Canonical Sobolev Projections of Weak Type $(1,1)$: 150 (Memoirs of the AMS) (Memoirs of the American Mathematical Society) - Image 1

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Brand: Amer Mathematical Society Canonical Sobolev Projections of Weak Type $(1,1)$: 150 (Memoirs of the AMS) (Memoirs of the American Mathematical Society)

Name: Canonical Sobolev Projections of Weak Type USD(1,1)USD: 150 (Memoirs of the AMS) (Memoirs of the American Mathematical Society)
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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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Brand: Brand: Amer Mathematical Society
Format: paperback
ASIN: 0821826654
Category: Books > Subjects > Science, Nature & Maths > Mathematics > Mathematical Analysis > Differential Equations
Domain: Amazon UK
Release Date: 15 March 2001
Listed Since: 10 December 2006

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Brand: Amer Mathematical Society Canonical Sobolev Projections of Weak Type $(1,1)$: 150 (Memoirs of the AMS) (Memoirs of the American Mathematical Society)

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