| Summary: | Let $\mathcal S$ be a second order smoothness in the $\mathbb{{R}}^n$ setting. We can assume without loss of generality that the dimension $n$ has been adjusted as necessary so as to insure that $\mathcal S$ is also non-degenerate. We describe how $\mathcal S$ must fit into one of three mutually exclusive cases, and in each of these cases we characterize by a simple intrinsic condition the second order smoothnesses $\mathcal S$ whose canonical Sobolev projection $P_{{\mathcal{{S}}}}$ is of weak type $(1,1)$ in the $\mathbb{{R}}^n$ setting. In particular, we show that if $\mathcal S$ is reducible, $P_{{\mathcal{{S}}}}$ is automatically of weak type $(1,1)$. We also obtain the analogous results for the $\mathbb{{T}}^n$ setting.We conclude by showing that the canonical Sobolev projection of every $2$-dimensional smoothness, regardless of order, is of weak type $(1,1)$ in the $\mathbb{{R}}^2$ and $\mathbb{{T}}^2$ settings. The methods employed include known regularization, restriction, and extension theorems for weak type $(1,1)$ multipliers, in conjunction with combinatorics, asymptotics, and real variable methods developed below. One phase of our real variable methods shows that for a certain class of functions $f\in L^{{\infty}}(\mathbb R)$, the function $(x_1,x_2)\mapsto f(x_1x_2)$ is not a weak type $(1,1)$ multiplier for $L^({{\mathbb R}}^2)$.
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