Improved $\ell^p$-Boundedness for Integral $k$-Spherical Maximal Functions
Theresa C. Anderson Brian Cook Kevin Hughes Angel Kumchev
math.CA math.NT 11, 42


[classification=text]














TA]Theresa Anderson
BC]Brian Cook
KH]Kevin Hughes
AK]Angel Kumchev



We improve the range of ℓ^p(ℤ^d)-boundedness of the integral k-spherical maximal functions introduced by Magyar. The previously best known bounds for the full k-spherical maximal function require the dimension d to grow at least cubically with the degree k. Combining ideas from our prior work with recent advances in the theory of Weyl sums by Bourgain, Demeter, and Guth and by Wooley, we reduce this cubic bound to a quadratic one. As an application, we improve upon bounds in the authors' previous work <cit.> on the ergodic Waring–Goldbach problem, which is the analogous problem of ℓ^p(ℤ^d)-boundedness of the k-spherical maximal functions whose coordinates are restricted to prime values rather than integer values.





§ INTRODUCTION


Our interest lies in proving ℓ^p(ℤ^d)-bounds for the integral k-spherical maximal functions when k ≥ 3. These maximal functions are defined in terms of their associated averages, which we now describe. Define a positive definite function 𝔣 on ℝ^d by 

    𝔣(𝐱) = 𝔣_d,k(𝐱) := |x_1|^k + … + |x_d|^k,

and note that when 𝐱∈ℝ_+^d, 𝔣(𝐱) is the diagonal form x_1^k + … + x_d^k. For λ∈ℕ, let R(λ) denote the number of integral solutions to the equation

    𝔣_d,k(𝐱) = λ.

When R(λ)>0, define the normalized arithmetic surface measure 

    σ_λ(𝐱) := 1/R(λ)1_{𝐲∈ℤ^d: 𝔣(𝐲)=λ}(𝐱).
 
We are interested in averages given by convolution with these measures:

    σ_λ * f(𝐱) = 1/R(λ)∑_𝔣 (𝐲) = λ f(𝐱 - 𝐲)

for functions f: ℤ^d →ℂ.

We know from the literature on Waring's problem that as λ→∞, one has the asymptotic 

    R(λ) ∼𝔖_d,k(λ) λ^d/k-1,
 
where 𝔖_d,k(λ) is a convergent product of local densities:

    𝔖_d,k(λ) = ∏_p ≤∞μ_p(λ).

Here μ_p(λ) with p < ∞ is related to the solubility of (<ref>) over the p-adic field ℚ_p, and μ_∞(λ) to solubility over the reals. It is known that when d is sufficiently large in terms of k, one has 

    1 ≲𝔖_d,k(λ) ≲ 1.
 
In particular, these bounds hold for d ≥ 4k when k ≥ 4 is a power of 2, and for d ≥3/2k otherwise (see Theorems 4.3 and 4.6 in Vaughan <cit.>). 

Throughout the paper we use the notation f(x) ≲ g(x) or g(x) ≳ f(x) to mean that there exists a constant C > 0 so that |f(x)| ≤ C|g(x)| for all sufficiently large x ≥ 0. 
The implicit constant C may depend on `inessential' or fixed parameters, but will be independent of `x';  below the implicit constants will often depend on the parameters k,d and p. 
For instance, (<ref>) means that there exists positive constants C_1 and C_2 depending on d and k so that C_1 ≤𝔖_d,k(λ) ≤ C_2. 

In view of (<ref>) and (<ref>), we may replace the convolution σ_λ * f above by the average

    A_λ f(𝐱) := λ^1-d/k∑_𝔣 (𝐲) = λ f(𝐱 - 𝐲).

Our k-spherical maximal function is then defined, for 𝐱∈ℤ^d, as the pointwise supremum of all averages

    A_* f(𝐱) := sup_λ∈ℕ |A_λ f(𝐱)|.

Variants of this maximal function were introduced by Magyar <cit.> and studied later in <cit.>. In particular,  Magyar, Stein and Wainger <cit.> considered the above maximal function when k=2 and d ≥ 5 and proved that it is bounded on ℓ^p(ℤ^d) when p > d/d-2. This result is sharp except at the endpoint, for which the restricted weak-type bound was proved later by Ionescu <cit.>. To the best of our knowledge, the sharpest results on the boundedness of A_* for degrees k ≥ 3 are those obtained by the third author in <cit.>. In the present paper, we give a further improvement.

For k ≥ 3, define 

    d_0(k) := k^2 - max_2 ≤ j ≤ k-1{k j - min(2^j+2,j^2+j)/k-j+1}.
 
Further, set τ_k = max{ 2^1-k,  (k^2 - k)^-1} and define the function δ_0(d,k) by 

    k  δ_0(d,k) :=  (d-d_0)/(k^2+k - d_0)    if  d_0(k) ≤ d ≤ k^2+k, 
     1 + (d-k^2-k)τ_k    if  d > k^2+k.

Finally, define

    p_0(d,k):=max{d/d-k, 1+1/1+2δ_0(d,k)}.

We remark that when d > d_0(k), p_0(d,k) always lies in the range (1,2). Our main result is the following.


If k ≥ 3, d > d_0(k) and p>p_0(d,k), then the maximal operator A_*, defined by (<ref>), is bounded on ℓ^p(ℤ^d): that is,

    A_*f _ℓ^p(ℤ^d)≲ f _ℓ^p(ℤ^d) .



Let d_0^*(k) := 1 + ⌊ d_0(k) ⌋ denote the least dimension in which Theorem <ref> establishes that A_* is bounded on ℓ^2(ℤ^d). We emphasize that for large k, we have d_0^*(k) = k^2 - k + O(k^1/2), whereas in previous results, such as the work of the third author <cit.>, one required d > k^3-k^2. 
While our results improve on these previous results by a factor of the degree k, the conjecture is that the maximal function is bounded on ℓ^2(ℤ^d) for d ≳ k (see <cit.> for details), but such a result appears to be way beyond the reach of present methods. 

