Weighted norm inequalities for the Fourier transform on connected locally compact groups (Q1113037)

In a paper by \textit{B. Muckenhoupt} [Trans. Am. Math. Soc. 276, 729-742 (1983; Zbl 0513.42002)] sufficient conditions are placed on nonnegative pairs of functions T, V, to imply the inequality: \[ (I)\quad (\int_{R^ n}| \hat f(x)|^ qT(x)dx)^{1/q}\leq c(\int_{R^ n}| f(x)|^ pV(x)dx)^{1/p} \] for all integrable functions f on \(R^ n\), with c a constant independent of f. In the paper under review the author extends this result in three different settings, for \(1<p\leq 2:\) (1) for an LCA, non-compact, connected group G with dual group \(\Gamma\) ; (2) for a compact connected abelian group G with dual group \(\Gamma\) ; (3) for a compact connected non-abelian group G with dual object \(\hat G.\) With respect to the inequality (I), the integration of the left hand side has to be taken over \(\Gamma\), or it becomes a sum over \(\hat G,\) while the integration on the right hand side is taken over G. For instance, in (1) the result becomes \[ (\int_{\Gamma}| \hat f(\gamma)|^{p'}T(\gamma)dm_{\Gamma}(\gamma))^{1/p'}\leq c(\int_{G}| f(x)|^ pV(x)dm_ G(x))^{1/p}, \] with \(p'=p/(p-1)\). For a more restricted class of groups, an analogous result is given with less restrictions on the exponents.