Pointwise Rectangular Lipschitz Regularities for Fractional Brownian Sheets and Some Sierpinski Selfsimilar Functions

Abstract

We consider pointwise rectangular Lipschitz regularity and pointwise level coordinate axes Lipschitz regularities for continuous functions f on the unit cube $I^2$ in ${\mathbb{R}}^2$. Firstly, we provide characterizations by simple estimates on the decay rate of the coefficients (resp. leaders) of the expansion of f in the rectangular Schauder system, near the point considered. We deduce that pointwise rectangular Lipschitz regularity yields pointwise level coordinate axes Lipschitz regularities. As an application, we refine earlier results in Ayache et al. (Drap brownien fractionnaire. Potential Anal. 2002, 17, 31–43) and Kamont (On the fractional anisotropic Wiener field. Probab. Math. Statist. 1996, 16, 85–98), where uniform rectangular Lipschitz regularity of the trajectories of the fractional Brownian sheet over the total $I^2$ (or any cube) was considered. Actually, we prove that fractional Brownian sheets are pointwise rectangular and level coordinate axes monofractal. On the opposite, we construct a class of Sierpinski selfsimilar functions that are pointwise rectangular and level coordinate axes multifractal.

1. Introduction

Many authors have studied several properties of anisotropic random fields, see for example [1,2,3,4,5,6,7,8,9,10,11,12,13], and the references therein for further information. Some of these fields are models for textures in images (see [11] and the references therein).

For a given two parameter index $\overline{H}=(H_1,H_2)\in {(0,1)}^2$, the fractional Brownian sheet $\{B^{\overline{H}}\left(x\right):x=(x_1,x_2)\}$, is a real-valued centered Gaussian field, introduced by Kamont in [7], then redefined by Ayache et al. in [2] through the following fractional integration with respect to the standard real-valued Brownian sheet W on ${\mathbb{R}}^2$

$$\forall x=(x_1,x_2)\in {\mathbb{R}}_{+}^2{B}^{\overline{H}}\left(x\right)={\int }_{{\mathbb{R}}^2}\prod _{i=1}^2{g}_{H_i}(x_i,u_i)dW\left(u\right),$$

(1)

where

$$u=(u_1,u_2),g_a(t,s)={(t-s)}_{+}^{a-1/2}-{(-s)}_{+}^{a-1/2}and{s}_{+}=max(s,0).$$

The realizations of the fractional Brownian sheet are continuous. The covariance is given by

$$\mathbb{E}\left[B^{\overline{H}}(t_1,t_2)B^{\overline{H}}(s_1,s_2)\right]=\prod _{i=1}^2{K}^{2H_i}(t_i,s_i)$$

where

$$K^{2a}(t,s)=\frac{1}{2}{\left(\right|t|}^{2a}+{\left|s\right|}^{2a}{-|t-s|}^{2a})$$

is the covariance kernel of the fractional Brownian motion $B_a$ in $\mathbb{R}$ with Hurst index a.

Recall that a corresponds to the critical uniform Lipschitz exponent of the sample paths of $B_a$ over any arbitrary compact interval I; one has, almost surely

$$sup\{\alpha \in (0,1):B_a\in Li{p}^{\alpha }\left(I\right)\}=a.$$

Actually, (see for example [14]), $B_a$ is monofractal of order a in the sense that

$$\forall tsup\{\alpha \in (0,1):B_a\in Li{p}^{\alpha }\left(t\right)\}=a.$$

Let us recall the notions of $Li{p}^{\alpha }\left(I\right)$ and $Li{p}^{\alpha }\left(t_0\right)$. Without any loss of generality, let I denotes the unit interval $[0,1]$.

Definition 1.

Let u be a continuous function on I (we write $u\in C\left(I\right)$). Let $t_0\in I$. Let $\alpha \in (0,1)$. We say that u is Lipschitz of order α at $t_0$ and we write $u\in Li{p}^{\alpha }\left(t_0\right)$, if there exists $C>0$ such that

$$\forall t\in I|u\left(t\right)-u\left(t_0\right)|\le C|t-t_0{|}^{\alpha }.$$

(2)

We say that u is Lipschitz of order α on I and we write $u\in Li{p}^{\alpha }\left(I\right)$, if the constant C in (2) is uniform on all $t_0\in I$.

We say that u is uniform Lipschitz on I if there exists $\delta >0$ such that $u\in Li{p}^{\delta }\left(I\right)$.

When $H_1\ne H_2$, the fractional Brownian sheet given in (1) has the following anisotropic operator-selfsimilarity:

$$\forall a_1,a_2>0a_1^{H_1}{a}_2^{H_2}{B}^{\overline{H}}(x_1/a_1,x_2/a_2)=B^{\overline{H}}\left(x\right)\mathrm{in} \mathrm{law}.$$

(3)

It is proved that the fractional Brownian sheet has stationary rectangular increments. In [2] (resp. [7]), it is also proved that for any cube $\mathcal{Q}\subset {\mathbb{R}}^2$, the restrictions $B_{\mathcal{Q}}^{\overline{H}}$ of realizations of $B^{\overline{H}}$ to $\mathcal{Q}$ are uniform rectangular Lipschitz with order ${\overline{H}}^{\prime }=(H_1^{\prime },H_2^{\prime })$ for $H_1^{\prime }<H_1$ and $H_2^{\prime }<H_2$, in the sense that

$$\exists C>0;\forall (x,y)\in {\mathcal{Q}}^2|{\square }_y{B}_{\mathcal{Q}}^{\overline{H}}\left(x\right)|\le C\prod _{i=1}^2{|y_i-x_i|}^{H_i^{\prime }},$$

(4)

where rectangular increments of a continuous function f are defined for $x=(x_1,x_2)$ and $y=(y_1,y_2)$ by

$${\square }_{y}f\left(x\right)=f\left(y\right)-f(x_1,y_2)-f(y_1,x_2)+f\left(x\right).$$

(5)

This paper is concerned with pointwise rectangular Lipschitz regularity and pointwise level coordinate axes Lipschitz regularities; without any loss of generality, we will take $\mathcal{Q}$ the unit cube $I^2={[0,1]}^2$.

Definition 2. 

(Pointwise rectangular Lipschitz regularity)

Let f be a continuous function on $I^2$ (we write $f\in C\left(I^2\right)$). Let $x\in I^2$. Let $\overline{\alpha }=({\alpha }_1,{\alpha }_2)\in {(0,1)}^2$. We say that f is rectangular Lipschitz of order $\overline{\alpha }$ at x and we write $f\in Li{p}^{\overline{\alpha }}\left(x\right)$, if there exists $C>0$ such that

$$\forall y\in I^2|{\square }_{y}f\left(x\right)|\le C\prod _{i=1}^2{|y_i-x_i|}^{{\alpha }_i}.$$

(6)

We say that f is rectangular Lipschitz of order $\overline{\alpha }$ on $I^2$ and we write $f\in Li{p}^{\overline{\alpha }}\left(I^2\right)$, if the constant C in (6) is uniform on all $x\in I^2$.

We say that f is uniform Lipschitz on $I^2$ if there exists $\delta >0$ such that $f\in Li{p}^{\delta }\left(I^2\right)$ in the sense that there exists $C>0$ such that

$$\forall x,y\in I^2|f\left(y\right)-f\left(x\right)|\le {C|x-y|}^{\delta }.$$

(7)

Let $(e_1,e_2)$ be the canonical basis of ${\mathbb{R}}^2$.

Definition 3. 

(Pointwise level coordinate axes Lipschitz regularities)

Let $s\in (0,1)$, $x=(x_1,x_2)\in I^2$ and $f\in C\left(I^2\right)$. We say that $f\in N^s(x,e_1)$ if there exists $C>0$ such that

$$\forall y=(y_1,y_2)\in I^2\left|f\left(y\right)-f(x_1,y_2)\right|\le C{\left|y_1-x_1\right|}^s.$$

We say that $f\in N^s(x,e_2)$ if there exists $C>0$ such that

$$\forall y=(y_1,y_2)\in I^2\left|f\left(y\right)-f(y_1,x_2)\right|\le C{\left|y_2-x_2\right|}^s.$$

Clearly

$$\forall \sigma \in (0,1)\bigcap _{i=1}^2{N}^{{\alpha }_i}(x,e_i)\subset Li{p}^{(\sigma {\alpha }_1,(1-\sigma ){\alpha }_2)}\left(x\right).$$

(8)

Let us mention that pointwise level coordinate axes Lipschitz regularities $N^s(x,e_i)$ are stronger then pointwise directional Lipschitz regularities $C^s(x,e_i)$ that have been studied in [15]. Let $f\in C\left(I^2\right),x=(x_1,x_2)\in I^2$ and $0<s<1$. Recall that $f\in C^s(x,e_1)$ (resp. $f\in C^s(x,e_2)$) if there exists a positive constant C such that $|f(y_1,x_2)-f\left(x\right)|\le C|y_1-x_1{|}^s\forall y_1\in I$ (resp. $|f(x_1,y_2)-f\left(x\right)|\le C|y_2-x_2{|}^s\forall y_2\in I$). Actually $N^s(x,e_i)\subset C^s(x,e_i)$. Of course there is no converse embedding between $C^s(x,e_i)$ and both pointwise rectangular Lipschitz regularity and pointwise level coordinate axes Lipschitz regularities.

