetilde{{\mathcal {L}}}}=\{ Q\in \varDelta : Q\subset {\widetilde{\varOmega }}\ \text { and } \ \ell ( Q)= {\widetilde{\rho }} (Q,\partial {\widetilde{\varOmega }})\}. \end{aligned}$$

Clearly,

$$\begin{aligned} {\widetilde{{\mathcal {L}}}}= \bigcup _{k\in {\mathbb {Z}}} {\widetilde{L}}_k, \ \ \text {where} \ \ {\widetilde{L}}_k=\{ Q\in {\widetilde{{\mathcal {L}}}}: \ell (Q)=2^{-k}\}. \end{aligned}$$

Fig. 3

A part of the partition ${\widetilde{\mathcal {L}}}$ for $n=2$ with the five layers ${\widetilde{L}}_{-1}$, ${\widetilde{L}}_0$,...,${\widetilde{L}}_{3}$

Similarly to the previous section, for every positive integer m the partition ${\widetilde{{\mathcal {L}}}}_m$ consists of dyadic cubes which are obtained by dividing each cube Q from ${\widetilde{{\mathcal {L}}}}$ into $2^{mn}$ dyadic cubes each of the side-length $2^{-m}\ell (Q)$ (Fig. 3). Set

$$\begin{aligned} {\widetilde{{\mathcal {D}}}}={\widetilde{\mathcal {L}}} \cup \bigcup _{m=1}^{\infty }{\widetilde{\mathcal {L}}}_{m}. \end{aligned}$$

Define the local maximal dyadic function $M_{\widetilde{\mathcal {D}}}$ and localized sharp function $f^{\#}_{\mathrm{loc},\,{\widetilde{\varOmega }}}$ associated with the Whitney covering ${\widetilde{{\mathcal {L}}}}$ of ${\widetilde{\varOmega }}$ as

$$\begin{aligned}&M_{\widetilde{\mathcal {D}}}f(x)=\sup _{x\in Q\in {\widetilde{\mathcal {D}}}} |f|_Q,\\&f^{\#}_{\mathrm{loc},\,{\widetilde{\varOmega }}}(x)=\sup _{x \in K \subset \{y \in {\widetilde{\varOmega }}: \frac{1}{4}{\widetilde{\rho }}(x,\partial {\widetilde{\varOmega }})\le {\widetilde{\rho }}(y,\partial {\widetilde{\varOmega }}) \le 4{\widetilde{\rho }}(x,\partial {\widetilde{\varOmega }})\}} \frac{1}{|K|}\int _{K}|f(y)-f_{K}|\,dy, \end{aligned}$$

where the supremum is taken over all cubes (not necessarily dyadic).

It turns out that the following theorem analogue to Theorem 2.1 holds.

Theorem 3.1

For every $1 \le p<\infty$ there is a constant $C>0$ such that for every locally integrable function f on ${\widetilde{\varOmega }}$ for which there exists $0<p_0\le p$ such that $M_{\widetilde{\mathcal {D}}} f\in ~L^{p_0}({\widetilde{\varOmega }})$ one has

$$\begin{aligned} \Vert M_{\widetilde{\mathcal {D}}}f\Vert _{L^p({\widetilde{\varOmega }})} \le C \Vert f^{\#}_{\mathrm{loc},\,{\widetilde{\varOmega }}}\Vert _{L^p(\widetilde{\varOmega })}. \end{aligned}$$

Corollary 3.2

For every $1<p<\infty$ there is a constant $C>0$ such that

$$\begin{aligned} \Vert f\Vert _{L^{p}({\widetilde{\varOmega }})} \le C \Vert f^{\#}_{\mathrm{loc},\,{\widetilde{\varOmega }}}\Vert _{L^{p}({\widetilde{\varOmega }})} \ \ \ \text {for} \ \ f\in L^p({\widetilde{\varOmega }}). \end{aligned}$$

The remaining part of this section is devoted to proving Theorem 3.1.

