Displaying similar documents to “The class Bpfor weighted generalized Fourier transform inequalities”

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. ...