Dependencies

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 \({\mathfrak u}\in {\mathfrak U}\) and 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

Given an open set \(O\ne X\), there exists a countable family of balls \(B_j = B(x_j, r_j)\) such that

and for \(B_j^* := B(x_j, 3r_j)\),

and for \(B_j^{**} := B(x_j, 7r_j)\),

and we have the bounded intersection property that each \(x\in O\) is contained in at most \(2^{6a}\) of the \(B_j^*\).

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 \(y\in Y_k\),

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 \(k,n\), the collection \({\mathfrak C}(k,n)\) is convex.

For each set \(\mathfrak {A} \subset \mathfrak {C}(k,n)\), we have

For each \(k,n,j\), the collection \({\mathfrak C}_1(k,n,j)\) is convex.

For each \(k,n,j\), the collection \({\mathfrak C}_2(k,n,j)\) is convex.

For each \(k,n,j\), the collection \({\mathfrak C}_3(k,n,j)\) is convex.

For each \(k,n,j\), the collection \({\mathfrak C}_4(k,n,j)\) is convex.

For each \(k,n,j\), the collection \({\mathfrak C}_5(k,n,j)\) is convex.

We have

Let \(f\) be a bounded, measurable function supported on a set of finite measure and let \(\alpha {\gt}\frac{1}{\mu (X)}\int |f|\, d\mu \). Then there exists a measurable function \(g\), a countable family of balls \(B_j^*\) (where we allow \(B_1^* = X\) in the special case that \(\mu (X){\lt}\infty \)) such that each \(x\in X\) is contained in at most \(2^{6a}\) of the \(B_j^*\), and a countable family of measurable functions \(\{ b_j\} _{j\in J}\) such that for all \(x \in X\)

and such that the following holds. For almost every \(x\in X\),

We have

For every \(j\)

For every \(j\)

and

We have

and

Let \(f\) be a \(2\pi \)-periodic complex-valued continuous function on \(\mathbb {R}\). Then for almost all \(x \in \mathbb {R}\) we have

where \(S_N f\) is the \(N\)-th partial Fourier sum of \(f\) defined in 1.0.2.

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 \(0{\lt}r\le R\) and \(x\in X\). Let \(g:X\to {\mathbb {C}}\) be a bounded measurable function supported on a set of finite measure. Then for all \(x'\in X\) with \(\rho (x,x')\le \frac{R}{4}\) we have

Assume that 10.0.3 holds. Let \(0{\lt}r\le R\) and \(x\in X\). Let \(g:X\to {\mathbb {C}}\) be a bounded measurable function supported on a set of finite measure. Then

Assume that 10.0.3 holds. Let \(0{\lt}r\le R\) and \(x\in X\). Let \(g:X\to {\mathbb {C}}\) be a bounded measurable function supported on a set of finite measure. Then the measure \(|F_1|\) of the set \(F_1\) of all \(x'\in B(x,\frac{R}4)\) such that

is less than or equal to \(\mu (B(x,\frac{R}{4}))/4\). Moreover, the measure \(|F_2|\) of the set \(F_2\) of all \(x'\in B(x,\frac{R}4)\) such that

is less than or equal to \(\mu (B(x,\frac{R}{4}))/4\).

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

For each \(-S\le k\le S\), the ball \(B(o, 4D^S-D^k)\) is contained in the union of the balls \(B(y,2D^k)\) with \(y\in Y_k\).

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

For all \(N\in {\mathbb {Z}}\) and \(x\in [-\pi ,\pi ] \setminus \{ 0\} \),

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

In a doubling metric measure space \((X,\rho ,\mu , a)\), every disjoint family of balls \(B_j = B(x_j, r_j)\), \(j\in J\), is countable.

For each \(I \in \mathcal{D}\), and \({\mathfrak p}_1, {\mathfrak p}_2\in {\mathfrak P}(I)\), if

then \({\mathfrak p}_1={\mathfrak p}_2\).

The sets \(E_j\), \(1 \le j \le 2^n\) are pairwise disjoint.

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.

We have

For \(F\) as defined in Lemma 10.2.7, we have

Let \(x\in X\setminus \Omega \). Then

where

Let \(0{\lt}r\) and \(x\in X\). Let \(g:X\to {\mathbb {C}}\) be a bounded measurable function supported on a set of finite measure. Then for all \(x'\) with \(\rho (x,x')\le r\).

We have

Let \(f\) be a \(2\pi \)-periodic complex-valued continuous function on \(\mathbb {R}\). For all \(\epsilon {\gt}0\), 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

Let \({\sigma _1},\sigma _2\colon X\to \mathbb {Z}\) be measurable functions with finite range and \({\sigma _1}\leq \sigma _2\). 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 \({\mathfrak u}\in {\mathfrak U}_3(k,n,j)\), the set \(\mathfrak {T}_2({\mathfrak u})\) satisfies the convexity condition 2.0.33.

For each \({\mathfrak u}\in {\mathfrak U}_3(k,n,j)\), the set \(\mathfrak {T}_2({\mathfrak u})\) satisfies 2.0.32.

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 for \(k\le n\) 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 each \(I \in \mathcal{D}\), we have

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 \(n, m \in \mathbb {Z}\)

For all real numbers \(x\ge 4\),

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

There exists a grid structure \((\mathcal{D}, c,s)\).

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

where

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

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.