It is also instructive to compare d_0^*(k) to the known bounds for the function G̃(k) in the theory of Waring's problem (defined as the least value of d for which the asymptotic formula (<ref>) holds). It transpires that the values of d_0^*(k) match the best known upper bounds on G̃(k) for all but a handful of small values of k, and even in those cases, we miss the best known bound on G̃(k) only by a dimension or two. For an easier comparison, we list the numerical values of d_0^*(k), k ≤ 10, their respective analogues in earlier work, and the bounds on G̃(k) in Table <ref>. 


A key ingredient in the proof of Theorem <ref> and its predecessors is an approximation formula generalizing (<ref>). First introduced in <cit.> when k=2, such approximations are obtained for the average's corresponding Fourier multiplier:

    A_λ() = λ^1-d/k∑_𝔣(𝐱)=λ  e(𝐱·ξ),

where ξ∈𝕋^d and e(z) = e^2π iz. 

We need to introduce some notation in order to state our approximation formula. Given an integer q ≥ 1, we write ℤ_q = ℤ/qℤ and ℤ_q^* for the group of units; we also write e_q(x) = e(x/q). The d-dimensional Gauss sum of degree k is defined as 

    G(q; a, 𝐛) = q^-d∑_𝐱∈ℤ_q^d e_q ( a𝔣(𝐱) + 𝐛·𝐱)

for a ∈ℤ_q and 𝐛∈ℤ_q^d. If Σ_λ denotes the surface in ℝ_+^d defined by (<ref>) and dS_λ(𝐱) denotes the induced Lebesgue measure on Σ_λ, we define a continuous surface measure dσ_λ(𝐱) on Σ_λ by  

    dσ_λ(𝐱) 
    := λ^1-d/kdS_λ(𝐱)/|∇𝔣(𝐱)|.

We note that dσ_λ is essentially a probability measure on Σ_λ for all λ. We also fix a smooth bump function ψ, which is 1 on [-1/8,1/8]^d and supported in [-1/4, 1/4]^d. Finally, we write μ for the ℝ^d-Fourier transform of a measure μ on ℝ^d and f for the ℤ^d-Fourier transform (which has domain 𝕋^d) of a function f on ℤ^d. 

 
If k ≥ 3, d > d_0(k) and λ∈ℕ is sufficiently large, then one has

    A_λ(ξ) = ∑_q=1^∞∑_a ∈ℤ_q^* e_q( -λ a) ∑_ w ∈{± 1 }^d∑_𝐛∈ℤ^d G(q; a,  w𝐛) ψ(qξ-𝐛) dσ_λ( w(ξ-q^-1𝐛)) + E_λ(ξ),

where the error terms E_λ are the multipliers of convolution operators satisfying the dyadic maximal inequality

    sup_λ∈ [Λ/2, Λ) |E_λ| _ℓ^2(ℤ^d) →ℓ^2(ℤ^d)≲Λ^-δ

for each Λ≥ 1 and all sufficiently small δ > 0.


Our Approximation Formula takes the same shape as those in <cit.> and <cit.>, but with an improved error term that relies on two recent developments: 


  * the underlying analytic methods were improved in the authors' previous work <cit.>,

  * and the recent resolution of the main conjecture about Vinogradov's mean value integral <cit.> and related refinements of classical mean value estimates <cit.>.

The most dramatic improvement follows from our improved analytic methods originating in <cit.> where we improve the range of ℓ^2(ℤ^d) by a factor of the degree k. 
In <cit.>, this sort of improvement - which also used the recent resolution of the Vinogradov mean values theorems <cit.> - was limited to maximal functions over sufficiently sparse sequences. 
Here, our bounds supersede those for integral k-spherical maximal functions over sparse sequences in <cit.> because our treatment of the minor arcs in the error term is more efficient. 
The reader may compare Lemmas <ref> and <ref> below to Lemmas 2.1 and 2.2 of <cit.> to determine the efficacy of our method here. 
Consequently, <cit.> and <cit.> allow us to further improve slightly upon a more direct application of the Vinogradov mean value theorems from <cit.> and <cit.>. 
One minor drawback is that in our method the ϵ-losses in <cit.> do not allow us to deduce endpoint bounds. 

As an application, we deduce that the maximal function of the "ergodic Waring–Goldbach problem" introduced in our recent work <cit.> is bounded on the same range of spaces as above. That maximal function is associated with averages where, instead of sampling over integer points, we sample over points where all coordinates are prime. To be precise, let R^*(λ) denote the number of prime solutions to the equation (<ref>) weighted by logarithmic factors: that is, 

    R^*(λ) := ∑_𝔣(𝐱) = λ1_ℙ^d(𝐱)(log x_1) ⋯ (log x_d),

where 1_ℙ^d is the indicator function of vectors 𝐱∈ℤ^d with all coordinates prime. When R^*(λ)>0, define the normalized arithmetic surface measure 

    ω_λ(𝐱) := 1/R^*(λ) 1_{𝐲∈ℙ^d: 𝔣(𝐲)=λ}(𝐱) (log x_1) ⋯ (log x_d)
 
and the respective convolution operators  

    W_λ f := ω_λ * f.

Similarly to (<ref>), we know that as λ→∞, one has the asymptotic 

    R^*(λ) = (𝔖_d,k^*(λ) + o(1)) λ^d/k-1,

where 𝔖_d,k^*(λ) is a product of local densities similar to 𝔖_d,k(λ) above. Moreover, when d > 3k and λ is restricted to a particular arithmetic progression Γ_d,k, we have 1 ≲𝔖_d,k^*(λ) ≲ 1, and the above estimate turns into a true asymptotic formula for R^*(λ) (see the introduction and references in <cit.>). By Theorem 6 of <cit.> and Theorem <ref> above, we immediately obtain the following result. Here, as in <cit.>, d_1(3) = 13 and d_1(k) = k^2+k+3.


If k ≥ 3, d ≥ d_1(k) and p > p_0(d,k), then the maximal function defined by

    W_* f(𝐱) := sup_λ∈Γ_d,k |W_λ f(𝐱)|,

is bounded on ℓ^p(ℤ^d).