Similarly to the previous section [see (2.1)] we set

$$\begin{aligned} {\widetilde{S}}f(x)=|f|_Q \ \ \text {and}\ \ {\widetilde{S}}'f(x)=f_Q , \end{aligned}$$

(3.1)

where Q is the unique cube from ${\widetilde{\mathcal {L}}}$ which contains x (such a Q is well-defined for almost every x). Let k be such that $Q\in {\widetilde{L}}_k$. Our goal is to define the successors $x'$ and $Q'\in {\widetilde{L}}_{k-1}$ of x and Q respectively in such a way that $x'\in Q'$ and

$$\begin{aligned} {\widetilde{S}} f(x)\le {\widetilde{S}} f(x')+ 2^{1-n}3^{2n} f^{\#}_{\mathrm{loc},\,{\widetilde{\varOmega }}}(x'). \end{aligned}$$

To this end, observe that there is a unique vector $q=(q_1,q_2,\ldots ,q_n)$, where $q_j$ are non-negative integers such that at least one $q_j$ equals 0, and $Q=2^{-k}Q_0+2^{-k}( q+\mathbf 1 )$, where here and subsequently, $\mathbf 1 =(1,1,\ldots ,1)$. Consider the coordinates $x_j$ of x for which $q_j=0$. There is no loss of generality if we assume these are the first m coordinates, $m\in \{1,2,\ldots ,n\}$. So $Q=2^{-k}Q_0+2^{-k}(1,1,\ldots ,1,1+q_{m+1},\ldots ,1+q_n)$, $q_{m+1},\ldots ,q_n\ge 1$. Define

$$\begin{aligned} x'=F(x)=(2x_1,2x_2,\ldots ,2x_m, x_{m+1},\ldots ,x_n) \ \text { for }\ x\in Q. \end{aligned}$$

(3.2)

Then, for almost every x, the point $x'$ belongs to the unique $Q'\in L_{k-1}$ and

$$\begin{aligned}&Q'=2^{-k+1}Q_0+2^{-k+1}( q'+\mathbf 1 ),\\&q'=(0,\ldots ,0,q_{m+1}',\ldots ,q_n'), \ \ \ q_j'=\lfloor (q_j-1)\slash 2\rfloor , \ j={m+1},\ldots ,n. \end{aligned}$$

Lemma 3.3

For every $f\in L^1_{\mathrm{loc}}({\widetilde{\varOmega }})$ and almost every $x \in {\widetilde{\varOmega }}$ one has

$$\begin{aligned} {\widetilde{S}}'f(x)\le & {} {\widetilde{S}}' f(x')+ 2^{-n}\cdot 3^{2n}f^{\#}_{\mathrm{loc},\,{\widetilde{\varOmega }}}(x'), \end{aligned}$$

(3.3)

$$\begin{aligned} {\widetilde{S}}f(x)\le & {} {\widetilde{S}} f(x')+ 2^{1-n}\cdot 3^{2n}f^{\#}_{\mathrm{loc},\,{\widetilde{\varOmega }}}(x'), \end{aligned}$$

(3.4)

$$\begin{aligned} {\widetilde{S}}'f(x')\le & {} {\widetilde{S}}' f(x)+ 2^{-n}\cdot 3^{2n}f^{\#}_{\mathrm{loc},\,{\widetilde{\varOmega }}}(x). \end{aligned}$$

(3.5)

Proof

Define (non-dyadic) cube $K''=[2^{-k},2^{-k+2}]^m\times I_{m+1}\times \cdots \times I_n\subset {\widetilde{\varOmega }}$, where

$$\begin{aligned} I_j=\left[ 2^{-k+1}q_{j}'+2^{-k+2}-3\cdot 2^{-k},2^{-k+1}q_j'+ 2^{-k+2}\right] . \end{aligned}$$

We have $\ell (K'')=3\cdot 2^{-k}$, $Q\cup F(Q) \cup Q' \subset K''$. Moreover, $K''$ is taken into account if we compute $f^{\#}_{\mathrm{loc},\,{\widetilde{\varOmega }}}(x')$ and $f^{\#}_{\mathrm{loc},\,{\widetilde{\varOmega }}}(x)$. Set $M=K''{\setminus } (Q\cup Q')$. We have [(cf. (2.4)]

$$\begin{aligned} \frac{|Q|}{|K''|}=\frac{1}{3^n},\ \ \frac{|Q'|}{|K''|}=\frac{2^n}{3^n}, \ \ \frac{|M|}{|K''|}=\frac{3^n-2^n-1}{3^n}, \end{aligned}$$

