Multiplier Criteria of Hörmander Type for Fourier Series and  Applications to Jacobi Series and Hankel Transforms.

Georg Gasper; Walter Trebls

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Under which conditions is the Jacobi space $L_{w^{(a, b)}}^{p} [- 1, 1]$ subset of $L_{w^{(α, β)}}^{1} [- 1, 1]$ ?

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Exact conditions for α, β, a, b > −1 and 1 ≤ p ≤ ∞ are determined under which the inclusion property $L_{w^{(a, b)}}^{p} [- 1, 1]$ ⊂ $L_{w^{(α, β)}}^{1} [- 1, 1]$ is valid. It is shown that the conditions characterize the inclusion property. The paper concludes with some results, in which the inclusion property can be detected in relation with estimates of Jacobi differential operators and with Muckenhoupt’s transplantation theorems and multiplier theorems for Jacobi series.