In the next section (respectively the third section), we characterize pointwise rectangular Lipschitz regularity (respectively pointwise level coordinate axes Lipschitz regularities) by simple estimates on the decay rate of the coefficients/leaders of the expansion of the function in the basis of tensor products of Schauder functions, near the point considered (see Theorem 1/Theorem 2 (respectively Theorem 3/Theorem 4)). We deduce that pointwise rectangular Lipschitz regularity yields pointwise level coordinate axes Lipschitz regularities (see Theorem 5).

In the fourth section, as an application, we refine result (4) by proving that fractional Brownian sheets are pointwise rectangular and level coordinate axes monofractal (see Theorem 6). A second application will be done to more general anisotropic deterministic selfsimilar functions that can modelize anisotropic turbulence or cascades. We construct a class of Sierpinski selfsimilar functions that are pointwise rectangular and level coordinate axes multifractal (see Theorem 7).

Finally, a short conclusion section is given.

2. Characterization of ${\mathit{Lip}}^{\overline{\mathit{\alpha }}}\left(\mathit{x}\right)$ in Rectangular Schauder Bases

2.1. Characterization with Rectangular Schauder Coefficients

The rectangular Schauder system ${\left\{{\Phi }_{\mathbf{m}=(m_1,m_2)}\right\}}_{\mathbf{m}\in {\mathbb{N}}_0^2}$ of $C\left(I^2\right)$ is obtained by tensor products ${\Phi }_{\mathbf{m}}\left(y\right)={\Phi }_{\mathbf{m}}(y_1,y_2)={\prod }_{i=1}^2{\varphi }_{m_i}\left(y_i\right)$, of classical (1-variable) Schauder functions $\{{\varphi }_m,m\ge 0\}$ on I, normed in $L^{\infty }$. Recall that ${\varphi }_0=1$, ${\varphi }_1\left(t\right)=t$, and for $m\ge 2$, $m=2^j+n$ with $j\ge 0$ and $1\le n\le 2^j$, ${\varphi }_m\left(t\right)=\varphi (2^{j+1}t-2n+1)$ with support $[(n-1)2^{-j},n{2}^{-j}]$, where $\varphi \left(t\right)=max(0,1-|t\left|\right)$.

Let $M={\mathbb{N}}_0\cup \{-2,-1\}$. For $j\in M$, let

$${\tilde{N}}_{-2}=\left\{0\right\},{\tilde{N}}_{-1}=\left\{1\right\},\mathrm{and}{\tilde{N}}_j=\{2^j+n:n=1,\cdots ,2^j\}\mathrm{for}j\ge 0.$$

(9)

It is known that if $u\in C\left(I\right)$, then ${\displaystyle u=\sum _{j\in M}\sum _{m\in {\tilde{N}}_j}{b}_m\left(u\right){\varphi }_m}$ with

$$b_0\left(u\right)=u\left(0\right),b_1\left(u\right)=u\left(1\right)-u\left(0\right)$$

and

$$\forall j\ge 0\forall m=2^j+n\in {\tilde{N}}_j{b}_m\left(u\right)=u\left(\frac{2n-1}{2^{j+1}}\right)-\frac{1}{2}\left(u\left(\frac{n-1}{2^j}\right)+u\left(\frac{n}{2^j}\right)\right).$$

(10)

For $\mathbf{j}=(j_1,j_2)\in M^2$, we put

$${\tilde{N}}_{\mathbf{j}}={\tilde{N}}_{j_1}\times {\tilde{N}}_{j_2}.$$

(11)

Denote by $\mathbf{0}$ and $\mathbf{1}$ respectively the vectors $(0,0)$ and $(1,1)$. If $\mathbf{a}=(a_1,a_2)$ and $\mathbf{b}=(b_1,b_2)$ belong to ${\mathbb{R}}^2$, we will write $\mathbf{a}\le \mathbf{b}$ if $a_i\le b_i$ for all $i\in \{1,2\}$, $\mathbf{a}<\mathbf{b}$ (resp. $\mathbf{a}>\mathbf{b}$) if $a_i<b_i$ (respectively $a_i>b_i$) for all $i\in \{1,2\}$, and $\mathbf{a}\nleqq \mathbf{b}$ if either $a_1>b_1$ or $a_2>b_2$.

Any $f\in C\left(I^2\right)$ can be written as

$$\begin{array}{ccc}\hfill f\left(y\right)& =& \sum _{\mathbf{j}\in M^2}\sum _{\mathbf{m}\in {\tilde{N}}_{\mathbf{j}}}{C}_{\mathbf{m}}{\Phi }_{\mathbf{m}}\left(y\right)\hfill \\ & =& C_{(0,0)}+C_{(0,1)}{y}_2+C_{(1,0)}{y}_1+C_{(1,1)}{y}_1{y}_2\hfill \\ & +& \sum _{j\ge 0}\sum _{m\in {\tilde{N}}_j}{C}_{(0,m)}{\varphi }_m\left(y_2\right)+\sum _{j\ge 0}\sum _{m\in {\tilde{N}}_j}{C}_{(m,0)}{\varphi }_m\left(y_1\right)\hfill \\ & +& y_1\sum _{j\ge 0}\sum _{m\in {\tilde{N}}_j}{C}_{(1,m)}{\varphi }_m\left(y_2\right)+y_2\sum _{j\ge 0}\sum _{m\in {\tilde{N}}_j}{C}_{(m,1)}{\varphi }_m\left(y_1\right)\hfill \\ & +& \sum _{\mathbf{j}\ge \mathbf{0}}\sum _{\mathbf{m}=(m_1,m_2)\in {\tilde{N}}_{\mathbf{j}}}{C}_{\mathbf{m}}{\varphi }_{m_1}\left(y_1\right){\varphi }_{m_2}\left(y_2\right),\hfill \end{array}$$

where

$$C_{(0,0)}=f(0,0),C_{(0,1)}=f(0,1)-f(0,0),C_{(1,0)}=f(1,0)-f(0,0),$$

$$C_{(1,1)}=f(1,1)+f(0,0)-f(1,0)-f(0,1),$$

$$\forall j\ge 0\forall m\in {\tilde{N}}_j{C}_{(0,m)}=f_m\left(0\right)$$

with

$$f_m\left(t\right)=f(t,\frac{2n-1}{2^{j+1}})-\frac{1}{2}\left(f(t,\frac{n-1}{2^j})+f(t,\frac{n}{2^j})\right),$$

(12)

$$\forall j\ge 0\forall m\in {\tilde{N}}_j{C}_{(m,0)}=g_m\left(0\right)$$

with

$$g_m\left(t\right)=f(\frac{2n-1}{2^{j+1}},t)-\frac{1}{2}\left(f(\frac{n-1}{2^j},t)+f(\frac{n}{2^j},t)\right),$$

(13)

$$\forall j\ge 0\forall m\in {\tilde{N}}_j{C}_{(1,m)}=f_m\left(1\right)-f_m\left(0\right),$$

$$\forall j\ge 0\forall m\in {\tilde{N}}_j{C}_{(m,1)}=g_m\left(1\right)-g_m\left(0\right),$$

and

$$\forall \mathbf{j}\ge \mathbf{0}\forall \mathbf{m}=(m_1,m_2)\in {\tilde{N}}_{\mathbf{j}}{C}_{\mathbf{m}}=f_{m_2}\left(\frac{2n_1-1}{2^{j_1+1}}\right)-\frac{1}{2}\left[f_{m_2}\left(\frac{n_1-1}{2^{j_1}}\right)+f_{m_2}\left(\frac{n_1}{2^{j_1}}\right)\right].$$

(14)

It is known that both uniform and pointwise Lipschitz regularities $Li{p}^{\alpha }\left(I\right)$ and $Li{p}^{\alpha }\left(t_0\right)$ (given in Definition 1) are characterized in Schauder bases (see [16] for example).

Proposition 1. 

Let $0<\alpha <1$. Let $u\in C\left(I\right)$.

1. 

$$u\in Li{p}^{\alpha }\left(I\right)\iff \exists C>0\forall j\ge 0\forall m\in {\tilde{N}}_j\left|b_m\left(u\right)\right|\le C{2}^{-\alpha j}.$$

(15)

2. 

Let $t_0\in I$. If $u\in Li{p}^{\alpha }\left(t_0\right)$ then there exists C such that

$$\forall j\ge 0\forall m=2^j+n\in {\tilde{N}}_j|b_m\left(u\right)|\le C(2^{-\alpha j}+|t_0-n{2}^{-j}{|}^{\alpha }).$$

(16)

Conversely, if u is uniform Lipschitz on I and (16) holds then $u\in Li{p}^{{\alpha }^{\prime }}\left(t_0\right)$ for all ${\alpha }^{\prime }<\alpha$.

Remark 1.

