The weak type inequality for maximal operator of Fej$ \acute{\mbox{e}} $r means for trigonometric system can be found in Zygmund [1], in Schipp [2] for Walsh system and in P$ \acute{\mbox{a}} $l, Simon [3] for bounded Vilenkin system. Later, Schipp [2] showed that maximal operator $ \sigma^{\ast}f:=\sup\limits_{n}| \sigma_{n}f| $ is of weak type (1, 1), from which the a.e. convergence follows by standard argument. Schipp's result implies by interpolation also the boundedness of $ \sigma^{\ast}: L^{p}\rightarrow L^{p}(1<p\leqslant \infty) $. This fails to hold for $ p=1 $, but Fujii [4] proved that $ \sigma^{\ast} $ is bounded from the dyadic Hardy space $ H^{1} $ to $ L^{1} $ (see also Simon [5]). Fujii's results were extended by Wesiz [6], [7] to $ H^p $ spaces for $ 1/2<p\leqslant 1 $, in the two-dimensional case, too. Simon [8] gave a counterexample, which shows that boundedness of $ \sigma^* $ does not hold for $ 0< p <1/2 $. The counterexample for $ \sigma^* $ when $ p= 1/2 $ is due to Goginava [9]. Goginava [10] proved that the maximal operator $ \tilde{\sigma}^* $ defined by

$ \tilde{\sigma}^*f=\sup\limits_{n\in {\bf{N}}}\frac{|\sigma_nf|}{\log^2(n+1)} $

is bounded from the Hardy space $ H^{1/2} $ to the space $ L^{1/2} $ for Walsh system. He also proved, that for any nondecreasing function $ \varphi:{\bf{N}}\rightarrow [1, \infty) $, satisfying the condition

$ \overline{\lim\limits_{n\rightarrow \infty}}\frac{\log^2(n+1)}{\varphi(n)}=+\infty, $

the maximal operator

$ \sup\limits_{n\in {\bf{N}}}\frac{|\sigma_nf|}{\varphi(n)} $

is not bounded from the Hardy space $ H^{1/2} $ to the space $ L^{1/2} $. Tephnadze [11] generalized this result and proved the boundedness of

$ \sup\limits_{n\in {\bf{N}}}\frac{|\sigma_nf|}{(n+1)^{1/p-2}} $

is bounded from the martingale Hardy space $ H^p $ to the space $ L^p $, where $ \sigma_nf $ is $ n $-th Fejér mean with respect to bounded Vilenkin system for $ 0<p<1/2 $.

In this paper the two-dimensional case will be investigated with respect to Vilenkin-like system. We show that the boundedness of some maximal operators. Throughout this paper, we denote the set of integers and the set of non-negative integers by $ {\bf{Z}} $ and $ {\bf{N}} $, respectively. We use $ c, c_p, C_p $ to denote constants and may denote different constants at different occurrences.

Let $ m:=(m_0, m_1, \cdots, m_k, \cdots) $ be sequence of natural numbers such that $ m_k \geq 2 (k\in {\bf{N}} ) $. For all $ k\in {\bf{N}} $ we denote by $ Z_{m_k} $ the $ m_k $-th discrete cyclic group. Let $ Z_{m_k} $ be represented by $ \{0, 1, \cdots, m_k-1 \} $. Suppose that each (coordinate) set has the discrete topology and the measure $ \mu_k $ which maps every singleton of $ Z_{m_k} $ to $ 1/m_k\ (u_k(Z_{m_k})=1) $ for $ k\in {\bf{ N}} $. Let $ G_m $ denote the complete direct product of $ Z_{m_k} $'s equipped with product topology and product measure $ \mu $, then $ G_m $ forms a compact Abelian group with Haar measure 1. The elements of $ G_m $ are sequences of the form $ (x_0, x_1, \cdots, x_k, \cdots), $ where $ x_k\in Z_{m_k} $ for every $ k\in {\bf{N}} $ and the topology of the group $ G_m $ is completely determined by the sets

$ I_n(0):=\{ (x_0, x_1, \cdots, x_k, \cdots)\in G_m:x_k=0\ (k=0, \cdots, n-1)\} $

$ (I_0(0):=G_m). $ The Vilenkin space $ G_m $ is said to be bounded if the generating system $ m $ is bounded. We assume $ q=\sup_{i}\{m_i\}<\infty. $

Let $ M_0:=1 $ and $ M_{k+1}:=m_kM_k $ for $ k\in {\bf{N}} $, it is so-called the generalized powers. Then every $ n\in {\bf{N}} $ can be uniquely expressed as $ n=\sum\limits_{k=0}^{\infty} n_kM_k, 0\leq n_k<m_k, n_k\in {\bf{N}}. $ The sequence $ (n_0, n_1, \cdots) $ is called the expansion of $ n $ with respect to $ m $. We often use the following notations: $ |n|:=\max\{k\in {\bf{N}}:n_k\neq0\} $ (that is, $ M_{|n|}\leq n<M_{|n|+1}) $ and $ n^{(k)}=\sum\limits_{j=k}^\infty n_jM_j$.

