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Moser's Inequality for a class of integral operators

Finbarr Holland, David Walsh (1995)

Studia Mathematica

Let 1 < p < ∞, q = p/(p-1) 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 non-negative and homogeneous of degree -1. A sufficient condition on K is found for the analogue...

Moser-Trudinger and logarithmic HLS inequalities for systems

Itai Shafrir, Gershon Wolansky (2005)

Journal of the European Mathematical Society

We prove several optimal Moser–Trudinger and logarithmic Hardy–Littlewood–Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere S 2 , on a bounded domain Ω ⊂ ℝ 2 and on all of ℝ 2 . In some cases we also address the question of existence of minimizers.

Multidimensional Opial inequalities for functions vanishing at an interior point

George A. Anastassiou, Gisèle Ruiz Goldstein, Jerome A. Goldstein (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this paper we generalize Opial inequalities in the multidimensional case over balls. The inequalities carry weights and are proved to be sharp. The functions under consideration vanish at the center of the ball.

New equivalent conditions for Hardy-type inequalities

Alois Kufner, Komil Kuliev, Gulchehra Kulieva, Mohlaroyim Eshimova (2024)

Mathematica Bohemica

We consider a Hardy-type inequality with Oinarov's kernel in weighted Lebesgue spaces. We give new equivalent conditions for satisfying the inequality, and provide lower and upper estimates for its best constant. The findings are crucial in the study of oscillation and non-oscillation properties of differential equation solutions, as well as spectral properties.

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