(3.6)

hence the proof of the lemma is the same as those of Lemmata 2.5 and 2.7. $\square$

Fig. 4

Cubes which contain points $x,F(x),F(F(x)),\ldots$ for $n=2$

Corollary 3.4

Assume that $f \in L^1_{\mathrm{loc}}({\widetilde{\varOmega }})$ and $\lim _{m\rightarrow \infty } {\widetilde{S}}f(F^{m}( x))=0$ for almost every $x\in {\widetilde{\varOmega }}$, where $F^m( x)=F(F^{m-1}( x))$ (Fig. 4). Then

$$\begin{aligned} {\widetilde{S}}f( x)\le {\widetilde{\mathbb {S}}}f(x), \end{aligned}$$

where

$$\begin{aligned} {\widetilde{\mathbb {S}}}f( x)=2^{1-n}\cdot 3^{2n}\sum _{k_1=0}^\infty \sum _{k_2=0}^\infty \cdots \sum _{k_n=0}^\infty \sum _{m=0}^\infty f^{\#}_{\mathrm{loc},\,{\widetilde{\varOmega }}}\left( 2^{k_1+m}x_1,2^{k_2+m}x_2,\ldots ,2^{k_n+m}x_n\right) . \end{aligned}$$

Proof

It suffices to apply Lemma 3.3 and note that

$$\begin{aligned} 2^{1-n}3^{2n}\sum \limits _{m=1}^\infty f^{\#}_{\mathrm{loc},\,{\widetilde{\varOmega }}}(F^m( x))\le ~{\widetilde{\mathbb {S}}}f( x). \end{aligned}$$

Remark 3.5

Let us note that $\lim _{m\rightarrow \infty } {\widetilde{S}} f(F^m( x))=0$ for $f\in L^p({\widetilde{\varOmega }})$. This is a consequence of the fact that ${\widetilde{\rho }}(F^m( x), \partial {\widetilde{\varOmega }}) \rightarrow \infty$ and $\ell (Q^m)\rightarrow \infty$, where $Q^m$ is the unique cube from ${\widetilde{{\mathcal {L}}}}$ such that $F^m( x)\in Q^m$.

Lemma 3.6

For every $1\le p<\infty$ there is a constant $C>0$ such that for every $f\in L^1_{\mathrm{loc}}({\widetilde{\varOmega }})$ one has

$$\begin{aligned} \Vert {\widetilde{\mathbb {S}}}f\Vert _{L^{p}({\widetilde{\varOmega }})}\le C\Vert f^{\#}_{\mathrm{loc},\,{\widetilde{\varOmega }}}\Vert _{L^{p}({\widetilde{\varOmega }})}. \end{aligned}$$

Proof

This follows from the Minkowski inequality and the summability of the series

$$\begin{aligned} \sum _{k_1=0}^{\infty }\sum _{k_2=0}^{\infty } \cdots \sum _{k_n=0}^{\infty }\sum _{m=0}^{\infty } 2^{-k_1/p-m/p}2^{-k_2/p-m/p}\cdot \ldots \cdot 2^{-k_n/p-m/p}. \end{aligned}$$

$\square$

Proposition 3.7

For every constant $0<b<1$ there is a constant $C>0$ such that for all $c,\alpha >0$, and every $f\in L^1_{\mathrm{loc}} ({\widetilde{\varOmega }})$ satisfying

$$\begin{aligned} \lim _{m\rightarrow \infty } {\widetilde{S}}f(F^m(x))=0 \end{aligned}$$

we have

$$\begin{aligned}&\left| \{x \in {\widetilde{\varOmega }} :M_{\widetilde{\mathcal {D}}}f(x)>\alpha , \,f^{\#}_{\mathrm{loc},\,{\widetilde{\varOmega }}}(x)<c\alpha \}\right| \\&\quad \le Cc\left| \{x \in {\widetilde{\varOmega }} :M_{\widetilde{\mathcal {D}}}f(x)>b\alpha \}\right| +\left| \{x \in {\widetilde{\varOmega }} : {\widetilde{{\mathbb {S}}}}f(x)>b\alpha \}\right| . \end{aligned}$$