As another application one may give analogous improvements of the ergodic theorems obtained in <cit.>, but we do not consider this here.

To establish our theorems, we follow the paradigm in <cit.> and strengthen the connection to Waring's problem as initiated in <cit.> by using a lemma from <cit.>; we then use recent work on Waring's problem to obtain improved bounds. 
We remark that <cit.> and <cit.> previously connected mean values (Hypothesis K^* and Vinogradov's mean value theorems respectively) to discrete fractional integration. 
In Section <ref>, we outline the proofs of Theorems <ref> and <ref>; we recall some results from <cit.> and state the key propositions required in the proofs. The remaining sections establish the propositions. 
In Section <ref> we deal with the minor arcs; particularly, in Section <ref>, we use the recent work of Bourgain, Demeter and Guth <cit.> on Vinogradov's mean value theorem and a method of Wooley <cit.> for estimation of mean values over minor arcs. 
In Section <ref>, we establish the relevant major arc approximations. Finally, in Section <ref>, we establish the boundedness of the maximal function associated with the main term in the Approximation Formula. 




§ OUTLINE OF THE PROOF


Since A_* is trivially bounded on ℓ^∞(ℤ^d), we may assume through the rest of the paper that p ≤ 2. 
Fix Λ∈ℕ and consider the dyadic maximal operator

    A_Λ f := sup_λ∈ [Λ/2,Λ) |A_λf|.

When λ≤Λ, we have

    A_λ f = λ^1-d/k∫_𝕋 (h_Λ(θ) * f)e(-λθ)   dθ,

where 

    h_Λ(θ) = h_Λ(θ; 𝐱) := e(θ𝔣(𝐱))1_[-N,N]^d(𝐱)   with   N = Λ^1/k.
 
This representation allows us to decompose A_λ into operators of the form

    A_λ^𝔅 f = λ^1-d/k∫_𝔅 (h_Λ(θ) * f)e(-λθ)   dθ,

for various measurable sets 𝔅⊆𝕋.

Our decomposition of A_λ, is inspired by the Hardy–Littlewood circle method. When q ∈ℕ and 0 ≤ a ≤ q, we define the major arc 𝔐(a/q) by

    𝔐(a/q) = [ a/q - 1/4kqN^k-1,a/q + 1/4kqN^k-1].

We then decompose 𝕋 into sets of major and minor arcs, given by

    𝔐 = 𝔐(Λ) = ⋃_q ≤ N⋃_a ∈ℤ_q^*𝔐(a/q)    and   𝔪(Λ) = 𝕋∖𝔐(Λ).

Since the major arcs 𝔐(a/q) are disjoint, this yields a respective decomposition of A_λ as 

    A_λ = ∑_q ≤ N∑_a ∈ℤ_q^* A_λ^a/q + A_λ^𝔪,

where A_λ^a/q := A_λ^𝔐(a/q). 

We will use the notations A_*^𝔅 and A_Λ^𝔅 to denote the respective maximal functions obtained from the operators A_λ^𝔅. For example, from (<ref>) and the trivial bound for the trigonometric polynomial h_Λ(θ) * f, we obtain the trivial ℓ^1-bound

    A_Λ^𝔅_ℓ^1(ℤ^d) →ℓ^1(ℤ^d)≲Λ |𝔅|.

In Section <ref>, we analyze the minor arc term and prove the following result.


If k ≥ 3, d > d_0(k), and Λ≳ 1, then

    A_Λ^𝔪_ℓ^2(ℤ^d) →ℓ^2(ℤ^d)≲Λ^-δ

for all δ∈ (0,δ_0(d,k)).


For 1 < p ≤ 2, interpolation between (<ref>) and (<ref>) yields 

    A_Λ^𝔪_ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲Λ^-α_p

with α_p = 2(1+δ)(1-1/p) - 1. When p_0(d,k) < p ≤ 2, we have α_p > 0, and hence,

    A_*^𝔪_ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≤∑_Λ = 2^j ≳ 1 A_Λ^𝔪_ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲∑_Λ = 2^j ≳ 1Λ^-α_p≲ 1.
  

The estimation of the major arc terms is more challenging, because an analogue of (<ref>) does not hold for A_Λ^𝔐. Still, it is possible to establish a slightly weaker version of Proposition <ref>. The following result was first established by Magyar <cit.>, for d ≥ 2^k, and then extended by the third author <cit.> in the present form.


If k ≥ 3, d > 2k, d/d-k < p ≤ 2, and Λ≳ 1, one has 

    A_Λ^𝔐_ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲ 1.



This proposition suffices to establish the ℓ^p-boundedness of the dyadic maximal functions A_Λ (this is the main result of Magyar <cit.>), but falls just short of what is needed for an equally quick proof of Theorem <ref>. Instead, we use Theorem <ref> to approximate A_*^𝔐 by a bounded operator. Let M_λ^a/q denote the convolution operator on ℓ^p(ℤ^d) with Fourier multiplier

    M_λ^a/q(ξ) :=  e_q( -λ a) ∑_ w ∈{± 1 }^d∑_𝐛∈ℤ^d G(q; a,  w𝐛) ψ(qξ-𝐛) dσ_λ( w(ξ-q^-1𝐛)),

and define

    M_λ := ∑_q ∈ℕ∑_a ∈ℤ_q^* M_λ^a/q   and    M_* := sup_λ∈ℕ |M_λ|.
 
In Section <ref>, we will handle the major arc approximations and prove the following proposition. 


If k ≥ 3, d > 2k, and d/d-k < p ≤ 2, then there exists an exponent β_p = β_p(d,k) > 0 such that 

    sup_λ∈ [Λ/2, Λ)| A_λ^𝔐-M_λ| _ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲Λ^-β_p.




Theorem <ref> is an immediate consequence of Propositions <ref> and <ref>. Moreover, since when p = 2 and d ≥5/2k, inequality (<ref>) holds for any β_2 < 1/(2k) (see inequality (<ref>) and the comments after it), the error bound (<ref>) holds for all δ with 0 < δ < min{δ_0(d,k), 1/(2k) }. 
  

