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Full Description
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 $L^({\mathbb R}^2)$.
Contents
Introduction and notation Some properties of weak type multipliers and canonical projections of weak type $(1,1)$ A class of weak type $(1,1)$ rational multipliers A subclass of $L^\infty(\mathbb{R}^2)\backslash M_1^{(w)}(\mathbb{R}^2)$ induced by $L^\infty(\mathbb{R})$ Some combinatorial tools Necessity proof for the second order homogeneous case: A converse to Corollary (2.14) Canonical projections of weak type $(1,1)$ in the $\mathbb{T}^n$ model: Second order homogeneous case The non-homogeneous case Reducible smoothnesses of order $2$ the canonical projection of every two-dimensional smoothness is of weak type $(1,1)$ References.