Proof

The proof is identical to that of Proposition 2.4, and uses Corollary 3.4 instead of Corollary 2.8. $\square$

Proof of Theorem 3.1

The theorem follows from Lemma 3.6 and Proposition 3.7. Its proof is identical to that of Theorem 2.1. $\square$

4 Localized sharp function for cube

In this section we consider the cube $(0,2)^n\subset {\mathbb {R}}^n$ and its Whitney decomposition ${\widetilde{{\mathcal {L}}}}''$ which is defined in the following way. Let ${\widetilde{ {\mathcal {L}}}}'$ be the restriction of the decomposition ${\widetilde{{\mathcal {L}}}}$ defined in the previous section into the unit cube $(0,1]^n$. Let us denote by ${\widetilde{{\mathcal {L}}}}''$ the set of cubes which is obtained from ${\widetilde{ {\mathcal {L}}}}'$ under the action of the group G of transformation generated by the reflections with respect to planes $x_j=1$. Let ${\widetilde{L_k}}$ be the set of cubes from ${\widetilde{{\mathcal {L}}}}''$ of the side-length $2^{-k}$. Clearly, ${\widetilde{{\mathcal {L}}}}''={\widetilde{L_1}}\cup {\widetilde{L_2}}\cup \cdots$ (Fig. 5).

Fig. 5

Cubes from the set ${\widetilde{\mathcal {L}}}''$ for $n=2$ (the picture presents cubes from the layers ${\widetilde{L}}_{1}$, ${\widetilde{L}}_{2}$,...,${\widetilde{L}}_{5}$)

We define the partition ${\widetilde{{\mathcal {L}}}}''_1$ by dividing each cube K from ${\widetilde{\mathcal {L}}}''$ into $2^n$ dyadic cubes each of the side-length $2^{-1} \ell (K)$. Inductively, ${\widetilde{{\mathcal {L}}}}''_{m+1}$ is defined by dividing each cube K from ${\widetilde{{\mathcal {L}}}}''_{m}$ into $2^n$ dyadic cubes of side-length $2^{-1}\ell (K)$. Set

$$\begin{aligned} {\widetilde{\mathscr {D}}}={\widetilde{{\mathcal {L}}}}''\cup \bigcup _{m=1}^\infty {\widetilde{{\mathcal {L}}}}''_m, \ \ \ M_{\widetilde{\mathscr {D}}}f(x) = \sup _{x\in K\in {\widetilde{\mathscr {D}}}} \frac{1}{|K|}\int _{K} |f(x)|\, dx. \end{aligned}$$

The localized sharp function is defined by

$$\begin{aligned} f^{\#}_{{\mathrm{loc}},\,(0,2)^n}(x)=\sup _{x \in K \subset {\mathcal {B}}_x}\frac{1}{|K|}\int _{K}|f(y)-f_{K}|\,dy, \end{aligned}$$

where

$$\begin{aligned} {\mathcal {B}}_x={\left\{ y \in (0,2)^n: \frac{1}{4}\text { dist}\big (x,\partial [0,2]^n\big )\le \text { dist}\big (y,\partial [0,2]^n\big ) \le 4\text { dist}\big (x,\partial [0,2]^n\big )\right\} } \end{aligned}$$

and the supremum is taken over all cubes K not necessarily dyadic.

Our aim of this section is to prove the following theorem.

Theorem 4.1

For every $1\le p<\infty$ there is a constant $C_p>0$ such that if $f\in L^1((0,2)^n)$, $\int _{(0,2)^n} f(x)\, dx =0$ and $M_{\widetilde{\mathscr {D}}} f\in L^1((0,2)^n)$, then

$$\begin{aligned} \Vert M_{\widetilde{\mathscr {D}}} f\Vert _{L^p((0,2)^n)}\le C_p\Vert f^{\#}_{{\mathrm{loc}},\,(0,2)^n}\Vert _{L^p((0,2)^n)}. \end{aligned}$$

The proof requires preparations.