When we sum (<ref>) over dyadic Λ = 2^j, we deduce that

    sup_λ≳ 1| A_λ^𝔐-M_λ| _ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲ 1.

Combining this bound and (<ref>), we conclude that the ℓ^p-boundedness of the maximal operator A_* follows from the ℓ^p-boundedness of M_*. The following proposition, which we establish in Section <ref>, then completes the proof of Theorem <ref>.


If k ≥ 3, d> 2k, and d/d-k < p ≤ 2, then M_* is bounded on ℓ^p(ℤ^d).  





§ MINOR ARC ANALYSIS



Our minor arc analysis splits naturally in two steps. The first step is a reduction to mean value estimates related to Waring's problem; for this we use a technique introduced in <cit.>. We then apply recent work on Waring's problem to estimate the relevant mean values and to prove Proposition <ref>.



 §.§ Reduction to mean value theorems



The reduction step is based on the following lemma, a special case of Lemma 7 in <cit.>. In the present form, the result is a slight variation of Lemma 4.2 in <cit.> and is implicit also in <cit.>. 


For λ∈ℕ, let T_λ be a convolution operator on ℓ^2(ℤ^d) with Fourier multiplier given by

    T_λ(ξ) := ∫_𝔅 K(θ; ξ) e(-λθ)   dθ,

where 𝔅⊆𝕋 is a measurable set and K( ·; ξ) ∈ L^1(𝕋) is a kernel independent of λ. Further, for Λ≥ 2, define the dyadic maximal functions  

    T_*f(𝐱) = T_*(𝐱; Λ) := sup_λ∈ [Λ/2,Λ) |T_λ f(𝐱)|.

Then 

    T_* _ℓ^2(ℤ^d) →ℓ^2(ℤ^d)≤∫_𝔅sup_ξ∈𝕋^d  | K(θ; ξ) |   dθ.



For a measurable set 𝔅⊂𝕋, we have

    A_λ^𝔅(ξ) 
    = λ^1-d/k∫_𝔅ℱ_N(θ; ξ) e(-λθ)   dθ,

where

    ℱ_N(θ; ξ) := ∏_j=1^d S_N(θ, ξ_j) 
      with   S_N(θ, ξ) := ∑_|n| ≤ N e(θ |n|^k + ξ n).

Thus, in the proof of Proposition <ref>, we apply (<ref>) with K = ℱ_N and 𝔅 = 𝔪. The supremum over ξ on the right side of (<ref>) then stands in the way of a direct application of known results from analytic number theory. Our next lemma overcomes this obstacle; its proof is a variant of the argument leading to (12) in Wooley <cit.>.


If k ≥ 2, l ≤ k-1 and s are natural numbers and 𝔅⊆𝕋 a measurable set, then

    ∫_𝔅sup_ξ∈𝕋 |S_N(θ, ξ) |^2s  dθ≲ N^l(l+1)/2∫_𝔅∫_𝕋^l| ∑_n=1^N e(θ n^k + ξ_l n^l + … + ξ_1 n) |^2s dξ dθ + 1.




Define

    S_N(θ, ξ) = ∑_1 ≤ n ≤ N e(θ n^k + ξ n).

Since

    sup_ξ∈𝕋 |S_N(θ, ξ)| ≤ 2sup_ξ∈𝕋 |S_N(θ, ξ)| + 1,

it suffices to establish (<ref>) with S_N in place of S_N. 

Set H_j = sN^j. For 𝐡 = (h_1, …, h_l) ∈ℤ^l, we define 

    a_𝐡(θ) = ∑_1 ≤ n_1, …, n_s ≤ Nδ(𝐧; 𝐡) e(θ𝔣_s,k(𝐧)),

where 

    δ(𝐧; 𝐡) =  1    if 𝔣_s,j(𝐧) = h_j  for all  j = 1, …, l, 
     0    otherwise.

We have

    S_N(θ, ξ)^s = ∑_h_1 ≤ H_1⋯∑_h_l ≤ H_l a_𝐡(θ)e(ξ h_1),

so by applying the Cauchy–Schwarz inequality we deduce that

    sup_ξ |S_N(θ, ξ)|^2s≤ H_1 ⋯ H_l ∑_h_1 ≤ H_1⋯∑_h_l ≤ H_l |a_𝐡(θ)|^2.

Hence, 

    ∫_𝔅sup_ξ |S_N(θ, ξ) |^2s  dθ≲ N^l(l+1)/2∫_𝔅∑_h_1 ≤ H_1⋯∑_h_l ≤ H_l a_𝐡(θ)a_𝐡(θ)  dθ.
 

We have 

    ∑_h_1 ≤ H_1⋯∑_h_l ≤ H_l a_𝐡(θ)a_𝐡(θ) = ∑_1 ≤𝐧, 𝐦≤ N e(θ( 𝔣_s,k(𝐧) - 𝔣_s,k(𝐦))) ∑_h_1 ≤ H_1⋯∑_h_l ≤ H_lδ(𝐧; 𝐡)δ(𝐦; 𝐡).

By orthogonality,   

    ∑_h_1 ≤ H_1⋯∑_h_l ≤ H_lδ(𝐧; 𝐡)δ(𝐦; 𝐡)    = ∑_h_1 ≤ H_1⋯∑_h_l ≤ H_lδ(𝐦; 𝐡) ∫_𝕋^l e( ∑_j=1^l ξ_j( 𝔣_s,j(𝐧) - h_j ) )   dξ
       = ∫_𝕋^l e( ∑_j=1^l ξ_j( 𝔣_s,j(𝐧) - 𝔣_s,j(𝐦)) )   dξ,

since for a fixed 𝐦, the sum over 𝐡 has exactly one term (in which h_j = 𝔣_s,j(𝐦)). Hence,  

    ∑_h_1 ≤ H_1⋯∑_h_l ≤ H_l  a_𝐡(θ)a_𝐡(θ) = ∫_𝕋^l| ∑_n=1^N e( θ n^k + ξ_ln^l + … + ξ_1n) |^2s dξ.
 
The lemma follows from (<ref>)–(<ref>).