Let $x=(x_1,x_2)$ and $y=(y_1,y_2)$. Clearly if $W\left(y\right)=u\left(y_1\right)v\left(y_2\right)$, $U\left(y\right)=u\left(y_1\right)$ and $V\left(y\right)=v\left(y_2\right)$ then

$${\square }_{y}W\left(x\right)=(u\left(y_1\right)-u\left(x_1\right))(v\left(y_2\right)-v\left(x_2\right)),$$

$${\square }_{y}U\left(x\right)=0and{\square }_{y}V\left(x\right)=0.$$

Thanks to Remark 1

• function

  $$g_0\left(y\right):=C_{(0,0)}+C_{(0,1)}{y}_2+C_{(1,0)}{y}_1+C_{(1,1)}{y}_1{y}_2$$

  (17)

  belongs to $Li{p}^{\overline{\alpha }}\left(I^2\right)$ for all $\overline{\alpha }=({\alpha }_1,{\alpha }_2)<\mathbf{1}$,
• function

  $${\displaystyle g_1\left(y\right):=\sum _{j\ge 0}\sum _{m\in {\tilde{N}}_j}{C}_{(0,m)}{\varphi }_m\left(y_2\right)=f(0,y_2)-f(0,0)-C_{(0,1)}{y}_2}$$

  (18)

  belongs to $Li{p}^{\overline{\alpha }}\left(x\right)$ for all $\overline{\alpha }=({\alpha }_1,{\alpha }_2)<\mathbf{1}$,
• function

  $${\displaystyle g_2\left(y\right):=\sum _{j\ge 0}\sum _{m\in {\tilde{N}}_j}{C}_{(m,0)}{\varphi }_m\left(y_1\right)=f(y_1,0)-f(0,0)-C_{(1,0)}{y}_1}$$

  (19)

  belongs to $Li{p}^{\overline{\alpha }}\left(I^2\right)$ for all $\overline{\alpha }=({\alpha }_1,{\alpha }_2)<\mathbf{1}$,
• function

  $${\displaystyle g_3\left(y\right):=y_1\sum _{j\ge 0}\sum _{m\in {\tilde{N}}_j}{C}_{(1,m)}{\varphi }_m\left(y_2\right)=y_1\left[f(1,y_2)-f(0,y_2)\right]-y_1{C}_{(1,0)}-y_1{y}_2{C}_{(1,1)}}$$

  (20)

  belongs to $Li{p}^{\overline{\alpha }}\left(x\right)$ if and only if the one variable function $f(1,t)-f(0,t)$ belongs to $Li{p}^{{\alpha }_2}\left(x_2\right)$,
• function

  $${\displaystyle g_4\left(y\right):=y_2\sum _{j\ge 0}\sum _{m\in {\tilde{N}}_j}{C}_{(m,1)}{\varphi }_m\left(y_1\right)=y_2\left[f(y_1,1)-f(y_1,0)\right]-y_2{C}_{(0,1)}-y_1{y}_2{C}_{(1,1)}}$$

  (21)

  belongs to $Li{p}^{\overline{\alpha }}\left(x\right)$ if and only if the one variable function $f(t,1)-f(t,0)$ belongs to $Li{p}^{{\alpha }_1}\left(x_1\right)$.

If f is uniform Lipschitz on $I^2$, then functions $f(1,t)-f(0,t)$ and $f(t,1)-f(t,0)$ are uniform Lipschitz on I, consequently $g_3$ and $g_4$ can be sharply characterized in $Li{p}^{\overline{\alpha }}\left(x\right)$ using the second result of Proposition 1. More precisely

Proposition 2.

Let$f\in C\left(I^2\right)$, $\mathbf{0}<\overline{\alpha }<\mathbf{1}$and$x\in I^2$.

1. 

If $g_3$ and $g_4$ belong to $Li{p}^{\overline{\alpha }}\left(x\right)$ then there exists $C>0$ such that

$$\forall j\ge 0\forall m=2^j+n\in {\tilde{N}}_j\left|C_{(1,m)}\right|\le C{(2^{-j}+\left|n{2}^{-j}-x_2\right|)}^{{\alpha }_2}$$

(22)

and

$$\forall j\ge 0\forall m=2^j+n\in {\tilde{N}}_j\left|C_{(m,1)}\right|\le C{(2^{-j}+\left|n{2}^{-j}-x_1\right|)}^{{\alpha }_1}.$$

(23)

2. 

Conversely, if f is uniform Lipschitz on $I^2$ and both (22) and (23) are satisfied then

$$\forall i\in \{3,4\}\forall {\overline{\alpha }}^{\prime }<\overline{\alpha }{g}_i\in Li{p}^{{\overline{\alpha }}^{\prime }}\left(x\right).$$

(24)

Remark 2.

Note that if $f\in Li{p}^{\overline{\alpha }}\left(x\right)$ then $g_3$ and $g_4$ belong to $Li{p}^{\overline{\alpha }}\left(x\right)$. In fact

$$\begin{array}{ccc}\hfill |f(1,t)-f(0,t)-f(1,x_2)+f(0,x_2)|& =& |{\square }_{(1,t)}f\left(x\right)-{\square }_{(0,t)}f\left(x\right)|\hfill \\ & \le & |{\square }_{(1,t)}f\left(x\right)|+|{\square }_{(0,t)}f\left(x\right)|\hfill \\ & \le & C|t-x_2{|}^{{\alpha }_2}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill |f(t,1)-f(t,0)-f(x_1,1)+f(x_1,0)|& =& |{\square }_{(t,1)}f\left(x\right)-{\square }_{(t,0)}f\left(x\right)|\hfill \\ & \le & |{\square }_{(t,1)}f\left(x\right)|+|{\square }_{(t,0)}f\left(x\right)|\hfill \\ & \le & C|t-x_1{|}^{{\alpha }_1}.\hfill \end{array}$$

To achieve the sharp characterization of $Li{p}^{\overline{\alpha }}\left(x\right)$ by Schauder coefficients, it remains to deal with the series

$${\displaystyle F\left(y\right)=\sum _{\mathbf{j}\ge \mathbf{0}}\sum _{\mathbf{m}=(m_1,m_2)\in {\tilde{N}}_{\mathbf{j}}}{C}_{\mathbf{m}}{\varphi }_{m_1}\left(y_1\right){\varphi }_{m_2}\left(y_2\right).}$$

(25)

Proposition 3.

Let $f\in C\left(I^2\right)$, $\mathbf{0}<\overline{\alpha }<\mathbf{1}$ and $x\in I^2$.

• If $f\in Li{p}^{\overline{\alpha }}\left(x\right)$ then there exists $C>0$ such that

  $$\forall \mathbf{j}\ge \mathbf{0}\forall \mathbf{m}=(m_1,m_2)\in {\tilde{N}}_{\mathbf{j}}\left|C_{\mathbf{m}}\right|\le C\prod _{i=1}^2{(2^{-j_i}+\left|n_i{2}^{-j_i}-x_i\right|)}^{{\alpha }_i}.$$

  (26)
• Conversely, if f is uniform Lipschitz on $I^2$ and (26) is satisfied, then the function F given in (25) satisfies

  $$\forall {\overline{\alpha }}^{\prime }<\overline{\alpha }F\in Li{p}^{{\overline{\alpha }}^{\prime }}\left(x\right).$$

  (27)

Proof of Proposition 3.

• Assume that $f\in Li{p}^{\overline{\alpha }}\left(x\right)$. Recall that coefficients $C_{\mathbf{m}}$ are given by (14) where $f_{m_2}$ is as in (12). Write

  $$\begin{array}{cc}\hfill f_{m_2}\left(t\right)& =f(t,\frac{2n_2-1}{2^{j_2+1}})-f(x_1,\frac{2n_2-1}{2^{j_2+1}})-f(t,x_2)+f\left(x\right)\hfill \\ \hfill & -\frac{1}{2}\left[f(t,\frac{n_2-1}{2^{j_2}})-f(x_1,\frac{n_2-1}{2^{j_2}})-f(t,x_2)+f\left(x\right)\right.\hfill \\ \hfill & +\left.f(t,\frac{n_2}{2^{j_2}})-f(x_1,\frac{n_2}{2^{j_2}})-f(t,x_2)+f\left(x\right)\right]\hfill \\ \hfill & +f_{m_2}\left(x_1\right)\hfill \\ \hfill & ={\square }_{(t,\frac{2n_2-1}{2^{j_2+1}})}f\left(x\right)-\frac{1}{2}\left[{\square }_{(t,\frac{n_2-1}{2^{j_2}})}f\left(x\right)+{\square }_{(t,\frac{n_2}{2^{j_2}})}f\left(x\right)\right]+f_{m_2}\left(x_1\right).\hfill \end{array}$$
  Thus

  $$\begin{array}{cc}\hfill C_{\mathbf{m}}& =f_{m_2}\left(\frac{2n_1-1}{2^{j_1+1}}\right)-\frac{1}{2}\left[f_{m_2}\left(\frac{n_1-1}{2^{j_1}}\right)+f_{m_2}\left(\frac{n_1}{2^{j_1}}\right)\right]\hfill \\ \hfill & ={\square }_{(\frac{2n_1-1}{2^{j_1+1}},\frac{2n_2-1}{2^{j_2+1}})}f\left(x\right)-\frac{1}{2}\left({\square }_{(\frac{2n_1-1}{2^{j_1+1}},\frac{n_2-1}{2^{j_2}})}f\left(x\right)+{\square }_{(\frac{2n_1-1}{2^{j_1+1}},\frac{n_2}{2^{j_2}})}f\left(x\right)\right)\hfill \\ \hfill & -\frac{1}{2}\left[{\square }_{(\frac{n_1-1}{2^{j_1}},\frac{2n_2-1}{2^{j_2+1}})}f\left(x\right)-\frac{1}{2}\left({\square }_{(\frac{n_1-1}{2^{j_1}},\frac{n_2-1}{2^{j_2}})}f\left(x\right)+{\square }_{(\frac{n_1-1}{2^{j_1}},\frac{n_2}{2^{j_2}})}f\left(x\right)\right)\right.\hfill \\ \hfill & +\left.{\square }_{(\frac{n_1}{2^{j_1}},\frac{2n_2-1}{2^{j_2+1}})}f\left(x\right)-\frac{1}{2}\left({\square }_{(\frac{n_1}{2^{j_1}},\frac{n_2-1}{2^{j_2}})}f\left(x\right)+{\square }_{(\frac{n_1}{2^{j_1}},\frac{n_2}{2^{j_2}})}f\left(x\right)\right)\right].\hfill \end{array}$$
  Since $f\in Li{p}^{\overline{\alpha }}\left(x\right)$ then (26) holds.
• Conversely, assume that f is uniform Lipschitz on $I^2$ and (26) is satisfied. Using (12)

  $$|f_m\left(t\right)|\le \frac{1}{2}|f(t,\frac{2n-1}{2^{j+1}})-f(t,\frac{n-1}{2^j})|+\frac{1}{2}|f(t,\frac{2n-1}{2^{j+1}})-f(t,\frac{n}{2^j})|\le C{2}^{-\delta j}.$$
  Using (7) and (14)

  $$\forall \mathbf{j}\ge \mathbf{0}\forall \mathbf{m}\in {\tilde{N}}_{\mathbf{j}}\left|C_{\mathbf{m}}\right|\le C{2}^{-\delta j_1}.$$