For $ k\in{\bf{N}} $ and $ x\in G_m $ denote $ r_k $ the $ k $-th generalized Rademacher function:

$ r_k(x):=\exp(2\pi\iota \frac{x_k}{m_k})\quad (x\in G_m, \iota:=\sqrt{-1}, k\in{\bf{N}}). $

It is known that for $ x\in G_m , n\in {\bf{N}} $

$ \begin{equation} \sum\limits_{i=0}^{m_n-1}r_n^i(x)=\left \{ \begin{array}{ll}0 & \mbox{if}\ x_n\neq 0, \\ m_n &\mbox{if}\ x_n=0. \end{array}\right. \end{equation} $

(2.1)

Now we define the $ \psi_n $ by

$ \psi_n:=\prod\limits_{k=0}^\infty r_k^{n_k}(n\in{\bf{N}}). $

Then $ \{ \psi_n: n\in {\bf{N}}\} $ is a complete orthonormal system with respect to $ \mu$.

We introduce the so-called Vilenkin-like $ ( $or $ \psi\alpha) $ system (see [12]). Let functions $ \alpha_n, \alpha_j^k : G_m\rightarrow \mathcal{C} $$ (n, j, k\in {\bf{N}}) $ satisfy for all $ x, y\in G_m $:

(1) $\alpha_j^k$ is measurable with respect t $\Sigma_j$ and $\alpha_j^k(x+y)=\alpha_j^k(x)\alpha_j^k(y)$;

(2) $|\alpha_j^k|=\alpha_j^k(0)=\alpha_0^k=\alpha_j^0=1\ \ (j, k\in {\bf{N}})$;

(3) $\alpha_n:=\prod\limits_{j=0}^\infty \alpha_j^{n^{(j)}}\ \ (n\in {\bf{N}})$.

Let $ \chi_n:=\psi_n\alpha_n\ (n\in {\bf{N}}) $. The system $ \chi:=\{\chi_n:n\in{\bf{N}}\} $ is called a Vilenkin-like $ ( $or $ \psi\alpha) $ system.

Define Dirichlet kernels and Fejér kernels with respect to Vilenkin-like system and Vilenkin system as follows.

$ \begin{eqnarray*} D_n(y, x)=\sum\limits_{k=0}^{n-1}\chi_k(y)\overline{\chi}_k(x), \quad D_n(x)=\sum\limits_{k=0}^{n-1}\psi_k(x), \\ \quad K_n(y, x)=\frac{1}{n}\sum\limits_{k=0}^{n-1}D_k(y, x), \quad K_n(x)=\frac{1}{n}\sum\limits_{k=0}^{n-1}D_k(x). \end{eqnarray*} $

It's well known that

$ \begin{equation} D_{M_n}(y, x)=D_{M_n}(y-x)=\left \{ \begin{array}{ll}M_n & \mbox{if}\ y-x\in I_n, \\ 0&\mbox{if}\ y-x\in G_m \backslash I_n. \end{array}\right. \end{equation} $

(2.2)

Moreover for $ y, x\in G_m $,

$ \begin{eqnarray} D_n(y, x)=\alpha_n(y)\bar{\alpha}_n(x)D_n(y-x) =\chi_n(y)\bar{\chi}_n(x)(\sum\limits_{j=0}^\infty D_{M_j}(y-x)\sum\limits_{k=m_j-n_j}^{m_j-1}r_j^k(y-x)). \end{eqnarray} $

(2.3)

Since $ \alpha_j^k(x+y)=\alpha_j^k(x)\alpha_j^k(y) $ and $ r_j(x+y)=r_j(x)r_j(y) $, we have

$ \begin{eqnarray} \chi_n(y)\bar{\chi}_n(x)&=&\chi_n(y-x+x)\bar{\chi}_n(x)=\chi_n(y-x)\chi_n(x)\bar{\chi}_n(x)\\&=&\chi_n(y-x)|\chi_n(x)|^2=\chi_n(y-x)\bar{\chi}_n(0). \end{eqnarray} $

(2.4)

Thus we obtain

$ \begin{equation} D_n(y, x)=D_n(y-x, 0)\quad \mbox{ and}\quad K_n(y, x)=K_n(y-x, 0). \end{equation} $

(2.5)

Now we define $ \chi_{n, m}(x, y):=\chi_{n}(x)\chi_{m}(y), (x, y\in G_m) $. If $ f\in L^1 $ then the number $ \hat{f}(n, m):=E(f\chi_{n, m}) $ is said to be the $ (n, m) $-th coefficient of $ f $ with respect to system $ \chi. $ Denote by $ S_{n, m}f $ the $ (n, m) $-th partial sum of the Fourier series of a martingale $ f $ with respect to character system $ \chi $, namely,

$ S_{n, m}f:=\sum\limits_{k=0}^{n-1}\sum\limits_{l=0}^{m-1}\hat{f}(k, l)\chi_{k, l}. $

It is easy to see that

$ S_{M_n, M_m}f=f_{n, m}. $

Let $ \mathcal{F}_{n, m}(n, m\in{\bf{N}}) $ be the $ \sigma $-algebra generated by the rectangles $ I_{n, m}(x, y):=I_n(x)\times I_m(y), $$ (x, y\in G_m). $ A sequence of integrable functions $ f=(f_{n, m};n, m\in{\bf{N}}) $ is said to be a martingale if $ f_{n, m} $ is $ \mathcal{F}_{n, m} $ measurable for all $ n, m\in{\bf{N}} $ and $ S_{M_{n}, M_{m}}f_{k, l}=f_{n, m} $ for all $ n, m, k, l\in{\bf{N}} $ such that $ n\leqslant k $ and $ m\leqslant l $.

