Canonical Sobolev Projections of Weak Type 1 1

quoted in Theorem (2.3), one can assume for these purposes that S is non-degenerate (if S is degenerate, ... Corollary 4,1]), while the canonical projection of every one-dimensional smoothness is of strong type (1,1) in both the R and T ...

Author: Earl Berkson

Publisher: American Mathematical Soc.

ISBN: 9780821826652

Category: Mathematics

Page: 75

View: 316


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 $L^1({\mathbb R}^2)$.

Projections of Educational Statistics to

Chart 1 . Summary of forecast assumptions to 2006 ix Figures 1. Past and projected trends in selected education statistics: 1966-67 to 2006-2007 x Births 2. Annual number of births, with projections: 1 946 to 2006 5 School-Age ...



ISBN: UOM:39015038901974

Category: Education

Page: 468

View: 483


Each issue provides a 10-year projection from edition date of statistics for elementary and secondary schools and institutions of higher education; includes enrollments, graduates, teachers, and expenditures.

National Environmental Impact Projection No 1

One logical energy future for which environmental impacts must be assessed is characterized by the continuation of ... Each set of forecasts is known as a Projection Series and has a letter reference of A through F. A Projection Series ...

Author: Mitre Corporation


ISBN: UCSD:31822024379968

Category: Pollution

Page: 114

View: 582


Population Projections

Figure 1 Total period fertility rates 1974-1995 assumed for six variant projections United Kingdom 2.4 2.4 2.2 2.2 2.0 2.0 1.8 1.8 1.6 1.8 1 4 r I I I I I I I I I I I I I I I I I I I I I 1 1.4 1 4 t_l ''''I 1 4 197S 1980 1985 1980 1999 ...

Author: Great Britain. Office of Population Censuses and Surveys


ISBN: STANFORD:36105043177430

Category: Conference. Population projections: trends, methods and uses

Page: 114

View: 243


Map Projection Methods

Tables Table Title Page 3.1.1 Geocentric and Geodetic Latitudes for the WGS-72 Spheriod.. 67 3.2.1 Radii as a Function Latitude for the WGS-72 Spheriod 73 3.3.1 Distance Along the Circle of Parallel for a Separation of V for the WGS-72 ...

Author: Frederick Pearson

Publisher: Sigma Scientific Incorporated

ISBN: UOM:39015011586586

Category: Science

Page: 292

View: 506


English Patents of Inventions Specifications

The serrated surface ( 1 ) is secured to a bar ( 13 ) which can oscillate 5 and slide on a pin ( 2 ) , and when the ... The cams 15 ( 7 and * ) are slided by the handle ( 16 ) so as to bring one tappet ( 1 ) between the projections upon ...



ISBN: DMM:057002675380


Page: 448

View: 951


Engineering Drawing

Draw a CP 1 HP, 430° VP circle of radius 10 and then use 30° set square to draw the CP. CP 10 from 0V (iii) Draw generators and locate common points in the other ?? the TV, SFV, TS views. (iv) Complete the projections.

Author: Shah, M. B.

Publisher: Pearson Education India

ISBN: 9788131742792


Page: 580

View: 140


The second edition of Engineering Drawing continues to cover all the fundamental topics of the field. This edition includes a new chapter on scales, the latest version of AutoCAD, and new pedagogy. Combining technical accuracy with readable explana

Population Projections for Pakistan 1960 2000

It was felt that these data should be utilized to project the population of Pakistan to determine the effect of data which show a rate of growth strikingly greater than what has appeared in the Perspective Plan for Pakistan [1 1 , p.24] ...

Author: Lee L. Bean


ISBN: STANFORD:36105038260480

Category: Pakistan

Page: 93

View: 935