Improving and maximal inequalities in discrete harmonic analysis
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Giannitsi, Christina
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Abstract
In this thesis we study various averaging operators of discrete functions, inspired by number theory, in order to show they satisfy $\ell^p$ improving and maximal bounds. The maximal bounds are obtained via sparse domination results for $p \in (1,2)$, which imply boundedness on $\ell ^p (w)$ for $p \in (1, \infty )$, for all weights $w$ in the Muckenhoupt $A_p$ class. The contents of this thesis have been published in \cite{bibcg2021-divisor,bibcg2021-progressions}.
We start by looking at averages along the integers using the divisor function $d(n)$ as defined in \eqref{eq-defin-divisor}:
\begin{equation*}
K_N f (x) = \frac{1}{D(N)} \sum _{n \leq N} d(n) \,f(x+n) ,
\end{equation*}
where $D(N) = \sum _{n=1} ^N d(n) $.
We obtain a uniform, scale free $\ell^p$-improving estimate for $p \in (1,2)$, that is for $f$ is supported on $[0,N]$,
\begin{equation*}
\left( \frac{1}{N} \sum |K_Nf|^{p'} \right)^{1/p'}
\lesssim _p
\left(\frac{1}{N} \sum |f|^p \right)^{1/p}.
\end{equation*}
We also show that the associated maximal function $K^*f = \sup_N |K_N f|$ satisfies $(p,p)$ sparse bounds for $p \in (1,2)$.
We move on to study averages along primes in arithmetic progressions. Let $ y < N$ are integers, and that $ (b,y) =1$.
Define an average along the primes in a progression of step $ y$,
\begin{align*}
A_{N,y,b} := \frac{\phi (y)}{N} \sum _{\substack{n <N\\n\equiv b\mod{y}}} \Lambda (n) f(x-n),
\end{align*}
where $\Lambda $ is the von Mangoldt function, \eqref{eq-defin-vonmangoldt}, and $\phi $ is the Euler totient function, \eqref{eq-defin-totient}.
We establish improving and maximal inequalities for these averages that are uniform in the choice of progression. The uniformity over progressions imposes several novel elements on our approach.
The main result is that for $ 1< r < \infty $ there is an integer $N _{y, r}$ so that for all $0<y<q$ with $(q,y)=1$, independently from the choice of progression,
\begin{align*}
\lVert \sup _{N>N _{y,r}} \lvert A_{N,y,b} f \rvert \rVert_{r}\lesssim _r
\lVert f\rVert_{r}.
\end{align*}
Lastly we generalize our setting in the context of number fields.
Let $\Lambda_i(n) $ be the Von Mangoldt function for the Gaussian integers and consider the norm function $N:\mathbb{Z}[i]\rightarrow \mathbb{Z}^+$, $\alpha + i \beta \mapsto \alpha ^2 + \beta ^2$. Define the averages
$$A_Nf(x)=\frac{1}{N}\sum_{N(n)<N}\Lambda_i(n)f(x-n)$$
with the associated maximal operator $A^{*} := \sup_{N} A_N$.
For $p \in (1,2)$,
\begin{equation*}
\left( \frac{1}{N} \sum |A_Nf|^{p'} \right)^{1/p'}
\lesssim _p
\left(\frac{1}{N} \sum |f|^p \right)^{1/p}.
\end{equation*}
In the last chapter we explore the connections of our work to number theory. More specifically, the tools to prove this are similar to those required to prove versions of Goldbach conjectures for Gaussian Primes in arbitrary sectors.
Fix an interval $\omega \subset \mathbb{T}$. There is an integer $N_\omega $, depending on $ \lvert \omega \rvert$, so that every odd integer $n$ with $N(n)>N_\omega $ is a sum of three Gaussian primes $n=p_1+p_2+p_3$, with
$\arg (p_j) \in \omega $, for $j=1,2,3$.
A density version of the binary Goldbach conjecture is proved; indeed
most even integers $n$ with $\arg (n)\in \omega $ and $N(n)<N$, for $N$ sufficiently large, can be written as a sum of two Gaussian primes $n=p_1+p_2$, with
$\arg (p_1), \arg (p_1) \in \omega $.
Those even integers which do not have this representation have density less than $\lesssim N/(\log N)^B$, for arbitrary integers $B$.
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Date
2023-06-15
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