# Moser's Inequality for a class of integral operators

Studia Mathematica (1995)

- Volume: 113, Issue: 2, page 141-168
- ISSN: 0039-3223

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topHolland, Finbarr, and Walsh, David. "Moser's Inequality for a class of integral operators." Studia Mathematica 113.2 (1995): 141-168. <http://eudml.org/doc/216166>.

@article{Holland1995,

abstract = {Let 1 < p < ∞, q = p/(p-1) and for $f ∈ L^p(0,∞)$ define $F(x) = (1/x) ʃ_0^x f(t)dt$, x > 0. Moser’s Inequality states that there is a constant $C_p$ such that $sup_\{a≤1\} sup_\{f∈B_\{p\}\} ʃ_\{0\}^\{∞\} exp[ax^\{q\}|F(x)|^\{q\} - x]dx= C_p$ 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.},

author = {Holland, Finbarr, Walsh, David},

journal = {Studia Mathematica},

keywords = {Moser's Inequality; integral operator; distribution function; Moser's inequality},

language = {eng},

number = {2},

pages = {141-168},

title = {Moser's Inequality for a class of integral operators},

url = {http://eudml.org/doc/216166},

volume = {113},

year = {1995},

}

TY - JOUR

AU - Holland, Finbarr

AU - Walsh, David

TI - Moser's Inequality for a class of integral operators

JO - Studia Mathematica

PY - 1995

VL - 113

IS - 2

SP - 141

EP - 168

AB - Let 1 < p < ∞, q = p/(p-1) and for $f ∈ L^p(0,∞)$ define $F(x) = (1/x) ʃ_0^x f(t)dt$, x > 0. Moser’s Inequality states that there is a constant $C_p$ such that $sup_{a≤1} sup_{f∈B_{p}} ʃ_{0}^{∞} exp[ax^{q}|F(x)|^{q} - x]dx= C_p$ 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.

LA - eng

KW - Moser's Inequality; integral operator; distribution function; Moser's inequality

UR - http://eudml.org/doc/216166

ER -

## References

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