Journe
In this post, we recall how the proof of Journe’s lemma goes.
Set up
We will be working on $\mathbb{R}^N$ and with two continuous foliations $F_1$ and $F_2$ which are transversely Holder continuous and which have $C^k$ leaves ($k \geq 1$). Denote the dimensions of these foliations by $N_1$ and $N_2$, so $N_1 + N_2 = N$. To help with notation later, we will view $\mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$ with the product metric, and will write $B_\varepsilon(0) = B_{\varepsilon}^{N_1}(0) \times B_{\varepsilon}^{N_2}(0)$. Recall that having two transverse foliations means that each $F_i$ induces equivalence relations on $\mathbb{R}^N$, where the equivalence classes are referred to as leaves, and that there exists a $\varepsilon > 0$ such that if $B_\varepsilon(0) \subseteq \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$ is the open ball in the product metric, then for each $x \in \mathbb{R}^N$ there is an open neighborhood $U_x$ and a homeomorphism $\Phi_x : B_\varepsilon(0) \rightarrow U_x$ such that
1) $\Phi_x(0,0) = x$,
2) for any fixed $(v,w) \in B_\varepsilon(0)$, the slice maps defined by $u \mapsto \Phi_x(u,w)$ and $q \mapsto \Phi_x(v, q)$ satisfy $\text{Im}(\Phi_x(\cdot, w)) \subseteq F_1(\Phi_x(0,w))$ and $\text{Im}(\Phi_x(v, \cdot)) \subseteq F_2(\Phi_x(v,0))$,
| 3) the map $x \mapsto \Phi_x$ is a Holder continuous map, meaning that there are constants $C > 0$ and $\alpha \in (0,1)$ such that $ | \Phi_x - \Phi_y | _{C^k} \leq C | x-y | ^\alpha$, and |
4) denoting the local leaves for $F_1$ and $F_2$ at $x$ by $F_1^{loc}(x) := {\Phi_x(v, 0) : v \in B_r^{N_1}(0) }$ and $F_2^{loc}(x) := {\Phi_x(0, w) : w \in B_{r}^{N_2}(0) }$ for $r > 0$ small enough so that this is a $C^k$ embedding, we have that $T_x F_1(x) \oplus T_x F_2(x) = \mathbb{R}^N$.
We call the family of maps ${\Phi_x \mid x \in \mathbb{R}^N}$ the foliation maps. It is not hard to see from the above definition that the leaves $F_i(x)$ are $C^k$ immersed submanifolds. It thus makes sense to talk about regularity of functions on the leaves of the foliations. In particular, given a function $\psi : \mathbb{R}^N \rightarrow \mathbb{R}$, we say that it is $C^{n,\alpha}$ on the foliation $F_1$ (where $n + \alpha < k$) if for each $x \in \mathbb{R}^N$ and $(v,w) \in B_\varepsilon(0)$ fixed, we have that the functions $\psi \circ \Phi_x(\cdot, w) : B_\varepsilon^{N_1}(0) \rightarrow \mathbb{R}$ are $C^{n,\alpha}$. Similarly, it is $C^{n,\alpha}$ on the foliation $F_2$ if the same holds for $\psi\circ \Phi_x(v, \cdot ) : B_{\varepsilon}^{N_2}(0) \rightarrow \mathbb{R}$. We denote the collection of functions which are uniformly $C^{n,\alpha}$ on the foliations $F_i$ by $C^{n,\alpha}_i$. If $\Sigma$ is an open subset, then restricting these maps it makes sense to write $C^{n,\alpha}_i(\Sigma)$ to indicate the family of functions whose domain is $\Sigma$. Finally, since on each leaf $\psi$ is $C^{n,\alpha}$, we can define
\[|\psi|_{C^{n,\alpha}_1 \cap C^{n,\alpha}_2} = \sup_{x \in \Sigma} \sup_{w \in B_{\varepsilon}^{N_2}(0)} |\psi \circ \Phi_x(\cdot, w)|_{C^{n,\alpha}(B^{N_1}_\varepsilon(0))} + \sup_{x \in \Sigma} \sup_{v \in B_{\varepsilon}^{N_1}(0)} |\psi \circ \Phi_x(v, \cdot)|_{C^{n,\alpha}(B^{N_2}_\varepsilon(0))}.\]As it will be important later, let’s also define the bracket of two points. If $y,z \in \Phi_x(B_\varepsilon(0))$ for some $x$, then the leaf bracket of these points is the unique point $[y,z] \in F_1^{loc}(y) \cap F_2^{loc}(z) \cap \Phi_x(B_\varepsilon(0))$. To see why there exists such a point, write $\Phi_x^{-1}(y) = (v_y, w_y)$ and $\Phi_x^{-1}(z) = (v_z, w_z)$. Using the above properties of $\Phi_x$, we have that $[y,z] := \Phi_x(v_y, w_z) \in F_1^{loc}(y) \cap F_2^{loc}(z) \cap \Phi_x(B_\varepsilon(0))$. Note that we have to use the local leaves, as the global leaves can have multiple intersections.
From now on, let $\Sigma$ be a precompact open subset. If the foliations were transversely $C^k$ instead of just Holder continuous, then it follows that $C^{n,\alpha}_1(\Sigma) \cap C^{n,\alpha}_2(\Sigma) \subseteq C^{n,\alpha}(\Sigma)$ when $n + \alpha < k$. For example, from standard calculus, we know that this is the case if one takes the foliations which are the coordinate axes (in fact, it is an equality in this case). However, the lack of regularity in the transverse direction makes this harder to evaluate in general. As noted by Journe, though, it turns out that transverse regularity should not prevent such a function from being smooth. Our main goal is to prove Journe’s theorem (see here).
Theorem: Let $\Sigma \subseteq \mathbb{R}^N$ be a precompact open subset. If $F_1$ and $F_2$ are two continuous foliations of $\mathbb{R}^N$ with $C^k$ leaves $(k \geq 1$) that are transversely Holder continuous, then for all integers $n \geq 0$ and $\alpha \in (0,1)$ such that $n + \alpha < k$, we have $C_1^{n,\alpha}(\Sigma) \cap C_2^{n,\alpha}(\Sigma) \subseteq C^{n,\alpha}(\Sigma)$.
