Discrete Fractional Integration Operators Along the Primes

TL;DR

This paper establishes boundedness properties of discrete fractional integration operators along primes, extending harmonic analysis techniques to number-theoretic settings with implications for understanding prime-related operators.

Contribution

It proves boundedness of fractional integration operators along primes in discrete settings, a novel extension of classical harmonic analysis to prime number sequences.

Findings

Operators are bounded on ^p to ^{p'} under specified conditions.

Provides new bounds for prime-based fractional operators.

Extends harmonic analysis techniques to prime number sequences.

Abstract

We prove that the discrete fractional integration operators along the primes \[ T^{\lambda}_{\mathbb{P}}f(x) := \sum_{p} \frac{f(x-p)}{p^{\lambda}} \cdot \log p \] are bounded $\ell^p\to \ell^{p'}$ whenever $ \frac{1}{p'} < \frac{1}{p} - (1-\lambda), \ p > 1.$ Here, the sum runs only over prime $p$.

Equations

$$T_{\mathbb{P}}^{\lambda }f(x):=\sum _p\frac{f(x-p)}{p^{\lambda }}\cdot \log p$$

$$J_{d,\lambda }f(x):=\sum _{0\ne m\in {\mathbb{Z}}^d}\frac{f(x-m)}{|m{|}^{d\cdot \lambda }}$$

$$I_{\lambda }^{s}f(x):=\sum _{m\ge 1}\frac{f(x-m^s)}{m^{\lambda }}.$$

$$\sum _{0\ne m\in {\mathbb{Z}}^d}\frac{f(x-m,y-Q(m))}{|m{|}^{d\cdot \lambda }},$$

$$T_{\mathbb{P}}^{\lambda }f(x):=\sum _p\frac{f(x-p)}{p^{\lambda }}\cdot \log p;$$

$$\varphi (q){\gtrsim }_{ϵ}{q}^{1-ϵ}$$

$$K_k(x):=\frac{1}{2^k}\sum _{p\le 2^k}{\delta }_p(x)\cdot \log p,$$

$$\Vert K_k\ast f{\Vert }_{{\ell }^{p^{\prime }}}{\lesssim }_{ϵ}{2}^{-k\cdot (1/p-1/p^{\prime }-ϵ)}\Vert f{\Vert }_{{\ell }^p}.$$

$$⟨K_k\ast 1_F,1_G⟩{\lesssim }_{ϵ}|Q|\cdot {\left(\frac{|F|}{|Q|}\right)}^{1-ϵ}\cdot {\left(\frac{|G|}{|Q|}\right)}^{1-ϵ}$$

$$\Vert {\left(m(\xi )f(\xi )\right)}^{\vee }{\Vert }_{L^p(\mathbb{R})}\le A\Vert f{\Vert }_{L^p(\mathbb{R})}.$$

$$\left\{\frac{a}{q}\text{ reduced}:q\le N\right\}\subset {\mathcal{U}}_N\subset \left\{\frac{a}{q}\text{ reduced}:q\le C_{\rho }{e}^{N^{\rho }}\right\},$$

$${\left(\sum _{\theta \in {\mathcal{U}}_{\mathcal{N}}}m(\alpha -\theta ){\eta }_N(\alpha -\theta )\hat{f}(\beta )\right)}^{\vee }$$

$${\left(\text{Number of Frequencies}\right)}^{|1/2-1/p|},$$

$$\hat{f}=\widehat{f_1}+\widehat{f_2},$$

$$\widehat{f_1}(\alpha ):=\sum _{\theta \in \mathcal{U}}{\eta }_k(\alpha -\theta )\cdot \hat{f}(\alpha ),$$

$$\Vert f_1{\Vert }_{{\ell }^p}\lesssim \log k\cdot \Vert f{\Vert }_{{\ell }^p}.$$

$$\widehat{K_k}(\alpha ):=\frac{1}{2^k}\sum _{p\le 2^k}e(-p\alpha )\cdot \log p.$$

$$L_k^{\prime }(\alpha ):=\sum _t{L}_{k,t}^{\prime }(\alpha )$$

$$L_{k,t}^{\prime }(\alpha ):=\sum _{q\approx 2^t}\frac{\mu (q)}{\varphi (q)}\sum _{(a,q)=1}{V}_k(\alpha -a/q){\chi }_t(\alpha -a/q),$$

$$L_{k,t}(\alpha ):=\sum _{q\approx 2^t}\frac{\mu (q)}{\varphi (q)}\sum _{(a,q)=1}{V}_k(\alpha -a/q){\phi }_k(\alpha -a/q),$$

$$\underset{\alpha }{\sup }|L_k(\alpha )-L_k^{\prime }(\alpha )|\lesssim 2^{-ϵk}.$$

