# 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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## Abstract

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Let 1 < p < ∞, q = p/(p-1) and for $f\in {L}^{p}\left(0,\infty \right)$ define $F\left(x\right)=\left(1/x\right){ʃ}_{0}^{x}f\left(t\right)dt$, x > 0. Moser’s Inequality states that there is a constant ${C}_{p}$ such that $su{p}_{a\le 1}su{p}_{f\in {B}_{p}}{ʃ}_{0}^{\infty }exp\left[a{x}^{q}{|F\left(x\right)|}^{q}-x\right]dx={C}_{p}$ 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.

## How to cite

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Holland, 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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1. [1] D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. 128 (1988), 385-398. Zbl0672.31008
2. [2] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
3. [3] L. Carleson and S. Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. (2) 110 (1986), 113-127. Zbl0619.58013
4. [4] S. Y. A. Chang, Extremal functions in a sharp form of Sobolev inequality, in: Proc. Internat. Congress of Mathematicians, Berkeley, Calif., 1986.
5. [5] I. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, translated from the fourth Russian ed., Scripta Technica, Academic Press, 1965. Zbl0918.65002
6. [6] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals I, Math. Z. 27 (1928), 565-606. Zbl54.0275.05
7. [7] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1967.
8. [8] M. Jodeit, An inequality for the indefinite integral of a function in ${L}^{q}$, Studia Math. 44 (1972), 545-554. Zbl0244.26010
9. [9] D. E. Marshall, A new proof of a sharp inequality concerning the Dirichlet integral, Ark. Mat. 27 (1989), 131-137. Zbl0692.30028
10. [10] P. McCarthy, A sharp inequality related to Moser's inequality, Mathematika 40 (1993), 357-366. Zbl0822.26012
11. [11] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971), 1077-1092. Zbl0203.43701
12. [12] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. Zbl0207.13501

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