$$\mathcal{L}(u) := \frac{\alpha - 2}{2 - \beta} \int\_{\partial \Omega} u^2 \Gamma^{\frac{\alpha - 2}{2 - \beta} - 1} \langle \tilde{\nabla} \Gamma, d\nu \rangle - \int\_{\partial \Omega} 2 \Gamma^{\frac{\alpha - 2}{2 - \beta}} u \langle \tilde{\nabla} u, d\nu \rangle.$$

Combining (12.66) and (12.67) we obtain

$$\begin{split} -2\int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}} u \mathcal{L} u d\nu + (\beta + \alpha - 4)(\alpha - 2) \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2 u^2 d\nu \\ = 2 \int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}} |\nabla\_X u|^2 d\nu + \mathcal{C}(u). \end{split} \tag{12.68}$$

By using (12.21) we have

$$\begin{split} & -2 \int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}} u \mathcal{L} u d\nu + (\beta + \alpha - 4)(\alpha - 2) \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2 |u|^2 d\nu \\ & \geq 2 \left( \frac{\beta + \alpha - 4}{2} \right)^2 \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2 |u|^2 d\nu \\ & + \frac{\beta + \alpha - 4}{\beta - 2} \int\_{\partial \Omega} \Gamma^{\frac{\alpha-2}{2-\beta} - 1} |u|^2 \langle \tilde{\nabla} \Gamma, d\nu \rangle + \mathcal{C}(u). \end{split}$$

It follows that

$$\begin{split} -\int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}} u \mathcal{L} u d\nu &\geq \left(\frac{\beta+\alpha-4}{2}\right) \left(\frac{\beta-\alpha}{2}\right) \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_{X} \Gamma^{\frac{1}{2-\beta}}|^{2} |u|^{2} \, d\nu \\ &+ \frac{\beta+\alpha-4}{2(\beta-2)} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} |u|^{2} \langle \widetilde{\nabla}\Gamma, d\nu \rangle + \frac{1}{2} \mathcal{C}(u). \end{split} \tag{12.69}$$

On the other hand, for any > 0, H¨older's and Young's inequalities give

$$\begin{split} -\int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}} u \mathcal{L} u d\nu &\leq \left( \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_{X} \Gamma^{\frac{1}{2-\beta}}|^{2} |u|^{2} d\nu \right)^{1/2} \left( \int\_{\Omega} \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|\nabla\_{X} \Gamma^{\frac{1}{2-\beta}}|^{2}} |\mathcal{L}u|^{2} d\nu \right)^{1/2} \\ &\leq \epsilon \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_{X} \Gamma^{\frac{1}{2-\beta}}|^{2} |u|^{2} d\nu + \frac{1}{4\epsilon} \int\_{\Omega} \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|\nabla\_{X} \Gamma^{\frac{1}{2-\beta}}|^{2}} |\mathcal{L}u|^{2} d\nu. \end{split} \tag{12.70}$$

Inequalities (12.70) and (12.69) imply that

$$\begin{split} \int\_{\Omega} \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|\nabla\_{X}\Gamma^{\frac{1}{2-\beta}}|^{2}} |\mathcal{L}u|^{2} d\nu &\geq \left( -4\epsilon^{2} + (\beta+\alpha-4)(\beta-\alpha)\epsilon \right) \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_{X}\Gamma^{\frac{1}{2-\beta}}|^{2} |u|^{2} d\nu \\ &\quad + \frac{2(\beta+\alpha-4)\epsilon}{\beta-2} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} |u|^{2} \langle \widetilde{\nabla}\Gamma, d\nu \rangle + 2\epsilon \mathcal{C}(u). \end{split}$$

Taking = (β+α−4)(β−α) 8 , we obtain

$$\begin{split} &\int\_{\Omega} \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|\nabla\_{X}\Gamma^{\frac{1}{2-\beta}}|^{2}} |\mathcal{L}u|^{2} d\nu \\ &\geq \frac{(\beta+\alpha-4)^{2}(\beta-\alpha)^{2}}{16} \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_{X}\Gamma^{\frac{1}{2-\beta}}|^{2} |u|^{2} \, d\nu \\ &+ \frac{(\beta+\alpha-4)^{2}(\beta-\alpha)}{4(\beta-2)} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} |u|^{2} \langle \widetilde{\nabla}\Gamma, d\nu \rangle + \frac{(\beta+\alpha-4)(\beta-\alpha)}{4} \mathcal{C}(u), \end{split}$$

which proves (12.63).