  (28)
  Since

  $$C_{\mathbf{m}}=f_{m_1}\left(\frac{2n_2-1}{2^{j_2+1}}\right)-\frac{1}{2}\left[f_{m_1}\left(\frac{n_2-1}{2^{j_2}}\right)+f_{m_1}\left(\frac{n_2}{2^{j_2}}\right)\right]$$

  then similarly

  $$\forall \mathbf{j}\ge \mathbf{0}\forall \mathbf{m}\in {\tilde{N}}_{\mathbf{j}}\left|C_{\mathbf{m}}\right|\le C{2}^{-\delta j_2}.$$

  (29)
  It follows that

  $$\forall \mathbf{j}\ge \mathbf{0}\forall \mathbf{m}\in {\tilde{N}}_{\mathbf{j}}\forall \theta \in (0,1)\left|C_{\mathbf{m}}\right|\le C{2}^{-\theta \delta j_1}{2}^{-(1-\theta )\delta j_2}.$$

  (30)
  Put ${\delta }_1=\delta \theta$ and ${\delta }_2=\delta (1-\theta )$. By (26) there exists $C>0$ such that

  $$\forall \sigma \in [0,1]\forall \mathbf{j}\ge \mathbf{0}\forall \mathbf{m}\in {\tilde{N}}_{\mathbf{j}}\left|C_{\mathbf{m}}\right|\le C\prod _{i=1}^2{2}^{-(1-\sigma ){\delta }_i{j}_i}{(2^{-j_i}+\left|n_i{2}^{-j_i}-x_i\right|)}^{\sigma {\alpha }_i}.$$
  Then

  $$\left|C_{\mathbf{m}}\right|\le C\prod _{i=1}^2{\mu }_{{\delta }_i,{\alpha }_i,j_i,m_i,x_i}$$

  (31)

  where for $0<\delta <1$, $0<\alpha <1$, $j\ge 0$, $m=2^j+n\in {\tilde{N}}_j$ and $t\in \mathbb{R}$

  $${\mu }_{\delta ,\alpha ,j,m,t}=2^{-(1-\sigma )\delta j}{(2^{-j}+\left|n{2}^{-j}-t\right|)}^{\sigma \alpha }.$$

  (32)
  For $h\in \mathbb{R}$, put

  $${\Delta }_h{\varphi }_m\left(t\right)={\varphi }_m(t+h)-{\varphi }_m\left(t\right)$$

  (33)

  and

  $$R_{\delta ,\alpha }(h,t)=\sum _{j=0}^{\infty }\sum _{m\in {\tilde{N}}_j}{\mu }_{\delta ,\alpha ,j,m,t}\left|{\Delta }_h{\varphi }_m\left(t\right)\right|.$$

  (34)
  Clearly, if $\mathbf{h}=(h_1,h_2)$ then

  $${\square }_{x+\mathbf{h}}{\Phi }_{\mathbf{m}}\left(x\right)=\prod _{i=1}^2{\Delta }_{h_i}{\varphi }_{m_i}\left(x_i\right)$$

  (35)

  and the function F given in (25) satisfies

  $$\left|{\square }_{x+\mathbf{h}}F\left(x\right)\right|\le C\prod _{i=1}^2{R}_{{\delta }_i,{\alpha }_i}(h_i,x_i).$$

  (36)
  Relation (36) together with the following lemma yield (27).

 □

Lemma 1.

There exists $C>0$ such that

$$\forall 0<\left|h\right|\le 1\forall t{R}_{\delta ,\alpha }(h,t)\le C{\left|h\right|}^{\sigma \alpha +(1-\sigma )\delta }.$$

(37)

Proof of Lemma 1. :

Clearly

$${\mu }_{\delta ,\alpha ,j,m,t}\le C{2}^{-(1-\sigma )\delta j}(2^{-j\sigma \alpha }+{\left|n{2}^{-j}-t\right|}^{\sigma \alpha }).$$

(38)

Remark 3.If $m=2^j+n\in {\tilde{N}}_j$, with $j\ge 0$ then ${\varphi }_m$ has support $[(n-1)2^{-j},n{2}^{-j}]$. It follows that for $t\in I$ and $j\ge 0$, there exists a unique value of $m=2^j+n$ for which $t\in [(n-1)2^{-j},n{2}^{-j})$.

On the other hand

$$\left|{\Delta }_h{\varphi }_m\left(t\right)\right|\le \left|{\varphi }_m\left(t\right)\right|+\left|{\varphi }_m(t+h)\right|.$$

(39)

Relation (38) together with Remark 3 and the triangle inequality $\left|n{2}^{-j}-(t+h)\right|\le \left|n{2}^{-j}-t\right|+\left|h\right|$ yield

$$\sum _{m\in {\tilde{N}}_j}{\mu }_{\delta ,\alpha ,j,m,t}\left|{\Delta }_h{\varphi }_m\left(t\right)\right|\le C{2}^{-(1-\sigma )\delta j}(2^{-\sigma \alpha j}+{\left|h\right|}^{\sigma \alpha }).$$

(40)

Since $0<\left|h\right|\le 1$, let $J\in {\mathbb{N}}_0$ such that $2^{-J}\le \left|h\right|<2.2^{-J}$. Split $R_{\delta ,\alpha }(h,t)$ as

$$R_{\delta ,\alpha }(h,t)=\sum _{j=0}^J\sum _{m\in {\tilde{N}}_j}{\mu }_{\delta ,\alpha ,j,m,t}\left|{\Delta }_h{\varphi }_m\left(t\right)\right|+\sum _{j=J+1}^{\infty }\sum _{m\in {\tilde{N}}_j}{\mu }_{\delta ,\alpha ,j,m,t}\left|{\Delta }_h{\varphi }_m\left(t\right)\right|.$$

(41)

Since $\alpha >0,0<\sigma <1$ and $\delta >0$ then relation (40) yields

$$\sum _{j=J+1}^{\infty }\sum _{m\in {\tilde{N}}_j}{\mu }_{\delta ,\alpha ,j,m,t}\left|{\Delta }_h{\varphi }_m\left(t\right)\right|\le C{\left|h\right|}^{\sigma \alpha +(1-\sigma )\delta }.$$

(42)

Let us bound the first sum in (41). Since $\varphi$ is Lipschitz then

$$\left|{\Delta }_h{\varphi }_m\left(t\right)\right|\le C{2}^j\left|h\right|.$$

An argument similar to that of (40) yields

$$\sum _{j=0}^J\sum _{m\in {\tilde{N}}_j}{\mu }_{\delta ,\alpha ,j,m,t}\left|{\Delta }_h{\varphi }_m\left(t\right)\right|\le C\sum _{j=0}^J{2}^{-(1-\sigma )\delta j}(2^{-\sigma \alpha j}+{\left|h\right|}^{\sigma \alpha })2^j\left|h\right|\le C{\left|h\right|}^{\sigma \alpha +(1-\sigma )\delta }.$$

(43)

Both (42) and (43) yield (37). □

Proposition 2 together with Remark 2 and Proposition 3 yield the following full characterization of $Li{p}^{\overline{\alpha }}\left(x\right)$.

Theorem 1.

Let $f\in C\left(I^2\right)$, $\mathbf{0}<\overline{\alpha }<\mathbf{1}$ and $x\in I^2$.

1. 

If $f\in Li{p}^{\overline{\alpha }}\left(x\right)$ then (22) together with (23) and (26) hold.

2. 

Conversely, if f is uniform Lipschitz on $I^2$ and (22) together with (23) and (26) hold, then

$$\forall {\overline{\alpha }}^{\prime }<\overline{\alpha }f\in Li{p}^{{\overline{\alpha }}^{\prime }}\left(x\right).$$

(44)

2.2. Characterization of $Li{p}^{\overline{\alpha }}\left(x\right)$ by Decay Conditions of Schauder Leaders

An equivalent characterization of $Li{p}^{\overline{\alpha }}\left(x\right)$ by decay conditions of Schauder leaders can also be obtained.

If $j\ge 0$ and $m=n+2^j\in {\tilde{N}}_j$, we will denote by $\lambda$ the dyadic interval

$${\lambda }_m=[(n-1)2^{-j},n{2}^{-j}).$$

(45)

Set

$$3{\lambda }_m=[(n-2)2^{-j},(n+1)2^{-j}).$$

(46)

If $t\in I$, denote by ${\lambda }_j\left(t\right)$ the dyadic interval at scale j that contains t.

