Yoram Sagher

Rearrangement-Function Inequalities and Interpolation Theory

Using the tools of Real Interpolation Theory, we develop a general method for proving rearrangement-function inequalities for important classes of operators.

BMO estimates for lacunary series

We proveBMO andLp norm inequalities inRn for lacunary Walsh and generalized trigonometric series.

What Pythagoras Could Have Done

Pythagoras allegedly proved that the square root of 2 is irrational. Platos dialogue Theaitetos tells of proofs by Theodorus of Kyrene of the irrationality of the square roots of all integers less than or equal to 17, with the obvious exceptions. Why Theodorus stopped at 17 is a subject of lively speculation, but the general tone of modern writers (see, e.g., [1]) is to sympathize with the giants of the classical age for not having had access to the properties of prime numbers. Had they but known that p divides a2 implies p divides a, they would have seen through the problem at a glance. In an issue of the MONTHLY [2], W. C. Waterhouse discusses a plausibility argument for the square root of k being either an irrational or an integer, based on the assumption that the properties of prime numbers are as mysterious to our students as they were to the Greeks. However, the plausibility argument is but a rewording of the usual idea. In preparing a talk to high school students, the following proof occurred to me. The main point of the proof is that it does not depend on properties of prime numbers, and so was fully accessible to Pythagoras and to Theodorus. It is also accessible to high school students after one year of algebra. When shown to a number of bright ninth-graders, it caused some excitement. Suppose vk = m/n, where m and n are integers with n > 0. If k is not a square, there exists an integer q so that q < m/n < q + 1. Now m2 = kn2 implies m(m qn) = n(kn qm) and, hence, m/n = (kn qm)/(m qn). From q < m/n < q + I we get 0 < m qn < n. Therefore we have:

The Appell transform and the semigroup property for temperatures

We prove that if u(x, t) is a solution of the one dimensional heat equation and if A u(x, t) is its Appell transform, then u(x, t) has the semi-group (Huygens) property in a domain D if and only if A u(x, t) has the semi-group property in a dual region. We apply this result to simplify and extend some results of Rosenbloom and Widder.

Homogeneous solutions of the heat equation

Abstract We complete a characterization of homogeneous solutions of the heat equation begun by D. V. Widder. We determine regions of convergence for expansions of temperature functions in terms of the homogeneous solutions.

Limits of sequences of operators on spaces of vector valued functions

We generalize the celebrated theorem of Stein on the maximal operator of a sequence of translation invariant operators, from the scalar case to vector valued functions.