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1 Correction to: The Journal of Geometric Analysis (2021) 31:7455–7512 https://doi.org/10.1007/s12220-020-00550-8
It is claimed in [1] that the proof of [1, Lemma 21] follows from an adaptation of the corresponding argument of [2], as presented in [4]. We recently realised that this is not the case and at this point we do not know whether [1, Lemma 21] holds as stated or not for \(d>1\). Therefore, [1, Lemma 21] should be disregarded for \(d>1\). Note that the one-dimensional case corresponds to [2, Theorem 3.1].
We remark that although [1, Lemma 21] was used in the proof of [1, Theorem 19], [1, Theorem 19] is correct via an alternative argument that we describe below. We also note that [1, Lemma 21] was not used in any other proof or argument in [1].
1.1 Correction of the proof of [1, Theorem 19]
We use the notation of [1].
The one-dimensional case is a consequence of [1, (7.5)] (for \(d=1\)) combined with [2, Theorem 3.1].
To establish the two-dimensional case, we shall argue as in [3]. Note that it follows from [1, (7.5)] for \(d=1\) that there exists an absolute constant \(C>0\) such that for all \(w \in A_2 ({\mathbb {R}})\), \(M , N \in {\mathbb {N}}\), and \(\Lambda \subset {\mathbb {N}}\) with \(\sigma _{\Lambda } < \infty \) one has
for all Schwartz functions \(g_1, \ldots , g_N\). It thus follows from (1) and [2, Theorem 3.1] that
for all \(p \in (1,2)\) and \(h_1, \ldots , h_N \in L^p ({\mathbb {R}})\), where \(A > 0\) is an absolute constant.
Fix \(p \in (1,2)\), \(N_1, N_2 \in {\mathbb {N}}\), and \(\Lambda _1, \Lambda _2 \subset {\mathbb {N}}\) with \(\sigma _{\Lambda _1}, \sigma _{\Lambda _2} < \infty \) and let f be a Schwartz function on \(\mathbb {R}^2\). Let I denote the one-dimensional identity operator. For \(y \in {\mathbb {R}}\), using (2) for \(\Lambda = \Lambda _1\), \(M=N_1\), \(N = N_2\), and \(h_j (x) = ( I \otimes \Delta _j^{(\Lambda _2)} ) (f) (x,y) \), \(j \in \{ 1, \ldots , N_2\}\), one gets
where \(S_{ N_2}^{( \Lambda _2)}\) acts on the second variable. Hence, if we raise (3) to the p-th power, integrate in the second variable, use Fubini’s theorem, and then employ (2) for \(\Lambda = \Lambda _2\), \(M=N_2\), \(N = 1\), we deduce that
The desired estimate follows from (4) and a standard limiting argument. The case \(d \ge 3\) is obtained in a completely analogous way.
Remark
In view of the correction presented above, to establish [1, Theorem 19] one only needs [1, (7.5)] for \(d=1\).
References
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Acknowledgements
The author would like to thank Dr. Ioannis Parissis for some useful discussions and for his suggestions that improved the presentation of this note. The author was supported by the ‘Wallenberg Mathematics Program 2018’, Grant No. KAW 2017.0425, financed by the Knut and Alice Wallenberg Foundation.
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Bakas, O. Correction to: On a Problem of Pichorides. J Geom Anal 31, 12637–12639 (2021). https://doi.org/10.1007/s12220-021-00733-x
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DOI: https://doi.org/10.1007/s12220-021-00733-x