We say that a martingale $ f=(f_{n, m};n, m\in{\bf{N}}) $ is $ L^{p} $-bounded if $ \|f\|_p:=\sup_{n, m}\parallel f_{n, m}\|_{p}<\infty $. The set of the $ L^{p} $-bounded martingales will be denoted by $ L^{p}(G_m^{2}) $.

The diagonal maximal function of a martingale $ f=(f_{n, m};n, m\in{\bf{N}}) $ is defined by

$ f^{\ast}:=\sup\limits_{n\in {\bf{N}}}| f_{n, n}|. $

It is easy to see that in case when $ f $ is an integrable real valued function given on $ G_m^{2} $, the above maximal functions can be computed for all $ x, y\in G_m $ by

$ f^{\ast}(x, y)=\sup\limits_{n\in {\bf{N}}}\frac{1}{| I_{n, n}(x, y)|}|\int_{I_{n, n}(x, y)}f|. $

Define the spaces $ H^{p}(G_m^{2}) $ of Hardy type as the set of martingales $ f $ such that

$ \| f\|_{H^{p}(G_m^2)}:=\| f^{\ast}\|_{p}<\infty . $

The martingale Hardy spaces $ H^p(G^2_m) (0<p\leqslant 1) $ have atomic characterizations. A bounded measurable function $ a $ defined on $ G_m^{2} $ is a $ p $-atom if $ a\equiv1 $ or there exists a dyadic square $ I $ such that

$ \mbox{supp}\ a\subset I, \| a\|_{\infty}\leqslant | I|^{-1/p}, \int\int a\equiv0. $

We shall say also that $ a $ is supported on $ I $. Then a martingale $ f=(f_{n, m};n, m\in{\bf{N}}) $ is in $ H^{p}(G_m^{2}) $ if there exists a sequence $ (a_{k}, k\in{\bf{N}}) $ of $ p $-atoms and a sequence $ (\lambda_{k}, k\in{\bf{N}}) $ of real numbers such that $ \sum\limits_{k=0}^{\infty}| \lambda_{k}|^{p}<\infty $ and

$ \begin{equation} \sum\limits_{k=0}^{\infty}\lambda_{k}S_{M_{n}, M_{n}}a_{k}=f_{n, n}\quad(n\in{\bf{N}}). \end{equation} $

(2.6)

Moreover, $ c_{p}\inf(\sum\limits_{k=0}^{\infty}|\lambda_{k}|^{p})^{1/p}\leqslant\| f\|_{H^{p}}\leqslant C_{p}\inf(\sum\limits_{k=0}^{\infty}|\lambda_{k}|^{p})^{1/p} $, where the infimum is taken over all decompositions of $ f $ of the form (2.6).

Next we will consider the boudedness of operator $ \tilde{\sigma}^*f $ and $ \hat{\sigma}^*f $ in the two-dimensional Vilenkin-like system, where $ \tilde{\sigma}^*f=\sup\limits_{2^{-\alpha}\leqslant\frac{n}{m}\leqslant 2^{\alpha}}\frac{|\sigma_{n, m}f|}{[(n+1)(m+1)]^{1/p-2}} $, $ \hat{\sigma}^*f=\sup\limits_{n, m\in {\bf{N}}}\frac{|\sigma_{n, m}f|}{[(n+1)(m+1)]^{1/2p-1}}. $

Lemma 3.1 ([13]) Suppose that the operator $ T $ is sublinear and for $ 0<p\leq 1 $, there exists a constant $ C_p>0 $ such that

$ \begin{equation} \int_{G_m\setminus I}| Ta|^{p}\leqslant C_{p}, \end{equation} $

(3.1)

for every $ p $-atom $ a\in H^{p} $ supported on the dyadic interval $ I $. If $ T $ is bounded from $ L^{s} $ into $ L^{s} $ for some $ 1\leqslant s\leqslant\infty $, then

$ \|Tf\|_p\leq C_p\|f\|_{H^p}\quad (f\in H^p\cap L^1). $

If (3.1) is true, $ T $ is called $ p $-quasi-local.

Lemma 3.2 ([13]) Let $ 0<p<1, 1<s\leq\infty $ and assume that the sublinear operator $ T $ is $ p $-quasi-local and $ (L^{s}, L^{s}) $-bounded. Then $ T:H^{u, v}\rightarrow L^{u, v} $ is bounded for all $ p<u<s $ and $ 0<v\leqslant\infty. $ Especially, $ T $ is of weak type (1, 1).

