Reverse generalized h�older and minkowski type inequalities and their applications Report as inadecuate




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In this paper we give a reverse generalization of the generalized H�older and Minkowski type inequalities and their applications to inverse source problems.

Tipo de documento: Artículo - Article

Palabras clave: H�older and Minkowski type inequalities, heat equation, Weierstrass transform.





Source: http://www.bdigital.unal.edu.co


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Bol.
Mat.
17(2), 137–142 (2010) 137 Reverse generalized Hölder and Minkowski type inequalities and their applications René Erlı́n Castillo1 Departamento de Matemáticas Universidad Nacional de Colombia Eduard Trousselot2 Departamento de Matemáticas Universidad de Oriente 6101 Cumaná, Estado de Sucre, Venezuela In this paper we give a reverse generalization of the generalized Hölder and Minkowski type inequalities and their applications to inverse source problems. Keywords: Hölder and Minkowski type inequalities, heat equation, Weierstrass transform. En este artı́culo damos una generalización de la desigualdad inversa generalizada de Hölder y Minkowski y sus aplicaciones a problemas inversos. Palabras claves: Desigualdades de tipo Hölder y Minkowski, ecuación del calor, transformada de Weierstrass. MSC: 44A35, 26D20. 1 2 recastillo@unal.edu.co eddycharles2007@hotmail.com 138 Castillo and Trousselot, Reverse generalized Hölder and Minkowski 1 Introduction It is well known that, for 0 p 1, f ∈ Lp(X), g ∈ Lq(X), Z |f g| dµ ≥ kf kp kgkq . (1) X Since q is negative in this case, we assume that g 0, µ− a.e.
on X. Also, if f ∈ Lp(X), g ∈ Lp(X), and 0 p 1, then it follows, see [1], by applying the result of (1), that kf gkp ≥ kf kp kgkp . (2) The following version of inequality (1) was proved in [3]; see also [2] and [4], pages 125–126. Theorem 1.1.
Suppose p, q 0 and positive function satisfying 0 m≤ on a set X.
Then Z 1-p Z p f dµ X 1 p 1 q = 1.
If f and g are two fp ≤ M ∞, gq 1-q   1 Z m − pq f g dµ , g dµ ≤ M X q X (3) if the right hand side integral converges. Under appropriate conditions, we prove a generalized version of inequalities (2) and (3).
Our estimates are based on Theorem 1.1. 2 Main results Theorem 2.1.
Suppose p, q, r 0 and positive functions such that i) 0 m ≤ f p-s g q-s ii) 0 m ≤ (f g)s hr 1 p 1 q 1 r = 1.
If f, g and h...






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