Weighted integrability of multiplicative Fourier transforms

We study the relationship between the weighted integrability of a function and that of its multiplicative Fourier transform (MFT). In particular, for the MFT we prove an analog of R. Boas' conjecture related to the Fourier sine and cosine transforms. In addition, we obtain a sufficient condition under which a contraction of an MFT is also an MFT. For the moduli of continuity $\omega$ satisfying N. K. Bari's condition, we present a criterion for determining whether a function with a nonnegative MFT belongs to the class $H^\omega$.