Let us now prove (12.64). From (12.68), if we use (12.22), we get

$$\begin{split} & -2 \int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}} u \mathcal{L} u d\nu + (\beta + \alpha - 4)(\alpha - 2) \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_{X} \Gamma^{\frac{1}{2-\beta}}|^{2} |u|^{2} d\nu \\ & \ge 2 \left( \frac{\beta + \alpha - 4}{2} \right)^{2} \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_{X} \Gamma^{\frac{1}{2-\beta}}|^{2} |u|^{2} \, d\nu \\ & + \frac{1}{2} \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_{X} \Gamma^{\frac{1}{2-\beta}}|^{2} \left( \ln \frac{R}{\Gamma^{\frac{1}{2-\beta}}} \right)^{-2} |u|^{2} d\nu \\ & + \frac{1}{(\beta - 2)} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} \left( \ln \frac{R}{\Gamma^{\frac{1}{2-\beta}}} \right)^{-1} |u|^{2} \langle \widetilde{\nabla} \Gamma, d\nu \rangle \\ & + \frac{\beta + \alpha - 4}{\beta - 2} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} |u|^{2} \langle \widetilde{\nabla} \Gamma, d\nu \rangle + \mathcal{C}(u). \end{split}$$

It follows that

$$\begin{split} -\int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}} u \mathcal{L} u d\nu &\geq \left(\frac{\beta+\alpha-4}{2}\right) \left(\frac{\beta-\alpha}{2}\right) \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_{X} \Gamma^{\frac{1}{2-\beta}}|^{2} |u|^{2} \, d\nu \\ &+ \frac{1}{4} \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_{X} \Gamma^{\frac{1}{2-\beta}}|^{2} \left(\ln \frac{R}{\Gamma^{\frac{1}{2-\beta}}}\right)^{-2} |u|^{2} \, d\nu \\ &+ \frac{1}{2(\beta-2)} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} \left(\ln \frac{R}{\Gamma^{\frac{1}{2-\beta}}}\right)^{-1} |u|^{2} \langle \widetilde{\nabla}\Gamma, d\nu \rangle \\ &+ \frac{\beta+\alpha-4}{2(\beta-2)} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} |u|^{2} \langle \widetilde{\nabla}\Gamma, d\nu \rangle + \frac{1}{2} \mathcal{C}(u). \end{split}$$

Inequalities (12.70) and (12.71) imply that

$$\begin{split} \int\_{\Omega} \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|\nabla\_{X}\Gamma^{1-\frac{\alpha}{2-\beta}}|^{2}} |\mathcal{L}u|^{2} d\nu &\geq \left(-4\epsilon^{2} + (\beta+\alpha-4)(\beta-\alpha)\epsilon\right) \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_{X}\Gamma^{\frac{1}{2-\beta}}|^{2} |u|^{2} d\nu \\ &+ \epsilon \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_{X}\Gamma^{\frac{1}{2-\beta}}|^{2} \left(\ln \frac{R}{\Gamma^{\frac{1}{2-\beta}}}\right)^{-2} |u|^{2} d\nu \\ &+ \frac{2\epsilon}{(\beta-2)} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} \left(\ln \frac{R}{\Gamma^{\frac{1}{2-\beta}}}\right)^{-1} |u|^{2} \langle \widetilde{\nabla}\Gamma, d\nu \rangle \\ &+ \frac{2(\beta+\alpha-4)\epsilon}{\beta-2} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} |u|^{2} \langle \widetilde{\nabla}\Gamma, d\nu \rangle + 2\epsilon \mathcal{C}(u). \end{split}$$

