Sharp Strichartz estimates for the Schrödinger equation on the sphere

Abstract

In this contribution we investigate the Schrördinger equation associated to the Laplacian on the sphere in the form of sharp Strichartz estimates. We will provided simple proofs for our main theorems using purely the \(L^2\rightarrow L^p\) spectral estimates for the operator norm of the spectral projections (associated to the spherical harmonics) proved in Kwon and Lee (RIMS Kokyuroku Bessatsu 70:33–58, 2018). A sharp index of regularity is established for the initial data in spheres of arbitrary dimension \(d\ge 2\).

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Notes

  1. 1.

    Observe that under the symmetric compromise \(p=q,\) \(\Vert u(t,z) \Vert _{L^{p}_z[\mathbb {S}^{d},\,L^q_t(\mathbb {T})]}=\Vert u(t,z) \Vert _{L^{p}(\mathbb {S}^{d}\times \mathbb {T})},\) and for \(p=4,\) \( \Vert u \Vert ^4_{ L^4(\mathbb {S}^d\times \mathbb {T} )}= \Vert u^2 \Vert ^2_{L^2(\mathbb {S}^d\times \mathbb {T})} .\) We use the notation \(A\lesssim B\) to indicate \(A\le cB,\) where \(c>0\) does not depend on fundamental quantities.

  2. 2.

    By following the usual nomenclature, such pair (pq) is called admissible.

  3. 3.

    As usually, defined by those measurable functions on \(\mathbb {S}^d,\) such that \(\Vert f\Vert _{L^2(\mathbb {S}^d)}:=\left( \,\int \limits _{\mathbb {S}^d}|f(\omega )|^2d\omega \right) ^{\frac{1}{2}}<\infty .\) In general, on a measure space (Xdx),  we use \(\Vert f\Vert _{L^p(X)}:=\left( \,\int \limits _{X}|f(x)|^pdx\right) ^{\frac{1}{p}},\) for all \(1\le p<\infty .\)

  4. 4.

    Here, \(W^{s,p}:=\text {Dom}(\Delta ^s_{\mathbb {S}^d})\) is considered as the completion of \(C^\infty (\mathbb {S}^d)\) with respect to the norm defined in (2.3).

  5. 5.

    Where we use \(\asymp \) because \(\ell \sim \lambda _\ell := [\ell (\ell +d-1)]^{\frac{1}{2}}\) for \(\ell \rightarrow \infty .\)

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Correspondence to Duván Cardona Sánchez.

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The first author was supported by the FWO Odysseus 1 Grant G.0H94.18N: Analysis and Partial Differential Equations of Professor Michael Ruzhansky. The second author was supported by Gran Sasso Science Institute, L’Aquila, Italy.

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Sánchez, D.C., Esquivel, L. Sharp Strichartz estimates for the Schrödinger equation on the sphere. J. Pseudo-Differ. Oper. Appl. 12, 23 (2021). https://doi.org/10.1007/s11868-021-00376-0

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Mathematics Subject Classification

  • Primary: 35Q40
  • Secondary: 42B35
  • 42C10
  • 35K15