Carleson maximal operator is strong $L^2$

Apply F2 at $r=3\mathrm{/}2$ and $s=3$, then F3 with $p=2$. These strict endpoint exponents give a finite-model strong $L^2$ constant independent of the family and selector. F1 gives $\mathrm{\parallel }{C}_{\mathbb{R}}u{\mathrm{\parallel }}_2\le K\mathrm{\parallel }u{\mathrm{\parallel }}_2$ for every complex Schwartz $u$, for a fixed finite $K$. This uses estimates on both sides of two; no restricted weak-$L^2$-to-strong-$L^2$ inference is made.

For Schwartz $u,v$, linearity of every cutoff and $\mathrm{\mid }\mathrm{\mid }a\mathrm{\mid }-\mathrm{\mid }b\mathrm{\mid }\mathrm{\mid }\le \mathrm{\mid }a-b\mathrm{\mid }$ imply pointwise $\mathrm{\mid }{C}_{\mathbb{R}}u-C_{\mathbb{R}}v\mathrm{\mid }\le C_{\mathbb{R}}(u-v)$. The suprema are finite because the Schwartz transform is integrable. Thus step 1.1 gives $\mathrm{\parallel }{C}_{\mathbb{R}}u-C_{\mathbb{R}}v{\mathrm{\parallel }}_2\le K\mathrm{\parallel }u-v{\mathrm{\parallel }}_2$. Also $C_{\mathbb{R}}(zu)=\mathrm{\mid }z\mathrm{\mid }{C}_{\mathbb{R}}u$ and $C_{\mathbb{R}}(u+v)\le C_{\mathbb{R}}u+C_{\mathbb{R}}v$ pointwise.

For each fixed $f\in L^2$, F5 and the countable choice supplied by F7 give Schwartz $u_j$ with $\mathrm{\parallel }{u}_j-f{\mathrm{\parallel }}_2<1\mathrm{/}j$. Step 2.1 makes $(C_{\mathbb{R}}{u}_j)$ Cauchy in $L^2$, so F6 gives a limit; define ${\bar{C}}_{\mathbb{R}}f$ to be this class. If $v_j$ is another such sequence, the same Lipschitz inequality bounds the distance between their output sequences by $K\mathrm{\parallel }{u}_j-v_j{\mathrm{\parallel }}_2\to 0$, so the limit is independent of the approximation. On a Schwartz class take the constant sequence to see agreement with the original operator. Passing norms to the limit gives $\mathrm{\parallel }{\bar{C}}_{\mathbb{R}}f{\mathrm{\parallel }}_2\le K\mathrm{\parallel }f{\mathrm{\parallel }}_2$.

Applying the same argument to approximants for $f,g$ gives $\mathrm{\parallel }{\bar{C}}_{\mathbb{R}}f-{\bar{C}}_{\mathbb{R}}g{\mathrm{\parallel }}_2\le K\mathrm{\parallel }f-g{\mathrm{\parallel }}_2$, so the extension is continuous and is the unique continuous extension from the dense Schwartz classes. It is nonnegative almost everywhere: F6 supplies an a.e.-convergent subsequence of its nonnegative approximating outputs. Homogeneity and subadditivity also pass from step 2.1; for subadditivity take subsequences along which the three output sequences for $u_j$, $v_j$ and $u_j+v_j$ converge a.e., successively using F6, and pass the pointwise inequality off the finite union of exceptional null sets. The zero class maps to zero. This is the required bounded maximal-operator extension on complex $L^2$, with precisely the inherited AC assumption and no simultaneous arbitrary-index choice of approximants.