Each of the sets \({\mathfrak L}_2(k,n,j)\) 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 supported on a set of finite measure. Then for \(\mu \) almost every \(x\), we have

where \(\{ B_n\} _{n\geq 1}\) is a sequence of balls with radii \(r_n{\gt}0\) such that \(x\in B_n\) for each \(n\geq 1\) and

Let \({\mathfrak p}\in {\mathfrak T}({\mathfrak u}_2) \setminus \mathfrak {S}\), \(J \in \mathcal{J}'\) and suppose that

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. Let \({Q}:X\to {\Theta }\) be a Borel function with finite range. Let \(K\) be a one-sided Calderón–Zygmund kernel on \((X,\rho ,\mu ,a)\). Assume that for every \({\vartheta }\in {\Theta }\) and every bounded measurable function \(g\) on \(X\) supported on a set of finite measure we have

where \(T_{{Q}}^{\vartheta }\) is defined in 1.0.21. Then for all bounded 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 \(T_{Q}\) defined in 1.0.22,

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 \(\eta {\gt}0\) and \(-2\pi +\eta \le x\le 2\pi -\eta \) with \(|x|\ge \eta \). Then

Let \(x\in X\). Then

Let \(f: X \to {\mathbb {C}}\) be bounded, measurable, supported on a set of finite measure, and let \(\alpha {\gt} 0\). Then

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

Assume 10.0.3 holds. Then, for every bounded measurable function \(g : X \to {\mathbb {C}}\) supported on a set of finite measure we have

For all bounded \(f\) with bounded supportand all \({\vartheta }\in {\Theta }\)

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 \({\mathfrak T}({\mathfrak u}_1) \subset \mathfrak {S}\).

For each \(k\), the collection \({\mathfrak P}(k)\) is convex.

If \({\mathfrak p}, {\mathfrak p}' \in {\mathfrak {M}}(k,n)\) and

then \({\mathfrak p}={\mathfrak p}'\).

Let \(g:{\mathbb {R}}\to {\mathbb {C}}\) be a measurable \(2\pi \)-periodic function such that for some \(\delta {\gt}0\) and every \(x\in {\mathbb {R}}\),

Then for every \(x\in [0,2\pi ]\) and \(N{\gt}0\),

Let \(g:{\mathbb {R}}\to {\mathbb {C}}\) be a measurable \(2\pi \)-periodic function such that for some \(\delta {\gt}0\) and every \(x\in {\mathbb {R}}\),

Then for every \(\epsilon {\gt}0\), there exists a measurable set \(E\subset [0,2\pi ]\) with \(|E|{\lt}\epsilon \) such that for every \(x\in [0,2\pi ]\setminus E\) and \(N{\gt}0\),

where

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

Let \(f\) be a bounded measurable function on \({\mathbb {R}}\). Then \(Tf\) as defined in 11.1.16 is measurable.

For \(x\in R\) and \(R{\gt}0\), the ball \(B(x,R)\) is the interval \((x-R,x+R)\)

We have for every \(x\in {\mathbb {R}}\) and \(R{\gt}0\)

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

Assume that 10.0.3 holds. For every \(r{\gt}0\) and every bounded measurable function \(g\) supported on a set of finite measure we have

where

Let \(f:X\to {\mathbb {C}}\) be a bounded measurable function supported on a set of finite measure. Let \(x\in X\) and \(R{\gt}0\). Then, for all \(\epsilon {\gt}0\), there exists some \(\delta {\gt}0\) such that

and

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 \(J \in \mathcal{J}'\) and all \(s \ge 0\), we have

Let \({\vartheta }\in {\Theta }\), \(N\ge 0\) and \(L\in \mathcal{D}\). Then

Let \(f:X\to {\mathbb {C}}\) be a bounded measurable function supported on a set of finite measure. For all \(x\in X\),

If \({\mathfrak p}\in {\mathfrak T}({\mathfrak u}_2) \setminus \mathfrak {S}\) 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 6.1.43 vanishes unless

Let \({\mathfrak p},{\mathfrak p}'\in \mathfrak {A}\). If there exists an \(x\in X\) with \(x\in E({\mathfrak p})\cap E({\mathfrak p}')\), then \({\mathfrak p}= {\mathfrak p}'\).

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

For a given grid structure \((\mathcal{D}, c,s)\), there exists a tile structure \(({\mathfrak P},{\mathcal{I}},{\Omega },{\mathcal{Q}},{\mathrm{c}},{\mathrm{s}})\).

We have for all \(x\in G\setminus G'\)

Let \({\mathfrak p}_1, {\mathfrak p}_2\in {\mathfrak P}\) with \(B({\mathrm{c}}({\mathfrak p}_1),5D^{{\mathrm{s}}({\mathfrak p}_1)}) \cap B({\mathrm{c}}({\mathfrak p}_2),5D^{{\mathrm{s}}({\mathfrak p}_2)}) \ne \emptyset \) and \({\mathrm{s}}({{\mathfrak p}_1})\leq {\mathrm{s}}({{\mathfrak p}_2})\). 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

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 two-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 and all \(r{\gt}0\) we have

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,

Let \(\alpha \le \beta \) be real numbers. Let \(g:{\mathbb {R}}\to {\mathbb {C}}\) be a measurable function and assume

Then for any \(\alpha \le \beta \) and \(n\in {\mathbb {Z}}\) we have

Let \(f:X\to {\mathbb {C}}\) be a bounded measurable function supported on a set of finite measure and assume for some \(r{\gt}0\) that for every bounded measurable function \(g:X\to {\mathbb {C}}\) supported on a set of finite measure,

Then for all \(\alpha {\gt}0\), 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 \(n, m \ge 1\) and \(k {\gt} 0\). 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