Canonical Sobolev Projections of Weak Type (1,1) - Couverture souple

American Mathematical Society

Couverture souple

ISBN 10 : 0821826654 ISBN 13 : 9780821826652

Editeur : American Mathematical Society, 2001

Synopsis

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 $L^({\mathbb R}^2)$.

Détails de l'édition

• Éditeur: American Mathematical Society
• Date d'édition: 2001
• Langue: anglais
• ISBN 10: 0821826654
• ISBN 13: 9780821826652
• Reliure: Broché
• Nombre de pages: 75
• Coordonnées du fabricant: non disponible
• Personne responsable: non disponible