Scattering above energy norm of a focusing size-dependent log energy-supercritical Schrödinger equation with radial data below ground state

Abstract

Given $n \in \{ 3,4,5 \}$ and $k > 1$ (resp. $\frac{4}{3}> k > 1$) if $n \in \{ 3,4 \}$ (resp. $n=5$), we prove scattering of the radial ${\tilde{H}}^{k} := {\dot{H}}^{k} ({\mathbb {R}}^{n}) \cap {\dot{H}}^{1} ({\mathbb {R}}^{n})-$ solutions of the focusing log energy-supercritical Schrödinger equation $i \partial _{t} u + \triangle u = -|u|^{\frac{4}{n-2}} u \log ^{\gamma } ( 2 + |u|^{2})$ for a range of positive $\gamma \, s$ depending on the size of the initial data, for critical energies below the ground states’, and for critical potential energies below. In order to control the barely supercritical nonlinearity in the virial identity and in the estimate of the growth of the critical energy for nonsmooth solutions, i.e solutions with data in ${\tilde{H}}^{k}$, $k \le \frac{n}{2}$, we prove some Jensen-type inequalities, in the spirit of Roy (Int Math Res Not 2020(8):2501–2541, 2020).

Appendixes

1.1 Appendix A: Equivalence of assumptions in [8]

Let $\delta > 0$. We denote by $Ass \; 0$, $Ass \; 1$, and $Ass \; 2$ the following assumptions

$$\begin{aligned} \begin{array}{ll} Ass \; 0: &{} E_{\mathrm{cr}}(u_{0})< (1- \delta ) E_{\mathrm{cr}}(W) \\ Ass \; 1: &{} \Vert \nabla u_{0} \Vert _{L^{2}}< \Vert \nabla W \Vert _{L^{2}} \\ Ass \; 2: &{} \Vert u_{0} \Vert _{L^{1_{2}^{*}}} < \Vert W \Vert _{L^{1_{2}^{*}}} \end{array} \end{aligned}$$

It was proved in [8] that if $Ass \; 0$ and $Ass \; 1$ hold, then u exists for all time and scatters as $t \rightarrow \pm \infty$.

Observe that $Ass \; 0$ and $Ass \; 1$ hold if and only if $Ass \; 0$ and $Ass \; 2$ hold. Indeed assume that $Ass \; 0$ and $Ass \; 1$ hold. Then, the sharp Sobolev inequality (see Remark 7) and (28) yield $K_{\mathrm{cr}}(u_{0}) \ge \Vert \nabla u_{0} \Vert ^{2}_{L^{2}} - \frac{1}{\Vert \nabla W \Vert ^{1_{2}^{*} -2}_{L^{2}}} \Vert \nabla u_{0} \Vert ^{1_{2}^{*}}_{L^{2}}$ with $K_{\mathrm{cr}}(f)$ defined in (27). Hence, elementary considerations show that $K_{\mathrm{cr}}(u_{0}) \ge 0$. Hence, from $E_{\mathrm{cr}}(u_{0}) = \left( \frac{1}{2} - \frac{1}{1_{2}^{*}} \right) \Vert u_{0} \Vert ^{1_{2}^{*}}_{L^{1_{2}^{*}}} + \frac{1}{2} K_{\mathrm{cr}}(u_{0})$ and (28) we see that $Ass \; 2$ holds. Conversely if $Ass \; 0$ and $Ass \; 2$ hold, then it is clear that $Ass \; 0$ and $Ass \; 1$ hold.

Hence, if $Ass \; 0$ and $Ass \; 2$ hold then u exists for all time and scatters as $t \rightarrow \pm \infty$.