Whitney-Campanato theorem
| As a preliminary, we need to understand when a function has regularity $C^{n,\alpha}$. This leads us to the following lemma, which can be thought of as a version of Campanato’s theorem. Let $\Sigma \subseteq \mathbb{R}^N$ be some subset. We say that a function $\psi : \Sigma \rightarrow \mathbb{R}$ is Whitney $C^{n,\alpha}$ if there is a constant $K > 0$ such that for each $x \in \Sigma$, there is a polynomial $P_x$ with degree at most $n$ where $P_x(x) = \psi(x)$ and for each $x,y \in \Sigma$ and multi-index $0 \leq | \beta | \leq n$, we have |
In essence, a function is Whitney $C^{n,\alpha}$ on a subset $\Sigma$ if it satisfies uniform Taylor remainder bounds globally across $\Sigma$. We call the above equation the Whitney compatibility condition. We note that the Whitney compatibility condition is deeply connected to another major analysis result, which is the Whitney extension theorem. We will just work with open subsets in our setting.
Theorem: Let $\Sigma \subseteq \mathbb{R}^N$ be an open subset. If $\psi$ is Whitney $C^{n,\alpha}$ on $\Sigma$, then $\psi$ is $C^{n,\alpha}$ on $\Sigma$.
A version of this theorem is referred to as Campanato’s lemma by Journe, and can be found here. We offer an elementary proof of this in the case where $N=1$ which can be generalized easily to higher dimensions (we remark on the differences between this and the actual Campanato lemma at the end).
Proof: To prove the elementary case $N=1$, let $P_x$ be the family of polynomials with degree at most $n$. We can write
\[P_x(y) = \sum_{k=0}^n a_k(x) \frac{(y-x)^k}{k!}.\]By definition, $\psi(x+h) = P_{x+h}(x+h)$, and applying the Whitney compatibility conditions we see that
\[|P_{x+h}(x+h) - P_x(x+h)| \leq K |h|^{n+\alpha}.\]In particular, we have
\[\psi'(x) = \lim_{h \rightarrow 0} h^{-1} [\psi(x+h) - \psi(x)] = \lim_{h \rightarrow 0} h^{-1}[P_x(x+h) - P_x(x) + O(|h|^{n+\alpha})] = P_x'(x).\]With our above notation,
\[P_x'(x) = a_1(x).\]Thus, we have shown that $\psi’(x) = a_1(x)$, which shows that $\psi$ is differentiable. This shows the base case, and we now proceed by an induction argument. We now assume that the $k$ derivative satisfies $\psi^{(k)}(x) = a_{k}(x)$, and our goal is to use the Whitney compatibility conditions to show that $\psi^{(k+1)}(x) = a_{k+1}(x)$. First, note
\[|\partial^k P_{x+h}(x+h) - \partial^k P_x(x+h)| \leq K |h|^{n+\alpha - k} \quad \text{and}\quad\partial^k P_{x+h}(x+h) = a_k(x+h) .\]Finally, note that
\[\partial^kP_{x}(x+h) = a_k(x) + a_{k+1}(x) h + O(h^2).\]Substituting this in,
\[|a_k(x+h) - a_k(x) + O(h^2)| \leq K |h|^{n+\alpha-k}.\]Thus, normalizing both sides by $h$ and taking a limit as $h$ tends to zero:
\[\lim_{h \rightarrow 0} |h^{-1}(a_k(x+h) - a_k(x)) - a_{k+1}(x)| = 0,\]provided $k < n$. Putting it all together, this shows us that
\[\psi^{(k+1)}(x) = \lim_{h \rightarrow 0} h^{-1}[\psi^{(k)}(x+h) - \psi^{(k)}(x)] = \lim_{h \rightarrow 0} h^{-1}[a_k(x+h) - a_k(x)] = a_{k+1}(x).\]We now need to show that $\psi^{(n)}(x) = a_n(x)$ is Holder continuous. But this simply follows from the fact that $\partial^n P_y(z) = a_n(y)$ for all $z$, so
\[| \partial^n P_x(x) - \partial^n P_y(x)| = |a_n(x) - a_n(y)| = |\psi^{(n)}(x) - \psi^{(n)}(y)| \leq K d(x,y)^\alpha. \square\]Remark: The actual statement and proof of Campanato’s lemma is much more technical. There, it is not assumed that $\Sigma$ is open, but rather that $\Sigma$ is a convex domain with compact closure. The space of functions defined is much more technical as well. One then has to use some machinery (i.e., the Campanato embedding theorem) to get the result. This is really the highlight of Campanato’s paper. $\square$
In our quest for Journe, we will be working with $\Sigma$ an open subset and we will be working with the Whitney $C^{n,\alpha}$ spaces.
Remark: Since $\Sigma$ is an open precompact set, we only need to know the Whitney compatibility conditions on small scales. More precisely, we only need to verify that for a uniform $\delta> 0$, the Whitney compatibility condition holds for all polynomials corresponding to $x$ and $y$ with $d(x,y) < \delta$. $\square$
Journe’s theorem
In light of the above, let’s now describe the strategy of the proof of Journe’s theorem. Let $\psi \in C^{n,\alpha}_1 \cap C^{n,\alpha}_2$. Given $x \in \Sigma$, we can use the inverse function theorem to find a $C^k$-diffeomorphism $\hat{\Phi}_x : V \rightarrow U$, where $U \subseteq \Sigma$ is an open subset of $x$ and $V \subseteq \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$ is an open subset of $(0,0)$, such that it flattens the local leaves to the coordinate axes, i.e., $\hat{\Phi}_x(0,0) = x$, $\hat{\Phi}_x((\mathbb{R}^{N_1} \times {0}) \cap V) \subseteq F_1^{loc}(x)$, and $\hat{\Phi}_x(({0} \times \mathbb{R}^{N_2}) \cap V) \subseteq F_2^{loc}(x)$.
