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For each \({\mathfrak p}\in {\mathfrak P}\), we have
For each \({\mathfrak u}\in {\mathfrak U}\) and each \({\mathfrak p}\in {\mathfrak T}({\mathfrak u})\), we have
We have for all bounded \(g\) with bounded support
For all bounded \(g\) with bounded support, we have that
We have that
For any antichain \(\mathfrak {A} \) and for all \(f:X\to {\mathbb {C}}\) with \(|f|\le \mathbf{1}_F\) and all \(g:X\to {\mathbb {C}}\) with \(|g| \le \mathbf{1}_G\)
Set \(p:=4a^4\) and let \(p'\) be the dual exponent of \(p\), that is \(1/p+1/p'=1\). For every \({\vartheta }\in {\Theta }\) and every subset \(\mathfrak {A}'\) of \(\mathfrak {A}\) we have
Let \(B' \subset X\) be a ball. Let \(r {\gt} 0\), \({\vartheta }\in {\Theta }\) and \(k \in \mathbb {N}\). Suppose that \(\mathcal{Z} \subset B_{B'}({\vartheta }, r2^k)\) satisfies that \(\{ B_{B'}(z,r)\mid z \in \mathcal{Z}\} \) is a collection of pairwise disjoint sets. Then
For each \(-S\le k\le S\) and \(1\le j\le 3\) the following holds.
If \(j\neq 2\) and for some \(x\in X\) and \(y_1,y_2\in Y_k\) we have
then \(y_1=y_2\).
If \(j\neq 1\), then
We have for each \(y\in Y_k\),
Let \(I=[s, t)\) and let \(n\) be a non-negative integer. Then the intervals in \(\textrm{ch}_n(I)\) are pairwise disjoint. For any \(J\in \textrm{ch}_n(I)\),
Further,
and
We have
Let \(\mathcal{L}({\mathfrak u})\) be as defined in 5.1.20. We have for each \({\mathfrak u}\in {\mathfrak U}_1(k,n,l)\),
For each \(-S\le k\le S\) and \(y\in Y_k\) and \(0{\lt}t{\lt}1\) with \(tD^k\ge D^{-S}\) we have
For all \({\mathfrak u}\in {\mathfrak U}\) and all bounded functions \(f\) with bounded support
For every cube \(I \in \mathcal{D}\), there exist at most \(2^{9a}\) cubes \(J \in \mathcal{D}\) with \(s(J) = s(I)\) and \(B(c(I), 16D^{s(I)}) \cap B(c(J), 16 D^{s(J)}) \ne \emptyset \).
For each set \(\mathfrak {A} \subset \mathfrak {C}(k,n)\), we have
For each \(k,n,j\), the collection \({\mathfrak C}_5(k,n,j)\) is convex.
Let \({\mathfrak u}\in {\mathfrak U}_3(k,n,j)\). If \({\mathfrak p}\le {\mathfrak p}' \le {\mathfrak p}''\) and \({\mathfrak p}, {\mathfrak p}'' \in \mathfrak {T}_2({\mathfrak u})\), then \({\mathfrak p}' \in \mathfrak {T}_2({\mathfrak u})\).
We have
Let \(f\) be a bounded, measurable function with bounded support. Let \(\alpha {\gt}0\) and \(\gamma \in (0, 1)\). Then there exists a measurable functions \(g\), a countable family of disjoint intervals \(I_j = [s_j, t_j)\), and a countable family of measurable functions \(\{ g_j\} _{j\geq 1}\) such that for almost every \(x \in {\mathbb {R}}\)
and such that the following holds. For almost every \(x\in \mathbb {R}\),
We have
For every \(j\)
For every \(j\)
and
We have
and
Let \(f\) be a \(2\pi \)-periodic complex-valued uniformly continuous function on \(\mathbb {R}\) satisfying the bound \(|f(x)|\le 1\) for all \(x\in \mathbb {R}\). For all \(0{\lt}\epsilon {\lt}1\), there exists a Borel set \(E\subset [0,2\pi ]\) with Lebesgue measure \(|E|\le \epsilon \) and a positive integer \(N_0\) such that for all \(x\in [0,2\pi ]\setminus E\) and all integers \(N{\gt}N_0\), we have
There is a set \(E \subset {\mathbb {R}}\) with Lebesgue measure \(|E|\le \epsilon \) such that for all
we have
There exists some \(N_0 \in {\mathbb {N}}\) such that for all \(N{\gt}N_0\) and \(x\in [0,2\pi ]\) we have
Let \(f:{\mathbb {R}}\to {\mathbb {C}}\) be \(2\pi \)-periodic and twice continuously differentiable. Then
as \(N \rightarrow \infty \).
