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 $L(p,q)$ .

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