Axis-parallel cubes, their averages, and cube maximal functions

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the principle already assumed by the published ball-based maximal functions.

For $x\in {\mathbb{R}}^n$ and $r>0$ let $Q(x,r):={\prod }_{i<n}(x_i-r,x_i+r)$ be the open axis-parallel cube with centre $x$ and side length $2r$, so that $Q(x,r)=\{y\in {\mathbb{R}}^n:\mathrm{\mid }{y}_i-x_i\mathrm{\mid }<r\text{ for every }i<n\}$. Its half-open counterpart is a box in the sense of Half-open boxes in ${\mathbb{R}}^n$ and their volume with the same real endpoints $a_i=x_i-r$ and $b_i=x_i+r$, and the box-measure theorem (A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included) gives $\mathrm{\mid }Q(x,r)\mathrm{\mid }=(2r{)}^n$; the face convention is immaterial because all boxes with the same endpoints have the same Lebesgue measure. For $\lambda >0$, $\lambda Q$ denotes the cube concentric with $Q$ whose side length is $\lambda$ times that of $Q$; thus $\lambda Q(x,r)=Q(x,\lambda r)$ and $\mathrm{\mid }\lambda Q(x,r)\mathrm{\mid }=(2\lambda r{)}^n={\lambda }^n\mathrm{\mid }Q(x,r)\mathrm{\mid }$ by the same box-measure computation.

For $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^n)$ (A locally integrable function on ${\mathbb{R}}^n$) and an axis-parallel cube $Q$ the average of $f$ over $Q$ is the finite number $⟨f{⟩}_Q:=\mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_{Q}fd\lambda$, and $⟨\mathrm{\mid }f\mathrm{\mid }{⟩}_Q$ is the average of the nonnegative function $\mathrm{\mid }f\mathrm{\mid }$. For any nonnegative measurable $g$, the same notation $⟨g{⟩}_Q:=\mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_{Q}gd\lambda$ denotes an extended average in $[0,\mathrm{\infty }]$, with value $+\mathrm{\infty }$ when the integral diverges. The centred and uncentred cube maximal functions of $f$ are $M_{c}f(x):={\sup }_{r>0}⟨\mathrm{\mid }f\mathrm{\mid }{⟩}_{Q(x,r)},M_c^{\ast }f(x):={\sup }_{Q\ni x}⟨\mathrm{\mid }f\mathrm{\mid }{⟩}_Q,$ the second supremum taken over all axis-parallel cubes $Q=Q(y,r)$ with $y\in {\mathbb{R}}^n$, $r>0$ that contain $x$. Both functions take values in $[0,\mathrm{\infty }]$.

These are the cube analogues of the published centred and uncentred ball-based Hardy-Littlewood maximal functions $Mf$ and $M^{\ast }f$ (The centered and uncentered Hardy-Littlewood maximal functions), whose averages are formed with the ball average operator (The average of a locally integrable function over a Euclidean ball). Every Euclidean ball between the inscribed and circumscribed cube of a fixed cube has comparable volume, so the ball and cube maximal functions are pointwise comparable by a constant depending only on $n$; that comparison is proved on this page. The two functions $M_{c}f$ and $M_c^{\ast }f$ are themselves pointwise comparable by $2^n$, since the centred cube $Q(x,r)$ is among the cubes containing $x$, and every cube $Q(y,r)\ni x$ lies in $Q(x,2r)$, whose volume is $2^n\mathrm{\mid }Q(y,r)\mathrm{\mid }$.