1.2 Appendix B: Scattering for small data

Let $J \subset {\mathbb {R}}$ be an interval containing 0 and such that $J \subsetneqq I_{\max }$. Observe from the embedding $\Vert f \Vert _{L^{\frac{2(n+2)}{n-2}}} \lesssim \Vert D f \Vert _{L^{\frac{2n(n+2)}{n+4}}}$ and the interpolation of $\Vert D u \Vert _{L_{t}^{\frac{2n(n+2)}{n^{2}+4}} L_{x}^{\frac{2n(n+2)}{n^{2}+4}} (J)}$ between $\Vert D u \Vert _{L_{t}^{\infty } L_{x}^{2}(J)}$ and $\Vert D u \Vert _{L_{t}^{\frac{2(n+2)}{n}} L_{x}^{\frac{2(n+2)}{n}} (J)}$ that $\Vert u \Vert _{L_{t}^{\frac{2(n+2)}{n-2}} L_{x}^{\frac{2(n+2)}{n-2}} (J)} \lesssim Q(J,u)$. (16), (22), and Proposition 8 allow to show that if $\Vert u_0 \Vert _{{\tilde{H}}^{k}} \ll 1$, then the solution u constructed by Proposition 1 satisfies the conclusions of Theorem  3. More precisely, there exists a positive constant C such that

$$\begin{aligned} \begin{array}{ll} Q(J,u) &{} \lesssim \Vert u_0 \Vert _{{\tilde{H}}^{k}} + \Vert D( |u|^{\frac{4}{n-2}} u g(|u|) ) \Vert _{L_{t}^{\frac{2(n+2)}{n+4}} L_{x}^{\frac{2(n+2)}{n+4}}(J)}\\ &{}\quad +\, \Vert D^{k}( |u|^{\frac{4}{n-2}} u g(|u|) ) \Vert _{L_{t}^{\frac{2(n+2)}{n+4}} L_{x}^{\frac{2(n+2)}{n+4}} (J)} \\ &{} \lesssim \Vert u_0 \Vert _{{\tilde{H}}^{k}} + \langle Q(J,u) \rangle ^{C} \Vert u \Vert ^{\frac{4}{n-2}-}_{L_{t}^{\frac{2(n+2)}{n-2}} L_{x}^{\frac{2(n+2)}{n-2}}(J)} \\ &{} \lesssim \Vert u_0 \Vert _{{\tilde{H}}^{k}} + \langle Q(J,u) \rangle ^{C} Q^{\frac{4}{n-2}-}(J,u) \cdot \end{array} \end{aligned}$$

A continuity argument applied to the estimate above shows that $I_{\max } = {\mathbb {R}}$ and that $Q ({\mathbb {R}},u) \lesssim \Vert u_{0} \Vert _{{\tilde{H}}^{k}} < \infty$.

Let $1 \gg \epsilon > 0$. There exists $A(\epsilon )$ large enough such that if $t_2 \ge t_1 \ge A(\epsilon )$ then $\Vert u \Vert _{L_{t}^{\frac{2(n+2)}{n-2}} L_{x}^{\frac{2(n+2)}{n-2}} ([t_1,t_2])} \ll \epsilon$. Hence, $\left\| e^{-it_{1} \triangle } u(t_1) - e^{-i t_{2} \triangle } u(t_2) \right\| _{{\tilde{H}}^{k}} \ll \epsilon$ by a similar estimate to (26). Hence, (13) holds.

1.3 Appendix C: Comments on paper [14]

1.3.1 Proposition 7 in [14]

The proposition stated below is a slight modification of Proposition 7 in [14]:

Proposition 11

Let $0 \le \alpha < 1$, $k'$ and $\beta$ be integers such that $k' \ge 2$ and $\beta > k'-1$, $(r ,r_{2}) \in (1,\infty )^{2}$, $(r_{1},r_{3}) \in (1, \infty ]^{2}$ be such that $\frac{1}{r}= \frac{\beta }{r_{1}} + \frac{1}{r_{2}} +\frac{1}{r_{3}}$. Let $F: {\mathbb {R}}^{+} \rightarrow {\mathbb {R}}$ be a ${\mathcal {C}}^{k'}-$ function, let ${\tilde{F}}: {\mathbb {R}}^{+} \rightarrow {\mathbb {R}}$ be a nondecreasing function, and let $G := {\mathbb {R}}^{2} \rightarrow {\mathbb {R}}^{2}$ be a ${\mathcal {C}}^{k^{'}}-$ function such that