Further we assume that for all $ n\in{\bf{N}} $ the kernel $ P_{n}\in L^{\infty} $ is given such that $ \sup\limits_{n}\| P_{n}\|_{1}<\infty $. If we consider the maximal operator

$ Tf:=\sup\limits_{n}| f\ast P_{n}| \quad(f\in L^{1}), $

then $ T:L^{\infty}\rightarrow L^{\infty} $ is evidently bounded. Therefore, if $ T $ is $ p $-quasi-local for some $ 0<p<1 $, then Lemma 3.2 can be applied to $ T $.

Lemma 3.3 If $ P_{n} $ is a $ summation $ $ kernel $, i.e. with suitable real coefficients $ \lambda_{n, k}(n, k\in{\bf{N}}) $

$ P_{n}(x, 0)=\sum\limits_{k=0}^{n}\lambda_{n, k}\chi_{k}(x, 0)\quad (n\in {\bf{N}}), $

then the assumption

$ \begin{equation} \int_{G_m\setminus I_{N}}(\sup\limits_{n\geqslant M_{N}}\int_{I_{N}}| P_n(x-t, 0)| dt)^{p}dx\leqslant C_{p}\frac{1}{M_{N}}\quad(n\in{\bf{N}}) \end{equation} $

(3.2)

implies the $ p $-quasi-locality of $ T $.

Proof Indeed, to prove (3.1) let $ a $ be a $ p $-atom supported on the interval $ I $. Without loss of generality we can assume that $ I=I_{N} $ for some $ N\in{\bf{N}} $. Then $ a\ast P_{n}=0 $ holds for all $ n=0, \ldots, M_{N}-1 $, since the functions $ \chi_{k}(k=0, \ldots, M_{N}-1) $ are constant on $ I $. Therefore, $ Ta=\sup\limits_{n\geqslant M_{N}}| a\ast P_{n}| $ and thus

$ \begin{eqnarray} \int_{G_m\setminus I_{N}}(Ta(x))^{p}dx&=&\int_{G_m\setminus I_{N}}(\sup\limits_{n\geqslant M_{N}}|\int_{I_{N}}a(t)P_n(x-t, 0)dt|)^{p}dx\\ &\leqslant& \| a\|_{\infty}^{p}\int_{G_m\setminus I_{N}}(\sup\limits_{n\geqslant M_{N}}\int_{I_{N}}| P_n(x-t, 0)| dt)^{p}dx\\ &\leqslant&M_{N}\int_{G_m\setminus I_{N}}(\sup\limits_{n\geqslant M_{N}}\int_{I_{N}}| P_n(x-t, 0)| dt)^{p}dx. \end{eqnarray} $

(3.3)

Hence, (3.1) follows from (3.2) and (3.3).

Lemma 3.4 ([14]) Let $ z\in I_{N}^{k, l}, k=0, \cdots, N-2, l=k+1, \cdots, N-1 $ and $ n\geqslant M_N. $ Then

$ \begin{equation} \int_{I_N}|K_n(z-t, 0)|d\mu(t)\leqslant \frac{cM_l M_k}{nM_N}. \end{equation} $

(3.4)

Let $ z\in I_{N}^{k, N}, k=0, \cdots, N-1 $ and $ n\geq M_N. $ Then

$ \begin{equation} \int_{I_N}|K_n(z-t, 0)|d\mu(t)\leqslant \frac{c M_k}{M_N}, \end{equation} $

(3.5)

where $ c $ is an absolute constant and

$ \begin{equation*} I_{N}^{k, l}=\left \{ \begin{array}{ll}I_N(0, \cdots, 0, x_k\neq 0, 0\cdots, 0, x_l\neq 0, x_{l+1}, \cdots x_{N-1}, \cdots) & \mbox{if}\ k<l<N, \\ I_N(0, \cdots, 0, x_k\neq 0, x_{k+1}=0, \cdots, x_{N-1}=0, x_N \cdots) &\mbox{if}\ l=N. \end{array}\right. \end{equation*} $

.

Lemma 3.5 ([14]) Let $ 2< A\in \mathbb{N}_+, k\leq s < A $, $ n^*_A:=M_{2A}+M_{2A-2}+\cdots+M_2+M_0. $ Then we have

$ n^*_{A-1}|K_{n^*_{A-1}}(z, 0)|\geq \frac{M_{2k}M_{2s}}{4}, $

for $ z\in I^{2k, 2s}_{2A}, k= 0, 1, \cdots, A-3, s=k+ 2, k+ 3, \cdots, A-1$.

If $ I:=I\times J $ is a dyadic square and let $ I^{r}:=I^{r}\times J^{r}. $ Then it is not hard to see that the definition of the $ p $-quasi-locality of $ T $ can be modified as follows: there exists $ r=0, 1\ldots $ such that

$ \begin{equation} \int_{G_m^2\setminus I^{r}}| Ta|^{p}\leqslant C_{p} \end{equation} $

(3.6)

holds for every $ p $-atom $ a $ supported on the dyadic square $ I $.

