Moser's Inequality for a class of integral operators

Holland, Finbarr; Walsh, David

Studia Mathematica, Tome 113 (1995), p. 141-168 / Harvested from The Polish Digital Mathematics Library

Résumé

Let 1 < p < ∞, q = p/(p-1) and for $f \in L^{p} (0, \infty)$ 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 $s u p_{a \leq 1} s u p_{f \in B_{p}} ʃ_{0}^{\infty} e x p [a x^{q} {| F (x) |}^{q} - x] d x = C_{p}$ 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 non-negative 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.

Publié le : 1995-01-01

Zbl 0843.26008

EUDML-ID : urn:eudml:doc:216166

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     author = {Finbarr Holland and David Walsh},
     title = {Moser's Inequality for a class of integral operators},
     journal = {Studia Mathematica},
     volume = {113},
     year = {1995},
     pages = {141-168},
     zbl = {0843.26008},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv113i2p141bwm}
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Holland, Finbarr; Walsh, David. Moser's Inequality for a class of integral operators. Studia Mathematica, Tome 113 (1995) pp. 141-168. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv113i2p141bwm/

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