Pitt's theorem (Pitt 1937), states that the one-dimensional Fourier tranform is well defined and continuous between the weighted spaces $L^p(\mathbb{R},|x|^{bp}dx)$ and $L^q(\mathbb{R},|x|^{\beta q}dx)$, where $1<p\leq q<+\infty$, $0<b<(p-1)/p$, and $$\beta:=1-\frac{1}{p}-\frac{1}{q}-b<0$$
I'm wondering if a similar results holds for a (slighty) more general class of operators.
More specifically, given $\alpha\in L^\infty(\mathbb{R})$, let's consider the following operator:
$$T_{\alpha}f(x):=\int_{\mathbb{R}}e^{-ixy}\alpha(x+y)f(y)dy$$
Can i expect (possibly under suitable hypothesis on $\alpha$) that the same conclusions of Pitt's theorem also hold for $T_{\alpha}$?
Thank you for any suggestions.