Let $ P_{n, m}(n, m\in{\bf{N}}) $ be the Kronecker product of $ P_{n} $ and $ P_{m} $, i.e. $ P_{n, m}(x_1, 0, x_2, 0):=P_{n}(x_1, 0)P_{m}(x_2, 0) $ and for a fixed $ \alpha\geqslant0 $ define $ T_\alpha $ by

$ T_{\alpha}f:=\sup\limits_{2^{-\alpha}\leqslant\frac{n}{m}\leqslant 2^{\alpha}}| f\ast P_{n, m}|. $

Theorem 4.1 Assume (3.2) for a given $ 0<p\leqslant 1 $. Then $ T_\alpha $ is $ p $-quasi-local.

Proof It is enough to prove (3.6) with a suitable $ r\in {\bf{N}} $. To this end let $ a\in L^{\infty}(G_m^{2}) $ be a $ p $-atom. We can assume that $ a $ is supported on the dyadic square $ I_{N}\times I_{N} $ for some $ N\in {\bf{N}} $. Furthermore, it follows that $ a\ast P_{n, m}=0 $ when $ n, m < M_{N}. $ Therefore, to compute $ T_\alpha a=\sup\limits_{ 2^{-\alpha}\leqslant\frac{n}{m}\leqslant 2^{\alpha}}| a\ast P_{n, m}| $ it can be assumed $ n\geqslant M_{N} $ or $ m \geqslant M_{N} $. In the first case $ m\geqslant M_{N-r} $, while in the second case $ n\geqslant M_{N-r} $ follows. In other words, we get the estimate

$ T_\alpha a\leqslant \sup\limits_{n, m\geqslant M_{N-r}}| a\ast P_{n, m}|, $

where $ r\in {\bf{N}} $ is determined by $ r-1\leqslant\alpha<r $. Here, $ \| a\|_{\infty} \leqslant M_{N}^{\frac{2}{p}} $ implies

$ \begin{eqnarray} T_\alpha a(x, y)&\leqslant& \sup\limits_{n, m\geqslant M_{N-r}}|\int_{I_{N}}\int_{I_{N}}a(u, v)P_n(u-x, 0)P_m(v-y, 0)dudv|\\ &\leqslant& M_{N}^{\frac{2}{p}} \sup\limits_{n, m\geqslant M_{N-r}}\int_{I_{N}}| P_n(u-x, 0)| du\int_{I_{N}}| P_m(v-y, 0)| dv. \end{eqnarray} $

(4.1)

Therefore, to verity (3.6) it is enough to show that

$ \begin{eqnarray} \int_{G_m^{2} \setminus (I_{N-r}\times I_{N-r})}(\sup\limits_{n, m\geqslant M_{N-r}}\int_{I_{N}}| P_n(u-x, 0)| du\int_{I_{N}}| P_m(v-y, 0)| dv)^{p}dxdy\leqslant\frac{ C_{p}}{M_{N}^{2}}. \end{eqnarray} $

(4.2)

To this end let us decompose the double integral in question as follows:

$ \begin{eqnarray} &&\int_{G_m^{2}\setminus(I_{N-r}\times I_{N-r})}(\sup\limits_{n, m\geqslant M_{N-r}}\int_{I_{N}}| P_n(u-x, 0)| du\int_{I_{N}}| P_m(v-y, 0)| dv)^{p}dxdy\\ &=&\int_{G_m\setminus I_{N-r}}\int_{I_{N-r}}(\sup\limits_{n, m\geqslant M_{N-r}}\int_{I_{N}}| P_n(u-x, 0)| du\int_{I_{N}}| P_m(v-y, 0)| dv)^{p}dxdy\\ &&+\int_{I_{N-r}}\int_{G_m\setminus I_{N-r}}(\sup\limits_{n, m\geqslant M_{N-r}}\int_{I_{N}}| P_n(u-x, 0)| du\int_{I_{N}}| P_m(v-y, 0)| dv)^{p}dxdy\\ &&+\int_{G_m\setminus I_{N-r}}\int_{G_m\setminus I_{N-r}}(\sup\limits_{n, m\geqslant M_{N-r}}\int_{I_{N}}| P_n(u-x, 0)| du\int_{I_{N}}| P_m(v-y, 0)| dv)^{p}dxdy\\ &=:&A_{1}+A_{2}+A_{3}. \end{eqnarray} $

(4.3)

Here $ A_{1} $ can be estimated in the following way:

