1 Erratum to: Monatsh Math (2012) 168:125–149 DOI 10.1007/s00605-011-0363-8
In order for the main theorem of the original paper to be true one needs the additional assumption on \(L^p\) contractivity of the heat semigroups of the investigated operators. We need to assume that for each \(n=1,\ldots ,d\) and all \(p\in [1,\infty ],\)
Since (1) implies item (ii) of Theorem 2.3, to draw the conclusion of the main theorem (Theorem 2.3) of the original paper (i.e. the \(L^p\) boundedness of the operator \(m(\mathcal L )\)), it now suffices to assume only item (i),
We kindly refer the reader to the original paper for the definitions of the quantities considered above. Condition (1) is not a serious restriction and it is satisfied in case of all applications presented in the original paper. We need to include (1) in order to obtain Theorem 2.4 for \(1<p<2,\) i.e. the \(L^p\) boundedness of the \(g\)-function \(g_N\) for \(1<p<2.\) The proof of Theorem 2.4 from the original paper, is incorrect for \(1<p<2.\) The correction we present here is a slight modification of the proof of [2, Theorem 1.5 ii)].
For the sake of simplicity we focus on \(d=2.\) The notations we use are from the original paper. Let \(N\in \mathbb{N }_+\) be fixed. Take a smooth function \(h\) on \(\mathbb{R }\), supported in \([-1,1],\) and such that
Then we set \(h_{k}(x)=h_{k_1}(x_1)h_{k_2}(x_2)=h(x_1-k_1)h(x_2-k_2).\) Next, for each \(j\in \mathbb{Z }^2\) we define the functions
where \(\Sigma _{\pi /2}=\{z\in \mathbb{C }\,:\, \text{ Re}(z)>0\},\) is the right complex half plane. Proceeding as in [2, p. 2207] we easily see that
Then from a two-dimensional variant of [2, Lemma 1.3] (which is easily proved by using a two-dimensional Khinchine inequality) and the product structure of \(b_{j,k}\) it follows that
Using the latter inequality and [2, Lemma 1.4] we easily adjust the final steps of [2, Theorem 1.5 ii)] to our situation, obtaining the desired bound
Note that assumption (1) is needed to justify the use of the crucial transference result of [1, Theorem 1 and Lemma 1.4].
References
Cowling, M.G.: Harmonic analysis on semigroups. Ann. Math. 117, 267–283 (1983)
Meda, S.: On the Littlewood–Paley–Stein g-function. Trans. Am. Math. Soc. (3) 110, 639–647 (1990)
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The online version of the original article can be found under doi:10.1007/s00605-011-0363-8.
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Wróbel, B. Erratum to: Multivariate spectral multipliers for tensor product orthogonal expansions. Monatsh Math 169, 113–115 (2013). https://doi.org/10.1007/s00605-012-0456-z
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DOI: https://doi.org/10.1007/s00605-012-0456-z