For each $K\in {\widetilde{{\mathcal {L}}}}''$ there is a unique $\sigma \in G$ such that $\sigma (K)\subset [0,1]^n$. Therefore in our considerations we shall deal with cubes contained in $[0,1]^n$ and then use the group action for other cubes.

From now on, let $Q_1=\frac{1}{2}Q_0+\frac{1}{2}{} \mathbf 1$.

For $x\in (0,1]^n$ let F(x) be defined by (3.2). Clearly, for every $R\in {\widetilde{{\mathcal {L}}}}'\cap {\widetilde{L_k}}$ with $k\ge 2$ there is a unique $K\in {\widetilde{{\mathcal {L}}}}'\cap {\widetilde{L_{k-1}}}$ such that $F(R)\subset K$. For $K\in {\widetilde{{\mathcal {L}}}}'$ we set

$$\begin{aligned} \text {Pre} (K)=\{ R\in {\widetilde{{\mathcal {L}}}}': F^j(R) \subset K \ \ \text { for a certain positive integer } j\}. \end{aligned}$$

Figure 6 shows examples of  $\bigcup \text { Pre}(K)$ for $n=2$. We have

$$\begin{aligned} \Big |\bigcup \text {Pre} (K)\Big |\le (2^{n}-1)|K|. \end{aligned}$$

(4.1)

Fig. 6

Cubes K and $\bigcup \text {Pre} (K)$ for $n=2$

Lemma 4.2

Let f be an integrable function on $(0,2)^n$. Assume that $K\in {\widetilde{{\mathcal {L}}}}'\cap {\widetilde{L}}_m$. Then

$$\begin{aligned} \begin{aligned} \Big |\bigcup \text { Pre}(K)\Big | f_K&\le \sum _{R\in {\widetilde{{\mathcal {L}}}}'\cap {\widetilde{L}}_{m+1}\cap \text { Pre}(K)} |R|f_R +3^{2n} \sum _{R\in {\widetilde{{\mathcal {L}}}}'\cap {\widetilde{L}}_{m+1}\cap \text { Pre}(K)} |R|\inf _{y\in R} f^{\#}_{{\mathrm{loc}},\,(0,2)^n}(y)\\&\quad + \sum _{R\in {\widetilde{{\mathcal {L}}}}'\cap {\widetilde{L}}_{m+1}\cap \text { Pre}(K)} \Big |\bigcup \text { Pre}(R)\Big |f_R. \end{aligned} \end{aligned}$$

Proof

Set $C_n=3^{2n}2^{-n}$. By the same arguments we used to prove (3.5), we get $f_K\le f_R+C_n\inf _{y\in R} f^{\#}_{{\mathrm{loc}},\,(0,2)^n}(y)$ for $R\in {\widetilde{L}}_{m+1}$ such that $F(R)\subset K$. Hence,

$$\begin{aligned} \begin{aligned}&\Big |\bigcup \text { Pre}(K)\Big |f_K \\&\quad =\sum _{R\in {\widetilde{{\mathcal {L}}}}'\cap {\widetilde{L}}_{m+1}\cap \text { Pre}(K)} \Big (|R|f_K + \Big |\bigcup \text { Pre}(R)\big | f_K\Big )\\&\quad \le \sum _{R\in {\widetilde{{\mathcal {L}}}}'\cap {\widetilde{L}}_{m+1}\cap \text { Pre}(K)} \Big (|R|f_R +C_n|R|\inf _{y\in R} f^{\#}_{{\mathrm{loc}},\,(0,2)^n}(y)\Big ) \\&\qquad + \sum _{R\in {\widetilde{{\mathcal {L}}}}'\cap {\widetilde{L}}_{m+1}\cap \text { Pre}(K)} \Big ( C_n \Big |\bigcup \text { Pre}(R)\Big | \inf _{y\in R}f^{\#}_{{\mathrm{loc}},\,(0,2)^n}(y) \Big )\\&\qquad + \sum _{R\in {\widetilde{{\mathcal {L}}}}'\cap {\widetilde{L}}_{m+1}\cap \text { Pre}(K)} \Big |\bigcup \text { Pre}(R)\Big |f_R,\\ \end{aligned} \end{aligned}$$