Let \(f:{\mathbb {R}}\to {\mathbb {C}}\) such that
Then
as \(N \rightarrow \infty \).
For each \({\mathfrak u}\in {\mathfrak U}\), we have
We have for all \({\mathfrak u}_1 \ne {\mathfrak u}_2 \in {\mathfrak U}\) with \({\mathcal{I}}({\mathfrak u}_1) \subset {\mathcal{I}}({\mathfrak u}_2)\) and all bounded \(g_1, g_2\) with bounded support
Let \(-S\le s_1\le s_2\le S\) and let \(x_1,x_2\in X\). Define
If \(\varphi (y)\neq 0\), then
Moreover, we have with \(\tau = 1/a\)
We have for all \({\mathfrak u}_1 \ne {\mathfrak u}_2 \in {\mathfrak U}\) with \({\mathcal{I}}({\mathfrak u}_1) \subset {\mathcal{I}}({\mathfrak u}_2)\) and all bounded \(g_1, g_2\) with bounded support
For any \({\mathfrak u}_1 \ne {\mathfrak u}_2 \in {\mathfrak U}\) and all bounded \(g_1, g_2\) with bounded support, we have
Let \(0{\lt}r{\lt}r_1{\lt}1\) and \(x\in \mathbb {R}\). Let \(g\) be a bounded measurable function with bounded support on \(\mathbb {R}\). Let \(Mg\) be the Hardy–Littlewood function defined in Proposition 2.0.6. Then for all \(x'\in {\mathbb {R}}\) with \(|x'-x|{\lt}\frac{r_1}4\) we have
Let \(0{\lt}r{\lt}r_1{\lt}1\) and \(x\in \mathbb {R}\). Let \(g\) be a bounded measurable function with bounded support on \(\mathbb {R}\). Let \(Mg\) and \(M(H_r g)\) be the respective Hardy–Littlewood maximal functions defined in Proposition 2.0.6. Then for all \(x\in {\mathbb {R}}\)
Let \(0{\lt}r{\lt}r_1{\lt}1\) and \(x\in \mathbb {R}\). Let \(g\) be a bounded measurable function with bounded support on \(\mathbb {R}\). Let \(Mg\) be the Hardy–Littlewood maximal function defined in Proposition 2.0.6. Let \(x\in {\mathbb {R}}\). Then the measure \(|F_1|\) of the set \(F_1\) of all \(x'\in [x-\frac{r_1}4,x+\frac{r_1}4]\) such that
is less than or equal to \(r_1/8\). Moreover, the measure \(|F_2|\) of the set \(F_2\) of all \(x'\in [x-\frac{r_1}4,x+\frac{r_1}4]\) such that
is less than \(r_1/8\).
Let \(-S\le k\le S\). Consider \(Y\subset X\) such that for any \(y\in Y\), we have
furthermore, for any \(y'\in Y\) with \(y\neq y'\), we have
Then the cardinality of \(Y\) is bounded by
Let \(-S\le l\le k\le S\) and \(y\in Y_k\). We have
For each \(r {\gt} 0\), there exists a countable collection \(C(r) \subset X\) of points such that
We have for every \(k\ge 0\) and \({\mathfrak P}'\subset {\mathfrak P}(k)\)
Set \(p:=4a^4\). We have
We have that
For each \(k\ge 0\), the union of all dyadic cubes in \(\mathcal{C}(G,k)\) has measure at most \(2^{k+1} \mu (G)\) .