Because the foliations lack transverse regularity, we pull our analysis back onto the flat space $V$, where the foliations are the standard coordinate axes. Consider the local function
\(\psi_x(v,w) := \psi \circ \hat{\Phi}_x(v,w) : V \rightarrow \mathbb{R}.\) Because $\psi \in C^{n,\alpha}_1 \cap C^{n,\alpha}_2$ and $\hat{\Phi}_x$ is $C^k$, the function $\psi_x$ inherits regularity in the separate variables. More precisely, the map $v \mapsto \psi_x(v,w)$ is uniformly $C^{n,\alpha}$ for every $w$ and $w \mapsto \psi_x(v,w)$ is uniformly $C^{n,\alpha}$ for each $v$.
The game is to construct a polynomial $Q_{(0,0)}$ on $V$ of degree at most $n$ which will be our candidate Taylor polynomial for $\psi_x$ at $(0,0)$. We will do so by interpolating what the Taylor polynomial for $\psi_x$ should be on a family of grids which are all converging to $(0,0)$ geometrically fast. We then push this polynomial to a polynomial $P_x$ on $U$ via Taylor’s theorem, and we then show that under appropriate compatible grids, these polynomials must satisfy the Whitney compatibility conditions. By the Whitney-Campanato theorem, this implies that $\psi \in C^{n,\alpha}(U)$, and we can make this uniform on all of $\mathbb{R}^N$. The goal of the rest of this post is to make this argument rigorous.
Step 1: Interpolating polynomials and stability
Before going into the construction of the polynomial, let’s recall how to interpolate polynomials. Let $\mathcal{N} = {\beta = (\beta_1, \ldots, \beta_N) \mid \beta_i \in {0, \ldots, n}}$ be the collection of admissible multi-indices, let $\mathcal{K} = { (k_1, \ldots, k_N) \mid 0 \leq k_i \leq n}$, let $\mathcal{B} := {x_k^i \mid 0 \leq k \leq n, 1 \leq i \leq N} \subseteq \mathbb{R}$, and let $\mathcal{P}_n(\mathbb{R}^N)$ be the collection of polynomials which have degree at most $n$ in each variable. Using multi-index notation, one can more precisely write
\[\mathcal{P}_n(\mathbb{R}^N) = \mathcal{P}_n = \left\{ \sum_{\beta \in \mathcal{N}} \alpha_\beta y^\beta \mid \alpha_\beta \in \mathbb{R} \right\}.\]Note that, as a vector space, $\dim(\mathcal{P}_n) = (n+1)^N$. Thus, a polynomial in $\mathcal{P}_n$ is uniquely determined by its values on $(n+1)^N$ points. For each multi-index $K = (k_1, \ldots, k_N) \in \mathcal{K}$, define
\[x_K = (x_{k_1}^1, \ldots, x_{k_N}^N) \in \mathbb{R}^N.\]The associated grid to $\mathcal{B}$ is given by
\[\mathcal{G}(\mathcal{B}) = \mathcal{G} = \left\{x_K \mid K \in \mathcal{K}\right\} \subseteq \mathbb{R}^N\]We define the norm of this grid by
\[|\mathcal{G}| = \sup_{K \in \mathcal{K}} |x_K|,\]| where here $ | \cdot | $ is some fixed norm on $\mathbb{R}^N$. This is a collection of $(n+1)^N$ points, and we want to construct a polynomial which takes prescribed values on these points. More precisely, given $K = (k_1, \ldots, k_N) \in \mathcal{K}$, we want a polynomial $L_K \in \mathcal{P}n(\mathbb{R}^N)$ such that for all $x{K’} \in \mathcal{G}$, we have |
Note that once we construct such a polynomial, it is unique. Constructing such a polynomial is done coordinate wise – we have
\[L_K(y_1, \ldots, y_N) = \prod_{i=1}^N \prod_{\substack{0 \leq j \leq n \\ j \neq k_i}} \frac{y_i - x_{j}^i}{x_{k_i}^i - x_{j}^i}\]Such a polynomial is referred to as a Lagrange basis polynomial, and we refer to the family of polynomials ${L_K \mid K \in \mathcal{K}}$ as the Lagrange basis associated to the grid $\mathcal{G}$. As it will be convenient, let’s also define the following Lagrange numerator polynomial for $K \in \mathcal{K}$:
\[N^i_{K}(y) = \prod_{\substack{ 0 \leq j \leq n \\ j \neq k_i}} (y_i - x_{j}^i),\]and let’s also define
\[L_k^i(y_1, \ldots, y_N) = \prod_{\substack{0 \leq j \leq n \\ j \neq k_i}} \frac{y_i - x_{j}^i}{x_{k_i}^i - x_{j}^i}.\]We emphasize that there is an ambient grid $\mathcal{G}$ in the background of $L_K$, $L_K^i$, and $N^i_K$. Denote the collection of functions taking values on $\mathcal{G}$ by
\[F(\mathcal{G}) = \{f : \mathcal{G} \rightarrow \mathbb{R}\}.\]Throughout this note, we are going to abuse notation. Given any kind of family of function $\mathcal{F} \subseteq {f : \mathbb{R}^N \rightarrow \mathbb{R}}$, there is a restriction operator
\[\eta_{\mathcal{F}, \mathcal{G}}:\mathcal{F} \rightarrow F(\mathcal{G}), \quad \eta_{\mathcal{F}, \mathcal{G}}(f)(x_K) = f(x_K).\]More precisely, we just restrict arbitrary functions on $\mathbb{R}^N$ to functions on $\mathcal{G}$ by just forgetting the values on the other points.