$$\begin{aligned} \begin{array}{l} (a): \; \left\{ \begin{array}{l} \tau \in [0,1]: \; \left| {\tilde{F}} \left( |\tau x + (1-\tau )y|^{2} \right) \right| \lesssim \left| {\tilde{F}}(|x|^{2}) \right| + \left| {\tilde{F}}( |y|^{2}) \right| , \, \text {and} \\ i \in \{0,\ldots ,k^{'} \}: \; F^{[i]}(x) = O \left( \frac{{\tilde{F}}(x)}{x^{i}} \right) \end{array} \right. \\ (b): \; i \in \{0,\ldots ,k^{'} \}: \; G^{[i]}(x,{\bar{x}}) = O \left( |x|^{\beta + 1 - i} \right) \cdot \end{array} \end{aligned}$$

Then

$$\begin{aligned} \begin{array}{ll} \left\| D^{ k' -1 + \alpha } \left( G(f,{\bar{f}}) F(|f|^{2}) \right) \right\| _{L^{r}}&\lesssim \Vert f \Vert ^{\beta }_{L^{r_{1}}} \Vert D^{k' -1 + \alpha } f \Vert _{L^{r_{2}}} \Vert {\tilde{F}}(|f|^{2}) \Vert _{L^{r_{3}}} \end{array} \end{aligned}$$

(57)

Here $F^{[i]}$ and $G^{[i]}$ denote the $i\mathrm{th}$-derivative of F and G, respectively.

Remark 10

We mention the differences between the statement of the proposition above and that of Proposition 7 in [14]:

• the ${\mathcal {C}}^{k}-$ regularity assumption of ${\tilde{F}}$ present in the statement of Proposition 7 in [14] has been removed from that of the proposition above. Indeed, while reading the proof of Proposition 7 in [14] we have observed that this assumption is not necessary to prove (57).

• one has added an extra assumption, namely the nondecreasing property of ${\tilde{F}}$. Indeed, recall that we prove Proposition 7 in [14] first in the particular case where ${\tilde{F}} := F$ and then in the general case. While reading the proof of Proposition 7 in [14] we have observed that this extra assumption²¹ is necessary to apply some Leibnitz-type rules²² and to prove (57).

Hence, taking into account the above remark, the proposition above was proved in [14]. Hence Proposition 7 holds for $\beta > k^{'}-1$. It remains to prove Proposition 7 for $\beta = k^{'} - 1$.

1.3.2 Proposition 7 in [14] and technical error in [14]

In this paragraph, we point out a technical error in [14]. We then explain how to fix the error. Recall that in order to apply use Proposition 7 in [14], we need to assume that $\beta > k' - 1$. In [14], we combine for $n \in \{ 3,4 \}$ this proposition (with $\beta := \frac{4}{n-2}$, $G(f, {\bar{f}}):= |f|^{\frac{4}{n-2}} f$, and ${\tilde{F}}(x) := F(x) := \log ^{c} \left( \log (10 + x) \right)$) with (16) to control norms at ${\dot{H}}^{k}-$ regularity of the nonlinearity on small intervals J for $\frac{n}{2}< k < \frac{n+2}{n-2}$: for example we use estimates in [14] such as $\left\| D^{k} \left( |u|^{\frac{4}{n-2}} u F(|u|^{2}) \right) \right\| _{L_{t}^{\frac{2(n+2)}{n+4}} L_{x}^{\frac{2(n+2)}{n+4}} (J) } \lesssim \Vert u \Vert ^{\frac{4}{n-2}}_{L_{t}^{\frac{2(n+2)}{n-2}} L_{x}^{\frac{2(n+2)}{n-2}} (J)} \Vert F(|u|^{2}) \Vert _{L_{t}^{\infty } L_{x}^{\infty } (J) } \cdot$ While writing this paper we have observed that for $n=3$ the constraint $\beta > k' - 1$ imposes that $k' \le 4$. Hence, in order to use (57) one must assume that $k< 4 - 1 + (1-) < 4$, which does not cover the full range $\frac{n}{2}< k < \frac{n+2}{n-2}$. We can fix this error by using Proposition 7 of this paper in Sect. 2: indeed the less restrictive constraint $\beta \ge k' - 1$ allows for $n \in \{ 3,4 \}$ to use (21) for $\frac{n}{2}< k < \frac{n+2}{n-2}$.