$$|L_{k,t}(\alpha )-L_{k,t}^{\prime }(\alpha )|\lesssim 2^{-(ϵ-1)t}\cdot 2^{-ϵk}.$$

$$|V_k(\alpha )|\lesssim (2^k|\alpha |{)}^{-1}\lesssim 2^{-ϵk}.$$

$$|K_k\ast f|\le M_{1}f+M_{2}f,f=1_F$$

$$\Vert M_{1}f{\Vert }_{p^{\prime }}\lesssim \frac{k^2}{2^{k\cdot (2/p-1)}}\cdot \Vert f{\Vert }_p$$

$$\Vert M_{2}f{\Vert }_2\lesssim k^{-C}\cdot \Vert f{\Vert }_2,$$

$$K_k\ast f=K_k\ast f_1+{\left(L_k\widehat{f_2}\right)}^{\vee }+{\left({\mathcal{E}}_k\widehat{f_2}\right)}^{\vee }.$$

$$\Vert K_k\ast f{\Vert }_{{\ell }^{p^{\prime }}}\lesssim \frac{k}{2^{k\cdot (2/p-1)}}\cdot \Vert f{\Vert }_{{\ell }^p},p>1$$

$$\sum _t\Vert L_{k,t}\widehat{f_2}{\Vert }_2=\sum _{t:2^t\ge k^{C_0}}\Vert L_{k,t}\widehat{f_2}{\Vert }_2;$$

Full text

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are bounded $\ell^{p}\to\ell^{p^{\prime}}$ whenever $\frac{1}{p^{\prime}}<\frac{1}{p}-(1-\lambda),\ p>1.$ Here, the sum runs only over prime $p$.

1. Introduction

The topic of this paper is discrete harmonic analysis, with a focus on discrete analogues of fractional integral operators. While elementary arguments link the discrete operators,

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to their continuous analogues – which leads to the expected range of norm estimates: $J_{d,\lambda}$ maps $\ell^{p}(\mathbb{Z}^{d})\to\ell^{p^{\prime}}(\mathbb{Z}^{d})$ when $1/p^{\prime}\leq 1/p-(1-\lambda)$ – the problem becomes much more subtle upon the introduction of radon behavior, where the primary objects of consideration (in the one-dimensional setting) are of the form

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While it is expected that $I_{\lambda}^{s}$ maps $\ell^{p}\to\ell^{q}$ when

• •

$1/p>1-\lambda$ and

• •

$\frac{1}{p^{\prime}}\leq\frac{1}{p}-\frac{1-\lambda}{s}$,

significant number-theoretic complications aside from the $s=2$ case, treated in [9, 10, 6, 2], have made it difficult to obtain these full range of exponents; see [7] for a discussion of these number-theoretic complications. Indeed, Pierce’s work on fractional radon transforms [8] along curves of the form (say)

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$Q$ a quadratic form, further established the link between the quadratic nature of the the curves in question and the ability to obtain wide range of $\ell^{p}$ estimates for fractional radon transforms.

In this short note, we explore the case of fractional radon transforms along the primes,

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these fractional radon transforms do not have a quadratic nature to the operator (the sum runs over prime $p$, and the presence of the logarithm is a normalizing factor, appearing from density considerations).

Nevertheless, drawing upon the techniques of [3], we prove the following theorem.

Theorem 1.3**.**

Suppose that $p>1$ and $1/p^{\prime}<1/p-(1-\lambda)$. Then (1.2) maps $\ell^{p}\to\ell^{p^{\prime}}$.

Remark 1.4**.**

The hard restriction $\frac{1}{p^{\prime}}<1/p-(1-\lambda)$ is an artifact of the real interpolation method we use; a soft inequality is expected, and would follow from complex methods. We do not pursue this issue here.

1.1. Notation

Here and throughout, $e(t):=e^{2\pi it}$. We let $\mu$ and $\phi$ denote the Möbius and totient functions, respectively. A key estimate is the lower bound

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valid for any $\epsilon>0$.

For $k\geq 1$ we let

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where the sum runs over only prime $p$.

We will make use of the modified Vinogradov notation. We use $X\lesssim Y$, or $Y\gtrsim X$, to denote the estimate $X\leq CY$ for an absolute constant $C$. We use $X\approx Y$ as shorthand for $Y\lesssim X\lesssim Y$. We also make use of big-O notation: we let $O(Y)$ denote a quantity that is $\lesssim Y$. If we need $C$ to depend on a parameter, we shall indicate this by subscripts, thus for instance $X\lesssim_{p}Y$ denotes the estimate $X\leq C_{p}Y$ for some $C_{p}$ depending on $p$. We analogously define $O_{p}(Y)$.

2. The Argument

By an appeal to the triangle inequality, Theorem 1.3 will follow from the following proposition.

