<p>In this paper we study boundedness properties of the short-time Fourier transform (STFT) on the Lorentz spaces <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_413_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$L^{p,q}(\mathbb {R}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The same results apply to the cases of the Wigner and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_413_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>-Wigner transforms. Reinterpreting these results in terms of operators we obtain boundedness properties for Weyl and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_413_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>-Weyl operators. We conclude with an application to the uncertainty principle of Donoho-Stark in the context of Lorentz spaces.</p>

Time-frequency representations on Lorentz spaces

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2025年3月6日

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Paolo Boggiatto,
Alessandro Oliaro,
Ayşe Sandıkçı

摘要

In this paper we study boundedness properties of the short-time Fourier transform (STFT) on the Lorentz spaces $L^{p,q}(\mathbb {R}^d)$ $L^{p, q} (R^{d})$ . The same results apply to the cases of the Wigner and $\tau $ $τ$ -Wigner transforms. Reinterpreting these results in terms of operators we obtain boundedness properties for Weyl and $\tau $ $τ$ -Weyl operators. We conclude with an application to the uncertainty principle of Donoho-Stark in the context of Lorentz spaces.