1.4 Appendix D: Proof of Proposition 7 for $\beta = k'-1$

1.4.1 A fractional Leibnitz-type rule

We write below a composition rule:

Proposition 12

Let $0< \alpha < 1$ and let $H \in {\mathcal {C}} \left( {\mathbb {R}}^{2} \right) \cap {\mathcal {C}}^{1} \left( {\mathbb {R}}^{2} - \{ (0,0) \} \right)$ that satisfies the following condition: for all $(x,y,\tau ) \in {\mathbb {R}}^{2} \times [0,1]$ such that $\tau x + (1- \tau ) y \ne 0$, $\left| H^{'} \left( \tau x + (1- \tau ) y \right) \right| \lesssim {\tilde{H}}(x) + {\tilde{H}}(y)$ for some ${\tilde{H}}: {\mathbb {R}}^{2} \rightarrow {\mathbb {R}}$ that is continuous at the origin O and such that ${\tilde{H}}(O) \lesssim {\tilde{H}}(z)$ for all $z \in {\mathbb {R}}^{2}$. Then,

$$\begin{aligned} \begin{array}{ll} \left\| D^{\alpha } H(f) \right\| _{L^{q}}&\lesssim \Vert {\tilde{H}}(f) \Vert _{L^{q_{1}}} \Vert D^{\alpha } f \Vert _{L^{q_{2}}} \end{array} \end{aligned}$$

(58)

if $(q,q_{2}) \in (1,\infty )^{2}$, $q_{1} \in ( 1 , \infty ]$, and $\frac{1}{q}= \frac{1}{q_{1}} + \frac{1}{q_{2}}$.

Remark 11

In (58) and in the proof of Proposition 12 we abuse notation: we write f, f(x), and f(y) for $(f,{\bar{f}})$, $\left( f(x), \overline{f(x)} \right)$, and $\left( f(y), \overline{f(y)} \right)$, respectively.

Proof

Assume that $(*): \left| H \left( f(y) \right) - H(f(x)) \right| \lesssim \left( {\tilde{H}}(f(x)) + {\tilde{H}}(f(y)) \right) \left| f(y) - f(x) \right|$ holds for all $(x,y) \in {\mathbb {R}}^{2} \times {\mathbb {R}}^{2}$. Then, rewriting the proof of Proposition 5.1, Chapter 2 of [11] we see that (58) holds. So it remains to prove $(*)$. To this end, we consider the line segment [f(x), f(y)]. If $O \notin [f(x),f(y)]$ then this follows from the fundamental theorem of calculus. Assume now that $O \in [f(x),f(y)]$. Let $\min \left( |f(x)|, |f(y)| \right)> \epsilon > 0$. Choose two points $( z_{1}, z_{2} ) \in [f(x),f(y)]$ such that $|z_{1}| \le \epsilon$ and $z_{2}$ symmetric of $z_{1}$ with respect to O such that for $Q \in \{ H, {\tilde{H}} \}$, $\left| Q(z_{1}) - Q(z_{2}) \right| \le \epsilon$. Swapping $z_{1}$ with $z_{2}$ if necessary, we see that $|f(x) - f(y)| = |f(x) - z_{2}| + |z_{2} - z_{1}| + |z_{1} - f(y)|$ and that (**) holds with

$$\begin{aligned} (**): \begin{array}{ll} \left| H \left( f(y) \right) - H ( z_{1} ) \right| &{} \lesssim \left( {\tilde{H}}(f(y)) + {\tilde{H}}(z_{1}) \right) | f(y) - z_{1}|, \; \text {and} \\ \left| H \left( f(x) \right) - H ( z_{2} ) \right| &{} \lesssim \left( {\tilde{H}}(f(x)) + {\tilde{H}}(z_{2}) \right) | f(x) - z_{2}| \cdot \end{array} \end{aligned}$$