Proposition 2.1**.**

For any $p>1$,

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Since our range of exponents is open, it suffices to prove a restricted weak-type estimate:

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whenever $F,G\subset Q:=Q_{k}:=\{|x|\lesssim 2^{k}\}$.

It is (2.3) to which we turn.

We next recall the following multi-frequency multiplier theorems, which in turn grew out of [2].

2.1. A Multi-Frequency Multiplier Theorem for Ionescu-Wainger Type Multipliers

The results of this section appear as the special one-dimensional case of [4, Theorem 5.1], a refinement of [2, Theorem 1.5] (which would also be adequate for our purposes).

Theorem 2.4** (Special Case).**

Suppose that $m(\xi)$ is an $L^{p}(\mathbb{R})$ multiplier with norm $A$:

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Let $\rho>0$ be arbitrary (for applications, we will take $0<\rho\ll_{p}1$). Then, for every $N$, there exists an absolute constant $C_{\rho}>0$ so that one may find a set of rational frequencies

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so that

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has $\ell^{p}$ norm $\lesssim_{\rho,p}A\cdot\log N$. Here, $\eta_{N}$ is a smooth bump function supported in a ball centered at the origin of radius $\leq e^{-N^{2\rho}}$.

Remark 2.6**.**

This result should be contrasted with the strongest analogous multiplier theorem for general frequencies, which accrues a norm loss of

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even in the special case when $m\in\mathcal{V}^{2}(\mathbb{R})$ has finite $2$-variation, see [1, Lemma 2.1].

2.2. Decompositions

Set $\mathcal{U}:=\mathcal{U}_{k^{C_{0}}}$ for some $C_{0}\gg_{p}1$, where $\mathcal{U}$ is as in Theorem 2.4, and choose $0<\rho=\rho_{p}\ll 1$. With these choices in mind, decompose $f=\mathbf{1}_{F}$ via the Fourier transform as

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where

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for $\eta$ a compactly supported bump function that is one in a neighborhood of the origin, and $\eta_{k}(t):=\eta(2^{k^{\rho^{\prime}}}t)$ for some $1\gg\rho^{\prime}\gg\rho$.

Note that for any $p>1$ so that

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We will also decompose that Fourier transform of $K_{k}$

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To do so, we recall the following approximation result of [5].

Lemma 2.10**.**

For any $A\gg 1$, there exists a $C=C(A)$ so that one may decompose $\widehat{K_{k}}(\alpha)=L_{k}^{\prime}(\alpha)+\mathcal{E}_{k}(\alpha)$, where

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with

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and $|\mathcal{E}_{k}|\lesssim k^{-A}$ pointwise. Here, $\chi_{t}(\alpha):=\chi(2^{Ct}\alpha)$ is a compactly supported bump function, and $V_{k}(\alpha):=\int_{0}^{1}e(-2^{k}t\cdot\alpha)\ dt$.

For our purposes, we will replace $L_{k}^{\prime}$ with $L_{k}:=\sum_{t}L_{k,t}$, where

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where $\varphi$ is a compactly supported bump function, and $\varphi_{k}(\alpha):=\varphi(2^{k(1-\epsilon)}\alpha)$.

Specifically, we have the following lemma.

Lemma 2.13**.**

The following estimate holds:

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Proof.

It suffices to show that

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The key point is that if $\varphi_{k}(\alpha)-\chi_{t}(\alpha)$ does not vanish, then $2^{(\epsilon-1)k}\lesssim|\alpha|$, so that

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The result then follows from the fact that $\{\chi_{t}(\cdot-a/q):(a,q)=1,\ q\approx 2^{t}\}$ are disjointly supported, taking into account the decay of the totient function, (1.5). ∎

With these decompositions in hand we turn to the proof.

Proof.

To prove (2.3) it suffices to bound

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where for any $p>1$

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and

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where $C$ may be adjusted to be as large as we wish.

Our decomposition is as follows:

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We set $M_{1}f:=K_{k}*f_{1}$; by interpolating between the $\ell^{1}\to\ell^{\infty}$ bound of $\frac{k}{2^{k}}$, and the trivial $\ell^{2}\to\ell^{2}$ bound, we see that

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which leads to the estimate (2.14), taking into account (2.8).

The contribution of the term involving $\mathcal{E}_{k}$ is absorbed into $M_{2}f$, but contributes a negligible bound, as we are free to choose $A$ in Lemma 2.10 as large as we wish; it suffices to show that $L_{k}\widehat{f_{2}}$ satisfies the $\ell^{2}$ estimate, (2.15).

In particular, we need to estimate

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using the decay of the totient function, (1.5), a bound of $k^{\epsilon-C_{0}}$ is obtained, which yields the result. ∎