Taking = (β+α−4)(β−α) 8 , we obtain

$$\begin{aligned} &\int\_{\Omega} \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2} |\mathcal{L}u|^2 d\nu \\ &\geq \frac{(\beta+\alpha-4)^2(\beta-\alpha)^2}{16} \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2 |u|^2 \, d\nu \end{aligned}$$

$$\begin{split} &+\frac{(\beta+\alpha-4)(\beta-\alpha)}{8} \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_{X}\Gamma^{\frac{1}{2-\beta}}|^{2} \left(\ln\frac{R}{\Gamma^{\frac{1}{2-\beta}}}\right)^{-2} |u|^{2}d\nu \\ &+\frac{(\beta+\alpha-4)(\beta-\alpha)}{4(\beta-2)} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} \left(\ln\frac{R}{\Gamma^{\frac{1}{2-\beta}}}\right)^{-1} |u|^{2} \langle\widetilde{\nabla}\Gamma, d\nu\rangle \\ &+\frac{(\beta+\alpha-4)^{2}(\beta-\alpha)}{4(\beta-2)} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} |u|^{2} \langle\widetilde{\nabla}\Gamma, d\nu\rangle + \frac{(\beta+\alpha-4)(\beta-\alpha)}{4} \mathcal{C}(u). \end{split}$$

This completes the proof of Theorem 12.7.1. -

By a modification and refinement of the proof of Theorem 12.7.1 we can obtain another alternative of an improved Rellich inequality with boundary terms.

Theorem 12.7.2 (Refined local Rellich inequalities for sums of squares). *Let* y ∈ M *be such that* (A+ y ) *holds with the fundamental solution* Γ=Γy *in* Ty*. Let* Ω ⊂ Ty *be a strongly admissible domain such that* y ∈ ∂Ω*. Let* α ∈ R*,* β>α> 8−β 3 *,* β > 2 *and* R ≥ e supΩΓ 1 2−*β . Then for all* u ∈ C2(Ω) C1(Ω) *we have*

$$\begin{split} \int\_{\Omega} \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|\nabla\_{X}\Gamma^{\frac{1}{2-\beta}}|^{2}} |\mathcal{L}u|^{2} d\nu &\geq \frac{(\beta-\alpha)^{2}}{4} \int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}} |\nabla\_{X}u|^{2} d\nu \\ &+ \frac{(\beta+3\alpha-8)(\beta+\alpha-4)(\beta-\alpha)}{8(\beta-2)} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} |u|^{2} \langle \widetilde{\nabla}\Gamma, d\nu \rangle \\ &+ \frac{(\beta+\alpha-4)(\beta-\alpha)}{4} \mathcal{C}(u), \end{split} \tag{12.72}$$

*and its further refinement*

$$\begin{split} &\int\_{\Omega} \frac{\Gamma^{\frac{\alpha-\beta}{2-\beta}}}{|\nabla\_{X}\Gamma^{\frac{1}{2-\beta}}|^{2}} |\mathcal{L}u|^{2} d\nu \\ &\geq \frac{(\beta-\alpha)^{2}}{4} \int\_{\Omega} \Gamma^{\frac{\alpha-\beta}{2-\beta}} |\nabla\_{X}u|^{2} d\nu \\ &\quad + \frac{(\beta+3\alpha-8)(\beta-\alpha)}{16} \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_{X}\Gamma^{\frac{1}{2-\beta}}|^{2} \left(\ln\frac{R}{\Gamma^{\frac{1}{2-\beta}}}\right)^{-2} |u|^{2} d\nu \\ &\quad + \frac{(\beta+3\alpha-8)(\beta-\alpha)}{8(\beta-2)} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} \left(\ln\frac{R}{\Gamma^{\frac{1}{2-\beta}}}\right)^{-1} |u|^{2} \langle\widetilde{\nabla}\Gamma, d\nu\rangle \\ &\quad + \frac{(\beta+3\alpha-8)(\beta+\alpha-4)(\beta-\alpha)}{8(\beta-2)} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} |u|^{2} \langle\widetilde{\nabla}\Gamma, d\nu\rangle \\ &\quad + \frac{(\beta+\alpha-4)(\beta-\alpha)}{4} \mathcal{C}(u), \end{split} \tag{12.73}$$

