1 Introduction

In [ , L. Carleson addressed a classical question regarding the convergence of Fourier series of continuous functions by proving their pointwise convergence almost everywhere, as stated below in Theorem 1.0.1.

Let \(f\) be a complex valued, \(2\pi \)-periodic bounded Borel measurable function on the real line, and for an integer \(n\), define the Fourier coefficient as

\begin{equation} \widehat{f}_n:=\frac{1}{2\pi } \int _0^{2\pi } f(x) e^{- i nx} dx . \end{equation}

1.0.1

Define the partial Fourier sum for \(N\ge 0\) as

\begin{equation} \label{eq:partial-fourier-sum} S_Nf(x):=\sum _{n=-N}^N \widehat{f}_n e^{i nx}\ . \end{equation}

1.0.2

Theorem 1.0.1 classical Carleson

✓

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

\begin{equation} \label{eq:fourier-limit} \lim _{N\to \infty }S_N f(x) = f(x), \end{equation}

1.0.3

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

The purpose of this paper is twofold. On the one hand, it prepares computer verification of Theorem 1.0.1 by presenting a very detailed proof as a blueprint for coding in Lean. We pass through a bound for a generalization of the so-called Carleson operator to doubling metric measure spaces. This generalization is new, and proving these bounds constitutes the second purpose of this paper. This generalization incorporates several results from the recent literature, most prominently bounds for the polynomial Carleson operator of V. Lie [ as well as its generalization [ . A computer verification of our theorem will also entail a computer verification for the bulk of the work in these results.

We proceed to introduce the setup for our general theorem. We carry a multi purpose parameter, a natural number

\begin{equation} a\ge 4 \end{equation}

1.0.4

in our notation that as it gets larger will allow more general applications but will worsen the constants in the estimates.

A doubling metric measure space \((X,\rho ,\mu , a)\) is a complete and locally compact metric space \((X,\rho )\) equipped with a \(\sigma \)-finite non-zero Radon–Borel measure \(\mu \) that satisfies the doubling condition that for all \(x\in X\) and all \(R{\gt}0\) we have

\begin{equation} \label{doublingx} \mu (B(x,2R))\le 2^a\mu (B(x,R))\, , \end{equation}

1.0.5

where we have denoted by \(B(x,R)\) the open ball of radius \(R\) centred at \(x\):

\begin{equation} \label{eq-define-ball} B(x,R):=\{ y\in X: \rho (x,y){\lt}R\} . \end{equation}

1.0.6

A collection \({\Theta }\) of real valued continuous functions on the doubling metric measure space \((X,\rho ,\mu ,a)\) is called compatible, if there is a point \(o\in X\) where all the functions are equal to \(0\), and if there exists for each ball \(B \subset X\) a metric \(d_B\) on \({\Theta }\), such that the following five properties 1.0.7, 1.0.8, 1.0.9, 1.0.10, and 1.0.11 are satisfied. For every ball \(B \subset X\)

\begin{equation} \label{osccontrol} \sup _{x,y\in B}|{\vartheta }(x)-{{\vartheta }(y)}- {\theta }(x)+{{\theta }(y)}| \le d_{B}({\vartheta },{\theta })\, . \end{equation}

1.0.7

For any two balls \(B_1=B(x_1,R)\), \(B_2= B(x_2,2R)\) in \(X\) with \(x_1\in B_2\) and any \({\vartheta },{\theta }\in {\Theta }\),

\begin{equation} \label{firstdb} d_{B_2}({\vartheta },{\theta })\le 2^a d_{B_1}({\vartheta },{\theta }) . \end{equation}

1.0.8

For any two balls \(B_1, B_2\) in \(X\) with \(B_1 \subset B_2\) and any \({\vartheta }, {\theta }\in {\Theta }\)

\begin{equation} \label{monotonedb} d_{B_1}({\vartheta },{\theta }) \le d_{B_2}({\vartheta }, {\theta }) \end{equation}

1.0.9

and for any two balls \(B_1=B(x_1,R)\), \(B_2= B(x_2,2^aR)\) with \(B_1\subset B_2\), and \({\vartheta },{\theta }\in {\Theta }\),

\begin{equation} \label{seconddb} 2d_{B_1}({\vartheta },{\theta }) \le d_{B_2}({\vartheta },{\theta }) . \end{equation}