With slight modifications, the argument of Lemma <ref> yields also the following estimate

    ∫_𝔅∫_𝕋 |S_N(θ, ξ) |^2s  dξ dθ≲ N^l(l+1)/2-1∫_𝔅∫_𝕋^l| ∑_n=1^N e(θ n^k + ξ_l n^l + … + ξ_1 n) |^2s dξ dθ + 1.





 §.§ Mean value theorems



We now recall several mean value estimates from the literature on Waring's problem. The first is implicit in the proof of Theorem 10 of Bourgain <cit.>, which is a variant of a well-known lemma of Hua (see Lemma 2.5 in <cit.>). The present result follows from eqn. (6.6) in <cit.>. We note that when l = k, the left side of (<ref>) turns into Vinogradov's integral J_s,k(N) and the lemma turns into the main result of Bourgain, Demeter and Guth <cit.>.


If k ≥ 3, 2 ≤ l ≤ k and s ≥1/2l(l+1) are natural numbers, then 

    ∫_𝕋∫_𝕋^l-1| ∑_n=1^N e(θ n^k + ξ_l-1 n^l-1 + … + ξ_1 n) |^2s dξ dθ≲ 
    N^2s - l(l+1)/2 + .



For small k, we will use another variant of Hua's lemma due to Brüdern and Robert <cit.>. The following is a weak form of Lemma 5 in <cit.>. 


If k ≥ 3 and 2 ≤ l ≤ k are natural numbers, then

    ∫_𝕋^2| ∑_n=1^N e(θ n^k + ξ n) |^2^l+2 dξ dθ≲ N^2^l-l+1+.

 

Note that by Remark <ref> (with l-1 in place of l) and Lemma <ref> we obtain a version of (<ref>) with l(l+1) in place of 2^l + 2. Together with Lemma <ref>, this observation yields the following bound.


If k ≥ 3 and 2 ≤ l ≤ k are natural numbers and r ≥min{ l(l+1),2^l+2 }, then

    ∫_𝕋^2| ∑_n=1^N e(θ n^k + ξ n) |^r dξ dθ≲ N^r-l-1+.



We also use a variant of Lemma <ref> that provides extra savings when the integration over θ is restricted to a set of minor arcs. Lemma <ref>, a slight modification of Theorem 1.3 in Wooley <cit.>, improves on (<ref>) in the case l = k-1. 
Here, 𝔪 is the set of minor arcs defined at the beginning of <ref>.


If k ≥ 3 and s ≥1/2k(k+1) are natural numbers, then

    ∫_𝔪∫_𝕋^k-2| ∑_n=1^N e(θ n^k + ξ_k-2 n^k-2 + … + ξ_1 n) |^2s dξ dθ≲ N^2s - k(k - 1)/2-2+.




The main point in the proof of Theorem 2.1 in Wooley <cit.> is the inequality (see p. 1495 in <cit.>)

    ∫_𝔪∫_𝕋^k-2| ∑_n=1^N e(θ n^k + ξ_k-2 n^k-2 + … + ξ_1 n) |^2s dξ dθ≲ N^k-2(log N)^2s+1 J_s,k(2N).

The lemma follows from this inequality and the Bourgain–Demeter–Guth bound for Vinogradov's integral J_s,k(2N) (the case l = k of (<ref>)). 


Now we interpolate between the above bounds. 


If k ≥ 3 and 2 ≤ l ≤ k-1 are natural numbers and r is real, with 

    r_1(k,l) := k^2-k l - min(l^2+l, 2^l+2)/k-l+1≤ r ≤ k^2+k,

then  

    I_r,k(N) := ∫_𝔪sup_ξ∈𝕋| S_N(θ, ξ) |^r dθ≲ N^r-k-δ(r) +,

where δ(r) is the linear function of r with values δ(r_1) = 0 and δ(k^2+k) = 1.



Let r_0 = min{ l^2+l, 2^l+2 }. The hypothesis on r implies that r_0 < r ≤ k^2+k, so we can find t ∈ [0,1) such that r = t r_0 + (1-t) (k^2 + k). By Lemma <ref> with l = 1 and Corollary <ref>, 

    I_r_0,k(N) ≲ N ∫_𝕋^2| ∑_n=1^N e(θ n^k + ξ n) |^r_0 dξ dθ≲ N^r_0 - l + .

On the other hand, Lemma <ref> with l = k-2 and Lemma <ref> yield

    I_k(k+1),k(N) ≲ N^(k-1)(k-2)/2 N^k(k+1)-k(k-1)/2-2+ = N^k^2-1+.

Using Hölder's inequality and the above bounds, we get

    I_r,k(N)    ≲ I_k(k+1),k(N)^1-t I_r_0,k(N)^t ≲ N^(1-t)(k^2-1+) N^t(r_0-l+)
       ≲ N^(1-t)(k^2+k) + t r_0 N^-(1-t)(k+1)-tl +  = N^r-k-1+t(k-l+1) + .

This inequality takes the form (<ref>) with δ(r) = 1 - t(k-l+1). Since t depends linearly on r and t = 0 when r = k^2+k, δ(r) is a linear function of r with δ(k^2+k) = 1. The value of r_1(k,l) in (<ref>) is the unique solution of the linear equation δ(r) = 0.




 §.§ Proof of Proposition <ref>


By Lemma <ref> and the arithmetic-geometric mean inequality,  

    sup_λ∈ [Λ/2,Λ) |A_λ^𝔪| _ℓ^2(ℤ^d) →ℓ^2(ℤ^d)≤ N^k-d∫_𝔪sup_ξ∈𝕋^d| ℱ_N(θ; ξ) |   dθ≤ N^k - dI_d,k(N),

where I_d,k(N) is the integral defined in (<ref>). Thus, the proposition will follow, if we prove the inequality

    I_d,k(N) ≲ N^d - k - δ +

with δ = kδ_0(d,k).

Let l_0(k) denote the value of l for which the maximum in the definition of d_0(k) is attained (recall (<ref>)). When d_0(k) < d ≤ k^2+k, we may apply (<ref>) with r=d and l = l_0 to deduce (<ref>) with δ = δ(d). We now observe that when d ≤ k^2+k, we have δ(d) = kδ_0(d,k) and that the hypothesis d > d_0(k) ensures that δ(d) > 0. 

