New generalizations of inequalities of Hardy and Levin-Cochran-Lee type for multidimensional balls

Čižmešija, Aleksandra; Pečarić, Josip

doi:10.7153/mia-05-63

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“…Analogously to the procedure established in [4,5], in this section we apply the previously obtained mixed-means inequality (4) to deduce the Hardy-type inequality for the operator S δ . Theorem 4.…”

Section: Hardy and Carleman-type Inequalitiesmentioning

confidence: 98%

“…In [5,Theorem 5], where integral means were taken over balls in R n centered at the origin, this form of functions f was the only possible for achieving equality. The proof presented there was simple since Jensen's inequality was applied to the angular part of polar coordinates so radiality of extremal functions was immediate.…”

Section: Remarkmentioning

confidence: 99%

“…Moreover, the family of all extremal functions for the related mixed-means inequality (cf. [5,Theorem 7]) was obtained from separability of extremal functions for Minkowski's inequality. The same holds also in our case here, but since the proof is lengthy and quite technical, it will be omitted.…”

Section: Remarkmentioning

confidence: 99%

“…The main aim of this paper is to give another approach (analogous to [5]) to the problem of obtaining the operator norm of weighted operators S δ . Namely, we shall consider the class of operators…”

Section: Introductionmentioning

confidence: 99%

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Mixed means over balls and annuli and lower bounds for operator norms of maximal functions

Čižmešija

Perić

2004

Journal of Mathematical Analysis and Applications

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In this paper we prove mixed-means inequalities for integral power means of an arbitrary real order, where one of the means is taken over the ball B(x, δ|x|), centered at x ∈ R n and of radius δ|x|, δ > 0. Therefrom we deduce the corresponding Hardy-type inequality, that is, the operator norm of the operator S δ which averages |f | ∈ L p (R n ) over B(x, δ|x|), introduced by Christ and Grafakos in Proc. Amer. Math. Soc. 123 (1995) 1687-1693. We also obtain the operator norm of the related limiting geometric mean operator, that is, Carleman or Levin-Cochran-Lee-type inequality. Moreover, we indicate analogous results for annuli and discuss estimations related to the HardyLittlewood and spherical maximal functions.

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Section: Hardy and Carleman-type Inequalitiesmentioning

confidence: 98%

Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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