1.0.10

For every ball \(B\) in \(X\) and every \(d_B\)-ball \(\tilde B\) of radius \(2R\) in \({\Theta }\), there is a collection \(\mathcal{B}\) of at most \(2^a\) many \(d_B\)-balls of radius \(R\) covering \(\tilde B\), that is,

\begin{equation} \label{thirddb} \tilde B\subset \bigcup \mathcal{B}. \end{equation}

1.0.11

Further, a compatible collection \({\Theta }\) is called cancellative, if for any ball \(B\) in \(X\) of radius \(R\), any Lipschitz function \(\varphi : X\to {\mathbb {C}}\) supported on \(B\), and any \({\vartheta },{\theta }\in {\Theta }\) we have

\begin{equation} \label{eq-vdc-cond} |\int _B e({\vartheta }(x)-{{\theta }(x)}) \varphi (x) d\mu (x)|\le 2^a \mu (B)\| \varphi \| _{\operatorname{\operatorname {Lip}}(B)} (1+d_B({\vartheta },{\theta }))^{-\frac{1}{a}}, \end{equation}

1.0.12

where \(\| \cdot \| _{\operatorname{\operatorname {Lip}}(B)}\) denotes the inhomogeneous Lipschitz norm on \(B\):

\[ \| \varphi \| _{\operatorname{\operatorname {Lip}}(B)} = \sup _{x \in B} |\varphi (x)| + R \sup _{x,y \in B, x \neq y} \frac{|\varphi (x) - \varphi (y)|}{\rho (x,y)}\, . \]

A one-sided Calderón–Zygmund kernel \(K\) on the doubling metric measure space \((X, \rho , \mu , a)\) is a measurable function

\begin{equation} \label{eqkernel0} K:X\times X\to \mathbb {C} \end{equation}

1.0.13

such that for all \(x,y',y\in X\) with \(x\neq y\), we have

\begin{equation} \label{eqkernel-size} |K(x,y)| \leq \frac{2^{a^3}}{V(x,y)} \end{equation}

1.0.14

and if \(2\rho (y,y') \leq \rho (x,y)\), then

\begin{equation} \label{eqkernel-y-smooth} |K(x,y) - K(x,y')| \leq \left(\frac{\rho (y,y')}{\rho (x,y)}\right)^{\frac{1}{a}}\frac{2^{a^3}}{V(x,y)}, \end{equation}

1.0.15

where

\[ V(x,y):=\mu (B(x,\rho (x,y))). \]

Define the maximally truncated non-tangential singular integral \(T_{*}\) associated with \(K\) by

\begin{equation} \label{def-tang-unm-op} T_{*}f(x):=\sup _{R_1 {\lt} R_2} \sup _{\rho (x,x'){\lt}R_1} \left|\int _{R_1{\lt} \rho (x',y) {\lt} R_2} K(x',y) f(y) \, \mathrm{d}\mu (y) \right|\, . \end{equation}

1.0.16

We define the generalized Carleson operator \(T\) by

\begin{equation} \label{def-main-op} Tf(x):=\sup _{{\vartheta }\in {\Theta }} \sup _{0 {\lt} R_1 {\lt} R_2}\left| \int _{R_1 {\lt} \rho (x,y) {\lt} R_2} K(x,y) f(y) e({\vartheta }(y)) \, \mathrm{d}\mu (y) \right|\, , \end{equation}

1.0.17

where \(e(r)=e^{ir}\).

Our main result is the following restricted weak type estimate for \(T\) in the range \(1{\lt}q\le 2\), which by interpolation techniques recovers \(L^q\) estimates for the open range \(1{\lt}q{\lt}2\).

Theorem 1.0.2 metric space Carleson

✓

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

\begin{equation} \label{nontanbound} \| T_{*}g\| _{2} \leq 2^{a^3} \| g\| _2\, , \end{equation}

1.0.18

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,

\begin{equation} \label{resweak} \left|\int _{G} T f \, \mathrm{d}\mu \right| \leq \frac{2^{450a^3}}{(q-1)^6} \mu (G)^{1-\frac{1}{q}} \mu (F)^{\frac{1}{q}}\, . \end{equation}

1.0.19

For a Borel function \({Q}\to {\Theta }\), and \({\vartheta }\in {\Theta }\) and \(x\in X\) define

\begin{equation} R_{{Q}}({\vartheta },x)=\sup \{ r{\gt}0:d_{B(x,r)}({\vartheta }, {Q}(x)){\lt}1\} \end{equation}

