Maximal Operators Associated to Multiplicative Characters

We show that the natural analogue of the Carleson-Hunt inequality fails for multiplicative characters.


Introduction
The Carleson-Hunt inequality states that there exists a finite constant C such that   1 0 max ℓ ℓ n=0 a n e(xn) for any sequence of complex numbers {a n } ∞ n=1 (denoting e(x) := e 2πix ). It is not hard to see that this is equivalent to the discretized claim that holds with a universal constant C, independent of N . This can be viewed as a natural analog of the Carleson-Hunt inequality in the family of additive groups, Z N . The fact that this inequality implies (1) follows from an easy approximation argument. The reverse implication is slightly more subtle but can be obtained, for instance, from Montgomery's maximal large sieve inequality [4] (Theorem 2). In light of (2), it is natural to consider the analogous maximal operator on the family of multiplicative groups, Z * N . That is, we consider the inequality  where the sum χ mod N is over all Dirichlet characters modulo N , φ(N ) is the Euler totient function, and ∆(N ) is the smallest value such that the inequality holds. We are interested in the growth of the function ∆(N ). It is easy to see that ∆(N ) ≪ log(N ) by the Rademacher-Menshov theorem. By comparison to (2), one might hope that ∆(N ) ≪ 1. In fact, we show in Section 2 that if one could take ∆(N ) = O(1) in (3), then the Carleson-Hunt inequality for the trigonometric system would be an easy corollary. We note that in the case that a n = 1 for all n it follows from work of Montgomery and Vaughan [5] that (3) holds with a universal constant independent of N .
Unfortunately, it turns out that ∆(N ) = O(1). We prove: There exists a subset S of primes of positive relative density such that for every p ∈ S, (log log(p)) 1/4 ≪ ∆(p).
Thus (log log(N )) 1/4 ≪ ∆(N ) holds infinitely often. It remains an interesting problem to establish sharp bounds on the growth of ∆(N ). In particular, any refinement of the upper bound ∆(N ) ≪ log(N ) (from the Rademacher-Menshov theorem) would be extremely interesting.

Connection with the Carleson-Hunt inequality
In this section we prove that if (3) with ∆(N ) = O(1) did hold then the classical Carleson-Hunt inequality (1) would easily follow. By a standard density argument, it suffices to prove (1) for finite sequences {a n } k n=1 as long as the constant C does not depend on k. Indeed, once k is fixed, we will show that a n e(xn/M ) holds for an infinite increasing sequence of integral M 's and a constant C independent of k.
Clearly, this is sufficient as the sum on the left will converge to the Riemann integral over the unit interval. Consider a large prime 2 k < p. Now we apply (3) with ∆(p) ≤ C, and associating the coefficient We let α be a generator of Z * p . For g ∈ Z * p , we define ν(g) to be the element of [p − 1] such that α ν(g) = g. We may express χ(g) = e(aν(g)/(p − 1)) (for some 0 ≤ a < p − 1). Thus ν(g) can be thought of as a permutation of [p − 1]. In addition, it follows from this definition that ν(2 i ) = iν(2). We thus have that (6) can be expressed as: We define coprime L, M ∈ Z by ν(2) p−1 = L M . We observe that M > log 2 (p). To see this, Since M |(p − 1) and e iLx M has period M as a function of x, this can be rewritten as:  We perform the change of variable Lx → y to obtain  for some M ≫ log(p). This completes the proof.
Remark 2. One could also deduce the Carleson-Hunt inequality for Walsh series from the claim that ∆(N ) = O(1). We briefly sketch the argument. Choose N to be the product of d distinct odd primes. Then Z * N will contain an isomorphic copy of the group Z d 2 . The characters of this group are distributionally equivalent with the first 2 d Walsh functions. The maximal operator on Z * N will induce some ordering on these functions other than the standard ordering. It follows, however, from a combinatorial lemma of Bourgain [1] (Lemma 2.3) that there is a function B(d) (tending to infinity) such that any ordering of the first 2 d Walsh functions must contain a subsequence of length B(d) distributionally equivalent to the first B(d) Walsh functions in the standard ordering.

