Events

COSAM Events 2022 September DMS Graduate Student Seminar

DMS Graduate Student Seminar

Time: Sep 21, 2022 (03:00 PM)
Location: 108 ACLC

Details:

Speaker: Professor Geraldo de Souza

Title: Computing some important Integrals using distribution or decreasing rearrangement functions and a note on the Lorentz Spaces \(L(p,1)\) for \(p>1\)

Nozara Sundus (PhD student at ASSMS-Pakistan), Professor Eddy Kwessi (Trinity University-USA) and
 Professor Emeritus Geraldo de Souza (Auburn University-USA) 

 

Abstract: In this talk we explore the famous relationship between the \(L_p\)-norm in terms of distribution and the decreasing rearrangement function to evaluate some important higher dimensional integrals. Indeed we use

\(\int |f(x)|^p d\mu(x) = p\int_0^\infty \alpha^{p-1}\mu_f(\alpha) d\alpha = \int_0^\infty (f^*(t))^p dt\)


The first integral on left is on \(X\); the measure space \((X, $\sigma$-algebra\,\, on\,\, X, $\mu$)\) for \(p > 0\); \(\mu_f\) the distribution function; and \(f^*\) the decreasing rearrangement of \(f\).

Finally we look at the Lorentz spaces \(L(p,1)\) for \(p \geq 1\) in 2-dimension, which says

\(f \in L(p,1)\) if and only if \(\| f \|_{L(p,1)} = \int_0^\infty f^*(t) t^{1/p - 1} dt < \infty\),

and we give a new characterization based on special atom spaces, and this allows us to study some operators on the Lorentz spaces \(L(p,q), \ p,q > 0\). We will comment on important facts about \(L(p,q)\).