For $\mathbf{j}=(j_1,j_2)\ge \mathbf{0}$ and $\mathbf{m}=(m_1,m_2)\in {\tilde{N}}_{\mathbf{j}}$, with $m_i=n_i+2^{j_i}$, define the Schauder leader of f at $\mathbf{m}$ by

$$d_{\mathbf{m}}=d_{{\lambda }_{m_1}\times {\lambda }_{m_2}}=\underset{{\lambda }_{m_1^{\prime }}\times {\lambda }_{m_2^{\prime }}\subset {\lambda }_{m_1}\times {\lambda }_{m_2}}{sup}\left|C_{{\mathbf{m}}^{\prime }}\right|,$$

(47)

where ${\lambda }_{m_i^{\prime }}=[(n_i^{\prime }-1)2^{-j_i},n_i^{\prime }{2}^{-j_i^{\prime }})$ for $m_i^{\prime }=n_i^{\prime }+2^{j_i^{\prime }}\in {\tilde{N}}_{j^{\prime }}$.

If $\mathbf{j}=(-1,j_2\ge 0)$ and $\mathbf{m}=(1,m_2)$, with $m_2=n_2+2^{j_2}\in {\tilde{N}}_{j_2}$, define the Schauder leader of f at $\mathbf{m}$ by

$$d_{\mathbf{m}}=\underset{{\lambda }_{m_2^{\prime }}\subset {\lambda }_{m_2}}{sup}\left|C_{(1,m_2^{\prime })}\right|.$$

(48)

If $\mathbf{j}=(j_1\ge 0,-1)$ and $\mathbf{m}=(m_1,1)$, with $m_1=n_1+2^{j_1}\in {\tilde{N}}_{j_1}$, define the Schauder leader of f at $\mathbf{m}$ by

$$d_{\mathbf{m}}=\underset{{\lambda }_{m_1^{\prime }}\subset {\lambda }_{m_1}}{sup}\left|C_{(m_1^{\prime },1)}\right|.$$

(49)

If $\mathbf{j}=(j_1,j_2)\ge \mathbf{0}$, set

$$d_{\mathbf{j}}\left(x\right)=\underset{{\lambda }_{m_1^{\prime }}\times {\lambda }_{m_2^{\prime }}\subset 3{\lambda }_{j_1\left(x_1\right)}\times 3{\lambda }_{j_2\left(x_2\right)}}{max}{d}_{{\mathbf{m}}^{\prime }}.$$

(50)

If $\mathbf{j}=(-1,j_2\ge 0)$, set

$$d_{\mathbf{j}}\left(x\right)=\underset{{\lambda }_{m_2^{\prime }}\subset 3{\lambda }_{j_2\left(x_2\right)}}{max}{d}_{(1,m_2^{\prime })}.$$

(51)

If $\mathbf{j}=(j_1\ge 0,-1)$, set

$$d_{\mathbf{j}}\left(x\right)=\underset{{\lambda }_{m_1^{\prime }}\subset 3{\lambda }_{j_1\left(x_1\right)}}{max}{d}_{(m_1^{\prime },1)}.$$

(52)

Theorem 2.

Let $f\in C\left(I^2\right)$, $\mathbf{0}<\overline{\alpha }<\mathbf{1}$ and $x\in I^2$.

1. 

If $f\in Li{p}^{\overline{\alpha }}\left(x\right)$ then there exists $C>0$ such that

$$\forall \mathbf{j}=(-1,j_2\ge 0)d_{\mathbf{j}}\left(x\right)\le C{2}^{-j_2{\alpha }_2},$$

(53)

$$\forall \mathbf{j}=(j_1\ge 0,-1)d_{\mathbf{j}}\left(x\right)\le C{2}^{-j_1{\alpha }_1}$$

(54)

and

$$\forall \mathbf{j}\ge \mathbf{0}{d}_{\mathbf{j}}\left(x\right)\le C\prod _{i=1}^2{2}^{-j_i{\alpha }_i}.$$

(55)

2. 

Conversely, if f is uniform Lipschitz on $I^2$ and (53) together with (54) and (55) hold, then

$$\forall {\overline{\alpha }}^{\prime }<\overline{\alpha }f\in Li{p}^{{\overline{\alpha }}^{\prime }}\left(x\right).$$

(56)

Proof of Theorem 2.

• Let $f\in Li{p}^{\overline{\alpha }}\left(x\right)$.
  • Since $g_3$ and $g_4$ belong to $Li{p}^{\overline{\alpha }}\left(x\right)$ then by Proposition 2

    $$\forall j^{\prime }\ge 0\forall m^{\prime }=2^{j^{\prime }}+n^{\prime }\in {\tilde{N}}_{j^{\prime }}\left|C_{(1,m^{\prime })}\right|\le C{(2^{-j^{\prime }}+\left|n^{\prime }{2}^{-j^{\prime }}-x_2\right|)}^{{\alpha }_2}$$

    (57)

    and

    $$\forall j^{\prime }\ge 0\forall m^{\prime }=2^{j^{\prime }}+n^{\prime }\in {\tilde{N}}_{j^{\prime }}\left|C_{(m^{\prime },1)}\right|\le C{(2^{-j^{\prime }}+\left|n^{\prime }{2}^{-j^{\prime }}-x_1\right|)}^{{\alpha }_1}.$$

    (58)
    Let $j\ge 1$, if ${\lambda }^{\prime }\subset 3\lambda$ then $j^{\prime }\ge j-1$ and $\left|n^{\prime }{2}^{-j^{\prime }}-x_i\right|\le C{2}^{-j}$ for all $i\in \{1,2\}$. Hence (57) and (58) yield (53) and (54).
  • Thanks to Proposition 3

    $$\forall {\mathbf{j}}^{\prime }\ge \mathbf{0}\forall {\mathbf{m}}^{\prime }=(m_1^{\prime },m_2^{\prime })\in {\tilde{N}}_{{\mathbf{j}}^{\prime }}\left|C_{{\mathbf{m}}^{\prime }}\right|\le C\prod _{i=1}^2{(2^{-j_i^{\prime }}+\left|n_i^{\prime }{2}^{-j_i^{\prime }}-x_i\right|)}^{{\alpha }_i}.$$

    (59)
    Let $\mathbf{j}\ge \mathbf{1}$, if ${\lambda }_i^{\prime }\subset 3{\lambda }_i$ then $j_i^{\prime }\ge j_i-1$ and $\left|n_i^{\prime }{2}^{-j_i^{\prime }}-x_i\right|\le C{2}^{-j_i}$ for all $i\in \{1,2\}$. Hence (59) yields (55).
• The converse part is reminiscent of that of [17] page 17. Assume that f is uniform Lipschitz on $I^2$ and (53) together with (54) and (55) hold.
  • Let $t\in I$. Let $j^{\prime }\ge 0$ be given. If ${\lambda }^{\prime }=[(n^{\prime }-1)2^{-j^{\prime }},n^{\prime }{2}^{-j^{\prime }})$, denote by $\lambda =[(n-1)2^{-j},n{2}^{-j})$ the dyadic interval defined by

    -

    if ${\lambda }^{\prime }\subset 3{\lambda }_{j^{\prime }}\left(t\right)$, then $\lambda ={\lambda }_{j^{\prime }}\left(t\right)$ and $j=j^{\prime }$,

    -

    else, if $j=sup\{l:{\lambda }^{\prime }\subset 3{\lambda }_l\left(t\right)\}$, then $\lambda ={\lambda }_j\left(t\right)$ and it follows that there exists $C>0$ such that ${\displaystyle \frac{1}{C}{2}^{-j}\le |n^{\prime }{2}^{-j^{\prime }}-t|\le C{2}^{-j}}$.

    Set $m^{\prime }=n^{\prime }+2^{j^{\prime }}$. In the first case, relation (53) implies that

    $$|C_{(1,m^{\prime })}|\le d_{(1,j)}\left(x\right)\le C{2}^{-j{\alpha }_2}=C{2}^{-j^{\prime }{\alpha }_2}.$$
    Similarly, relation (54) implies that

    $$|C_{(m^{\prime },1)}|\le d_{(j,1)}\left(x\right)\le C{2}^{-j{\alpha }_1}=C{2}^{-j^{\prime }{\alpha }_1}.$$
    In the second case,

    $$|C_{(1,m^{\prime })}|\le d_{(1,j)}\left(x\right)\le C{2}^{-j_2{\alpha }_2}\le C{|n^{\prime }{2}^{-j^{\prime }}-x_2|}^{{\alpha }_2}$$

    and

    $$|C_{(m^{\prime },1)}|\le d_{(j,1)}\left(x\right)\le C{2}^{-j_1{\alpha }_1}\le C{|n^{\prime }{2}^{-j^{\prime }}-x_1|}^{{\alpha }_1}.$$
    The conclusion of the converse part of Proposition 2 holds.
    If ${\mathbf{j}}^{\prime }\ge \mathbf{0}$. With the same notations as above

    $$|C_{(m_1^{\prime },m_2^{\prime })}|\le C\prod _{i=1}^2(2^{-j_i^{\prime }{\alpha }_i}+|n_i^{\prime }{2}^{-j_i^{\prime }}-x_i{|}^{{\alpha }_i}).$$
    The conclusion of the converse part of Theorem 3 holds. □

3. Pointwise Level Coordinate Axes Lipschitz Regularities

Remark 4.