which, by (4.1), finishes the proof. $\square$

Corollary 4.3

Assume that f is an integrable function on $(0,2)^n$. Then

$$\begin{aligned} f_{Q_1} \le \int _{(0,1]^n} f(y)\, dy + 3^{2n} \int _{(0,1]^n} f^{\#}_{{\mathrm{loc}},\,(0,2)^n}(y)\, dy. \end{aligned}$$

(4.2)

Proof

Observe that $1=|Q_1|+\Big |\bigcup \text { Pre}(Q_1)\Big |$. Thus

$$\begin{aligned} f_{Q_1}=|Q_1|f_{Q_1}+\Big |\bigcup \text { Pre}(Q_1)\Big | f_{Q_1}. \end{aligned}$$

By iterating Lemma 4.2 we obtain that for every positive integer $m \ge 2$ one has

$$\begin{aligned} f_{Q_1}&\le \sum _{j=1}^{m}\sum _{K \in {\widetilde{\mathcal {L}}}' \cap \widetilde{L}_j}|K|f_{K}+3^{2n}\sum _{j=2}^{m}\sum _{K \in {\widetilde{\mathcal {L}}}' \cap {\widetilde{L}}_j}|K| \inf _{y \in K}f^{\#}_{{\mathrm{loc}},\,(0,2)^n}(y)\\ {}&\quad + \sum _{K \in {\widetilde{\mathcal {L}}}' \cap \widetilde{L}_m} \Big |\bigcup \text { Pre}(K)\big |f_{K}. \end{aligned}$$

Letting $m\rightarrow \infty$, we obtain the corollary, since the last summand tents to 0. $\square$

Corollary 4.4

There is a constant $C'>0$ such that for every integrable function f on $(0,2)^n$ such that $\int _{(0,2)^n}f(x)\, dx =0$ one has

$$\begin{aligned}&\sum _{\sigma \in G} f_{\sigma (Q_1)} \le C'\int _{(0,2)^n} f^{\#}_{{\mathrm{loc}},\,(0,2)^n}(y)\, dy, \end{aligned}$$

(4.3)

$$\begin{aligned}&\big | f_{\bigcup _{\sigma \in G}\sigma (Q_1)} \big |=2^{-n}\big | \sum _{\sigma \in G} f_{\sigma (Q_1)} \big | \le C'\int _{(0,2)^n} f^{\#}_{{\mathrm{loc}},\,(0,2)^n}(y)\, dy, \end{aligned}$$

(4.4)

$$\begin{aligned}&|f|_{\bigcup _{\sigma \in G}\sigma (Q_1)} \le C'\int _{(0,2)^n} f^{\#}_{{\mathrm{loc}},\,(0,2)^n}(y)\, dy, \end{aligned}$$

(4.5)

$$\begin{aligned}&|f|_{\sigma (Q_1)} \le C'\int _{(0,2)^n} f^{\#}_{{\mathrm{loc}},\,(0,2)^n}(y)\, dy. \end{aligned}$$

(4.6)

Proof

Clearly, $\sum _{\sigma \in G} \int _{\sigma ((0,1]^n)} f\,dy=\int _{(0,2)^n}f\,dy=0$. Hence (4.3) follows from (4.2). The inequality (4.4) is a direct consequence of (4.3). Further we write

$$\begin{aligned} \int \limits _{\bigcup \sigma (Q_1)}|f|\,dy\le & {} \int \limits _{\bigcup \sigma (Q_1)}|f-f_{\bigcup \sigma (Q_1)}|\,dy+ |f_{\bigcup \sigma (Q_1)}|\\\le & {} \inf _{x \in \bigcup \sigma (Q_1)} f_{{\mathrm{loc}},\,(0,2)^n}^{\#} (x)+|f_{\bigcup \sigma (Q_1)}| \end{aligned}$$

