Radial and nontangential maximal functions of a tempered distribution

Fix an integer $n\ge 1$, work with $\mathcal{S}({\mathbb{R}}^n)$ and ${\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^n)$ of Schwartz space and its seminorms and Tempered distribution, and let $\phi \in \mathcal{S}({\mathbb{R}}^n)$ satisfy ${\int }_{{\mathbb{R}}^n}\phi \mathrm{\ne }0$. For $t>0$ write ${\phi }_t(x)=t^{-n}\phi (x\mathrm{/}t),x\in {\mathbb{R}}^n\mathrm{.}$ For $f\in {\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^n)$ define the radial maximal function $M_{\phi }^{0}f(x)={\sup }_{t>0}\mathrm{\mid }(f\ast {\phi }_t)(x)\mathrm{\mid },x\in {\mathbb{R}}^n,$ and, for an aperture $a\ge 1$, the nontangential maximal function $M_{\phi }^{\ast ,a}f(x)={\sup }_{t>0}{\sup }_{\mathrm{\mid }y-x\mathrm{\mid }\le at}\mathrm{\mid }(f\ast {\phi }_t)(y)\mathrm{\mid },x\in {\mathbb{R}}^n\mathrm{.}$

Each convolution is the distributional convolution of Convolution of a tempered distribution with a schwartz function. It is well defined: $x\mapsto t^{-n}\phi (x\mathrm{/}t)$ again lies in $\mathcal{S}({\mathbb{R}}^n)$ by Dilations and their normalisations preserve Schwartz space, with scaling identities, so the pairing of $f$ with the reflected translate of ${\phi }_t$ exists at every point. Its values are smooth and of polynomial growth by Tempered convolution is smooth with polynomial growth, so each function $x\mapsto (f\ast {\phi }_t)(x)$ is finite at every point and the suprema displayed above are suprema of a nonempty family of real numbers; the radial case is the diagonal $y=x$ of the nontangential case, so $M_{\phi }^{0}f\le M_{\phi }^{\ast ,a}f$ for every $a\ge 1$.

The aperture convention is $\mathrm{\mid }y-x\mathrm{\mid }\le at$ with the closed cone, and the normalisation $t^{-n}$ is the one of the sources. Under Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), each ${\phi }_t$ has the same integral as $\phi$ by Dilations and their normalisations preserve Schwartz space, with scaling identities. This additional mass identity uses that supplier's stated choice premise. No measurability of $M_{\phi }^{0}f$ or $M_{\phi }^{\ast ,a}f$ is asserted here, and the pointwise convolution and maximal-function definitions use no choice principle; the finiteness of the supremum at a point is not claimed, since the family $\{(f\ast {\phi }_t)(y)\}$ need not be bounded a priori. Apertures $a>1$ and the grand maximal function are treated in the following items.