Clearly if $W\left(y\right)=u\left(y_1\right)v\left(y_2\right)$ then $W\left(y\right)-W(x_1,y_2)=(u\left(y_1\right)-u\left(x_1\right))v\left(y_2\right)$ and $W\left(y\right)-W(y_1,x_2)=u\left(y_1\right)(v\left(y_2\right)-v\left(x_2\right))$.

Thanks to Remark 4, we have the following results.

• Function $g_0$ given in (17) belongs to ${\bigcap }_{i=1}^2{N}^s(x,e_i)$ for all $0<s<1$.
• Function $g_1$ given in (18) belongs to $N^s(x,e_1)$ for all $0<s<1$. It belongs to $N^s(x,e_2)$ if and only if the one variable function $f(0,t)$ belongs to $Li{p}^s\left(x_2\right)$.
• Function $g_2$ given in (19) belongs to $N^s(x,e_2)$ for all $0<s<1$. It belongs to $N^s(x,e_1)$ if and only if the one variable function $f(t,0)$ belongs to $Li{p}^s\left(x_1\right)$.
• Function $g_3$ given in (20) belongs to $N^s(x,e_1)$ for all $0<s<1$. It belongs to $N^s(x,e_2)$ if and only if the one variable function $f(1,t)-f(0,t)$ belongs to $Li{p}^s\left(x_2\right)$.
• Function $g_4$ given in (21) belongs to $N^s(x,e_2)$ for all $s<1$. It belongs to $N^s(x,e_1)$ if and only if the one variable function $f(t,1)-f(t,0)$ belongs to $Li{p}^s\left(x_1\right)$.

3.1. Characterization of $N^s(x,e_i)$ by Decay Conditions of Schauder Coefficients

If f is uniform Lipschitz on $I^2$, then $g_2$ and $g_4$ (resp. $g_1$ and $g_3$) can be sharply characterized in $N^s(x,e_1)$ using the second result of Proposition 1. More precisely

Proposition 4.

Let $f\in C\left(I^2\right)$, $s\in (0,1)$ and $x\in I^2$.

1. 

If $g_2$ and $g_4$ belong to $N^s(x,e_1)$ then there exists $C>0$ such that

$$\forall j\ge 0\forall m=2^j+n\in {\tilde{N}}_j\left|C_{(m,0)}\right|\le C{(2^{-j}+\left|n{2}^{-j}-x_1\right|)}^s$$

(60)

and

$$\forall j\ge 0\forall m=2^j+n\in {\tilde{N}}_j\left|C_{(m,1)}\right|\le C{(2^{-j}+\left|n{2}^{-j}-x_1\right|)}^s.$$

(61)

2. 

Conversely, if f is uniform Lipschitz on $I^2$ and both (60) and (61) are satisfied then

$$\forall i\in \{2,4\}\forall s^{\prime }<s{g}_i\in N^{s^{\prime }}(x,e_1).$$

(62)

3. 

If $g_1$ and $g_3$ belong to $N^s(x,e_2)$ then there exists $C>0$ such that

$$\forall j\ge 0\forall m=2^j+n\in {\tilde{N}}_j\left|C_{(0,m)}\right|\le C{(2^{-j}+\left|n{2}^{-j}-x_2\right|)}^s$$

(63)

and

$$\forall j\ge 0\forall m=2^j+n\in {\tilde{N}}_j\left|C_{(1,m)}\right|\le C{(2^{-j}+\left|n{2}^{-j}-x_2\right|)}^s.$$

(64)

4. 

Conversely, if f is uniform Lipschitz on $I^2$ and both (63) and (64) are satisfied then

$$\forall i\in \{1,3\}\forall s^{\prime }<s{g}_i\in N^{s^{\prime }}(x,e_2).$$

(65)

Remark 5.

Note that if $f\in N^s(x,e_1)$ (resp. $f\in N^s(x,e_2)$) then $g_2$ and $g_4$ belong to $N^s(x,e_1)$ (resp. $g_1$ and $g_3$ belong to $N^s(x,e_2)$).

To achieve the sharp characterization of both $N^s(x,e_1)$ and $N^s(x,e_2)$, it remains to deal with the series F given in (25).

Proposition 5.

Let $f\in C\left(I^2\right)$, $0<s<1$ and $x\in I^2$.

1. 

(a) 

If $f\in N^s(x,e_1)$ then there exists $C>0$ such that

$$\forall \mathbf{j}\ge \mathbf{0}\forall \mathbf{m}=(m_1,m_2)\in {\tilde{N}}_{\mathbf{j}}\left|C_{\mathbf{m}}\right|\le C{(2^{-j_1}+\left|n_1{2}^{-j_1}-x_1\right|)}^s.$$

(66)

(b) 

Conversely, if f is uniform Lipschitz on $I^2$ and (66) is satisfied, then

$$\forall s^{\prime }<sF\in N^{s^{\prime }}(x,e_1).$$

(67)

1. 

(a) 

If $f\in N^s(x,e_2)$ then there exists $C>0$ such that

$$\forall \mathbf{j}\ge \mathbf{0}\forall \mathbf{m}=(m_1,m_2)\in {\tilde{N}}_{\mathbf{j}}\left|C_{\mathbf{m}}\right|\le C{(2^{-j_2}+\left|n_2{2}^{-j_2}-x_2\right|)}^s.$$

(68)

(b) 

Conversely, if f is uniform Lipschitz on $I^2$ and (68) is satisfied, then

$$\forall s^{\prime }<sF\in N^{s^{\prime }}(x,e_2).$$

(69)

Proof of Proposition 5.

• (a)

  Let $\mathbf{j}\ge \mathbf{0}$ and $\mathbf{m}=(m_1,m_2)\in {\tilde{N}}_{\mathbf{j}}$. Using (14)

  $$\begin{array}{ccc}\hfill |C_{\mathbf{m}}|& =& |f_{m_2}\left(\frac{2n_1-1}{2^{j_1+1}}\right)-f_{m_2}\left(x_1\right)\hfill \\ & -& \frac{1}{2}\left(f_{m_2}\left(\frac{n_1-1}{2^{j_1}}\right)-f_{m_2}\left(x_1\right)+f_{m_2}\left(\frac{n_1}{2^{j_1}}\right)-f_{m_2}\left(x_1\right)\right)|.\hfill \end{array}$$

  Since $f\in N^s(x,e_1)$ then using (12) there exists $C>0$ such that (66) holds.

  (b)

  Conversely, assume that f is uniform Lipschitz on $I^2$ and (66) is satisfied.

  As in the beginning of the proof of the converse part in Proposition 3

  $$\left|C_{\mathbf{m}}\right|\le C{\mu }_{{\delta }_1,s,j_1,m_1,x_1}{2}^{-(1-\sigma ){\delta }_2{j}_2}.$$

  (70)

  So

  $$\begin{array}{ccc}\hfill |F\left(y\right)-F(x_1,y_2)|& \le & C{R}_{{\delta }_1,s}(y_1-x_1,x_1)\sum _{j_2\ge 0}\left(2^{-(1-\sigma ){\delta }_2{j}_2}\sum _{m_2}\left|{\varphi }_{m_2}\left(y_2\right)\right|\right)\hfill \\ & \le & C{R}_{{\delta }_1,s}(y_1-x_1,x_1)\hfill \\ & \le & C|y_1-x_1{|}^{\sigma s+(1-\sigma ){\delta }_1}.\hfill \end{array}$$

• The proof is similar.

 □

Proposition 4 together with both Remark 5 and Proposition 5 yield the following full characterization of both $N^s(x,e_1)$ and $N^s(x,e_2)$.

Theorem 3.

Let $f\in C\left(I^2\right)$, $0<s<1$ and $x\in I^2$.

• (a)

  If $f\in N^s(x,e_1)$ then there exists $C>0$ such that (60), (61) and (66) hold.

  (b)

  Conversely, if f is uniform Lipschitz on $I^2$ and (60) together with (61) and (66) hold, then

  $$\forall s^{\prime }<sf\in N^{s^{\prime }}(x,e_1).$$

  (71)

• (a)

  If $f\in N^s(x,e_2)$ then there exists $C>0$ such that (63), (64) and (68) hold.

  (b)

  Conversely, if f is uniform Lipschitz on $I^2$ and (63) together with (64) and (68) hold, then

  $$\forall s^{\prime }<sf\in N^{s^{\prime }}(x,e_2).$$

  (72)

3.2. Characterization of $N^s(x,e_i)$ by Decay Conditions of Schauder Leaders

If $\mathbf{j}=(-2,j_2\ge 0)$ and $\mathbf{m}=(0,m_2)$, with $m_2=n_2+2^{j_2}\in {\tilde{N}}_{j_2}$, then define the Schauder leader of f at $\mathbf{m}$ by

$$d_{\mathbf{m}}=\underset{{\lambda }_{m_2^{\prime }}\subset {\lambda }_{m_2}}{sup}\left|C_{(0,m_2^{\prime })}\right|.$$

(73)

If $\mathbf{j}=(j_1\ge 0,-2)$ and $\mathbf{m}=(m_1,0)$, with $m_1=n_1+2^{j_1}\in {\tilde{N}}_{j_1}$, then define the Schauder leader of f at $\mathbf{m}$ by

$$d_{\mathbf{m}}=\underset{{\lambda }_{m_1^{\prime }}\subset {\lambda }_{m_1}}{sup}\left|C_{(m_1^{\prime },0)}\right|.$$

(74)

If $\mathbf{j}=(-2,j_2\ge 0)$, set

$$d_{\mathbf{j}}\left(x\right)=\underset{{\lambda }_{m_2^{\prime }}\subset 3{\lambda }_{j_2\left(x_2\right)}}{max}{d}_{(0,m_2^{\prime })}.$$

(75)

If $\mathbf{j}=(j_1\ge 0,-2)$, set

$$d_{\mathbf{j}}\left(x\right)=\underset{{\lambda }_{m_1^{\prime }}\subset 3{\lambda }_{j_1\left(x_1\right)}}{max}{d}_{(m_1^{\prime },0)}.$$

(76)

Using the same arguments as in the proof of Theorem 2, we obtain the following results.