Let \({\mathfrak u}\in {\mathfrak U}\). Then for all \(f,g\) bounded with bounded support
If \(|f| \le \mathbf{1}_F\), then we have
Let \(0{\lt}r{\lt}1\). Let \(N\) be the smallest integer larger than \(\frac1r\). There is a \(2\pi \)-periodic continuous function \({L’}\) on \({\mathbb {R}}\) that satisfies for all \(-\pi \le x\le \pi \) and all \(2\pi \)-periodic bounded measurable functions \(f\) on \({\mathbb {R}}\)
and
We have for every \(2\pi \)-periodic bounded measurable \(f\) and every \(N\ge 0\)
where \(K_N\) is the \(2\pi \)-periodic continuous function of \({\mathbb {R}}\) given by
We have for \(e^{ix'}\neq 1\) that
Let \((\mathcal{D}, c, s)\) be a grid structure and
a tile structure for this grid structure. Define for \({\mathfrak p}\in {\mathfrak P}\)
and
Then there exists a Borel set \(G'\) with \(2\mu (G') \leq \mu (G)\) such that for all Borel functions \(f:X\to {\mathbb {C}}\) with \(|f|\le \mathbf{1}_F\) we have
We have that
We have
For each \(\mathfrak {S} \subset {\mathfrak P}\), we have
and
Let \(-S\le l\le k\le S\) and \(y\in Y_k\) and \(y'\in Y_l\) with \(I_3(y',l)\cap I_3(y,k)\neq \emptyset \). Then
For each \(x\in A(\lambda ,k,n)\), there is a dyadic cube \(I\) that contains \(x\) and is a subset of \(A(\lambda ,k,n)\).
For each \(k,n,j\), the relation \(\sim \) on \({\mathfrak U}_2(k,n,j)\) is an equivalence relation.
Let \(0{\lt}r{\lt}1\) and \(x\in \mathbb {R}\). Let \(g\) be a bounded measurable function with bounded support on \(\mathbb {R}\). Let \(Mg\) be the Hardy–Littlewood function defined in Proposition 2.0.6. Then for all \(x'\) with \(|x-x'|{\lt}r\).
We have
Let \({\sigma _1},\sigma _2\colon X\to \mathbb {Z}\) be measurable functions with finite range and \({\sigma _1}\leq \sigma _2\). Let \({Q}\colon X\to {\Theta }\) be a measurable function with finite range. Let \(F,G\) be bounded Borel sets in \(X\). Then there is a Borel set \(G'\) in \(X\) with \(2\mu (G')\leq \mu (G)\) such that for all Borel functions \(f:X\to {\mathbb {C}}\) with \(|f|\le \mathbf{1}_F\).
For all finite sets \(\tilde{{\Theta }}\subset {\Theta }\)
We have
For all \({\mathfrak u}\in {\mathfrak U}\), all \(L \in \mathcal{L}({\mathfrak T}({\mathfrak u}))\), all \(x, x' \in L\) and all bounded \(f\) with bounded support, we have
Let
For all \(f:X\to {\mathbb {C}}\) with \(|f|\le \mathbf{1}_F\) we have
For each \({\mathfrak u}\in {\mathfrak U}_3(k,n,j)\) and each \({\mathfrak p}\in \mathfrak {T}_2({\mathfrak u})\) we have
For any \(n\ge 0\) and any \(n\)-forest \(({\mathfrak U},{\mathfrak T})\) we have for all \(f: X \to \mathbb {C}\) with \(|f| \le \mathbf{1}_F\) and all bounded \(g\) with bounded support
Let \(({\mathfrak U}, {\mathfrak T})\) be an \(n\)-forest. Then there exists a decomposition
such that for all \(j = 1, \dotsc , 2^n\) the pair \(({\mathfrak U}_j, {\mathfrak T}|_{{\mathfrak U}_j})\) is an \(n\)-row.
