Let Δk,a and Δk, be the (k, a)-generalized Laguerre operator and the Dunkl Laplacian operator on ℝn, respectively. The aim of this article is twofold. First, we prove a restriction theorem for the Fourier-Δk,a transform. Next, as an application of the restriction problem, we establish Strichartz estimates for orthonormal families of initial data for the Schrödinger propagator <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2762_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$e^{-it\Delta_{k,a}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>i</mi> <mi>t</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>a</mi> </mrow> </msub> </mrow> </msup> </math></EquationSource> </InlineEquation> associated with the operator Δk,a. Further, using the classical Strichartz estimates for the free Schrödinger propagator <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2762_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$e^{-it\Delta_{k,a}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>i</mi> <mi>t</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>a</mi> </mrow> </msub> </mrow> </msup> </math></EquationSource> </InlineEquation> for orthonormal systems of initial data and the kernel relation between the semigroups <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2762_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$e^{-it\Delta_{k,a}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>i</mi> <mi>t</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>a</mi> </mrow> </msub> </mrow> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2762_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$e^{i {t\over a}\parallel x \parallel^{2-a}\Delta_{k}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mrow> <mfrac> <mi>t</mi> <mi>a</mi> </mfrac> </mrow> <mo>∥</mo> <mi>x</mi> <msup> <mo>∥</mo> <mrow> <mn>2</mn> <mo>−</mo> <mi>a</mi> </mrow> </msup> <msub> <mi mathvariant="normal">Δ</mi> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </msup> </math></EquationSource> </InlineEquation>, we prove Strichartz estimates for orthonormal systems of initial data associated with the Dunkl operator Δk on ℝn. Finally, we present some applications to our aforementioned results.

Orthonormal Strichartz inequalities for the (k, a)-generalized Laguerre operator and Dunkl operator

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发表日期：

2025年5月9日

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Shyam Swarup Mondal,
Manli Song

摘要

Let Δ_k,a and Δ_k, be the (k, a)-generalized Laguerre operator and the Dunkl Laplacian operator on ℝⁿ, respectively. The aim of this article is twofold. First, we prove a restriction theorem for the Fourier-Δ_k,a transform. Next, as an application of the restriction problem, we establish Strichartz estimates for orthonormal families of initial data for the Schrödinger propagator $e^{-it\Delta_{k,a}}$ $e^{- i t Δ_{k, a}}$ associated with the operator Δ_k,a. Further, using the classical Strichartz estimates for the free Schrödinger propagator $e^{-it\Delta_{k,a}}$ $e^{- i t Δ_{k, a}}$ for orthonormal systems of initial data and the kernel relation between the semigroups $e^{-it\Delta_{k,a}}$ $e^{- i t Δ_{k, a}}$ and $e^{i {t\over a}\parallel x \parallel^{2-a}\Delta_{k}}$ $e^{i \frac{t}{a} ∥ x ∥^{2 - a} Δ_{k}}$ , we prove Strichartz estimates for orthonormal systems of initial data associated with the Dunkl operator Δ_k on ℝⁿ. Finally, we present some applications to our aforementioned results.