Theorem 4.

Let $f\in C\left(I^2\right)$, $0<s<1$ and $x\in I^2$.

• (a)

  If $f\in N^s(x,e_1)$ then there exists $C>0$ such that

  $$\forall \mathbf{j}=(j_1\ge 0,j_2\in \{-2,-1\})d_{\mathbf{j}}\left(x\right)\le C{2}^{-j_{1}s}$$

  (77)

  and

  $$\forall \mathbf{j}\ge \mathbf{0}{d}_{\mathbf{j}}\left(x\right)\le C{2}^{-j_{1}s}.$$

  (78)

  (b)

  Conversely, if f is uniform Lipschitz on $I^2$ and both (77) and (78) hold, then

  $$\forall s^{\prime }<sf\in N^{s^{\prime }}(x,e_1).$$

  (79)

• (a)

  If $f\in N^s(x,e_2)$ then there exists $C>0$ such that

  $$\forall \mathbf{j}=(j_1\in \{-2,-1\},j_2\ge 0)d_{\mathbf{j}}\left(x\right)\le C{2}^{-j_{2}s}$$

  (80)

  and

  $$\forall \mathbf{j}\ge \mathbf{0}{d}_{\mathbf{j}}\left(x\right)\le C{2}^{-j_{2}s}.$$

  (81)

  (b)

  Conversely, if f is uniform Lipschitz on $I^2$ and both (80) and (81) hold, then

  $$\forall s^{\prime }<sf\in N^{s^{\prime }}(x,e_2).$$

  (82)

3.3. Relationship between $Li{p}^{\overline{\alpha }}\left(x\right)$ and Pointwise Level Coordinate Axes Lipschitz Regularities

We have already relation (8). In addition, both Theorems 2 and 4 imply that rectangular Lipschitz regularity yields pointwise level coordinate axes Lipschitz regularities.

Set $g\left(y\right)=f\left(y\right)-f(y_1,0)$ and $h\left(y\right)=f\left(y\right)-f(0,y_2)$. Define

$$N_f(x,e_1):=sup\{s\in (0,1):f\in N^s(x,e_1)\}$$

(83)

and

$$N_f(x,e_2):=sup\{s\in (0,1):f\in N^s(x,e_2)\}.$$

(84)

Theorem 5.

If $f\in Li{p}^{\overline{\alpha }}\left(x\right)$ and f is uniform Lipschitz on $I^2$ then $g\in N^{{\alpha }_1-\epsilon }(x,e_1)$ and $h\in N^{{\alpha }_2-\epsilon }(x,e_2)$ for all $\epsilon >0$.

If f is uniform Lipschitz on $I^2$, then

$$N_g(x,e_1)=sup\{{\alpha }_1\in (0,1):\exists {\alpha }_2\in (0,1)f\in Li{p}^{\overline{\alpha }}\left(x\right)\}$$

(85)

and

$$N_h(x,e_2)=sup\{{\alpha }_2\in (0,1):\exists {\alpha }_1\in (0,1)f\in Li{p}^{\overline{\alpha }}\left(x\right)\}.$$

(86)

4. Applications

4.1. The Fractional Brownian Sheets

The following result refines result (4).

Theorem 6.

The fractional Brownian sheet $B_{I^2}^{\overline{H}}$ is pointwise rectangular and level coordinate axes monofractal. More precisely, with probability 1,

$$\forall {\overline{H}}^{\prime }<\overline{H}\forall x\in I^2{B}_{I^2}^{\overline{H}}\in Li{p}^{{\overline{H}}^{\prime }}\left(x\right),$$

$$\forall {\overline{H}}^{\prime }\nleqq \overline{H}\forall x\in I^2{B}_{I^2}^{\overline{H}}\notin Li{p}^{{\overline{H}}^{\prime }}\left(x\right)$$

and

$$\forall x\in I^2\forall i\in \{1,2\}{N}_f(x,e_i)=H_i.$$

Proof of Theorem 6.

The coefficients of the fractional Brownian sheet $B_{I^2}^{\overline{H}}$ in the tensor product Schauder basis were obtained in [7]; in fact

$$B_{I^2}^{\overline{H}}=\sum _{\mathbf{j}\in M^2}\sum _{\mathbf{m}\in {\tilde{N}}_{\mathbf{j}}}{C}_{\mathbf{m}}{\Phi }_{\mathbf{m}}$$

(87)

where ${\left(C_{\mathbf{m}}\right)}_{\mathbf{m}\ge \mathbf{0}}$ is a Gaussian sequence, with $E\left(C_{\mathbf{m}}\right)=0$, and the variance given by the formula

$$E\left(\right|C_{\mathbf{m}}{|}^2)=\prod _{i=1}^2{a}_{m_i}$$

(88)

with

$$a_0=0,a_1=1\mathrm{and}{a}_{m_i}=(2^{-2H_i}-2^{-2})2^{-j_{i}2H_i}\mathrm{for}{m}_i\in {\tilde{N}}_{j_i}{j}_i\ge 0.$$

(89)

As mentioned in [2],

$$\forall y_1{B}_{I^2}^{\overline{H}}(y_1,0)=0\mathrm{and}\forall y_2{B}_{I^2}^{\overline{H}}(0,y_2)=0$$

(this remark follows also from the fact that $a_0=0$).

In [7], it is mentioned that if $m_1=0$ or $m_2=0$ then $C_{\mathbf{m}}=0$ almost surely. For $\mathbf{m}>\mathbf{0}$, put $g_{\mathbf{m}}=\frac{C_{\mathbf{m}}}{\sqrt{E\left(\right|C_{\mathbf{m}}{|}^2)}}$. The following lemma can be obtained using same arguments as Corollary 4.6 in [7].

Lemma 2.There exists $C>0$ such that, with probability 1,

$$C\le \underset{\mathbf{j}>\mathbf{0},\mathbf{m}\in {\tilde{N}}_{\mathbf{j}}}{sup}\left(\underset{{\mathbf{j}}^{\prime }\ge \mathbf{j}}{sup}\left(\underset{{\mathbf{m}}^{\prime }\in {\tilde{N}}_{{\mathbf{j}}^{\prime }},{\lambda }_{m_1^{\prime }}\times {\lambda }_{m_2^{\prime }}\subset {\lambda }_{m_1}\times {\lambda }_{m_2}}{sup}\frac{|g_{{\mathbf{m}}^{\prime }}|}{\sqrt{1+|{\mathbf{j}}^{\prime }-\mathbf{j}|ln2}}\right)\right)<\infty .$$

Since $2^{-j^{\prime }{H}_i}\le 2^{-j{H}_i}$ for all $j^{\prime }\ge j$, then Theorem 6 follows directly from (88), (89), Theorem 2 and Theorem 5. □

4.2. Sierpinski Selfsimilar Functions

We will construct a class of Sierpinski selfsimilar functions that will be pointwise rectangular and level coordinate axes multifractal. Let s and t be two integers with $s<t$. Choose $A\subset \{0,1,\cdots ,s-1\}\times \{0,1,\cdots ,t-1\}$. For $\omega =(a,b)\in A$, the contraction ${\displaystyle S_{\omega }(x_1,x_2)=\left(\frac{x_1}{s}+\frac{a}{s},\frac{x_2}{t}+\frac{b}{t}\right)}$ maps $I^2$ into the rectangle

$${\mathfrak{R}}_{\omega }=[\frac{a}{s},\frac{a+1}{s}]\times [\frac{b}{t},\frac{b+1}{t}].$$

(90)

The (general) Sierpinski carpet K (see [18,19,20]) and references therein) is the unique non-empty compact set (see [21,22]) satisfying

$$K=\bigcup _{\omega \in A}{S}_{\omega }\left(K\right).$$

(91)

It is given by

$$\begin{array}{ccc}\hfill K& =& \{x\in I^2:{(S_{{\omega }_1}\circ \cdots \circ S_{{\omega }_l})}^{-1}\left(x\right)\in \bigcup _{\omega \in A}{\mathfrak{R}}_{\omega }\forall \omega =({\omega }_1,\cdots ,{\omega }_l)\in A^l\forall l\in \mathbb{N}\}\hfill \\ & =& {\displaystyle \bigcap _{l\in \mathbb{N}}\left(\bigcup _{\omega \in A^l}{\mathfrak{R}}_{\omega }\right)}\hfill \end{array}$$

where

$${\mathfrak{R}}_{\omega }=(S_{{\omega }_1}\circ \cdots \circ S_{{\omega }_l})\left(I^2\right)\mathrm{for}\omega =({\omega }_1,\cdots ,{\omega }_l).$$

Put $\Gamma (x_1,x_2)=\mathsf{\Lambda }\left(x_1\right)\mathsf{\Lambda }\left(x_2\right)$, where $\mathsf{\Lambda }\left(t\right)=min(t,1-t)$ if $t\in [0,1]$ and 0 elsewhere. Clearly ${\displaystyle \mathsf{\Lambda }\left(t\right)=\frac{1}{2}{\Phi }_2\left(t\right)}$. The Sierpinski selfsimilar function adapted to the subdivision A satisfies

$$\forall x\in I^{2}f\left(x\right)=\sum _{\omega \in A}{\gamma }_{\omega }f\left(S_{\omega }^{-1}\left(x\right)\right)+\Gamma (x_1,x_2).$$

(92)

Define

$${\displaystyle {\left|\gamma \right|}_{max}=\underset{\omega \in A}{max}|{\gamma }_{\omega }{|,|\gamma |}_{min}=\underset{\omega \in A}{min}\left|{\gamma }_{\omega }\right|,and{H}_{min}=-\frac{{log\left|\gamma \right|}_{max}}{logt}.}$$

The following result was obtained in [18].