By the triangle inequality and by letting $\epsilon \rightarrow 0$, we see that $(*)$ holds. Assume now that $f(x)=O$ and that $f(y) \ne O.$ Let $|f(y)|> \epsilon >0$ we choose one point $z_1\in [O,f(y)]$ such that $|z_1|\ge \epsilon .$ Hence the first inequality of (**) holds. By letting $\epsilon \rightarrow$ 0 we see that (*) holds. A similar proof shows that (*) also holds if $f(x) \ne$ O and $f(y)=O$. $\square$

We then recall a product rule (see [11] and references therein). Let $\alpha ' \ge 0$. Then

$$\begin{aligned} \begin{array}{ll} \left\| D^{\alpha '} (fg) \right\| _{L^{q}}&\lesssim \Vert D^{\alpha '} f \Vert _{L^{q_{1}}} \Vert g \Vert _{L^{q_{2}}} + \Vert f \Vert _{L^{q_{3}}} \Vert D^{\alpha '} g \Vert _{L^{q_{4}}} \end{array} \end{aligned}$$

if $(q,q_{1},q_{4}) \in (1,\infty )^{3}$ and $(q_{2},q_{3}) \in (1,\infty ]^{2}$ .

1.4.2 Proof.

We slightly modify an argument in [14].

Let $k^{'} = 2$. Then (see [14]) $\left\| D^{2-1 + \alpha } \left( G(f,{\bar{f}}) F(|f|^{2}) \right) \right\| _{L^{r}} \lesssim A_{1} + A_{2} + A_{3}$ with $A_{1} := \left\| D^{\alpha } \left( \partial _{z} G(f,{\bar{f}}) \nabla f F(|f|^{2}) \right) \right\| _{L^{r}}$, $A_{2} := \left\| D^{\alpha } \left( \partial _{{\bar{z}}} G(f,{\bar{f}}) \overline{\nabla f} F(|f|^{2} \right) ) \right\| _{L^{r}}$, and $A_{3} := \left\| D^{\alpha } \left( F^{'}(|f|^{2}) \Re \left( {\bar{f}} \nabla f \right) G(f,{\bar{f}}) \right) \right\| _{L^{r}}$.

Let $H (x) : = \partial _{z} G(x,{\bar{x}}) F(|x|^{2})$ and ${\tilde{H}}(x) := \left| {\tilde{F}} \left( |x|^{2} \right) \right|$. Then $\left| H^{'}(\tau x + (1- \tau ) y ) \right| \lesssim {\tilde{H}}(x) + {\tilde{H}}(y)$. The composition rule and the product rule show that

$$\begin{aligned} A_{1}\lesssim & {} \left\| D^{\alpha } \left( \partial _{z} G(f,{\bar{f}}) F(|f|^{2}) \right) \right\| _{L^{r_{4}}} \Vert D f \Vert _{L^{r_{5}}} \\&+ \left\| \partial _{z} G(f,{\bar{f}}) F(|f|^{2}) \right\| _{L^{r_{6}}} \left\| D^{(2-1) + \alpha } f \right\| _{L^{r_{2}}} \\\lesssim & {} \left\| {\tilde{F}}(|f|^{2}) \right\| _{L^{r_{3}}} \left\| D^{\alpha } f \right\| _{L^{r_{8}}} \left\| D f \right\| _{L^{r_{5}}} + \Vert f \Vert _{L^{r_{1}}} \left\| D^{(2-1) + \alpha } f \right\| _{L^{r_{2}}} \left\| {\tilde{F}}(|f|^{2}) \right\| _{L^{r_{3}}} \cdot \end{aligned}$$

Here $\frac{1}{r} = \frac{1}{r_{4}} + \frac{1}{r_{5}}$, $\frac{1}{r} = \frac{1}{r_{6}} + \frac{1}{r_{2}}$, $\frac{1}{r_{4}} =\frac{1}{r_{3}} + \frac{1}{r_{8}}$, $\frac{1}{r_{5}} = \frac{1 - \theta _{1}}{r_{1}} + \frac{\theta _{1}}{r_{2}}$, and $\theta _{1} = \frac{1}{1+ \alpha }$. Notice that these relations imply that $\frac{1}{r_{8}} = \frac{\theta _{1}}{r_{1}} + \frac{1 - \theta _{1}}{r_{2}}$. Hence

