A new (non-Muckenhoupt type) weight characterization for the boundedness of the general Hardy-Steklov operator is obtained in the case 1 < p ≤ q < ∞. The estimates obtained for the norm of the Hardy-Steklov operator allow the limiting procedure and as a result the boundedness of the corresponding geometric Steklov operator is investigated.
Abstract. A generalization of the Lizorkin theorem on Fourier multipliers is proved. The proofs are based on using the so-called net spaces and interpolation theorems. An example is given of a Fourier multiplier which satisfies the assumptions of the generalized theorem but does not satisfy the assumptions of the Lizorkin theorem.Mathematics subject classification (2010): 42B15, 42B35.
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