Displaying similar documents to “Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions”

Weighted weak type inequalities for certain maximal functions

Hugo Aimar, Liliana Forzani (1991)

Studia Mathematica

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We give an A_p type characterization for the pairs of weights (w,v) for which the maximal operator Mf(y) = sup 1/(b-a) ʃ_a^b |f(x)|dx, where the supremum is taken over all intervals [a,b] such that 0 ≤ a ≤ y ≤ b/ψ(b-a), is of weak type (p,p) with weights (w,v). Here ψ is a nonincreasing function such that ψ(0) = 1 and ψ(∞) = 0.

Some weighted inequalities for general one-sided maximal operators

F. Martín-Reyes, A. de la Torre (1997)

Studia Mathematica

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We characterize the pairs of weights on ℝ for which the operators M h , k + f ( x ) = s u p c > x h ( x , c ) ʃ x c f ( s ) k ( x , s , c ) d s are of weak type (p,q), or of restricted weak type (p,q), 1 ≤ p < q < ∞, between the Lebesgue spaces with the coresponding weights. The functions h and k are positive, h is defined on ( x , c ) : x < c , while k is defined on ( x , s , c ) : x < s < c . If h ( x , c ) = ( c - x ) - β , k ( x , s , c ) = ( c - s ) α - 1 , 0 ≤ β ≤ α ≤ 1, we obtain the operator M α , β + f = s u p c > x 1 / ( c - x ) β ʃ x c f ( s ) / ( c - s ) 1 - α d s . For this operator, under the assumption 1/p - 1/q = α - β, we extend the weak type characterization to the case p = q and prove that in the case of equal...

The one-sided minimal operator and the one-sided reverse Holder inequality

David Cruz-Uribe, SFO, C. Neugebauer, V. Olesen (1995)

Studia Mathematica

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We introduce the one-sided minimal operator, m + f , which is analogous to the one-sided maximal operator. We determine the weight classes which govern its two-weight, strong and weak-type norm inequalities, and show that these two classes are the same. Then in the one-weight case we use this class to introduce a new one-sided reverse Hölder inequality which has several applications to one-sided ( A p + ) weights.

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. ...