$$\begin{aligned} \begin{array}{l} \Vert D^{\alpha } f \Vert _{L^{r_{8}}} \lesssim \Vert f \Vert ^{\theta _{1}}_{L^{r_{1}}} \Vert D^{(2-1) + \alpha } f \Vert ^{1- \theta _{1}}_{L^{r_{2}}}, \; \text {and} \\ \Vert D f \Vert _{L^{r_{5}}} \lesssim \Vert f \Vert ^{1 - \theta _{1}}_{L^{r_{1}}} \Vert D^{(2-1) + \alpha } f \Vert ^{\theta _{1}}_{L^{r_{2}}} \cdot \end{array} \end{aligned}$$

Hence, $A_{1} \lesssim \text {R.H.S of}$ (21). Similarly, $A_{2} \lesssim \text {R.H.S of}$ (21) . We also have

$$\begin{aligned} A_{3}&\lesssim \sum \limits _{{\tilde{f}} \in \{ f ,{\bar{f}}\}} \left\| D^{\alpha } \left( F^{'} (|f|^{2}) {\tilde{f}} G(f, {\bar{f}}) \right) \right\| _{L^{r_{4}}} \Vert D f \Vert _{L^{r_{5}}} \\&\quad + \sum \limits _{{\tilde{f}} \in \{ f ,{\bar{f}}\}} \Vert D^{(2-1) + \alpha } f \Vert _{L^{r_{2}}} \Vert F^{'}(|f|^{2}) {\tilde{f}} G(f,{\bar{f}}) \Vert _{L^{r_{6}}} \\&\lesssim A_{3,1} + A_{3,2} \end{aligned}$$

We have $A_{3,2} \lesssim \text {R.H.S of}$ (21). The composition rule shows that $A_{3,1} \lesssim \text {R.H.S of}$ (21).

Assume that Proposition 7 holds for $k'$. Let us prove that Proposition 7 also holds for $k' + 1$. Following [14], we have $\left\| D^{k^{'} + \alpha } \left( G(f,{\bar{f}}) F(|f|^{2}) \right) \right\| _{L^{r}} \lesssim A^{'}_{1} + A^{'}_{2} + A^{'}_{3}$ with $A^{'}_{1} := \left\| D^{k^{'} - 1 + \alpha } \left( \partial _{z} G(f,{\bar{f}}) \nabla f F(|f|^{2}) \right) \right\| _{L^{r}}$, $A^{'}_{2} := \left\| D^{k^{'} -1 + \alpha } \left( \partial _{{\bar{z}}} G(f,{\bar{f}}) \overline{\nabla f} F(|f|^{2}) \right) \right\| _{L^{r}}$, and $A^{'}_{3} := \left\| D^{k^{'} - 1 + \alpha } \left( \Re ( {\bar{f}} \nabla f ) G(f,{\bar{f}}) \right) \right\| _{L^{r}}$. We estimate $A^{'}_{3}$. Applying the induction assumption to the functions ${\check{F}}(x) := x F^{'}(x)$, $G_{1} (x, {\bar{x}}) := \frac{G(x,{\bar{x}})}{x}$, $G_{2} (x, {\bar{x}}) := \frac{G(x,{\bar{x}})}{x},$ and F, we see that

$$\begin{aligned} A^{'}_{3}\lesssim & {} \sum \limits _{{\tilde{f}} \in \{f,{\bar{f}}\}} \left\| D^{k^{'}-1 + \alpha } \left[ G(f,{\bar{f}}) F^{'}(|f|^{2}) {\tilde{f}} \right] \right\| _{L^{r^{'}_{4}}} \Vert D f \Vert _{L^{r^{'}_{5}}}\\&+ \Vert D^{k^{'} + \alpha } f \Vert _{L^{r_{2}}} \Vert G(f,{\bar{f}}) F^{'}(|f|^{2}) {\tilde{f}} \Vert _{L^{r_{6}}} \\\lesssim & {} \Vert f \Vert ^{\beta - 1}_{L^{r_{1}}} \Vert D^{k^{'}-1 + \alpha } f \Vert _{L^{r_{8}^{'}}} \Vert {\tilde{F}}(|f|^{2}) \Vert _{L^{r_{3}}} \Vert D f \Vert _{L^{r_{5}^{'}}}\\&+ \Vert D^{k^{'} + \alpha } f \Vert _{L^{r_{2}}} \Vert f \Vert ^{\beta }_{L^{r_{1}}} \Vert {\tilde{F}}(|f|^{2}) \Vert _{L^{r_{3}}} \\\lesssim & {} \Vert f \Vert ^{\beta }_{L^{r_{1}}} \Vert D^{k^{'} + \alpha } f \Vert _{L^{r_{2}}} \Vert {\tilde{F}}(|f|^{2}) \Vert _{L^{r_{3}}}, \end{aligned}$$