With the Lagrange basis polynomials, if we have a function $f \in F(\mathcal{G})$, then it naturally induces a polynomial of degree at most $n$ in each variable:
\[\mathcal{I}_{\mathcal{G}} : F(\mathcal{G}) \rightarrow \mathcal{P}_n, \quad \mathcal{I}_\mathcal{G}(f)(y) = \sum_{K \in \mathcal{K}} f(x_K) L_K(y).\]We call $\mathcal{I}{\mathcal{G}}$ the *Lagrange interpolation operator*. Broadly speaking, an *interpolation operator* is just a map $\Phi : F(\mathcal{G}) \rightarrow \mathcal{P}_n$ such that $\Phi(f)(x) = f(x)$ for all $x \in \mathcal{G}$; the Lagrange interpolation operator is just a particular example of such an operator. Note that all interpolation operators will satisfy the property that $\eta{\mathcal{P}_n, \mathcal{G}} \circ \Phi = I$, where $I$ will indicate the identity throughout. We also introduce the following norm on $F(\mathcal{G})$:
\[|f| = \sup_{v \in \mathcal{G}} |f(v)|.\]| The main question now is how stable this map is. Namely, we want to interpolate our function $f$ in such a way so that the new polynomial $\Phi_\mathcal{G}$ has good derivatives so that it can form a Whitney approximation just based on the values of $f$. We have the following stability lemma where we make our constants explicit. To help with notation, let $C_N > 0$ be a universal constant such that $ | v | _{\ell^1} \leq C_N | v | $. We say that a grid $\mathcal{G}$ is $c$-separated for some $c > 0$ if $ | \Delta(\mathcal{G}) | \geq c | \mathcal{G} | $. |
| Lemma: For every $R > 0$, $n \geq 1$ an integer, $0 \leq | \beta | \leq n$ a multi-index, and $\mathcal{G}$ a grid which is $c$-separated, we have that for all $y \in B_{R | \mathcal{G} | }(0)$, |
| Proof: First, let’s observe that for all $y = (y_1, \ldots, y_N) \in \mathbb{R}^N$ such that $ | y | \leq R | \mathcal{G} | $, we have |
For any grid point $x_K = (x_{k_1}^1, \ldots, x_{k_N}^N) \in \mathcal{G}$, we also have
\[|x_{k_i}^i| \leq C_N |x_K| \leq C_N |\mathcal{G}|.\]| Consequently, for all such $ | y | \leq R | \mathcal{G} | $, |
and thus, writing our multi-index as $\beta = (\beta_1, \ldots, \beta_N)$, we have
\[\left|\frac{d^{\beta_i}}{dy_i^{\beta_i}} N_K^i(y)\right| \leq \frac{n!}{(n - \beta_i)!} (C_N (R+1) |\mathcal{G}|)^{n - \beta_i}.\]By construction, the denominator has the bound
\[\prod_{\substack{0 \leq j \leq n \\ k_i \neq j}} |x^i_{k_i} - x^i_j| \geq \Delta(\mathcal{G})^n \geq c^n |\mathcal{G}|^n.\]This gives us the upper bound
\[\left| \frac{d^{\beta_i}}{dy_i^{\beta_i}} L_K^i(y) \right| \leq \frac{n!}{c^n (n-\beta_i)!} (C_N (R+1))^{n-\beta_i} |\mathcal{G}|^{-\beta_i}.\]For every multi-index $\beta$ and $K \in \mathcal{K}$, this then gives the bound
\[|\partial^\beta L_K(y)| \leq \frac{(n!)^N}{c^{nN} \prod_{i=1}^N (n - \beta_i)!} (C_N (R+1))^{Nn - |\beta|} |\mathcal{G}|^{-|\beta|}.\]The interpolating polynomial satisfies
\[|\partial^\beta \mathcal{I}_{\mathcal{G}}(f)(y)| \leq |f| \sum_{K \in \mathcal{K}} |\partial^\beta L_K(y)|,\]and the result follows. $\square$
Write
\[\mathcal{I}_\mathcal{G}(f)(y) = \sum_{0 \leq |\beta| \leq n} \alpha_\beta y^\beta,\]here using multi-index notation. Observe that the coefficients of the polynomial take the form
\[\alpha_\beta = \frac{1}{\beta!} \partial^\beta \mathcal{I}_\mathcal{G}(f)(0).\]We have the following corollary on the coefficients of the polynomial.
| Lemma: For every $n \geq 1$ an integer, $0 \leq | \beta | \leq Nn$ a multi-index, and $\mathcal{G}$ a grid which is $c$-separated, we have |
For our purposes, the constant $C_N$ will be fixed, the constant $n$ will be fixed, and the constant $R > 0$ will be fixed. The main issue with this stability estimate is that it is dependent on what $c$ is, which a priori cannot be fixed. It is clear that worse grids will make this constant $c$ go to zero, hence our estimates on the right will blow up and be useless. The game now is to understand the stability of our interpolated polynomial under good grids, i.e., grids which are well enough distributed so that we do not have this blow up phenomenon. For $R > 0$ and $\eta > 1$, we say that a grid $\mathcal{G}$ is $(R,\eta, \varepsilon)$-quasi-uniform in $B_{|\mathcal{G}|R}(0)$ if it is $\eta \varepsilon$-dense and $\varepsilon$-separated, meaning
-
for every $x \in B_{ \mathcal{G} R}(0)$, there is a $y \in \mathcal{G}$ such that $d(x,y) < \eta \varepsilon$, and - for every $x,y \in \mathcal{G}$, we have $d(x,y) \geq \varepsilon$. If our grid is nice, then our upper bound simplifies significantly after absorbing fixed constants, since the corresponding grid is automatically separated.
| Lemma: For every $\eta > 1$ and $R > 0$, there is a constant $C > 0$ depending only on $N$, $n$, $R$, and $\eta$ such that for all $\varepsilon > 0$, if $\mathcal{G}$ is an $(R,\eta, \varepsilon)$-quasi-uniform grid, then for all multi-indices $0 \leq | \beta | \leq n$ and all functions $f : \mathcal{G} \rightarrow \mathbb{R}$, we have |
Furthermore, the polynomial is uniquely determined.
| Proof: Simply note that $\Delta(\mathcal{G}) \geq \varepsilon$ from separation and $ | \mathcal{G} | R \leq (n+1) \eta \varepsilon$ from density. The constant $c$ now depends only on $N$, $n$, $R$, and $\eta$. The last part follows from the fact that a polynomial in $\mathcal{P}_n(\mathbb{R}^N)$ is completely determined by $(n+1)^N$-points. $\square$ |
This is the extent that one can push the Lagrange interpolation operator. Furthermore, this interpolation operator does not respect any regularity along the leaves. Journe took advantage of the Lagrange interpolation operator to define a new interpolation operator which does respect regularity. However, this interpolation operator is defined more geometrically and analytically as opposed to algebraically, and takes some time to define.
