Displaying similar documents to “Weighted generalized weak type inequalities for modified Hardy operators.”

Weighted L Φ integral inequalities for operators of Hardy type

Steven Bloom, Ron Kerman (1994)

Studia Mathematica

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Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for Φ 2 - 1 ( ʃ Φ 2 ( w ( x ) | T f ( x ) | ) t ( x ) d x ) ≤ Φ 1 - 1 ( ʃ Φ 1 ( C u ( x ) | f ( x ) | ) v ( x ) d x ) to hold when Φ 1 and Φ 2 are N-functions with Φ 2 ∘ Φ 1 - 1 convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored.

Two weighted inequalities for convolution maximal operators.

Ana Lucía Bernardis, Francisco Javier Martín-Reyes (2002)

Publicacions Matemàtiques

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Let φ: R → [0,∞) an integrable function such that φχ = 0 and φ is decreasing in (0,∞). Let τf(x) = f(x-h), with h ∈ R {0} and f(x) = 1/R f(x/R), with R > 0. In this paper we characterize the pair of weights (u, v) such that the operators Mf(x) = sup|f| * [τφ](x) are of weak type (p, p) with respect to (u, v), 1 < p < ∞.

First and second order Opial inequalities

Steven Bloom (1997)

Studia Mathematica

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Let T γ f ( x ) = ʃ 0 x k ( x , y ) γ f ( y ) d y , where k is a nonnegative kernel increasing in x, decreasing in y, and satisfying a triangle inequality. An nth-order Opial inequality has the form ʃ 0 ∞ ( ∏ i = 1 n | T γ i f ( x ) | q i | ) | f ( x ) | q 0 w ( x ) d x ≤ C ( ʃ 0 ∞ | f ( x ) | p v ( x ) d x ) ( q 0 + … + q n ) / p . Such inequalities can always be simplified to nth-order reduced inequalities, where the exponent q 0 = 0 . When n = 1, the reduced inequality is a standard weighted norm inequality, and characterizing the weights is easy. We also find necessary and sufficient conditions on the weights for second-order reduced Opial inequalities to hold. ...