Weighted norm inequalities for generalized Hankel conjugate transformations

K. Andersen; R. Kerman

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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 $ʃ_{0}^{\infty} (\prod_{i = 1}^{n} | T_{γ_{i}} {f (x) |}^{q_{i}} {|) | f (x) |}^{q_{0}} w (x) d x \leq C (ʃ_{0}^{\infty} {| f (x) |}^{p} {v (x) d x)}^{(q_{0} + \dots + q_{n}) / p}$ . 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. ...

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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.