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## Studia Mathematica

1995 | 113 | 2 | 141-168
Tytuł artykułu

### Moser's Inequality for a class of integral operators

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
141-168
Opis fizyczny
Daty
wydano
1995
otrzymano
1994-03-02
poprawiono
1994-09-27
Twórcy
autor
• Department of Mathematics, University College, Cork, Ireland
autor
• Department of Mathematics, Maynooth College, Co. Kildare, Ireland
Bibliografia
• [1] D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. 128 (1988), 385-398.
• [2] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
• [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.
• [4] S. Y. A. Chang, Extremal functions in a sharp form of Sobolev inequality, in: Proc. Internat. Congress of Mathematicians, Berkeley, Calif., 1986.
• [5] I. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, translated from the fourth Russian ed., Scripta Technica, Academic Press, 1965.
• [6] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals I, Math. Z. 27 (1928), 565-606.
• [7] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1967.
• [8] M. Jodeit, An inequality for the indefinite integral of a function in $L^q$, Studia Math. 44 (1972), 545-554.
• [9] D. E. Marshall, A new proof of a sharp inequality concerning the Dirichlet integral, Ark. Mat. 27 (1989), 131-137.
• [10] P. McCarthy, A sharp inequality related to Moser's inequality, Mathematika 40 (1993), 357-366.
• [11] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971), 1077-1092.
• [12] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.
Typ dokumentu
Bibliografia
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