Carleson maximal operator is strong ltwo

Apply F2 at r=3/2 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.