Throughout, we will let $\mathcal{G}$ be an $(R,\eta,\varepsilon)$-quasi-uniform grid which is closed, meaning that if $x, y \in \mathcal{G}$, then $[x,y] \in \mathcal{G}$. Let $F_1$ and $F_2$ be two transverse foliations on $\mathbb{R}^N$ and let ${\hat{\Phi}x \mid x \in \mathbb{R}^N}$ be the associated “flattening maps” described. Recall that there is a uniform $r > 0$ such that the domain of $\hat{\Phi}_x$ is $B_r(0$). If we define $\mathcal{G}* = \hat{\Phi}_x^{-1}(\mathcal{G})$, then we have an induced map
\[\hat{\Phi}_x^* : F(\mathcal{G}) \rightarrow F(\mathcal{G}_*), \quad \hat{\Phi}_x^*(f) = f \circ \hat{\Phi}_x.\]Note that the foliated map preserves a grid being an $(R, \eta, \varepsilon)$-quasi-uniform grid, up to possible multiplicative constants. Furthermore, it will also preserve being closed. In light of this, we can now study our grid relative to the domain of $B_r(0)$. Without loss of generality, we will also assume that $\mathcal{G}$ is a grid such that $x \in \mathcal{G}$, which implies that $0 \in \mathcal{G}_*$. The advantage of this viewpoint is that we may assume that our foliation leaves are precisely the coordinate axes. More precisely, given $x \in \mathbb{R}^N$, we define
\[\mathcal{G}^1(x) = \{ [x, x_K] \mid x_K \in \mathcal{G} \} \quad \text{and} \quad \mathcal{G}^2(x) = \{[x_K, x] \mid x_K \in \mathcal{G} \}.\]We then see that $\hat{\Phi}x^{-1}(\mathcal{G}^1(x)) = \mathcal{G}^1* \subseteq \mathbb{R}^{N_1} \times {0}$ and $\hat{\Phi}x^{-1}(\mathcal{G}^2(x)) = \mathcal{G}^2* \subseteq {0} \times \mathbb{R}^{N_2}$. It is easy to check that, since the grid is closed under brackets, we have that the projected grids $\mathcal{G}_*^i$ are also $(R, \eta, \varepsilon)$-quasi-uniform (up to possible multiplicative constants, which we ignore). We now want to view everything relative to the foliation map. Let
\[\Pi_i : \{1, \ldots, N\} \rightarrow \{1, \ldots, N_i\}, \quad \Pi_i(k) = \begin{cases} k & \text{if } 1 \leq k \leq N_1, \\ k - N_1 & \text{if } N_1 + 1 \leq k \leq N. \end{cases}\]| We then define $\mathcal{K}_1 = \Pi_1(\mathcal{K})$ and $\mathcal{K}_2 = \Pi_2(\mathcal{K})$. Provided $ | \mathcal{G} | $ is small enough, we see that every $z_K \in \mathcal{G}*$ is uniquely of the form $z_K = [y{K_2}, x_{K_1}]$, where $K_i \in \mathcal{K}i$, $x{K_1} \in \mathcal{G}^1*$, and $y{K_2} \in \mathcal{G}^2_*$. Thus, we have a well-defined bijective map |
Furthermore, because the distributions defining the foliations are continuous and are actually the coordinate axes at $0$, one can show that we have
\[| z_K - \zeta(z_K)| = o(|\mathcal{G}|).\]The constants can be made uniform. With this in mind, we have Lagrange interpolation bases ${L_{K_i} \mid K_i \in \mathcal{K}i}$ for the two foliations based on the $\mathcal{G}^i$ grids. We can then define the *tensorial Lagrange interpolation operator by
\[\mathcal{I}_{\mathcal{G}^1_*, \mathcal{G}^2_*} : F(\mathcal{G}^1_* \times \mathcal{G}_*^2) \rightarrow \mathcal{P}_n, \quad \mathcal{I}_{\mathcal{G}_*^1, \mathcal{G}_*^2}(f)(u,v) = \sum_{\substack{K_1 \in \mathcal{K}_1 \\ K_2 \in \mathcal{K}_2}} f(\zeta^{-1}(x_{K_1}, y_{K_2})) L_{K_1}(u) L_{K_2}(v).\]This can be thought of as an iterated interpolation polynomial which would be the “correct” polynomial if the foliated map respected the axes. To help with notation, we will drop the subscripts when it is understood from context. Journe’s interpolation operator uses the fact that this is a contraction for small enough $\varepsilon$ in order to construct an interpolation operator $\Psi: F(\mathcal{G}*) \rightarrow \mathcal{P}_n.$ More precisely, viewing $\mathcal{I}(F(\mathcal{G}^1 \times \mathcal{G}_^2)) \subseteq F(\mathcal{G}_*)$, Journe observed that our above stability estimates along with the mean value theorem yield
\[|\mathcal{I}(f)(z_K) - f(z_K)| \leq |\mathcal{I}(f)(z_K) - \mathcal{I}(f)(\zeta(z_K)))| \leq C |f| | \mathcal{G}|^{-1} |z_K - \zeta(z_K)|.\]| Note that this is essentially just mean value theorem with the Lagrange interpolation bases. Thus, as long as $ | \mathcal{G} | $ is small enough, we have |
Putting it all together, this shows that
\[|\mathcal{I}(I - \mathcal{I})^k| \leq C2^{-k},\]where $C > 0$ is some constant. We define the Journe interpolation operator by
\[\mathcal{J} : F(\mathcal{G}_*) \rightarrow \mathcal{P}_n, \quad \mathcal{J}(f) = \sum_{k=0}^\infty \mathcal{I}(I - \mathcal{I})^k(f).\]For $M \geq 1$ an integer, let’s also define