For each \({\mathfrak u},{\mathfrak u}'\in {\mathfrak U}_3(k,n,j)\) with \({\mathfrak u}\neq {\mathfrak u}'\) and each \({\mathfrak p}\in {\mathfrak T}_2({\mathfrak u})\) with \({\mathcal{I}}({\mathfrak p})\subset {\mathcal{I}}({\mathfrak u}')\) we have
It holds that
Let
For all \(f:X\to {\mathbb {C}}\) with \(|f|\le \mathbf{1}_F\) we have
Let \(f:{\mathbb {R}}\to {\mathbb {C}}\) be \(2\pi \)-periodic and differentiable, and let \(n \in {\mathbb {Z}}\setminus \{ 0\} \). Then
For any \(x,x'\in {\mathbb {R}}\) and \(R{\gt}0\) with \(x\in B(x',2R)\) and any \(n,m\in \mathbb {Z}\), we have
For any \(x,x'\in {\mathbb {R}}\) and \(R{\gt}0\) with \(B(x,R)\subset B(x',2R)\) and any \(n,m\in \mathbb {Z}\), we have
For each \(I \in \mathcal{D}\), it holds that
For every \({\mathfrak p}\in {\mathfrak P}\), it holds that
For every \(R {\gt} 0\) and \(x \in X\), the function \(d_{B(x,R)}\) is a metric on \({\Theta }\).
For any \(x, x' \in X\) and \(R, R' {\gt} 0\) with \(B(x,R) \subset B(x, R')\), and for any \(n, m \in \mathbb {Z}\)
Let \({\vartheta }\in {\Theta }\) and let \(N\ge 0\) be an integer. Then we have
Let \({\mathfrak C}= {\mathfrak T}({\mathfrak u}_1)\) or \({\mathfrak C}= {\mathfrak T}({\mathfrak u}_2) \cap \mathfrak {S}\). Then for each \(J \in \mathcal{J}'\) and all bounded \(g\) with bounded support, we have
and for all \(y,y' \in B(J)\)
We have for all \(J \in \mathcal{J}'\) and all bounded \(g\) with bounded support
Let \(\mathcal{B}\) be a finite collection of balls in \(X\). If for some \(\lambda {\gt}0\) and some measurable function \(u:X\to [0,\infty )\) we have
for each \(B\in \mathcal{B}\), then
For every measurable function \(v\) and \(1\le p_1{\lt}p_2\) we have
Moreover, given any measurable bounded function \(w: X \to {\mathbb {C}}\) there exists a measurable function \(Mw: X \to [0, \infty )\) such that the following 2.0.45 and 2.0.46 hold. For each ball \(B \subset X\) and each \(x \in B\)
and for all \(1 \le p_1 {\lt} p_2 \le \infty \)
For \(x,y\in {\mathbb {R}}\) with \(x\neq y\) we have
For \(x,y,y'\in {\mathbb {R}}\) with \(x\neq y,y'\) and
we have
Let \(0{\lt}r{\lt}1\). Let \(f\) be a bounded, measurable function on \(\mathbb {R}\) with bounded support. Then
Let \(f\) be a bounded measurable function on \(\mathbb {R}\) with bounded support. Let \(\alpha {\gt}0\). Then for all \(r\in (0, 1)\), we have
Let \({\mathfrak u}\in {\mathfrak U}\) and \({\mathfrak p}\in {\mathfrak T}({\mathfrak u})\). Then for all \(y, y' \in X\) and all bounded \(g\) with bounded support, we have
We have for all \(J \in \mathcal{J}'\) that
Let \(z\in X\) and \(R{\gt}0\) and set \(B=B(z,R)\). Let \(\varphi : X \to \mathbb {C}\) by supported on \(B\) and satisfy \(\| {\varphi }\| _{C^\tau (B)}{\lt}\infty \). Let \({\vartheta }, {\theta }\in {\Theta }\). Then
For every \(x\in {\mathbb {R}}\) and \(R{\gt}0\) and every \(n\in \mathbb {Z}\) and \(R'{\gt}0\), there exist \(m_1, m_2, m_3\in \mathbb {Z}\) such that
where
and for \(j=1,2,3\)
Let \(g,f\) be bounded measurable \(2\pi \)-periodic functions. Let \(0{\lt}r{\lt}\pi \). Assume we have for all \(0\le x\le 2\pi \)
Let
Then
Let \(I=[s, t)\subset \mathbb {R}\) be a bounded, right-open interval. Let \(I_1, I_2\in \textrm{bi} (I)\) with \(I_1\neq I_2\). Then
Further,
and the intervals \(I_1\) and \(I_2\) are disjoint.