$ \begin{eqnarray} A_{1}&\leqslant&\int_{G_m\setminus I_{N-r}}(\sup\limits_{n\geqslant M_{N-r}}\int_{I_{N}}| P_n(u-x, 0)| du)^{p}dx\int_{I_{N}}(\sup\limits_{m}\int_{G}| P_m(v-y, 0)| dv)^{p}dy\\ &\leqslant& \int_{G_m\setminus I_{N-r}}(\sup\limits_{n\geqslant M_{N-r}}\int_{I_{N}}| P_n(u-x, 0)| du)^{p}dx| I_{N}|(\sup\limits_{m}\|P_m\|_{1})^{p}\\ &\leqslant& C_{p}\frac{1}{M_{N}}\int_{G_m\setminus I_{N-r}}(\sup\limits_{n\geqslant M_{N-r}}\int_{I_{N-r}}| P_n(u-x, 0)| du)^{p}dx. \end{eqnarray} $

(4.4)

Thus we get

$ A_{1}\leqslant C_{p}\frac{1}{M_{N}}\frac{1}{M_{N-r}}\leqslant \frac{C_{p}}{M_{N}^{2}}. $

The estimate $ A_{2}\leqslant \frac{C_{p}}{M_{N}^{2}} $ can be derived similarly. Finally, applying (3.2) twice the estimation

$ \begin{eqnarray} A_{3}\leqslant(\int_{G_m\setminus I_{N-r}}(\sup\limits_{k\geqslant M_{N-r}}\int_{I_{N}}| P_n(u-x, 0)| du)^{p}dx)^{2}\leqslant C_{p}\frac{1}{M_{N}^{2}} \end{eqnarray} $

(4.5)

follows, which proves Theorem 4.1.

Theorem 4.2 Let $ \tilde{\sigma}^*f=\sup\limits_{2^{-\alpha}\leqslant\frac{n}{m}\leqslant 2^{\alpha}}\frac{|\sigma_{n, m}f|}{[(n+1)(m+1)]^{1/p-2}}. $ Then for all $ 0<p<1/2 $ we have

$ \|\tilde{\sigma}^*f\|_{p}\leqslant C_{p}\| f\|_{p}\quad (f\in L^{p}(G_m^{2})). $

Proof Let $ P_{n}(x, 0)=\sum\limits_{k=0}^{n}\frac{1}{(n+1)^{1/p-2}}K_n(x, 0) $. By Theorem 4.1, it is enough to prove (3.2) for $ P_{n}(x, 0) $. Let $ z\in I_{N}^{k, l} $, $ 0\leqslant k<l\leqslant N $. From Lemma 3.4 and $ 1/p-2>0 $ we get

$ \begin{eqnarray} \sup\limits_{n\geqslant M_N}\frac{1}{(n+1)^{1/p-2}}\int_{I}|K_n(z-t, 0)|dt \leq c \frac{1}{M_N^{1/p-2}} \frac{M_l M_k}{nM_N} \leq c\frac{ M_l M_k}{M_N^{1/p}} . \end{eqnarray} $

(4.6)

Thus we obtain

$ \begin{eqnarray} &&\int_{G_m\setminus I_{N}}(\sup\limits_{n\geqslant M_{N}}\int_{I_{N}}| P_n(x-t, 0)| dt)^{p}dx\\ &=&\int_{G_m\setminus I_{N}}(\sup\limits_{n\geqslant M_{N}}\frac{1}{(n+1)^{1/p-2}}\int_{I_{N}}| K_n(x-t, 0)| dt)^{p}dx\\ &=&\sum\limits_{k=0}^{N-2}\sum\limits_{l=k+1}^{ N-1}\sum\limits_{x_j=0, j\in \{l+1, \cdots, N-1\}}^{m_j-1} \int_{I_N^{k, l}}(\sup\limits_{n\geqslant M_{N}}\frac{1}{(n+1)^{1/p-2}}\int_{I_{N}}| K_n(x-t, 0)| dt)^pd\mu(z)\\ &&+\sum\limits_{k=0}^{N-1}\int_{I_N^{k, N}}(\sup\limits_{n\geqslant M_{N}}\frac{1}{(n+1)^{1/p-2}}\int_{I_{N}}| K_n(x-t, 0)| dt)^pd\mu(z)\\ &\leq & c\sum\limits_{k=0}^{N-2}\sum\limits_{l=k+1}^{ N-1}\frac{m_l\cdots m_N-1}{M_N}(\frac{M_lM_k}{M_N^{1/p}})^p+\sum\limits_{k=0}^{N-1}\frac{1}{M_N}(\frac{M_NM_k}{M_n^{1/p}})^p\\&\leq & c\sum\limits_{k=0}^{N-2}\sum\limits_{l=k+1}^{ N-1}\frac{(M_lM_k)^p}{M_lM_N}+\sum\limits_{k=0}^{N-1}\frac{1}{M^2_N}(M_NM_k)^p\\ &= & c\frac{1}{M_N}(\sum\limits_{k=0}^{N-2}\sum\limits_{l=k+1}^{ N-1}\frac{1}{M_l^{1-2p}}\frac{(M_lM_k)^p}{M^{2p}_l}+\sum\limits_{k=0}^{N-1}\frac{1}{M_N^{1-2p}}\frac{(M_NM_k)^p}{M_N^{2p}})\\ &= & c\frac{1}{M_N}(\sum\limits_{k=0}^{N-2}\sum\limits_{l=k+1}^{ N-1}\frac{1}{2^{(1-2p)l}}+\sum\limits_{k=0}^{N-1}\frac{1}{2^{N(1-2p)}}) \quad \\ &= & c\frac{1}{M_N}(\sum\limits_{k=0}^{N-2}\frac{1}{2^{(1-2p)k}}+\frac{N}{2^{N(1-2p)}})\\&\leq &\frac{c}{M_N}, \end{eqnarray} $

(4.7)

which complete the proof of Theorem 4.2.

