Correction to: On a Problem of Pichorides

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

$$\begin{aligned} \begin{aligned}&\Big ( \int _{{\mathbb {R}}} \sum _{j=1}^{N} \big [ S_M^{(\Lambda )} (g_j) (x) \big ]^2 w (x) dx \Big )^{1/2} \le \\&\quad \quad C \sigma _{\Lambda }^{1/2} [w]^{3/2}_{A_2 } \Big ( \int _{{\mathbb {R}}} \sum _{j=1}^{N } | g_j (x) |^2 w (x) dx \Big )^{1/2} \end{aligned} \end{aligned}$$

(1)

for all Schwartz functions $g_1, \ldots , g_N$. It thus follows from (1) and [2, Theorem 3.1] that

$$\begin{aligned}&\Big ( \int _{{\mathbb {R}}} \Big ( \sum _{j=1}^{N} \big [ S_M^{(\Lambda )} (h_j) (x) \big ]^2 \Big )^{p/2} dx \Big )^{1/p} \le \nonumber \\&\quad A \sigma _{\Lambda }^{1/2} (p-1)^{-3/2} \Big ( \int _{{\mathbb {R}}} \Big ( \sum _{j=1}^{N} \big | h_j (x) |^2 \Big )^{p/2} dx \Big )^{1/p} \end{aligned}$$

(2)

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

$$\begin{aligned}&\Big ( \int _{{\mathbb {R}}} \big [ S_{N_1, N_2}^{(\Lambda _1, \Lambda _2)} (f) (x,y) \big ]^p dx \Big )^{1/p} \le \nonumber \\&\quad A \sigma _{\Lambda _1}^{1/2} (p-1)^{-3/2} \Big ( \int _{{\mathbb {R}}} \big [ S_{N_2}^{( \Lambda _2)} (f) (x,y) \big ]^p dx \Big )^{1/p} , \end{aligned}$$

(3)

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

$$\begin{aligned}&\Vert S_{N_1, N_2}^{(\Lambda _1, \Lambda _2)} (f) \Vert _{L^p ({\mathbb {R}}^2)} \le \nonumber \\&\quad A^2 \sigma _{\Lambda _1}^{1/2} \sigma _{\Lambda _2}^{1/2} (p-1)^{-3} \Vert f \Vert _{L^p ({\mathbb {R}}^2)} . \end{aligned}$$

(4)

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 $d=1$.