For all integers \(k,n,\lambda \ge 0\), we have
Let \(-S\le s\le S\) and \(x,y,y'\in X\). If \(K_s(x,y)\neq 0\), then we have
We have
and
We have that
where each \({\mathfrak L}_0(k,n,l)\) is an antichain.
Each of the sets \({\mathfrak L}_1(k,n,j,l)\) and \({\mathfrak L}_3(k,n,j,l)\) is an antichain.
Let \(1\le p{\lt} \infty \). Then for any measurable function \(u:X\to [0,\infty )\) on the measure space \(X\) relative to the measure \(\mu \) we have
Let \(f\) be a bounded measurable function with bounded support. Then for \(\mu \) almost every \(x\), we have
where \(\{ I_n\} _{n\geq 1}\) is a sequence of intervals such that \(x\in I_n\) for each \(n\geq 1\) and
Let \({\mathfrak p}\in {\mathfrak T}_2 \setminus \mathfrak {S}\), \(J \in \mathcal{J}'\) and suppose that
Then
Let \(\sigma _1,\sigma _2\colon X\to \mathbb {Z}\) be measurable functions with finite range \([-S,S]\) and \(\sigma _1\leq \sigma _2\). Let \({Q}\colon X\to {\tilde{{\Theta }}}\) be a measurable function. Then we have
with
Let \(z\in X\) and \(R{\gt}0\). Let \(\varphi : X \to \mathbb {C}\) be a function supported in the ball \(B:=B(z,R)\) with finite norm \(\| \varphi \| _{C^\tau (B)}\). Let \(0{\lt}t \leq 1\). There exists a function \(\tilde\varphi : X \to \mathbb {C}\), supported in \(B(z,2R)\), such that for every \(x\in X\)
and
There exists a family of functions \(\chi _J\), \(J \in \mathcal{J}'\) such that
and for all \(J \in \mathcal{J}'\) and all \(y,y' \in {\mathcal{I}}({\mathfrak u}_1)\)
Let \({\vartheta }\in {\Theta }\) and \(N\) be an integer. Let \({\mathfrak p}_{{\vartheta }}\) be a tile with \({\vartheta }\in {\Omega }({\mathfrak p}_{{\vartheta }})\). Then we have
Let \({\mathfrak u}\in {\mathfrak U}\) and \(L \in \mathcal{L}({\mathfrak T}({\mathfrak u}))\). Then
Let \(J \in \mathcal{J}({\mathfrak T}({\mathfrak u}))\) be such that there exist \({\mathfrak q}\in {\mathfrak T}({\mathfrak u})\) with \(J \cap {\mathcal{I}}({\mathfrak q}) \ne \emptyset \). Then
For all \(J \in \mathcal{J}'\) and all bounded \(g\) with bounded support
For all \(J \in \mathcal{J}'\), we have that
Let \(n\in \mathbb {Z}\) with \(n\neq 0\), then
For all integers \(a \ge 4\) and real numbers \(1{\lt}q\le 2\) the following holds. Let \((X,\rho ,\mu ,a)\) be a doubling metric measure space. Let \({\Theta }\) be a cancellative compatible collection of functions and let \(K\) be a one-sided Calderón–Zygmund kernel on \((X,\rho ,\mu ,a)\). Assume that for every bounded measurable function \(g\) on \(X\) supported on a set of finite measure we have
where \(T_{*}\) is defined in 1.0.16. Then for all Borel sets \(F\) and \(G\) in \(X\) and all Borel functions \(f:X\to {\mathbb {C}}\) with \(|f|\le \mathbf{1}_F\), we have, with \(T\) defined in 1.0.17,
If \(J, J' \in \mathcal{J'}\) with
then \(|s(J) - s(J')| \le 1\).