\[\mathcal{J}^M : F(\mathcal{G}_*) \rightarrow \mathcal{P}_n, \quad \mathcal{J}^M(f) = \sum_{k=0}^M \mathcal{I}(I - \mathcal{I})^k(f).\]Since $\mathcal{P}n$ is a finite dimensional vector space, we note all norms are equivalent. Specifically, we consider the norm coming from the grid $\mathcal{G}*$:
\[|P| = \sup_{K \in \mathcal{K}} |P(x_K)|.\]| Claim: For every $\eta > 1$ and $R > 0$, if $\mathcal{G}$ is a closed $(R, \eta, \varepsilon)$-quasi-uniform grid with $ | \mathcal{G} | $ sufficiently small, then the Journe interpolation operator is well-defined. |
Proof: We will prove this by showing that ${\mathcal{J}^M}_{M=1}^\infty$ is a Cauchy sequence. Notice that if $M’ > M$, then
\[|\mathcal{J}^{M'}(f) - \mathcal{J}^M(f)| \leq |f| \sum_{k=M+1}^{M'} \left| \mathcal{I}(I - \mathcal{I})^k \right| \leq C |f| \sum_{k=M+1}^\infty 2^{-k}.\]We see that this tends to zero as $M \rightarrow \infty$. Furthermore, we need to check that this is an interpolation operator. Since the grid $\mathcal{G}$ is fixed, we see that
\[\mathcal{J}(f)(x_K) = \lim_{m \rightarrow \infty} \mathcal{J}^m(f)(x_K) = \lim_{m \rightarrow \infty} f(x_K) = f(x_K),\]as desired. $\square$
The real content of Journe’s lemma 1 is the following stability estimate, which is a consequence of our earlier stability estimate. To make this precise, let’s again write
\[\mathcal{J}(f)(y) = \sum_{0 \leq |\beta| \leq n} \alpha_\beta y^\beta.\]| Lemma: For every $\eta > 1$ and $R > 0$, there is a constant $C > 0$ depending only on $N$, $n$, $R$, and $\eta$ such that if $\mathcal{G}$ is a closed $(R, \eta, \varepsilon)$-quasi-uniform grid, then for all multi-indices $\beta = (\beta_1, \beta_2)$ corresponding to $\mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$ with $0 \leq | \beta | \leq n$ and all functions $f \in F(\mathcal{G}_*)$, we have |
In particular, there is a uniform constant $C > 0$ so that
\[\sum_{0 \leq |\beta| \leq nN}|\alpha_\beta| |\mathcal{G}|^{|\beta|} \leq C |f|. \quad \square\]To recap, Journe’s interpolation operator is an approximation of the interpolated polynomial for $f$ only along the grids projected to the corresponding axes. In particular, the Journe interpolation operator will respect the regularity along leaves better, as it was a tensorial operation.
Finally, note that this is a polynomial defined on the domain of the interpolation space. At the end of the day, we really wanted a polynomial on the codomain. Given a sufficiently smooth function $f$, we denote the $n$-Taylor polynomial of $f$ at a point $x$ by $\tau_{x,n}$. By Taylor’s theorem, this operator is bounded. With this in mind, we define the Journe-Taylor interpolation operator by
\[\mathcal{T}_{x,n} = \tau_{x,n} \circ (\hat{\Phi}_x^{-1})^*\circ \mathcal{J} \circ \hat{\Phi}_x^*.\]To reiterate: We push our function to flat coordinates with $\hat{\Phi}_x^*$, then we use the Journe interpolation operator to get a polynomial, then we pull back this polynomial to our original coordinates, and we finish by using Taylor’s theorem to find a true polynomial.
Step 2: Sequences of compatible grids
To recap, the previous step showed that if we have $\psi : \mathbb{R}^N \rightarrow \mathbb{R}$ an arbitrary function as well as a closed $(R, \eta, \varepsilon)$-quasi-uniform grid with $x \in \mathcal{G}$ and $|\mathcal{G}|$ sufficiently small, then there was a well-defined polynomial $P = \mathcal{T}{x,n}(\psi)$. So far, we cannot say much about this polynomial beyond some stability estimates involving either the Journe interpolation operator or the Lagrange interpolation operator. In this step, we assume that we have a function $\psi \in C^{n,\alpha}_1 \cap C^{n,\alpha}_2$, we have a fixed $x \in \Sigma$, and we have a family of closed $(R, \eta, \varepsilon_m)$-quasi-uniform grids ${\mathcal{G}_m(x)}{m=1}^\infty$ which satisfy the following for each $m \geq 1$: 1) $x \in \mathcal{G}m(x)$, 2) $\mathcal{G}_m(x)$ has $(n+1)^N$ points, 3) $|\mathcal{G}{m+2}(x)| \leq 2^{-1}|\mathcal{G}m(x)|$, where $\max{|\mathcal{G}_1(x)|, |\mathcal{G}_2(x)|} = r < 1$ and $r \geq 2^{-1}$ 4) there is a constant $c > 0$ such that $c |\mathcal{G}_m(x)| \leq \varepsilon_m$, and 5) if $m$ is even, then for every $z \in \mathcal{G}{m+1}(x) \setminus \mathcal{G}m(x)$ there is a $y \in \mathcal{G}_m(x)$ such that $z \in F_2(y)$, and if $m$ is odd then for every $z \in \mathcal{G}{m+1}(x) \setminus \mathcal{G}_m(x)$ there is a $y \in \mathcal{G}_m(x)$ such that $z \in F_1(y)$.
