Conditional Multipliers and Essential Norm of U[C.Sub.[Psi]] Between [L.Sup.P] Spaces (Report‪)‬

1. INTRODUCTION AND PRELIMINARIES Let (X, [summation], [mu]) be a sigma finite measure space. By [L.sup.0]([summation]), we denote the linear space of all [summation]-measurable functions on X. For any complete sigma finite sub-algebra A [subset or equal to] [summation] with 1 [less than or equal to] p [less than or equal to] [infinity] the [L.sup.p]-space [L.sup.p](X, A, [mu]|A) is abbreviated by [L.sup.p](A), and its norm is denoted by [[parallel]x[parallel].sub.p]. We understand [L.sup.p](A) as a Banach subspace of [L.sup.p]([summation]). All comparisons between two functions or two sets are to be interpreted as holding up to a [mu]-null set. The support of a measurable function f is defined as [sigma](f) = {x [member of] X; f(x) [not equal to] 0}. A [summation]-measurable function u on X for which u f [member of] [L.sup.q]([summation]) for each f [member of] [L.sup.p](A), is called a conditional multiplier.