When d > k^2+k, we enhance our estimates with the help of the L^∞-bound for S_N(θ, ξ) on the minor arcs: by combining a classical result of Weyl (see Lemma 2.4 in Vaughan <cit.>) and Theorem 5 in Bourgain <cit.>, we have

    sup_(θ, ξ) ∈𝔪×𝕋| S_N (θ, ξ) | ≲
    N^1-τ_k+,

τ_k being the quantity that appears in the definition of δ_0(d,k). Thus, when d > k^2+k, we have

    I_d,k(N) ≲ N^(d-k^2-k)(1-τ_k) + I_k(k+1),k(N) ≲ N^d-k-1 - (d-k^2-k) τ_k + .

We conclude that (<ref>) holds with δ = 1 + (d - k^2-k)τ_k. 




§ MAJOR ARC ANALYSIS



We will proceed through a series of successive approximations to A_λ^a/q, which we will define by their Fourier multipliers. Our approximations are based on the following major arc approximation for exponential sums that appears as part of Theorem 3 of Brüdern and Robert <cit.>. In this result and throughout the section, we write

    G(q; a, b) := q^-1∑_x ∈ℤ_q e_q( ax^k + bx ) 
       and    
    v_N(θ, ξ) := ∫_0^N e( θ t^k + ξ t)   dt.



Let θ, ξ∈𝕋, q ∈ℕ, a ∈ℤ_q^*, and b ∈ℤ, and suppose that

    | θ - a/q| ≤1/4kqN^k-1,   | ξ - b/q| ≤1/2q.

Then

    ∑_n ≤ N e(θ n^k + ξ n) = G(q; a,b)v_N(θ - a/q, ξ - b/q) + O( q^1-1/k+).



Recall the definition of S_N(θ,ξ) from Section <ref>. When a/q + θ lies on a major arc 𝔐(a/q) and ξ = b/q + η, with |η| ≤ 1/(2q), the above lemma yields

    S_N(a/q+θ, ξ) = G(q; a,b)v_N(θ, η) + G(q; a,-b)v_N(θ, -η) + O( q^1-1/k+).

We will use this approximation in conjunction with the following well-known bounds (see Theorems 7.1 and 7.3 in Vaughan <cit.>):

    G(q; a, b) ≲ q^-1/k+,    v_N(θ, η) ≲ N(1 + N|η| + N^k|θ|)^-1/k.

We will make use also of the inequality

    v_N(θ, η) ≲ N(1 + N|η|)^-1/2,

which follows from the representation

    v_N(θ, η) = 1/k∫_0^N^k u^1/k-1 e(θ u + η u^1/k)   du

and the second-derivative bound for oscillatory integrals (see p. 334 in <cit.>).



 §.§ Proof of Proposition <ref>


The bulk of the work concerns the case p=2 of the proposition. 
Since

    A_λ^a/q(ξ) =
    λ^1-d/k e_q(-λ a) ∫_𝔐(0/q)ℱ_N(a/q + θ; ξ) e(-λθ)   dθ,

the asymptotic (<ref>) suggests that the following multiplier should provide a good approximation to A_λ^a/q(ξ):

    B_λ^a/q(ξ) := λ^1-d/k e_q(-λ a) ∫_𝔐(0/q)𝒢_N(θ; ξ)e(-λθ)   dθ,

where 

    𝒢_N(θ; ξ) := ∏_j=1^d { G(q; a, b_j)v_N(θ, η_j) + G(q; a, -b_j)v_N(θ, -η_j)},

with b_j the unique integer such that -1/2≤ b_j - qξ_j < 1/2 and η_j = ξ_j - b_j/q. Let B_λ^a/q denote the operator on ℓ^2(ℤ^d) with the above Fourier multiplier. To estimate the ℓ^2-error of approximation of A_λ^a/q by B_λ^a/q, we use that when θ∈𝔐(0/q), (<ref>) and (<ref>) yield

    | ℱ_N(a/q+θ; ξ) - 𝒢_N(θ; ξ) | ≲ q^1-d/k+N^d-1(1 + N^k|θ|)^(1-d)/k.

Thus, if d > k+1, we have

    ∫_𝔐(0/q)sup_ξ∈𝕋^d | ℱ_N(a/q+θ; ξ) - 𝒢_N(θ; ξ) |   dθ   ≲∫_ℝq^1-d/k+N^d-1  dθ/(1 + N^k|θ|)^(d-1)/k
       ≲ q^1-d/k+N^d-k-1.

Lemma <ref> then yields

    sup_λ∈ [Λ/2,Λ)| A_λ^a/q - B_λ^a/q| _ℓ^2(ℤ^d) →ℓ^2(ℤ^d)≲ q^1-d/k+N^-1.


Next, we approximate B_λ^a/q by the operator C_λ^a/q with multiplier

    C_λ^a/q(ξ) := λ^1-d/k e_q(-λ a) ∑_𝐛∈ℤ^dψ(qξ - 𝐛)  ∫_𝔐(0/q)𝒢_N(θ; ξ)e(-λθ)   dθ.

By the localization of ψ, the above sum has at most one term in which 𝐛 matches the integer vector that appears in the definition of 𝒢_N(θ; ξ). Hence, 

    𝒢_N(θ; ξ)( 1 - ∑_𝐛∈ℤ^dψ(qξ - 𝐛) )

is supported on a set where 1/8≤ |qξ_j - b_j| ≤1/2 for some j. For such j, by (<ref>),

    v_N(θ, ξ_j - b_j/q) ≲ (qN)^1/2,

and we conclude that

    |𝒢_N(θ; ξ)|( 1 - ∑_𝐛∈ℤ^dψ(qξ - 𝐛) ) ≲ q^1/2-d/k+N^d-1/2(1 + N^k|θ|)^(1-d)/k.