We have for every bounded measurable \(2\pi \)-periodic function \(g\)
Let \((\mathcal{D}, c, s)\) be a grid structure. Denote for cubes \(I \in \mathcal{D}\)
Let \(I, J \in \mathcal{D}\) with \(I \subset J\). Then for all \({\vartheta }, {\theta }\in {\Theta }\) we have
and if \(I \ne J\) then we have
For every bounded measurable function \(g\) with bounded support we have
where
For all bounded \(f\) with bounded support
For every \(R {\gt} 0\) and \(x \in X\), and for all \(n, m \in \mathbb {Z}\), we have
Let \({\mathfrak u}_1 \ne {\mathfrak u}_2 \in {\mathfrak U}\) with \({\mathcal{I}}({\mathfrak u}_1) \subset {\mathcal{I}}({\mathfrak u}_2)\). If \({\mathfrak p}\in {\mathfrak T}({\mathfrak u}_1) \cup {\mathfrak T}({\mathfrak u}_2)\) with \({\mathcal{I}}({\mathfrak p}) \cap {\mathcal{I}}({\mathfrak u}_1) \ne \emptyset \), then \({\mathfrak p}\in \mathfrak {S}\). In particular, we have \({\mathfrak T}({\mathfrak u}_1) \subset \mathfrak {S}\).
If \({\mathfrak p}, {\mathfrak p}' \in {\mathfrak {M}}(k,n)\) and
then \({\mathfrak p}={\mathfrak p}'\).
Let \(f\) be a bounded \(2\pi \)-periodic measurable function. Then, for all \(N\ge 0\)
We have for any \(2\pi \)-periodic bounded measurable \(g,f\) that
Let \(f\) be a bounded \(2\pi \)-periodic function. We have for any \(0 \le x\le 2\pi \) that
Let \({\mathfrak u}\in {\mathfrak U}\) and \(L \in \mathcal{L}({\mathfrak T}({\mathfrak u}))\). Let \(x, x' \in L\). Then for all bounded functions \(f\) with bounded support
For all integers \(R{\gt}0\)
where
Let \(F,G\) be Borel subsets of \({\mathbb {R}}\) with finite measure. Let \(f\) be a bounded measurable function on \({\mathbb {R}}\) with \(|f|\le \mathbf{1}_F\). Then
where
We have for every \(x\in {\mathbb {R}}\) and \(R{\gt}0\)
The measure \(\mu \) is a sigma-finite non-zero Radon-Borel measure on \({\mathbb {R}}\).
The space \(({\mathbb {R}},\rho )\) is a complete locally compact metric space.
For any \(x\in {\mathbb {R}}\) and \(R{\gt}0\) and any function \(\varphi : X\to {\mathbb {C}}\) supported on \(B'=B(x,R)\) such that
is finite and for any \(n,m\in \mathbb {Z}\), we have
If \({\mathfrak u}\sim {\mathfrak u}'\), then \({\mathcal{I}}(u) = {\mathcal{I}}(u')\) and
For each \(1 \le j \le 2^n\) and each bounded \(g\) with bounded support, we have
and
For all \(1 \le j {\lt} j' \le 2^n\) and for all bounded \(g_1, g_2\) with bounded support, it holds that
For all integers \(S{\gt}0\)
where \(T_{1, s_1,s_2,{\vartheta }}\) is defined in 3.0.4.
Let \({\mathfrak C}= {\mathfrak T}({\mathfrak u}_1)\) or \({\mathfrak C}= {\mathfrak T}({\mathfrak u}_2) \cap \mathfrak {S}\). Then for each \(J \in \mathcal{J}'\) and \({\mathfrak p}\in {\mathfrak C}\) with \(B({\mathcal{I}}({\mathfrak p})) \cap B(J) \neq \emptyset \), we have \({\mathrm{s}}({\mathfrak p}) \ge s(J)\).
