Improved $\ell^p$-Boundedness for Integral $k$-Spherical Maximal Functions

Improved $\ell^p$-Boundedness for Integral $k$-Spherical Maximal Functions, Discrete Analysis 2018:10, 18pp.

An important role in harmonic analysis is played by the notion of a maximal function (which is actually a non-linear operator on a space of functions). The best-known example is the Hardy-Littlewood maximal function, which takes a function $f:\mathbb R^d\to\mathbb C$ and replaces it by the function $Mf:\mathbb R^d\to\mathbb R$, which is defined by the formula
$$Mf(x)=\sup_{r>0}\frac 1{|B_r(x)|}\int_{B_r(x)}|f(x)|dx,$$
where $B_r(x)$ is the ball of radius $r$ about $x$. In other words, $Mf$ is the largest average of $|f|$ over any ball centred at $x$. Particularly useful are inequalities bounding norms of $Mf$ in terms of norms of $f$: for example, it is known that if $1<p\leq\infty$, then there is a constant $A_{p,d}$ such that $\|Mf\|_p\leq A_{p,d}\|f\|_p$ for every function $f\in L_p(\mathbb R^d)$.

This paper concerns discrete maximal functions, where $f$ is now defined on $\mathbb Z^d$. They are also spherical maximal functions, meaning that the averages are over spheres rather than balls. And finally, the spheres are not (necessarily) Euclidean spheres but spheres in $\ell_k^d$ for more general $k$ than just 2. To be precise, if $k$ is a positive integer, $\lambda$ is a positive real, and $x$ is a point in $\mathbb Z^d$, define $S(x,k,\lambda)$ to be the set $\{y\in\mathbb Z^d:\sum_{i=1}^d|x_i-y_i|^k=\lambda\}$. The authors look at the corresponding maximal function, which takes $f$ to the function $Mf$ defined by
$$Mf(x)=\mathop{\sup}_{S(x,k,\lambda)\ne\emptyset}\frac 1{|S(x,k,\lambda)|}\sum_{y\in S(x,k,\lambda)}|f(y)|.$$
Their object is to understand, given $d$ and $k$, for which $p$ this maximal function is bounded: that is, for which $p$ there exists a constant $A_{p,k,d}$ such that $\|Mf\|_p\leq A_{p,d,k}\|f\|_p$, where $\|.\|_p$ is the $\ell_p$ norm for functions defined on $\mathbb Z^d$. It turns out that the larger the value of $p$, the easier it is for a maximal function to be bounded (an indication that this is to be expected is that it is trivially bounded when $p=\infty$), so the aim is to prove boundedness with $p$ as small as possible. The authors manage to prove boundedness when $p$ is at least a certain value $p_0(d,k)$, which is given by a slightly complicated expression that can be found on the second page of their paper. However, the key point is that $p_0(d,k)<2$ when $d$ is sufficiently large in terms of $k$, and the “sufficiently large” they require is only quadratic, whereas the previous best known results required $d$ to be at least cubic in $k$.

Not surprisingly, there are connections between this question and Waring’s problem, since both involve writing an integer as a sum of a certain number of $k$th powers. In particular, if $d$ is sufficiently large compared with $k$, good estimates are available for the sizes of the spheres $S(x,k,\lambda)$, which allow one to replace the maximal function given above by a simpler one that is equivalent (in the sense that it will be bounded for the same set of $p$).

However, the connections are deeper than this. In particular, the authors make use of the Hardy-Littlewood circle method, and in order to obtain an improvement on earlier results (which also used the circle method), they find an interesting way of exploiting recent breakthroughs by Bourgain, Demeter and Guth, and by Wooley, in the theory of Weyl sums.

As an application of their results, they obtain improved bounds for a closely related problem, the ergodic Waring–Goldbach problem, where one considers just points in $\mathbb Z^d$ for which all the coordinates are prime.