So, when d > k+1, Lemma <ref> gives

    sup_λ∈ [Λ/2,Λ)| B_λ^a/q - C_λ^a/q| _ℓ^2(ℤ^d) →ℓ^2(ℤ^d)   ≲∫_ℝq^1/2-d/k+N^k-1/2  dθ/(1 + N^k|θ|)^(d-1)/k
       ≲ q^1/2-d/k+N^-1/2.


In our final approximation, we replace C_λ^a/q by the operator D_λ^a/q with multiplier

    D_λ^a/q(ξ) := λ^1-d/k e_q(-λ a) ∑_ w ∈{± 1 }^d∑_𝐛∈ℤ^dψ(qξ - 𝐛) G(q; a,  w𝐛) J_λ( w(ξ - q^-1𝐛)),

where w𝐛 = (w_1b_1, …, w_db_d) and

    J_λ(η) := ∫_ℝ{∏_j=1^d v_N(θ; η_j) } e(-λθ)   dθ.

We remark that C_λ^a/q(ξ) can be expressed in a matching form, with J_λ(ξ - q^-1 w𝐛) replaced by the analogous integral over 𝔐(0/q). 
Thus, when d > k, we deduce from Lemma <ref> and (<ref>) that

    sup_λ∈ [Λ/2,Λ)| C_λ^a/q - D_λ^a/q| _ℓ^2(ℤ^d) →ℓ^2(ℤ^d)   ≲∫_𝔐(0/q)^cq^-d/k+N^k  dθ/(1 + N^k|θ|)^d/k
       ≲ q^-1+N^1-d/k.

Here, 𝔐(0/q)^c denotes the complement of the interval 𝔐(0/q) in ℝ. 

Finally, we note that D_λ^a/q is really M_λ^a/q.  Indeed, by the discussion on p. 498 in Stein <cit.> (see also Lemma 5 in  Magyar <cit.>), we have 

    J_λ(η)    = ∫_ℝ∫_ℝ^d1_[0,N]^d(𝐭)e(η·𝐭) e(θ(𝔣(𝐭) - λ))   d𝐭 dθ
       = λ^d/k-1∫_ℝ^d1_[0,N]^d(𝐭)e(η·𝐭)   dσ_λ(𝐭) = λ^d/k-1dσ_λ(η),

since the surface measure dσ_λ is supported in the cube [0,N]^d. Combining this observation and (<ref>)–(<ref>), and summing over a,q, we conclude that when d > 2k, 

    ∑_q ≤ N∑_a ∈ℤ_q^*sup_λ∈ [Λ/2,Λ)| A_λ^a/q - M_λ^a/q| _ℓ^2(ℤ^d) →ℓ^2(ℤ^d)≲ N^-γ+,
 
with γ = min(d/k-2, 1/2) > 0. In particular, when d ≥5/2 k, we have γ = 1/2. 

We can now finish the proof of Proposition <ref>. For brevity, we write p_1 = d/d-k. From (<ref>), we obtain that

    sup_λ∈ [Λ/2,Λ)| A_λ^𝔐 - M_λ| _ℓ^2(ℤ^d) →ℓ^2(ℤ^d)≲ N^-γ+.

On the other hand, we know from Propositions <ref> and <ref> that both A_Λ^𝔐 and M_* are bounded on ℓ^p(ℤ^d) when p_1 < p ≤ 2; hence,

    sup_λ∈ [Λ/2, Λ)| A_λ^𝔐-M_λ| _ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲ 1.

When p ∈ (p_0(d,k), 2], we interpolate between (<ref>) and (<ref>) with p = p_1 + for a sufficiently small > 0. The interpolation yields (<ref>) with β_p > 0 that can be chosen arbitrarily close to 2γ(p-p_1)/ (kp(2-p_1)). 




§ MAIN TERM CONTRIBUTION



In this section, we prove Proposition <ref>. First, we obtain L^p(ℝ^d)-bounds for the maximal function of the continuous surface measures dσ_λ.


If k ≥ 2, d>3/2k and p > 2d-k/2d - 2k, then for all f ∈ L^p(ℝ^d), 

    sup_λ > 0 |f * dσ_λ| _L^p(ℝ^d)≲f_L^p(ℝ^d).




We will deduce the lemma from a result of Rubio de Francia – Theorem A in <cit.> – which reduces (<ref>) to bounds for the Fourier transform of the measure dσ_λ. First, we majorize the measure dσ_λ by a smooth one. By the choice of normalization of dσ_λ, we have

    f * dσ_λ(𝐱) = ∫_ℝ^d f(𝐱 - 𝐲)   dσ_λ(𝐲) = ∫_ℝ^d f(𝐱 - t𝐲)   dσ_1(𝐲),

where t = λ^1/k. Let ϕ be a smooth function supported in [-1/2, 3/2] and such that 1_[0,1](x) ≤ϕ(x), and write ϕ(𝐱) = ϕ(x_1) ⋯ϕ(x_d). Since dσ_1 is supported inside the unit cube [0,1]^d, we have

    |∫_ℝ^d f(𝐱 - t𝐲)   dσ_1(𝐲) | ≤∫_ℝ^d |f(𝐱 - t𝐲)| ϕ(𝐲)   dσ(𝐲) =: ∫_ℝ^d |f(𝐱 - t𝐲)|   dμ(𝐲),

where dσ is the surface measure on the smooth manifold x_1^k + … + x_d^k = 1. By Rubio de Francia's theorem, the maximal function

    𝒜_tf(𝐱) := sup_t > 0∫_ℝ^d |f(𝐱 - t𝐲)|   dμ(𝐲)

is bounded on L^p(ℝ^d), provided that

    μ(ξ) ≲ (|ξ| + 1)^-a  for some  a > min{1/2p-2,1/2}.
 
Thus, the lemma will follow, if we establish (<ref>) with a = d/k - 1 and d > k+1 because a>1/2 once d>3/2k.

We now turn to (<ref>). Similarly to (<ref>), we have

    μ(ξ) = ∫_ℝ∫_ℝ^dϕ(𝐱)e(ξ·𝐱) e(θ(𝔣(𝐱)-1))   d𝐱 dθ = ∫_ℝ{∏_j=1^d v_ϕ(θ, ξ_j) }e(-θ)   dθ,

where

    v_ϕ(θ, ξ) := ∫_ℝϕ(x)e(θ x^k + ξ x)   dx.