Proposition 6.

Suppose that ${\displaystyle \sum _{\omega \in A}\left|{\gamma }_{\omega }\right|<st}$, then the series

$$f\left(x\right)=\Gamma \left(x\right)+\sum _{l=1}^{\infty }\sum _{({\omega }_1,\cdots ,{\omega }_l)\in A^l}{\gamma }_{{\omega }_1}\cdots {\gamma }_{{\omega }_l}\Gamma \left(S_{{\omega }_l}^{-1}\cdots S_{{\omega }_1}^{-1}\left(x\right)\right).$$

(93)

is a unique solution in $L^1\left(I^2\right)$ for equation (92).

If furthermore ${\displaystyle \frac{1}{t}<{\left|\gamma \right|}_{max}<1}$, then $f\in Li{p}^{H_{min}}\left(I^2\right)$ with $0<H_{min}<1$.

The Sierpinski selfsimilar function is written as the superposition of similar anisotropic structures at different scales, reminiscent of some possible modelization of turbulence or cascade models. In [18], we proved that some Sierpinski selfsimilar functions don’t satisfy the thermodynamic formalism.

Clearly if ${\omega }_l=(a_l,b_l)$ then

$$\Gamma \left(S_{{\omega }_l}^{-1}\cdots S_{{\omega }_1}^{-1}\left(x\right)\right)=\mathsf{\Lambda }(s^l{x}_1-s^{l-1}{a}_1-\cdots -s{a}_{l-1}-a_l)\mathsf{\Lambda }(t^l{x}_2-t^{l-1}{b}_1-\cdots -t{b}_{l-1}-b_l).$$

Consider the “separated open set condition”

$$\forall (\omega ,{\omega }^{\prime })\in A^2\omega \ne {\omega }^{\prime }\Rightarrow {\mathfrak{R}}_{\omega }\cap {\mathfrak{R}}_{{\omega }^{\prime }}=\varnothing .$$

(94)

If $x\notin K$ then there exists a neighborhood $\vartheta \left(x\right)$ of x and $L\in \mathbb{N}$ such that

$$\forall y\in \vartheta \left(x\right)f\left(y\right)=\Gamma \left(y\right)+\sum _{l=1}^L\sum _{({\omega }_1,\cdots ,{\omega }_l)\in A^l}{\gamma }_{{\omega }_1}\cdots {\gamma }_{{\omega }_l}\Gamma \left(S_{{\omega }_l}^{-1}\cdots S_{{\omega }_1}^{-1}\left(y\right)\right).$$

(95)

It follows that $f\in Li{p}^{\overline{\alpha }}\left(x\right)$ for all $\overline{\alpha }<\mathbf{1}$.

On the other hand, from the “separated open set condition” (94), any $x\in K$ has a unique expansion

$${\displaystyle x=\left(\sum _{l=1}^{\infty }\frac{a_l}{s^l},\sum _{l=1}^{\infty }\frac{b_l}{t^l}\right)with(a_l,b_l)=(a_l\left(x\right),b_l\left(x\right))={\omega }_l={\omega }_l\left(x\right)\in A.}$$

(96)

For $L\ge 1$, denote by

$$\omega (L,x)=({\omega }_1,\cdots ,{\omega }_L)and{\gamma }_{\omega (L,x)}={\gamma }_{{\omega }_1}\cdots {\gamma }_{{\omega }_L}.$$

Let S and T be two positive integers. Assume that $s=2^S$ and $t=2^T$. Set

$$r\left(x\right)=\underset{L\to \infty }{lim\; inf}\frac{log|{\gamma }_{\omega (L,x)}|}{log{2}^{-L}}.$$

Theorem 7.

Let ${\displaystyle \frac{1}{t}<{\left|\gamma \right|}_{max}<1}$. Assume that the “separated open set condition”(94) holds. Assume furthermore that each column and each row of the grid contains at most one box ${\Re }_{\omega }$, with $\omega \in A$. If ${\left|\gamma \right|}_{min}<{\left|\gamma \right|}_{max}$, then the obtained class of Sierpinski selfsimilar functions f are pointwise rectangular and level coordinate axes multifractal. More precisely, if $x\in K$ and $r\left(x\right)<S$ then

$$f\in Li{p}^{\overline{\alpha }}\left(x\right)\forall S{\alpha }_1+T{\alpha }_2<r\left(x\right),$$

$$f\notin Li{p}^{\overline{\alpha }}\left(x\right)\forall S{\alpha }_1+T{\alpha }_2>r\left(x\right)$$

and

$$(N_f(x,e_1),N_f(x,e_2))=(\frac{r\left(x\right)}{S},\frac{r\left(x\right)}{T}).$$

Proof of Theorem 7.

Clearly Schauder leaders of f satisfy $d_{(2,2)}=C_{(2,2)}=\frac{1}{4}$, and if $L\ge 1$,$\mathbf{j}=(LS,LT)$, $\mathbf{m}=(m_1,m_2)\in {\tilde{N}}_{\mathbf{j}}$ with $(m_1,m_2)=(n_1+2^{LS},n_2+2^{LT})$ and ${\displaystyle ((n_1-1)2^{-LS},(n_2-1)2^{-LT})=(\sum _{l=1}^L\frac{a_l}{s^l},\sum _{l=1}^L\frac{b_l}{t^l})}$, then $d_{\mathbf{m}}=C_{\mathbf{m}}=\frac{1}{4}|{\gamma }_{{\omega }_1}|\cdots |{\gamma }_{{\omega }_L}|$ (because ${\left|\gamma \right|}_{max}<1$).

Let $x\in K$ and $\mathbf{j}=(j_1,j_2)\ge \mathbf{0}$ with $\mathbf{j}\ne \mathbf{0}$. Write $(L_1-1)S<j_1\le L_{1}S$ and $(L_2-1)T<j_2\le L_{2}T$. Put $L=max(L_1,L_2)$. Then (47), (50), (94) and the assumption that each column and each row of the grid contains at most one box ${\Re }_{\omega }$, with $\omega \in A$, imply that

$$\frac{1}{4}|{\gamma }_{\omega (L,x)}|\le d_{\mathbf{j}}\left(x\right)\le \frac{1}{4}\left|{\gamma }_{\omega (L-1,x)}\right|.$$

(97)

Hence Theorem 2 and Theorem 5 yield Proposition 7. Indeed

$$\forall \epsilon >0\exists L_{\epsilon }\forall L\ge L_{\epsilon }\left|{\gamma }_{\omega (L,x)}\right|\le 2^{-L\left(r\right(x)-\epsilon )}$$

(98)

and

$$\epsilon >0\exists L_n\nearrow \infty \forall n\left|{\gamma }_{\omega (L_n,x)}\right|>2^{-L_n(r\left(x\right)+\epsilon )}.$$

(99)

If $S{\alpha }_1+T{\alpha }_2<r\left(x\right)$ then for $\epsilon =r\left(x\right)-S{\alpha }_1-T{\alpha }_2$ relations (97) and (98) imply that

$$\exists L_{\epsilon }\forall L>L_{\epsilon }{d}_{\mathbf{j}}\left(x\right)\le C{2}^{-LS{\alpha }_1-LT{\alpha }_2}\le C{2}^{-j_1{\alpha }_1-j_2{\alpha }_2}.$$

If $S{\alpha }_1+T{\alpha }_2>r\left(x\right)$ then for $2\epsilon =S{\alpha }_1+T{\alpha }_2-r\left(x\right)$ relations (97) and (99) imply that

$$\forall n{d}_{(L_{n}S,L_{n}T)}\left(x\right)>\frac{1}{4}{2}^{-L_{n}S{\alpha }_1-L_{n}T{\alpha }_2+\epsilon L_n}.$$

 □

5. Conclusions

In [2,7], it was proved that for any cube $\mathcal{Q}\subset {\mathbb{R}}^2$, the restrictions $B_{\mathcal{Q}}^{\overline{H}}$ of realizations of fractional Brownian sheets $B^{\overline{H}}$ to $\mathcal{Q}$ are uniform rectangular Lipschitz with order ${\overline{H}}^{\prime }<\overline{H}$. In this paper, we first improved that result pointwisely, namely $B_{\mathcal{Q}}^{\overline{H}}$ are pointwise rectangular (respectively level coordinate axes) monofractal with order ${\overline{H}}^{\prime }<\overline{H}$ (respectively $H_i$). The proof is based on some criteria of the latest pointwise regularities in terms of the rectangular Schauder system, obtained in this paper. A second application of these criteria was done, namely we constructed a class of Sierpinski selfsimilar functions that are pointwise rectangular and level coordinate axes multifractal.