By Lemma 3.2 and Theorem 4.2, we easily get Theorem 4.3, we omit the proof.

Theorem 4.3 Let $ 0<p<1/2 $. Then $ \tilde{\sigma}^*:H^{u, v}(G_m^2)\rightarrow L^{u, v}(G_m^2) $ is bounded for all $ p<u<\infty $ and $ 0<v\leqslant\infty. $ Especially, $ \tilde{\sigma}^* $ is of weak type (1, 1).

Theorem 4.4 Let $ 0<p< 1/2 $. Then the two dimensional maximal operator $ \hat{\sigma}^* $ defined by $ \hat{\sigma}^*f=\sup\limits_{n, m\in {\bf{N}}}\frac{|\sigma_{n, m}f|}{[(n+1)(m+1)]^{1/2p-1}} $ is not bounded from $ H^{p}(G_m^2) $ to $ L^{p}(G_m^2) $.

Proof Let $ A\in \mathbb{N} $ and

$ f_A(x, y):=(D_{M_{2A+1}}(x, 0)-D_{M_{2A}}(x, 0))(D_{M_{2A+1}}(y, 0)-D_{M_{2A}}(y, 0)). $

It is simple to calculate

$ \begin{eqnarray} \hat{f}_A(i, j)=\left \{ \begin{array}{ll}1, & \mbox{if}\ i, j=M_{2A}, M_{2A}+1, \cdots, M_{2A+1}-1\\ 0, &\mbox{}\ \mbox{otherwise} \end{array}\right. \end{eqnarray} $

(4.8)

and

$ \begin{eqnarray} S_{k, l}(f_A;x, y)) &=&\left \{ \begin{array}{lll}(D_k(x, 0)&-D_{M_{2A}}(x, 0))(D_l(y, 0)-D_{M_{2A}}(y, 0)), \\& \mbox{if}\ k, l=M_{2A}, M_{2A}+1, \cdots, M_{2A+1}-1\\ f_A(x, y), & \mbox{if}\ k, l\geq M_{2A+1} \\ 0, & \ \mbox{otherwise}. \end{array}\right. \end{eqnarray} $

(4.9)

We have

$ \begin{eqnarray*} f_A^*&=&\sup\limits_k|S_{M_k, M_k}(f_A;x, y)|=|f_A(x, y)|, \\ \|f_A\|_{H^p}&=&\|f^*_A\|_p=\|f_A\|_p\\ &=&\biggl(\int_{G_m}\bigl(D_{M_{2A+1}}(x, 0)-D_{M_{2A}}(x, 0)\bigl)^pdx\int_{G_m}\bigl(D_{M_{2A+1}}(y, 0)-D_{M_{2A}}(y, 0)\bigl)^pdy\biggl)^{1/p}\\ &=&\biggl(\int_{G_m}\bigl(D_{M_{2A+1}}(x)-D_{M_{2A}}(x)\bigl)^pdx\biggl)^{2/p}\\ &=& \biggl(\int_{I_{2A+1}}\bigl(D_{M_{A+1}}(x)-D_{M_{2A}}(x)\bigl)^pdx+\int_{I_{2A}\backslash I_{2A+1}}\bigl(D_{M_{2A+1}}(x)-D_{M_{2A}}(x)\bigl)^pdx\biggl)^{2/p}\\ &\leq&[\frac{m_{2A-1}}{M_{2A+1}}M_{2A}^p+\frac{(m_{2A}-1)^p M_{2A}^p}{M_{2A+1}}]^{2/p}\\ &\leq& cM_{2A}^{2(1-1/p)}. \end{eqnarray*} $

Since

$ \begin{equation} D_{i+M_A}(x, 0)-D_{M_A}(x, 0)=\chi_{M_A}(x)D_k(x, 0) \end{equation} $