By the corollary on p. 334 of Stein <cit.>, we have

    v_ϕ(θ, ξ) ≲ (1 + |θ|)^-1/k,
 
uniformly in ξ. On the other hand, if k2^k|θ| ≤ |ξ|, we have

    | d/dx( ξ^-1θ x^k + x ) | ≥1/2,   | d^j/dx^j( ξ^-1θ x^k + x ) | ≲_j 1

on the support of ϕ. Hence, Proposition VIII.1 on p. 331 of Stein <cit.> yields

    v_ϕ(θ, ξ) ≲_M (1 + |ξ|)^-M
 
for any fixed M ≥ 1. 

We now choose an index j, 1 ≤ i ≤ d, such that |ξ| ≤ d|ξ_i| and set θ_0 = |ξ_i|/(k2^k). We apply (<ref>) to the trigonometric integrals v_ϕ(θ,ξ_j), j  i, and to v_ϕ(θ,ξ_i) when |θ| > θ_0; we apply (<ref>) to v_ϕ(θ,ξ_i) when |θ| ≤θ_0. From these bounds and the integral representation for μ(ξ), we obtain

    μ(ξ)    ≲∫_|θ| ≤θ_0(1 + |ξ_i|)^-M/(1 + |θ|)^(d-1)/k  dθ + ∫_|θ| > θ_0dθ/(1 + |θ|)^d/k
       ≲ (1 + |ξ_i|)^-M + (1 + |ξ_i|)^1-d/k≲ (1+|ξ|)^1-d/k,

provided that M ≥ d/k - 1 and d > k+1. 
This completes the proof.




 §.§ Proof of Proposition <ref>


To prove Proposition <ref>, it suffices to prove that (uniformly in a and q)

    sup_λ∈ℕ| M_λ^a/q| _ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲ 
    q^-d/k(2-2/p)+,

for all d/d-k < p ≤ 2 and d>k. The proposition then follows by summing over a and q (the hypothesis on p ensures that the resulting series over q is convergent). 

Fix q ∈ℕ and a ∈ℤ_q^* and write ψ_1(ξ) = ψ(ξ/2) (so that ψ = ψψ_1). We borrow a trick from Magyar, Stein and Wainger <cit.> to express M_λ^a/q (ξ) as a linear combination of Fourier multipliers that separate the dependence on λ and from the dependence on a/q: 

    M_λ^a/q(ξ)    = e_q(-λ a) ∑_ w ∈{± 1 }^d∑_𝐛∈ℤ^dψ(qξ - 𝐛) G(q; a,  w𝐛) dσ_λ( w(ξ - q^-1𝐛)) 
       = e_q(-λ a) ∑_ w ∈{± 1 }^d( ∑_𝐛∈ℤ^dψ_1(qξ - 𝐛) G(q; a,  w𝐛) ) ( ∑_𝐛∈ℤ^dψ(qξ - 𝐛) dσ_λ( w(ξ - q^-1𝐛)) ) 
        =: e_q(-λ a) ∑_ w ∈{± 1 }^dS_ w^a/q(ξ) T_λ, w^q(ξ).

Since w takes on precisely 2^d values for each a/q, it suffices to prove that 

    sup_λ∈ℕ|T_λ, w^q∘ S_ w^a/q| _ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲ q^-d/k(2-2/p)+,

uniformly for w ∈{-1,1}^d. To prove (<ref>), we will first bound the maximal function over the `Archimedean' multipliers T_λ, w^q, and then we will bound the non-Archimedean multipliers S_ w^a/q. 
This is possible because S_ w^a/q is independent of λ∈ℕ. 

For d > 3/2k and p>2d-k/2d-2k, Lemma <ref> and Corollary 2.1 of <cit.> (the `Magyar–Stein–Wainger transference principle') give the bound

    sup_λ∈ℕ| T_λ, w^qg | _ℓ^p(ℤ^d)≲g_ℓ^p(ℤ^d),

with an implicit constant independent of q and w. We now observe that S_ w^a/q does not depend on λ and apply (<ref>) with g = S_ w^a/qf to find that 

    sup_λ∈ℕ|T_λ, w^q∘ S_ w^a/q| _ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲ S_ w^a/q_ℓ^p(ℤ^d) →ℓ^p(ℤ^d),

under the same assumptions on d and p which we note are weaker than the hypotheses of the proposition. Finally, observe that G(q; a, b) is a q-periodic function with ℤ_q-Fourier transform equal to 

    ∑_b ∈ℤ_q e_q(-mb) G(q; a, b) = 
    q^-1∑_x ∈ℤ_q e_q( ax^k ) ∑_b ∈ℤ_q e_q( b(x-m) ) = e_q( am^k ),

for each m ∈ℤ_q. Hence, we may apply Proposition 2.2 in <cit.> and the bound (<ref>) for G(q; a, b) to deduce that 

    S_ w^a/q_ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲ q^-d/k(2-2/p)+.

The desired inequality (<ref>) follows immediately from (<ref>) and (<ref>). 



§ ACKNOWLEDGMENTS

The first author was supported by NSF grant DMS-1502464. The second author was supported by NSF grant DMS-1147523 and by the Fields Institute. Last but not least, the authors thank Alex Nagel and Trevor Wooley for several helpful discussions.

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[TA]
Theresa C. Anderson

Department of Mathematics
 
University of Wisconsin–Madison
 
Madison, WI 53705, U.S.A.

tcandersonmathwiscedu  

[BC]
Brian Cook

Department of Mathematicl Sciences
 
Kent State University
 
Kent, OH 44242

U.S.A.

bcook25kentedu 

[KH]
Kevin Hughes

School of Mathematics
	
The University of Bristol
	
Bristol, BS8 1TW, UK

Kevin.Hughesbristolacuk 

[AK]
Angel Kumchev

Department of Mathematics
	
Towson University
	
Towson, MD 21252, U.S.A.

akumchevtowsonedu