We have
For all \({\mathfrak u}\in {\mathfrak U}\), all \(L \in \mathcal{L}({\mathfrak T}({\mathfrak u}))\), all \(x, x' \in L\) and all bounded \(f\) with bounded support, we have
For every \(0{\lt}r{\lt}1\) and every bounded measurable function \(g\) with bounded support we have
where
Let \(K = 2^{4a+1}\). For each \(-S+K\le k\le S\) and \(y\in Y_k\) we have
Let \(K = 2^{4a+1}\) and let \(n\ge 0\) be an integer. Then for each \(-S+nK\le k\le S\) we have
The function \(f_0\) is \(2\pi \)-periodic. The function \(f_0\) is smooth (and therefore measurable). The function \(f_0\) satisfies for all \(x\in {\mathbb {R}}\):
Let \(f\) be a bounded \(2\pi \)-periodic measurable function. Then, for all \(N\ge 0\)
For each \(J \in \mathcal{J}'\), we have
Let \({\vartheta }\in {\Theta }\), \(N\ge 0\) and \(L\in \mathcal{D}\). Then
Let \(f\) be a bounded, measurable function with bounded support on \(\mathbb {R}\). Let \(\alpha {\gt}0\). Then there exists \(A\subset \mathbb {R}\) such that the following properties 10.5.8, 10.5.9, 10.5.10, 10.5.11, and 10.5.12 are satisfied. For all \(x\in A\)
The set \(\mathbb {R}\setminus A\) can be decomposed into a countable union of intervals
such that
For each \(j\),
Further,
If \({\mathfrak p}\in {\mathfrak T}_2 \setminus \mathfrak {S}\) and \(J \in \mathcal{J'}\) with \(B({\mathcal{I}}({\mathfrak p})) \cap B(J) \ne \emptyset \), then
We have
For all \({\mathfrak u}\in {\mathfrak U}\), all \(L \in \mathcal{L}({\mathfrak T}({\mathfrak u}))\), all \(x, x' \in L\) and all bounded \(f\) with bounded support, we have
Let \({\mathfrak p}, {\mathfrak p}'\in {\mathfrak P}\) with \({\mathrm{s}}({{\mathfrak p}’})\leq {\mathrm{s}}({{\mathfrak p}})\). Then
Moreover, the term 6.1.42 vanishes unless
For each \({\mathfrak p}\in {\mathfrak P}\), and each \(y\in X\), we have that
implies
Let \({\vartheta }\in {\Theta }\) and \(N\ge 0\) be an integer. Let \({\mathfrak p}, {\mathfrak p}'\in {\mathfrak P}\) with
Assume \({\mathcal{I}}({\mathfrak p})\subset {\mathcal{I}}({\mathfrak p}')\) and \({\mathrm{s}}({\mathfrak p}){\lt}{\mathrm{s}}({\mathfrak p}')\). Then
We have for all \(x\in G\setminus G'\)
Let \({\mathfrak p}_1, {\mathfrak p}_2\in {\mathfrak P}\) with \({\mathrm{s}}({{\mathfrak p}_1})\leq {\mathrm{s}}({{\mathfrak p}_2})\). For each \(x_1\in E({\mathfrak p}_1)\) and \(x_2\in E({\mathfrak p}_2)\) we have
We have
Assume \(-S\le k''{\lt} k'{\lt} k\le S\) and \(y''\in Y_{k''}\), \(y'\in Y_{k'}\), \(y\in Y_k\). Assume there is \(x\in X\) such that
If \((y'',k''|y,k)\), the also \((y'',k''|y',k')\) and \((y',k'|y,k)\)
Let \(k,n,j\ge 0\). We have for every \(x\in X\)
Let \({\mathfrak u}\in {\mathfrak U}\). Then we have for all \(f, g\) bounded with bounded support
Let \(\alpha {\lt}\beta \) be real numbers. Let \(g:{\mathbb {R}}\to {\mathbb {C}}\) be a measurable function and assume
Then for any \(0\le \alpha {\lt}\beta \le 2\pi \) we have
If \(n{\mathfrak p}\lesssim m{\mathfrak p}'\) and \(n' \ge n\) and \(m \ge m'\) then \(n'{\mathfrak p}\lesssim m'{\mathfrak p}'\).
Let \(n, m \ge 1\). If \({\mathfrak p}, {\mathfrak p}' \in {\mathfrak P}\) with \({\mathcal{I}}({\mathfrak p}) \ne {\mathcal{I}}({\mathfrak p}')\) and
then
The following implications hold for all \({\mathfrak q}, {\mathfrak q}' \in {\mathfrak P}\):
Let \(f\) and \(g\) be two bounded non-negative measurable \(2\pi \)-periodic functions on \({\mathbb {R}}\). Then