LSU Digital Commons LSU Digital Commons Averages along the Primes: Improving and Sparse Bounds Averages along the Primes: Improving and Sparse Bounds

: Consider averages along the prime integers P given by These averages satisfy a uniform scale-free (cid:96) p -improving estimate. For all 1 < p < 2, there is a constant C p so that for all integer N and functions f supported on [0, N ], there holds The maximal function A * f = sup N | A N f | satisfies ( p , p ) sparse bounds for all 1 < p < 2. The latter are the natural variants of the scale-free bounds. As a corollary, A * is bounded on (cid:96) p ( w ), for all weights w in the Muckenhoupt A p class. No prior weighted inequalities for A * were known.


Introduction
Let P = { , , , . . . , } be the odd primes and de ne the logarithmically weighted averages along the primes by We prove scale-free p improving bounds for these averages, and sparse bounds for the associated maximal function We prove that the averages along the primes improve integrability, uniformly over all scales. where p = p p− .
We turn to the sparse inequalities. They are the natural extensions of the p improving inequalities above for the maximal function (1.1). We say that a sublinear operator B has sparse type (r, s), for < r, s < ∞ if there is a constant C so that for all nitely supported functions f , g there are a sparse collection of intervals S so that Therefore, the sparse bounds hold for the maximal function on the left. Our argument for the xed scale inequalities (1.3) requires the logarithmic averages.
Following Bourgain's work on arithmetic ergodic theorems 1], Wierdl 20] showed that A * is bounded on p for all < p < ∞. At the time, this was the rst arithmetic example for which this fact was known for all < p < . Bourgain's work 3] gave a comprehensive approach to the p theory of arithmetic averages. The subject continues to be under development, with important contributions by 8,15,16]. We point to the work of Mirek-Trojan and Trojan 17,18] also focused on the primes. The methods therein are di erent from those of this paper.
Our subject, developing the p -improving properties and sparse bounds started with 4], and continued in 6,14]. It now encompasses the discrete spherical maximal operators 9, 10, 12], as well as the square integers 5].
We use the High Low Method 5,7,11]. This depends upon e cient use of -methods, followed by a ne analysis of certain -type expressions. The latter are frequently the most intricate part. In this argument, they depend upon a relatively accessible property of Ramanujan sums, Lemma 3.4. Our argument is new, even if one is only interested in the p → p bounds for A * .

Preliminaries
Throughout, let ϕ(q) be the Euler totient function, let µ(q) be the Möbius function. The following estimate for ϕ(q) is well known: (2.1) We count primes in the standard logarithmic fashion. Put By the prime number theorem holds for some constant c, C > . This obviously implies ϑ(N) ∼ N. We now rede ne the averaging operators A N , by setting As this is a positive operator, there is no harm in this new de nition. The Fourier transform of a measure σ on Z is given by where e(ζ ) = e πiζ throughout. The inverse Fourier transform is denoted η. Occasionally, we may also denote the Fourier transform by F, and inverse Fourier transform by F − . We further set eq(ζ ) = e πiζ /q . Recall that Ramanujan sums are de ned by A ner property of Ramanujan sums is recalled in Lemma 3.4 below.

Approximating Multipliers
We de ne the approximating multipliers. Let [− / , / ] ≤ η ≤ [− / , / ] be a Schwartz function. For an integer s, let ηs(ξ ) = η( s ξ ). De ne the Fourier transform of the usual averages by The building blocks of the approximating multipliers are Throughout, q and s have the relationship above, although this will be suppressed in the notation. (This is a useful convention in the application of the multi-frequency maximal function inequality in the proof of the sparse bounds, see (5.7).) This is a consequence of standard facts in the number theory literature, and is very similar to how these facts are used in 20]. We recall them here.