| We call this a sequence of compatible grids centered at $x$. Since this is focused at a particular $x$, we can work in the flat coordinates above without loss of generality, and thus just work with the Journe operator as opposed to the Taylor-Journe operator. To help with notation, we will denote these operators by $\mathcal{J}_m$. As a warning, this is not the same thing as $\mathcal{J}^m$ defined above, which was the finitary version of Journe’s operator. Furthermore, we can suppose without loss of generality that $\psi$ vanishes on the axes and $ | \psi | _{C^{n,\alpha}_1 \cap C^{n,\alpha}_2} \leq 1$. Thus, in the flat coordinates, we get a family of polynomials $P^m_x = \mathcal{J}_m(\psi)$, and our goal is to show that this family is Cauchy in the appropriate topology. To make our lives easier, for $(v,w) \in \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$, let’s write |
where $\beta = (\beta_1, \beta_2)$ (we will use such notation throughout). The Cauchy convergence will be with respect to the coefficients ${\alpha^m_\beta}$ in the $\ell^1$ norm. More precisely:
Claim: There exists an $m_0$ and a uniform constant $C > 0$ such that for all $m > m_0$, we have
\[|\alpha^m_\beta| \leq C \sum_{j=m_0}^{m-1} r^{j(n+\alpha-|\beta|)/2}.\]Proof: First, we will assume that $m_0$ is large enough so that the Journe operator is well defined. For $z \in \mathcal{G}_{m+1}(x)$ and $m > m_0$, we then have
\[|P^m_x(z) - P^{m+1}_x(z)| = |P^m_x(z) - \psi(z)| = |\mathcal{J}_m(\psi)(z) - \psi(z)|.\]| Without loss of generality, let’s suppose that $z \in F_1(y)$ for some $y \in \mathcal{G}_m(x)$. Furthermore, since $\mathcal{G}_m(x)$ is closed, we may assume that $y$ lies on the coordinate axes in the flat coordinates, so we can write $y = (0,w)$ for some $w \in \mathbb{R}^{N_2}$. For $v \in \mathbb{R}^{N_1}$, we can parameterize $F_1(y)$ by $(\phi_y(v), v)$, where $\phi_y$ is a $C^k$-function satisfying $\phi_y(0) = 0$. Consequently, there is a $1$-dimensional smooth path $\gamma :[0,1] \mapsto (\phi_y(tv),w + tv)$ such that $\gamma(0) = y$, $\gamma(1) = z$, and $\gamma(t) \in F_1(y)$ for all $t \in [0,1]$. Furthermore, $ | v | \leq C | \mathcal{G}_{m}(x) | \leq C 2^{-m} r$, where here $C > 0$ can be made uniform. By composing, we have |
is a smooth function, allowing us to consider this as a $1$-dimensional problem. In light of this perspective, enlarging $C$ if necessary, we may use Taylor’s theorem along with the chain rule to get
\[|P^m_x(z) - P^{m+1}_x(z)| \leq C 2^{-m(n+\alpha)} \left| \frac{\partial^n}{\partial t^n} (\mathcal{J}_m(\psi) - \psi) \circ \gamma \right|_{C^\alpha([0,1])}.\]Since $\psi$ is Holder continuous along each leaf, we can rewrite this upper bound as
\[|P^m_x(z) - P^{m+1}_x(z)| \leq C 2^{-m(n+\alpha)}\left[ 1 + \left| \frac{d^n}{d t^n} (\mathcal{J}_m(\psi) \circ \gamma) \right|_{C^\alpha([0,1])} \right].\]We have
\[P^m_x(\gamma(t)) = \mathcal{J}_m(\psi)(\gamma(t)) = \sum_{\substack{0 \leq |\beta_1 + \beta_2|\leq nN}} \alpha_{\beta}^m \phi_y(tv)^{\beta_1} (w+tv)^{\beta_2}.\]| By Leibniz’s rule and the fact that $\phi_y(0) = 0$, we deduce that as long as $ | \beta | > n$, $\beta_1 \neq 0$, and $\beta_2 \neq 0$, we have that there is a uniform constant $C > 0$ such that |
where $\delta_m \rightarrow 0$ as $m \rightarrow \infty$. See, for example, Journe’s Lemma 2. Using our assumption that $\psi$ vanishes on the axes, we are able to ignore the problematic terms. Hence, by enlarging $C > 0$ if necessary, we can conclude
\[|P^m_x(z) - P^{m+1}_x(z)|\leq C \left[2^{-m(n+\alpha)} + \delta_m\sum_{n < |\beta| \leq nN} |\alpha_\beta^m| 2^{-m |\beta|}\right] .\]Since this holds for all $z$, we deduce from the Journe stability lemma that
\[\sum_{0 \leq |\beta| \leq nN} |\alpha^m_\beta - \alpha^{m+1}_\beta| 2^{-m|\beta|} r^{|\beta|} \leq C \left[2^{-m(n+\alpha)} + \delta_m\sum_{n < |\beta| \leq nN} |\alpha_\beta^m| 2^{-m |\beta|}\right]\]| Now, if $m_0$ is large enough, then it follows that for all $0 \leq | \beta | \leq nN$ we have $ | \alpha^{m_0}_\beta | \leq C.$ Assume that up to $m > m_0$, we know that |
We claim the result holds up to $m+1$, and hence by induction for all $m > m_0$. Indeed:
\[\begin{split} \sum_{0 \leq |\beta| \leq nN} |\alpha^{m+1}_\beta - \alpha^m_\beta| 2^{-m |\beta|} r^{|\beta|} & \leq C \left[2^{-m(n+\alpha)} + \delta_m\sum_{n < |\beta| \leq nN} |\alpha_\beta^m| 2^{-m |\beta|}\right] \\ & \leq C \left[2^{-m(n+\alpha)} + \delta_m\sum_{n < |\beta| \leq nN} \sum_{j=m_0}^{m-1} r^{j(n+\alpha-|\beta|)/2} 2^{-m |\beta|}\right]. \end{split}\]Since $\delta_m \rightarrow 0$ as $m \rightarrow \infty$ and the series on the right is convergent (recalling again that $\alpha \in (0,1)$), we can absorb terms:
\[\sum_{0 \leq |\beta| \leq nN} |\alpha^{m+1}_\beta - \alpha^m_\beta| 2^{-m |\beta|} r^{|\beta|} \leq C 2^{-m(n+\alpha)},\]hence
\[| \alpha^{m+1}_\beta - \alpha^m_\beta| \leq C 2^{-m(n+\alpha - |\beta|)} \leq C r^{ m(n+\alpha-|\beta|)/2},\]where again we are enlarging $C > 0$ if necessary. This completes the induction hypothesis, since
\[|\alpha^{m+1}_\beta| \leq |\alpha^{m+1}_\beta - \alpha^m_\beta| + |\alpha^m_\beta| \leq C \left[ \sum_{j=m_0}^{m-1} r^{j(n+\alpha-|\beta|)/2} + r^{ m(n+\alpha-|\beta|)/2} \right] = C \sum_{j=m_0}^m r^{j(n+\alpha-|\beta|)/2}. \quad \square\]We can now define the Journe polynomial in flat coordinates as
\[Q_x(v,w) = \sum_{0 \leq |\beta| \leq nN} \alpha_\beta v^{\beta_1} w^{\beta_2},\]where
\(\alpha_\beta = \begin{cases} \lim_{m \rightarrow \infty} \alpha_\beta^m& \text{if } |\beta| \leq n \\ 0 & \text{if } |\beta| > n.\end{cases}\) As described earlier, we can then pull this back and take the Taylor polynomial to get a corresponding polynomial in the non-flat coordinates, which we denote by $P_x$. This is our Journe polynomial associated to our decreasing family of grids.