1.0.20

and define further

\begin{equation} \label{def-lin-star-op} T_{{Q}}^{\vartheta }f(x):=\sup _{R_1{\gt}0} \ \sup _{\rho (x,x'){\lt}R_1} \left|\int _{R_1{\lt} \rho (x',y) {\lt} R_{{Q}}({\vartheta },x')} K(x',y) f(y) \, \mathrm{d}\mu (y) \right| \end{equation}

1.0.21

Define further the linearized generalized Carleson operator \(T_{Q}\) by

\begin{equation} \label{def-lin-main-op} T_{Q}f(x):= \sup _{0 {\lt} R_1 {\lt} R_2}\left| \int _{R_1 {\lt} \rho (x,y) {\lt} R_2} K(x,y) f(y) e({Q}(x)(y)) \, \mathrm{d}\mu (y) \right|\, , \end{equation}

1.0.22

where again \(e(r)=e^{ir}\).

Theorem 1.0.3 linearised metric Carleson

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

\begin{equation} \label{linnontanbound} \| T_{{Q}}^{\vartheta }g\| _{2} \leq 2^{a^3} \| g\| _2\, , \end{equation}

1.0.23

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,

\begin{equation} \label{linresweak} \left|\int _{G} T_{Q}f \, \mathrm{d}\mu \right| \leq \frac{2^{450a^3}}{(q-1)^6} \mu (G)^{1-\frac{1}{q}} \mu (F)^{\frac{1}{q}}\, . \end{equation}

1.0.24

We note that in Chapter ?? we formulate a variant of this theorem where \({Q}\) has countable range.

In the one-dimensional Euclidean setting, with \(K\) representing the Hilbert kernel:

\begin{equation*} K(x,y)=(x-y)^{-1} \end{equation*}

and \({\Theta }\) denoting the class of linear functions, the operator 1.0.17 is the classical Carleson operator, which plays a crucial role in proving the almost everywhere convergence of Fourier series [ , [ , [ . The supremum in \(R_1\) and \(R_2\) is often omitted in classical treatments, but considering the maximal truncations can easily be reduced to the case without these truncations.

By replacing \({\Theta }\) with the class of polynomials vanishing at \(0\) up to some fixed but arbitrary degree, we obtain the polynomial Carleson operator of Lie [ (quadratic case) and [ . The case of the class of polynomials with vanishing linear coefficient is simpler and was estimated in [ . The polynomial Carleson operator was generalized to the high-dimensional Euclidean setting in [ for \(K\) being a Calderón-Zygmund kernel with some Hölder regularity.

Doubling metric measure spaces are instances of spaces of homogeneous type. Indeed, by changing from a quasi-metric to an equivalent metric, every space of homogeneous type can be viewed as a doubling metric measure space (cf. [ ). Spaces of homogeneous type were introduced by [ as a natural setting for Calderón-Zygmund theory. We refer to the textbook [ for an account of these spaces.

Our concept of a compatible collection \({\Theta }\) as a natural class of phase functions on a doubling metric measure space does not appear in [ but is implicitly anticipated in [ and subsequent work of [ , who proves a Carleson-type theorem for the Malmquist-Takenaka series, which leads to classes of phases related to Blaschke products. A generalization of 1.0.17 from the previously mentioned Euclidean setting into the anisotropic setting that was suggested in [ is included in our theory. The polynomial Carleson operator also plays a role in the study of maximally modulated singular Radon transforms along the parabola, see [ and [ .

For the proof of Theorem 1.0.2, we largely follow [ , which in turn was inspired by [ . We make suitable modifications to adapt to our more general setting and have made a few technical improvements in the proof. In particular, in Chapter 2, we explicitly divide the main work of the proof into mutually independent sections 3, 4, 5, 6, 7, 8, and 9. Some of these sections follow a similar pattern, starting with a subsection dividing the proof into further mutually independent subsections. This modularization of our proof was strongly endorsed in personal communication by the author of [ .

Acknowledgement. L.B., F.v.D., R.S., and C.T. were funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813. L.B. , R.S., and C.T. were also supported by SFB 1060. A.J. is funded by the TÜBITAK (Scientific and Technological Research Council of Türkiye) under Grant Number 123F122.