and apply (4.4) to obtain (4.5) and then (4.6). $\square$

Assume that $f\in L^1((0,2)^n)$. For $x\in (0,1]^n$ we define the function ${\mathbf {S}} f(x)$ as follows

$$\begin{aligned} {\mathbf {S}} f(x)=|f|_{Q_1}+ 2^{1-n}3^{2n}\sum _{j=0}^{k-1} f^{\#}_{{\mathrm{loc}},\,(0,2)^n}(F^j(x)) \ \ \text {for} \ x\in K\in {\widetilde{L}}_k\cap {\widetilde{{\mathcal {L}}}}'. \end{aligned}$$

Proposition 4.5

For every constant $0<b<1$ there is a constant $C>0$ such that for all $c,\alpha >0$, and every $f \in L^1((0,2)^n)$ we have

$$\begin{aligned}&\left| \{x \in (0,1]^n :M_{\widetilde{\mathscr {D}}}f(x)>\alpha , f^{\#}_{{\mathrm{loc}},\,(0,2)^n}(x)<c\alpha \}\right| \\&\quad \le Cc\left| \{x \in (0,1]^n:M_{\widetilde{\mathscr {D}}}f(x)>b\alpha \}\right| +\left| \{x \in (0,1]^n : {\mathbf {S}}f(x) \ge b\alpha \}\right| . \end{aligned}$$

Proof

For $x\in (0,1]^n$ let ${\widetilde{S}}f(x)$ be defined by (3.1). The same arguments we used to prove Lemma 3.3 give

$$\begin{aligned} {\widetilde{S}}f(x)\le {\widetilde{S}}(F(x))+2^{1-n}3^{2n}f^{\#}_{{\mathrm{loc}},\, (0,2)^n}(F(x)) \ \ \text { for} \ x\in (0,1]^n{\setminus } Q_1. \end{aligned}$$

(4.7)

Iteration of (4.7) leads to ${\widetilde{S}}f(x)\le {\mathbf {S}}f(x)$ for $x\in (0,1]^n$. Now the proof is the same as that of Proposition 2.4. $\square$

Proof of Theorem 4.1

For $f\in L^1_{{\mathrm{loc}},\,(0,2)^n}((0,2)^n)$ and $\sigma \in G$ let $f_\sigma (x)=f(\sigma (x))$. Since $M_{\widetilde{\mathscr {D}}} f_\sigma = (M_{\widetilde{\mathscr {D}}} f)_\sigma$, $(f_\sigma )_{{\mathrm{loc}},\,(0,2)^n}^{\#}=(f_{{\mathrm{loc}},\,(0,2)^n}^{\#})_\sigma$, and $(0,2)^n=\bigcup _{\sigma \in G} \sigma ((0,1]^n)$, it suffices to prove that

$$\begin{aligned} \Vert M_{\widetilde{\mathscr {D}}}f\Vert _{L^p((0,1]^n)} \le C_p \Vert f^{\#}_{{\mathrm{loc}},\,(0,2)^n}\Vert _{L^p((0,2)^n)} \end{aligned}$$

(4.8)

for $f\in L^1((0,2)^n)$, $\int _{(0,2)^n} f=0$. Repeating the proof of Theorem 2.1 with the use of Proposition 4.5 we arrive at

$$\begin{aligned} \begin{aligned} \Vert M_{\widetilde{\mathscr {D}}}f\Vert _{L^p((0,1]^n)}&\le C\Vert {\mathbf {S}}f\Vert _{L^p((0,1]^n)} + C\Vert f^{\#}_{{\mathrm{loc}},\,(0,2)^n}\Vert _{L^p((0,1]^n)}\\&\le C' |f|_{Q_1} +C' \Vert f^{\#}_{{\mathrm{loc}},\,(0,2)^n}\Vert _{L^p((0,1]^n)}.\\ \end{aligned} \end{aligned}$$

(4.9)

Recall that the integral of f is zero. Hence, applying (4.6), we obtain the desired inequality (4.8). $\square$