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 \(p\in [1,\infty ],\)

$$\begin{aligned} \Vert e^{-t_n\mathcal L _n}f\Vert _{L^p(A_n,\nu _n)}\le \Vert f\Vert _{L^p(A_n,\nu _n)},\quad t_n>0,\quad f\in L^p(A_n,\nu _n) \cap L^2(A_n,\nu _n). \nonumber \\ \end{aligned}$$
(1)

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),

$$\begin{aligned} \int _{\mathbb{R }^d}\sup _{T\in (0,\infty )^d}|\mathcal M (m_{N,T})(u_1,\ldots ,u_d)|\,\Vert \mathcal L ^{iu_1,\ldots ,iu_d}\Vert _{p\rightarrow p}\,du<\infty . \end{aligned}$$

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

$$\begin{aligned} \sum _{l\in \mathbb{Z }} h(x-l)=1,\quad x\in \mathbb{R }. \end{aligned}$$

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

$$\begin{aligned} b_{j,k}(\xi )&=\int _{\mathbb{R }^2}h_{k}(u)\Gamma (N-iu_1)\Gamma (N-iu_2)e^{-ij_1 u_1}e^{-ij_2u_2}\xi _1^{iu_1}\xi _2^{iu_2}\,du\\&=\int _\mathbb{R }h_{k_1}(u_1)\Gamma (N-iu_1)e^{-ij_1 u_1}\xi _1^{iu_1}\,du_1\\&\qquad \times \int _\mathbb{R }h_{k_2}(u_2)\Gamma (N-iu_2)e^{-ij_2 u_2}\xi _2^{iu_2}\,du_2,\quad \xi _1, \xi _2 \in \Sigma _{\pi /2}, \end{aligned}$$

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

$$\begin{aligned} \Vert g_N(f)\Vert _p\le \sum _{k\in \mathbb{Z }^2}\left\Vert\left(\sum _{j\in \mathbb{Z }^2}|b_{j,k}(L_1,L_2)f|^2\right)^{1/2}\right\Vert_p. \end{aligned}$$

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

$$\begin{aligned} \Vert g_N(f)\Vert _p\le \sum _{k\in \mathbb{Z }^2}\sup _{|a_{j_1}|\le 1,|a_{j_2}|\le 1}\left\Vert\left(\sum _{j_1\in \mathbb{Z }}a_{j_1} b_{j_1,k_1}(L_1)\right)\left(\sum _{j_2\in \mathbb{Z }}a_{j_2} b_{j_2,k_2}(L_2)\right)f\right\Vert_p. \end{aligned}$$

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

$$\begin{aligned} \Vert g_N(f)\Vert _p\le C_p \Vert f\Vert _p,\quad 1<p<2. \end{aligned}$$

Note that assumption (1) is needed to justify the use of the crucial transference result of [1, Theorem 1 and Lemma 1.4].