Auxiliary Results
In this section we collect some auxiliary results that will be needed in the proof of Theorem 1. We first note the following result from [2]: Then, for all m ∈ N, we have that We will also need the following version of Weil's character sum estimate. This can be found, for instance, on page 45 of [8].
Proposition 5. (Weil) Let p be a prime and g(x) = g n x n + . . . + g 0 a degree n polynomial (0 < n < p) with integer coefficients such that p ∤ g n . Then, We will also use the following quantitative form of Kolmogorov's rearrangement theorem due to Nakata [6] (Lemma 4). a n e(σ(n)x) .
We remark that Nakata has a slightly stronger refinement of Kolmogorv's rearrangement theorem [7] where there are some additional iterated logarithmic factors. However, the result there is formulated in a slightly different way and it would require some additional work to derive a statement sufficient for our purposes from it. For the sake of simplicity, we will not pursue this modification here. A simple averaging argument gives the following discrete version of Proposition 6. From these results, we obtain Proposition 8. We will denote the fractional part of a ∈ R by {a}. We let p, q denote primes such that q|p − 1 and q ≥ Bp .6687 . We let A denote the subgroup of order q in Z * p (i.e. A = g p−1 q : g ∈ Z * p ). There exists a universal constant δ > 0 such that for any s < δ log 1/2 (p), and any permutation σ : [s] → [s], there exists an x ∈ A such that Proof. We let g 1 , . . . , g q denote the elements of A inside Z * p in the order induced by Z * p . For each i from 1 to q, we define y i ∈ [0, 1] s as y i = . We then divide [0, 1] s into (3s) s boxes of equal measure by dividing each coordinate into 3s equal intervals in the obvious way. The conclusion will now follow if we show that there exists a point y i in each of the (3s) s boxes. To see this, consider the 3s intervals in each coordinate as being s groups of 3 intervals each, and let I k j denote the "middle" interval of the j th group in the k th coordinate. Note that I k j and I k j ′ for j = j ′ do not intersect. Given a permutation σ, it suffices to obtain a point in the box whose k th side is equal to I k σ(k) .
To establish the existence of a point y i in every box, it suffices to show D(y 1 , y 2 , . . . , y q ) < (3s) −s . Invoking Proposition 4, we have that D(y 1 , y 2 , . . . , y q ) ≤ 2s 2 3 s+1 We define We then have: We note that for some constant C. Whenever some h i is not divisible by p, we may apply Proposition 5 to bound the quantity p−1 x=0 e(f h (x)/p) . We will choose m < p so that all h's will have this property. We may thus bound the right hand side of (13) by when m < p (for some new value of C). Here, we have applied Proposition 5 to polynomials of degree ≤ s p−1 q .
Fixing δ 2 such that s −δ 2 s ≤ 1 2 (3s) −s for all s > 1 say (note that the Proposition is trivial for s = 1), we may then require that s satisfy p ≥ s δ 3 s for δ 3 sufficiently large so that δ 3 > δ 1 and the quantity in (15) is < (3s) −s . We observe that s δ 3 s ≤ p is equivalent to δ 3 s log(s) ≤ log(p), which can be guaranteed by s ≤ δ log 1/2 (p) for a suitable choice of δ.

Proof of Theorem 1
We define S to be the set of primes p such that there exists a prime q dividing p − 1 with q ≥ Bp .6687 . By Proposition 3, this is an infinite set of positive relative density in the primes. For each p ∈ S, we let A denote the subgroup of order q in Z * p . Our goal is to define suitable coefficients supported on A to show that ∆(p) is ≫ (log log(p)) 1 4 . We enumerate the elements of A in the natural way (that is so their smallest representatives in Z + are ordered in increasing order), say {g n } q n=1 . Next, we let α be a generator of A. We define ν(g n ) to be the element of [q] such that α ν(gn) = g n . By restricting the coefficients in (3) to A, we see that the quantity in (3) to be bounded is: a n e(ν(g n )x/q) This follows because restricting χ mod p to A yields a character on A. Let s = ⌊δ log 1/2 (p)⌋ and σ : [s] → [s] be the permutation in Corollary 7, along with coefficients b 1 , . . . , b s such that s m=1 |b m | 2 = 1. By Proposition 8, we have a g ∈ A such that Of course g σ(1) , g σ(2) , . . . , g σ(s) ∈ A. By restricting the support of our coefficients to these terms in (16) and using b 1 , . . . , b s as our coefficients, it suffices to consider the quantity (17) where we have exploited the fact that ν(g i ) = iν(g). Finally, by the change of variables xν(g) → y, we have Applying Corollary 7 with M = q, we see that this quantity is ≫ (log(s)) 1/4 ≫ (log(log(p))) 1/4 . This completes the proof.

Conclusion
There is some flexibility in the techniques applied in the proof of Theorem 1, and variants of these arguments should give lower bounds on ∆(N ) for some more general N . However, a more delicate analysis will be needed to obtain a uniform lower bound in N .
It is consistent with our knowledge that one might be able to replace the log 1/4 (N ) in Proposition 6 with a log(N ). This would allow one to strengthen the conclusion of Theorem 1 to log log(p) ≪ ∆(N ). However, the Rademacher-Menshov theorem prevents the conclusion of Proposition 6 from holding with any function growing faster than log(N ). Thus a lower bound of log log(N ) would be the limitation of the approach developed here.
One can interpret the proof of Theorem 1 as showing that the permutation of [q] defined by ν(·) is sufficiently pseudorandom that it contains the same increasing subsequences that could be found in a random permutation (with large probability). In connection with this interpretation, we note that Bourgain [1] has shown that the L 2 norm of the maximal function of a randomly ordered bounded orthonormal system is at most log log(N ) (with large probability). Perhaps this is some indication that the correct bound on ∆(N ) may be near log log(N ), or at least somewhat smaller than the trivial bound of log(N ).