where at the last line we used

$$\begin{aligned} \begin{array}{l} \Vert D f \Vert _{L^{r_{5}^{'}}} \lesssim \Vert f \Vert ^{1 - \theta _{1}^{'}}_{L^{r_{1}}} \Vert D^{(k^{'}+1)-1 + \alpha } f \Vert ^{\theta _{1}^{'}}_{L^{r_{2}}}, \; \text {and} \\ \Vert D^{k' -1+ \alpha } f \Vert _{L^{r_{8}^{'}}} \lesssim \Vert f \Vert ^{\theta _{1}^{'}}_{L^{r_{1}}} \Vert D^{(k^{'}+1)-1 + \alpha } f \Vert ^{1 - \theta _{1}^{'}}_{L^{r_{2}}} \cdot \end{array} \end{aligned}$$

Here $\frac{1}{r_{5}^{'}} = \frac{1- \theta _{1}^{'}}{r_{1}} + \frac{\theta _{1}^{'}}{r_{2}}$ with $\theta _{1}^{'} = \frac{1}{k^{'} + \alpha }$, $\frac{1}{r} = \frac{1}{r_{4}^{'}} + \frac{1}{r_{5}^{'}}$, and $\frac{1}{r_{4}^{'}} = \frac{\beta -1 }{r_{1}} + \frac{1}{r_{8}^{'}} + \frac{1}{r_{3}}$. Observe that these relations imply that $\frac{1}{r_{8}^{'}} = \frac{\theta _{1}^{'}}{r_{1}} + \frac{1- \theta _{1}^{'}}{r_{2}}$. We also have

$$\begin{aligned} A_{1}^{'}\lesssim & {} \Vert D^{k' + \alpha } f \Vert _{L^{r_{2}}} \Vert \partial _{z} G(f,{\bar{f}}) F(|f|^{2}) \Vert _{L^{r_{6}}}\\&+ \left\| D^{k'-1 + \alpha } \left( \partial _{z} G(f,{\bar{f}}) F(|f|^{2}) \right) \right\| _{L^{r_{4}^{'}}} \Vert D f \Vert _{L^{r_{5}^{'}}} \\\lesssim & {} \Vert f \Vert ^{\beta }_{L^{r_{1}}} \Vert D^{k^{'} + \alpha } f \Vert _{L^{r_{2}}} \Vert {\tilde{F}}(|f|^{2}) \Vert _{L^{r_{3}}} \\&+ \Vert f \Vert ^{\beta -1}_{L^{r_{1}}} \Vert D^{k'-1 + \alpha } f \Vert _{L^{r_{8}^{'}}} \Vert {\tilde{F}}(|f|^{2}) \Vert _{L^{r_{3}}} \Vert D f \Vert _{L^{r_{5}^{'}}} \\\lesssim & {} \Vert f \Vert ^{\beta }_{L^{r_{1}}} \Vert D^{k^{'} + \alpha } f \Vert _{L^{r_{2}}} \Vert {\tilde{F}}(|f|^{2}) \Vert _{L^{r_{3}}} \cdot \end{aligned}$$

Similarly, $A^{'}_{2} \lesssim \Vert f \Vert ^{\beta }_{L^{r_{1}}} \Vert D^{k^{'} + \alpha } f \Vert _{L^{r_{2}}} \Vert {\tilde{F}}(|f|^{2}) \Vert _{L^{r_{3}}}$.