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October 12, 2025Open Access

On the Fourier transform of measures in Besov spaces

Key Points

The Fourier transform of the Riesz potential of measures in homogeneous Besov spaces decays with specific rates.
Estimates show that ext{||} ext{I_{1}3}0} ext{||}_{L^{p, 0}} ext{≤} ext{C} ext{||} ext{μ} ext{||}_{M_b}^{rac{1}{2}}.
The approach applies to Morrey spaces, including Frostman measures associated with compact sets.
Results extend existing work by Herz and Ko–Lee for specific measures and parameters.

Abstract

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* \|I_\|₋^₏, C \|\|₌₁^1{2} (ₓ>₀ t^d-{2}\|pₓ \|_) ^1{2}, align* where p=2d2+ with (0, d) and I_ is the Riesz potential of of order ( (d-) /2, d-/2). Our results are naturally applicable to the Morrey space M^, including for example the Frostman measure K of any compact set K with 0<H^ (K) <+ for some (0, d]. When =DE for E *BV (Rᵈ), =1, and =d-1, our results extend the work of Herz and Ko--Lee. We provide examples which show the sharpness of our results.

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