Fourier Inequalities and Moment Subspaces in Weighted Lebesgue Spaces

Abstract

For $k \in {\Bbb N}, k \not= 0,$ define ${\cal F}_kf(\gamma) = \int_{{\Bbb R}^n} f(t)R_k(-2i \pi \gamma.t) \, dt, n\geq 1,$ where $R_k(i\lambda) = e^{i\lambda} - \sum^{k-1}_{j=0} \left(i \lambda \right)^{j} / \left(j~!\right).$ Pointwise estimates and weighted inequalities describing the local Lipschitz continuity of ${\cal F}_kf$ are established. Sufficient conditions are found for the boundedness of ${\cal F}_k$ from $L^p_v$ into $L^q_\mu,$ and a spherical restriction property is proved. A study of the moment subspaces of $L^p_v$ is next developed in the one-variable case, for $1 < p < \infty, v$ locally integrable, $v > 0$ a.e. It includes a decomposition theorem and a complete classification of all possible sequences of moment subspaces in $L^p_v.$ Characterizations are also given for each class. Applications related to the approximation and decomposition of ${\cal F}_k$ are discussed.