On the Fourier transform of measures in Besov spaces

We prove quantitative estimates for the decay of the Fourier transform of the Riesz potential of measures that are in homogeneous Besov spaces of negative exponent: {align*} \|{Iαμ}\|_{Lp, ∞} ≤ C \|μ\|M_b1/2(t>0 td-β/2\|pₜ μ\|∞)1/2, {align*} where p=2d/2α+β with β ∈ (0,d) and I_α μ is the Riesz potential of μ of order α ∈ ((d-β)/2,d-β/2). Our results are naturally applicable to the Morrey space Mβ, including for example the Frostman measure μK of any compact set K with 0<H^β(K)<+∞ for some β ∈ (0,d]. When μ=DχE for χE ∈ *BV(Rᵈ), α =1, and β=d-1, our results extend the work of Herz and Ko--Lee. We provide examples which show the sharpness of our results.