Douglas S. Kurtz

Littlewood-Paley and multiplier theorems on weighted ^{} spaces

The Littlewood-Paley operator y(f), for functions f defined on RX, is shown to be a bounded operator on certain weighted LP spaces. The weights satisfy an AP condition over the class of all n-dimensional rectangles with sides parallel to the coordinate axes. The necessity of this class of weights demonstrates the 1-dimensional nature of the operator. Results for multipliers are derived, including weighted versions of the Marcinkiewicz Multiplier Theorem and Hormanders Multiplier Theorem.

Weighted weak (1,1) and weighted ^{} estimates for oscillating kernels

Weak type (1, 1) and strong type (p,p) inequalities are proved for operators defined by oscillating kernels. The techniques are sufficiently general to derive versions of these inequalities using weighted norms. 0. Introduction. Given a positive real number a > O, a 7& 1, define the oscillating kernel Ka by Ka(x) = (1 + |X|)-leilxla and consider the convolution operator Ka * f. In an earlier paper [2], we studied the boundedness properties of such operators on weighted LP spaces, 1 1, we define \ l/p f 1l P w = A E f(x)lpw(x) dx) R and say f E LP (R) if llfllP,w 1. The multiplier operator T, associated to Ka is defixled by (T:d) (() = 0(()1(l :/ e 1tl f((), where 0 is a C°° function defined by (i) 0(4) = (0 if :(: > 1/ when p < 0,

Covering lemmas and the sharp function

We present some new measure-theoretic inequalities for families of cubes covering open sets in RW and use these inequalities to estimate the Hardy-Littlewood maximal function Mf in terms of the sharp function f#.

Student perceptions of projects in learning calculus

This study presents a summary of perceptions expressed by students in response to university courses in calculus and differential equations that featured implementation of a discovery learning pilot programme. The programme used multi‐stage, multi‐task ‘projects’, with extended periods allocated for completion and requiring formal written reports of solutions. Project assignments included both independent and group work. Positive responses to the project approach were more common in more advanced courses, with students more frequently expressing appreciation of applications, enjoyment of problem‐solving, and willingness to enroll in future project courses. The most common objection to projects was the perceived greater time required by project work. For students in all courses, the most frequently‐cited difficulties were related to organization rather than to mathematical content of the projects. Responses concerning sources of help for project work suggested that students were largely unsuccessful in ide...