1 Erratum to: Constr Approx (2014) 40:141–171 DOI 10.1007/s00365-014-9235-5
Several forms of uncertainty principles on the unit sphere are established in [1]. When stated in term of the vector
of \({\mathbb R}^d\) (normalization constant \(1/{\omega }_d\) was missing in [1]), our main result is in:
Corollary 4.4
Let \(f\in W_2^1(\mathbb {S}^{d-1})\) be such that \(\int _{\mathbb {S}^{d-1}} f(y)\, d{\sigma }(y)=0\) and \(\Vert f\Vert _2=1\). If \(d \ge 2\), then
Here \(C_d\) is a constant given in Theorem 4.1. We next attempted to remove the condition that \(\int _{\mathbb {S}^{d-1}} f(y)\, d{\sigma }(y)=0\) and stated:
Theorem 4.5
Assume that \(d\ge 2\), and let \(f\in W_2^1(\mathbb {S}^{d-1})\) be such that \(\Vert f\Vert _{2}=1\). Then
This theorem, however, is incorrect. This was pointed out to us by Stefan Steinerberger, who showed that the inequality (4.14) does not hold for the function \(f(\cos {\theta }, \sin {\theta }) = 1+ \varepsilon \sin {\theta }\) for small enough \(\varepsilon \) when \(d=2\). The mistake in the proof appeared on the line 6 of page 166, which states that \(\Vert \tau (f)\Vert \le (2 |m_f| +1) \Vert g||_2^2\), but it should have been \(\Vert \tau (f)\Vert \le \Vert g\Vert _2^2 + 2 |m_f| \Vert g\Vert _2\). As a consequence, the right-hand side of (4.14) has to be replaced by \(c_d \Vert \tau (f)\Vert ^2\). Since \(\Vert \tau (f) \Vert \le \Vert f\Vert _2^2\), the resulted inequality is then equivalent to
which was already known in the literature; see the discussion in [1] and references therein.
Since (4.14) no longer holds, an immediate question is whether the uncertainty principle in (4.11) and that in (1) are equivalent, assuming \(\int _{\mathbb {S}^{d-1}} f(y)\, d{\sigma }(y)=0\). That they are not equivalent is shown in the following proposition.
Proposition 1
For \(n \ge 3\), let \(Y \in {\mathcal H}_n^d\), a spherical harmonic of degree \(n\) on \(\mathbb {S}^{d-1}\), and let \(Q\) be a polynomial of degree at most \(n-2\) such that \(\int _{\mathbb {S}^{d-1}} Q(x) d{\sigma }=0\). Assume that both \([Y(x)]^2\) and \([Q(x)]^2\) are even in every coordinate. Let
Then \(\tau (f) = 0\). In particular, (1) becomes the trivial inequality \(\Vert \nabla _0 f\Vert _2^2 \ge 0\), whereas (4.11) shows that \(\Vert \nabla _0 f\Vert _2^2 \ge c > 0\).
Proof
Since the degree of \(Q\) is at most \(n-2\), it follows from the orthogonality of \(Y\) and the even parity of \(Y^2\) and \(Q^2\) that
for \(1 \le k \le d\). Hence, \(\tau (f) =0\). By its definition, \(\Vert f\Vert _2 =1\), and, by the orthogonality of \(Y\) and the zero mean of \(Q\), we see that \(\int _{\mathbb {S}^{d-1}} f(x) d{\sigma }=0\) so that (4.11) is applicable on \(f\). \(\square \)
As a simple example of the function \(f\), we can choose \(Q(x) = x_1^k\) and \(Y(x) = C_n^{\lambda }(x_1)\) for \(x =(x_1,\ldots , x_d) \in \mathbb {S}^{d-1}\), where \({\lambda }= (d-2)/2\) and \(1 \le k \le n -2\).
Reference
Dai, F., Xu, Y.: The Hardy–Rellich inequality and uncertainty principle on the sphere. Constr. Approx. 40, 141–171 (2014)
Acknowledgments
The authors thank Stefan Steinerberger for pointing out the mistake in Theorem 4.5. of [1].
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The online version of the original article can be found under doi:10.1007/s00365-014-9235-5.
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Dai, F., Xu, Y. Erratum to: The Hardy–Rellich Inequality and Uncertainty Principle on the Sphere. Constr Approx 42, 181–182 (2015). https://doi.org/10.1007/s00365-015-9278-2
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DOI: https://doi.org/10.1007/s00365-015-9278-2