| Remark: The defining property of the Journe polynomial in flat coordinates is the following relationship with the sequence of Journe polynomials: there is a uniform constant $C > 0$ such that if $ | v | \leq C r^{m/2}$, then |
| $$ | Q_x(v) - P^m_x(v) | \leq C r^{m(n+\alpha)/2}.$$ | ||||||
| This can be seen by breaking up the difference into a sum involving $ | \alpha_\beta - \alpha^m_\beta | $ with $ | \beta | \leq n$ (where we have the Cauchy relationship) and a sum with $ | \alpha^m_\beta | $ for $ | \beta | > n$ (where we can use the stability estimates). $\square$ |
Remark: The limiting polynomial $Q_x$ is independent of the choice of sequence of compatible grids. One can see this by mixing the two sequences. $\square$
Step 3: Uniqueness of Journe’s polynomial
In this step, we now assume that for each $x \in \Sigma$, we can construct a compatible sequence of grids ${\mathcal{G}m(x)}{m=1}^\infty$. Motivated by Lemma 4.4 in the paper by Nicol and Torok and an implicit step in the paper by Journe, we define the vertical cone at the origin by
\[\mathcal{C}^v(\kappa) = \{(v,w) \in \mathbb{R}^{N_1} \times \mathbb{R}^{N_2} \mid |w| \leq \kappa |v|\}.\]Similarly, we have the horizontal cone at the origin:
\[\mathcal{C}^h(\kappa) = \{(v,w) \in \mathbb{R}^{N_1} \times \mathbb{R}^{N_2} \mid |v| \leq \kappa |w|\}.\]We will work in flat coordinates around a point $x \in \Sigma$. Let $\hat{\Phi}_x : V_x \rightarrow U_x$ be the flat coordinate map (see Step 1), and As described in the last step, we have the Journe polynomial $Q_x$ in flat coordinates centered at $x$, and this defines a polynomial $P_x$ in the non-flat coordinates.
| Lemma: There exists a uniform constant $C > 0$ such that for all $y \in U_x \cap \hat{\Phi}x(\mathcal{C}^v(2) \cap B{R | \mathcal{G}_1(x) | }(0,0))$, we have |
Sketch of Proof: We can work in flat coordinates without loss of generality. Thus, we just need to show that
\[|Q_x(v,w) - \psi_x(v,w)| \leq C |x-y|^{n+\alpha} \quad \text{for all } (v,w) \in V_x \cap \mathcal{C}^v(2) \cap B_{R|\mathcal{G}_1(x)|}(0,0).\]From our above estimates, it is straightforward to see that if we enumerate ${z_r}{r=1}^\infty = \bigcup{m =1}^\infty [(0,0), \mathcal{G}_m(x)]$, then
\[(v,w) \in \bigcup_{r=1}^\infty (F_1^{loc}(z_r) \cap \mathcal{C}^v(2) \cap V_x \cap B_{R|\mathcal{G}_1(x)|}(0,0)) \implies |Q_x(v,w) - \psi_x(v,w)| \leq C |x-y|^{n+\alpha}.\]Using the uniqueness of the flat Journe polynomial $Q_x$ along any sequence of compatible grids, we are able to adjust our compatible grids off of finitely many points so that we have the following:
\[(v,w) \in \mathcal{C}^v(2) \cap V_x \cap B_{R|\mathcal{G}_1(x)|}(0,0)\implies |Q_x(v,w) - \psi_x(v,w)| \leq C |x-y|^{n+\alpha}.\]The result follows. $\square$
One can play the same game with the horizontal cone in place of the vertical cone, and so our estimates hold everywhere in this small neighborhood. sIn particular, as shown in Lemma 4.10, this proves the following.
Theorem: If for every $x \in \Sigma$ there is a compatible sequence of grids ${\mathcal{G}m(x)}{m=1}^\infty$, then $\psi$ is Whitney $C^{n,\alpha}$ on $\Sigma$.
Step 4: Constructing the sequence of grids
We give a brief sketch of this (hopefully I’ll come back to fill in the details later!) The grids can be constructed by choosing appropriate geometric sequences along the coordinate axes and then taking the closure under the bracket. In particular, one can show that these are closed quasi-uniform grids with the correct properties and which control the distortion.
| More precisely, working in the flat coordinates $V_x$ around $x$, for each coordinate direction, choose $n+1$ points spaced geometrically scaling down by $2^{-1}$ as $m$ increases. Denote these grids by $\Lambda_m^1 \subseteq \mathbb{R}^{N_1} \times {0}$ and $\Lambda_m^2 \subseteq {0} \times \mathbb{R}^{N_2}$, and let $\mathcal{G}_m(x)$ be the closure of $\Lambda_m^1 \cup \Lambda_m^2$ under the bracket. While the foliations are only transversely Holder, the displacement from the flat bracket and the foliation bracket is bounded by $o( | \mathcal{G}_m(x) | )$. This guarantees quasi-uniformity. The alternating condition can be constructed by alternating the order we append new points to our decaying sequences. This builds finer and finer grids satisfying all of the necessary conditions (see Journe. $\square$ |