(4.10)

we have

$ \begin{eqnarray} \hat{\sigma}^*f &=&\sup\limits_{n, m\in {\bf{N}}}\frac{|\sigma_{n, m}f|}{[(n+1)(m+1)]^{1/2p-1}}\geq |f_{A}\ast\frac{ K_{n_A^*, n_A^*}}{(n_A^*+1)^{(1/p-2)}}|\notag\\ &=&\frac{1}{(n_{A}^*)^2(n_{A}^*+1)^{(1/p-2)}}|\sum\limits_{i=0}^{n_{A}^*-1}\sum\limits_{j=0}^{n_{A}^*-1}S_{i, j}f_{A}|\\ &=&\frac{1}{(n_{A}^*)^2(n_{A}^*+1)^{(1/p-2)}}|\sum\limits_{i=M_{2A}+1}^{n_{A}^*-1}\sum\limits_{j=M_{2A}+1}^{n_{A}^*-1}S_{i, j}f_{A}|\notag \\ &=&\frac{1}{(n_{A}^*)^2(n_{A}^*+1)^{(1/p-2)}}|\sum\limits_{i=M_{2A}+1}^{M_{2A+1}-1}\sum\limits_{j=M_{2A}+1}^{M_{2A+1}-1}(D_i(x, 0)-D_{M_A}(x, 0))(D_j(y, 0)-D_{M_A}(y, 0))|\notag\\ &=&\frac{1}{(n_{A}^*)^2(n_{A}^*+1)^{(1/p-2)}}|\sum\limits_{i=1}^{n_{A-1}^*-1}\sum\limits_{j=1}^{n_{A-1}^*-1}(D_{i+M_A}(x, 0)-D_{M_A}(x, 0))(D_{j+M_A}(y, 0)-D_{M_A}(y, 0 ))|\\ &=&\frac{(n_{A-1}^*)^2}{(n_{A}^*)^2(n_{A}^*+1)^{(1/p-2)}}|K_{n_{A-1}^*}(x, 0)K_{n_{A-1}^*}(y, 0)|.\qquad\qquad\qquad\qquad\qquad\qquad\qquad \end{eqnarray} $

(4.11)

Let $ q=\sup_{i}\{m_i\} $. For every $ l=1, \cdots, [\frac{1}{4}\log_q(\sqrt{A^{1/2p}})]-1 $ ($ A $ is supposed to be large enough) let $ k_l $ be the smallest natural numbers, for which $ M_{2A}\sqrt{A^{1/2p}}\frac{1}{q^{2l/p}}\leq M^2_{2k_l}<M_{2A}\sqrt{A^{1/2p}}\frac{1}{q^{(2l-2)/p}} $ holds.

Suppose $ x, y \in I_{2A}^{k_l, k_l+1}:= I_{2A}(0, \cdots, 0, z_{2k_l}\neq 0, z_{2k_l+1}\neq 0, z_{2s+1}, \cdots, z_{2A-1}) $, then by Lemma 3.5 we have

$ \begin{eqnarray*} \hat{\sigma}^*f&\geq& \frac{(n_{A-1}^*)^2}{(n_{A}^*)^2(n_{A}^*+1)^{(1/p-2)}}|K_{n_{A-1}^*}(x, 0)K_{n_{A-1}^*}(y, 0)|\\ &\geq& \frac{(M_{2k_l}M_{2k_l+1})^2}{(n_{A}^*)^2(n_{A}^*+1)^{(1/p-2)}}\geq c \frac{(M_{2k_l}M_{2k_l+1})^2}{(M_{2A})^2(M_{2A})^{(1/p-2)}} \geq\frac{1}{(M_{2A})^{(1/p-2)} } \frac{A^{1/2p}}{q^{4l/p}}. \end{eqnarray*} $

Thus

$ \begin{eqnarray} \|\hat{\sigma}^*f\|^{p}_p&\geq& (\frac{1}{( M_{2A} )^{(1-2p)} } \frac{\sqrt{A^{}}}{q^{4l}})(\sum\limits_{l=1}^{[\frac{1}{4}\log_q(\sqrt{A^{1/2p}})]-1}\sum\limits_{x_{2k_l+3}=0}^{m_{2k_l+3}-1 }\cdots \sum\limits_{ x_{2A-1}=0}^{m_{2A-1}-1}|I_{2A}^{k_l, k_l+1}|)^2\\ &\geq &(\frac{1}{( M_{2A} )^{(1-2p)} } \frac{\sqrt{A}}{q^{4l}}) (\sum\limits_{l=1}^{[\frac{1}{4}\log_q(\sqrt{A^{1/2p}})]-1} \frac{m_{2k_l+3}\cdots m_{2A-1 }}{M_{2A}})^2\\ &\geq& \frac{1}{( M_{2A} )^{(1-2p)} } \frac{A}{q^{4l}} (\sum\limits_{l=1}^{[\frac{1}{4}\log_q(\sqrt{A^{1/2p}})]-1} \frac{1}{M_{2k_l}})^2\\ &\geq& \frac{1}{( M_{2A} )^{(1-2p)} } (\frac{\log_qA}{\sqrt{M_{2A}}})^2 =\ \frac{1}{ M_{2A} ^{2-2p} } (\log_qA)^2. \end{eqnarray} $

(4.12)

Then

$ \begin{eqnarray} \frac{\|\hat{\sigma}^*f\|^{p}_p}{\|f_A\|^p_{H^p}}\geq \frac{\frac{1}{ M_{2A} ^{2-2p} } (\log_qA)^2}{ cM_{2A}^{2p-2} }=(\log_qA)^2\rightarrow \infty. \end{eqnarray} $

(4.13)

Thus the proof of Theorem 4.4 is complete.