Holland, Finbarr and Walsh, David
(1995)
Moser's Inequality for a class of integral operators.
Studia Mathematica, 113 (2).
pp. 141166.
ISSN 00393223
Abstract
Let 1 < p < ∞, q = p/(p1) and for f ∈ L p ( 0 , ∞ ) define F ( x ) = ( 1 / x ) ʃ 0 x f ( t ) d t , x > 0. Moser’s Inequality states that there is a constant C p such that s u p a ≤ 1 s u p f ∈ B p ʃ 0 ∞ e x p [ a x q  F ( x )  q  x ] d x = C p where B p is the unit ball of L p . Moreover, the value a = 1 is sharp. We observe that F = K 1 f where the integral operator K 1 has a simple kernel K. We consider the question of for what kernels K(t,x), 0 ≤ t, x < ∞, this result can be extended, and proceed to discuss this when K is nonnegative and homogeneous of degree 1. A sufficient condition on K is found for the analogue of Moser’s Inequality to hold. An internal constant ψ, the counterpart of the constant a, arises naturally. We give a condition on K that ψ be sharp. Some applications are discussed.
Repository Staff Only(login required)

Item control page 
Downloads per month over past year