If ξ does not meet any of the hypotheses of the prior two conditions, then
The points (1), (2) and (3)  Proof of Theorem 3.1. We note that by construction, the multipliers { L q,N : s ≤ q < s+ } are supported on disjoint intervals around the rationals a/q, with a ∈ Aq, and s ≤ q < s+ . From this, it follows from (2.1) that Above, A is the integer in Theorem 3.1.
It su ces to argue that for B = A + because we can use (3.8) to complete the proof of (3.3). We note that the intervals of ξ that appear in the conditions 1 and 2 of Lemma 3.2 are pairwise disjoint. Let us assume that ξ meets the condition 2, so |ξ − a/q| < (log N) B N for (a, q) = and < q < (log N) B . To prove (3.3) in this case, we need to see that, and furthermore by (3.5), We also need to see that all the other L q ,N (ξ ) are small. Indeed, for < q ≠ q ≤ (log N) B , and a ∈ A q , we Summing the estimates for L q ,N over ≤ q ≠ q ≤ (log N) B and using (2.1), we have Putting (3.10) and (3.11) together, we have veri ed (3.9) in this case. If ξ meets condition 1 of Lemma 3.2, the proof is completely analogous. We now assume that ξ does not meet the rst or second condition of Lemma 3.2. Then, (3.6) holds. And, similar to (3.11), we have Combining (3.6) with (3.11), we have completed the proof of (3.9).
The building blocks of the approximating multipliers have explicit inverse Fourier transforms. Lemma 3.3. With the notation of (3.2), there holds where we are using the notation of Ramanujan sums (2.5). Above ηs is understood as ηs,per, where ηs,per is the -periodic extension of ηs. For q = , (3.12) holds since c (x) ≡ .
The term on the right in (3.12) includes an average γ N . It also includes a Ramanujan sum term. One should note that cq( ) = ϕ(q), but this is far from typical behavior. This crude estimate shows that for most x, cq(x) is about one.

Lemma 3.4. For any ϵ > , and integer k > , uniformly in M
The implied constant depends upon k and ε.
Sketch of Proof. We will not give a complete proof. It follows from work of Bourgain 2, (3.43), page 126] that we have, under the assumptions above, that for any integer P, This is given a stand-alone proof in 12, Lemma 3.13]. Using the well known lower bound ϕ(q) q −ϵ/ , we see that (3.14) Finally, let integer m be such that m ≤ Q < m + . We have where we used (3.14). This proves the claimed result.

Fixed Scale
The xed scale result has fewer complications than the sparse bound. We show that for any < p < , there holds where E is an interval of length N, and the inequality is independent of N. Since the condition is open with respect to p, it su ces to consider the case of p ∈ N, with f = F supported on E and g = G supported on E. We trivially have We assume that this fails, thus min{ f E, , g E, } > (log N) −p .
The implied constants depend upon p. The term H is the High term, and it satis es a quanti ed estimate, while L satis es something close to the → ∞ endpoint. It consists of the 'low frequency' terms. From this, it follows that The two sides are equal provided that Here note that |γ N * ηs(y)| ≤ N ηs N . (4.7) Above, we have appealed to Hölder inequality and (3.13), with appropriate choice of parameters. Note this (3.13) only applies for N > Np, for a choice of Np that is only depending on p. After that, we simplify the expression, since f is an indicator set. This completes the proof.

Sparse Bound
We prove the sparse bound in Theorem 1.2. The sparse bound is stronger for smaller choices of (r, s), and so it su ces to prove the (p, p) sparse bound for all < p < . Again, by openness of the condition we are proving, it su ces to restrict attention to functions f , g that are indicator sets. The sparse bound is proved by recursion, which depends upon the following de nition. Let E be an interval of length n . Let f = F be supported on E, and g = G be supported on E. Let τ : E → { n : ≤ n ≤ n } be a choice of stopping time. We say that τ is admissible if for any interval We will have direct recourse to this at the end of the proof of the Lemma below.
The conclusion of (5.2) is very similar to the earlier argument in (4.6), and we omit the details.
We proceed with the construction of the High and Low terms. We begin with the trivial bound, following from admissibility, The point of this last line is that the inequality below is a direct consequence of Bourgain's multi-frequency maximal inequality, and the bound ϕ(q) p q − /( p ) : Summing this estimate over k ≥ j completes the analysis of the High term.
Remark 5.8. One of the main results of Bourgain 3] is the multi-frequency maximal inequality, a key aspect of discrete Harmonic Analysis. In the form that we have used it in (5.7), see for instance 13, Prop. 5.11].
The term that remains is the Low term below. We appeal to (3.12), to see that We need the following simple Lemma concerning γτ * ηs. In the last inequality we used s ≤ q ≤ J < Dp J /p < τ(x), due to (5.6) with a proper choice of Dp.