Grand maximal test class of order N and the grand maximal function

Fix an integer $n\ge 1$ and let $\mathcal{S}({\mathbb{R}}^n)$ carry the seminorms and topology of Schwartz space and its seminorms and Schwartz topology and convergence, with multi-indices as in $C^k$ maps and multi-index derivative notation in Euclidean space. For each integer $N\ge 1$ and $\phi \in \mathcal{S}({\mathbb{R}}^n)$ define the Schwartz test seminorm of order $N$ $P_N(\phi )={\sup }_{x\in {\mathbb{R}}^n}(1+\mathrm{\mid }x\mathrm{\mid }{)}^N{\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le N+1}\mathrm{\mid }{\mathrm{\partial }}^{\alpha }\phi (x)\mathrm{\mid }$ and the grand maximal test class of order $N$ ${\mathcal{F}}_N=\{\phi \in \mathcal{S}({\mathbb{R}}^n):P_N(\phi )\le 1\}\mathrm{.}$

For $f\in {\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^n)$ the grand maximal function of order $N$ is $M_{N}f(x)={\sup }_{\phi \in {\mathcal{F}}_N}{\sup }_{t>0}{\sup }_{\mathrm{\mid }y-x\mathrm{\mid }\le t}\mathrm{\mid }(f\ast {\phi }_t)(y)\mathrm{\mid },x\in {\mathbb{R}}^n,$ where ${\phi }_t(u)=t^{-n}\phi (u\mathrm{/}t)$ and convolution is the distributional convolution of Convolution of a tempered distribution with a schwartz function. The dilated test ${\phi }_t$ is Schwartz for every $t>0$ (Dilations and their normalisations preserve Schwartz space, with scaling identities), so each displayed convolution is defined even when $\int \phi =0$. Thus the full class ${\mathcal{F}}_N$, including its zero-integral tests, is used. Each convolution has a finite scalar value and the supremum is a well-defined function with values in $[0,\mathrm{\infty }]$; the value $+\mathrm{\infty }$ is allowed and no measurability is asserted here. The class ${\mathcal{F}}_N$ contains the zero function, is symmetric under $\phi \mapsto -\phi$ and under complex conjugation, and is nonempty for every $N$. No choice principle is used in this definition.

The order $N$ is a parameter. The characterisation theorem on this page fixes a finite admissible order $N_0(n,p,\phi )<\mathrm{\infty }$ depending only on $n$, $p$ and the fixed kernel $\phi$ and works for every $N\ge N_0(n,p,\phi )$; the value of $N_0$ is whatever the accumulated comparison estimates of that proof require, and its existence, not an explicit formula, is what the page uses. The sources record the explicit sufficient choices $N\ge \lfloor n\mathrm{/}p\rfloor +1$ for the nontangential class ${\mathcal{F}}_N$ in [DKKP, Proposition 1, p. 60], $N>1+n\mathrm{/}p$ for the radial class $B_N$ (with derivatives through order $N$) in [MSV, section 1, p. 16], and $N>n\mathrm{/}p+n+1$ in [CUW, Theorem 3.1, p. 8]; these recorded choices are not used as the definition of $N_0$ below. If $N^{\mathrm{\prime }}\ge N$ then $(1+\mathrm{\mid }x\mathrm{\mid }{)}^{N^{\mathrm{\prime }}}{\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le N^{\mathrm{\prime }}+1}\mathrm{\mid }{\mathrm{\partial }}^{\alpha }\phi (x)\mathrm{\mid }\ge (1+\mathrm{\mid }x\mathrm{\mid }{)}^N{\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le N+1}\mathrm{\mid }{\mathrm{\partial }}^{\alpha }\phi (x)\mathrm{\mid }$ pointwise, so ${\mathcal{F}}_{N^{\mathrm{\prime }}}\subseteq {\mathcal{F}}_N$ and hence $M_{N^{\mathrm{\prime }}}f\le M_{N}f\text{pointwise on }{\mathbb{R}}^n$ for every $f\in {\mathcal{S}}^{\mathrm{\prime }}$: the grand maximal functions are monotone in the order. The aperture is fixed to one; the comparison with larger apertures is the subject of the domination lemma on this page.