*where*

$$\mathcal{L}(u) := \frac{\alpha - 2}{2 - \beta} \int\_{\partial \Omega} u^2 \Gamma^{\frac{\alpha - 2}{2 - \beta} - 1} \langle \widetilde{\nabla} \Gamma, d\nu \rangle - 2 \int\_{\partial \Omega} \Gamma^{\frac{\alpha - 2}{2 - \beta}} u \langle \widetilde{\nabla} u, d\nu \rangle.$$

*Proof of Theorem* 12.7.2*.* Let us first prove (12.72). Let us rewrite (12.68) in the form

$$\begin{split} \frac{1}{2}\mathcal{C}(u) + \int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}} |\nabla\_X u|^2 d\nu \\ = -\int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}} u \mathcal{L}u d\nu + \frac{(\beta+\alpha-4)(\alpha-2)}{2} \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2 |u|^2 d\nu. \end{split} \tag{12.74}$$

Also recalling (12.70) we have

$$-\int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}} u \mathcal{L} u d\nu \le \epsilon \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2 |u|^2 d\nu + \frac{1}{4\epsilon} \int\_{\Omega} \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2} |\mathcal{L}u|^2 d\nu. \tag{12.75}$$

Inequalities (12.75) and (12.74) imply that

$$\begin{split} &\frac{1}{2}\mathcal{C}(u) + \int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}} |\nabla\_X u|^2 d\nu \\ &\qquad \le \left(\frac{(\beta+\alpha-4)(\alpha-2)}{2} + \epsilon\right) \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2 |u|^2 d\nu \\ &\qquad + \frac{1}{4\epsilon} \int\_{\Omega} \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2} |\mathcal{L}u|^2 d\nu. \end{split} \tag{12.76}$$

The already obtained inequality (12.63) can be rewritten as

$$\begin{split} &\frac{16}{(\beta+\alpha-4)^{2}(\beta-\alpha)^{2}} \int\_{\Omega} \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|\nabla\_{X}\Gamma^{\frac{1}{2-\beta}}|^{2}} |\mathcal{L}u|^{2} d\nu - \frac{4}{(\beta+\alpha-4)(\beta-\alpha)} \mathcal{C}(u) \\ & - \frac{4}{(\beta-\alpha)(\beta-2)} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} |u|^{2} \langle \widetilde{\nabla}\Gamma, d\nu \rangle \geq \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_{X}\Gamma^{\frac{1}{2-\beta}}|^{2} |u|^{2} \, d\nu. \end{split}$$

Combining this with (12.76) we obtain

$$\begin{split} \frac{1}{2}\mathcal{C}(u) &+ \left(\frac{(\beta+\alpha-4)(\alpha-2)}{2} + \epsilon\right) \frac{4}{(\beta+\alpha-4)(\beta-\alpha)}\mathcal{C}(u) \\ &+ \left(\frac{(\beta+\alpha-4)(\alpha-2)}{2} + \epsilon\right) \frac{4}{(\beta-\alpha)(\beta-2)} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} |u|^2 \langle \widetilde{\nabla}\Gamma, d\nu \rangle \\ &+ \int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}} |\nabla\_X u|^2 d\nu \\ &\leq \left(\frac{16\epsilon}{(\beta+\alpha-4)^2(\beta-\alpha)^2} + \frac{8(\alpha-2)}{(\beta+\alpha-4)(\beta-\alpha)^2} + \frac{1}{4\epsilon}\right) \\ &\times \int\_{\Omega} \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|\nabla\_X \Gamma^{\frac{1-\beta}{2-\beta}}|^2} |\mathcal{L}u|^2 d\nu. \end{split}$$

Taking = (β+α−4)(β−α) 8 this implies

$$\begin{split} & \frac{(\beta+\alpha-4)(\beta-\alpha)}{4} \mathcal{C}(u) + \frac{(\beta+3\alpha-8)(\beta+\alpha-4)(\beta-\alpha)}{8(\beta-2)} \int\_{\partial\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}-1} |u|^2 \langle \widetilde{\nabla}\Gamma, d\nu \rangle \\ & + \frac{(\beta-\alpha)^2}{4} \int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}} |\nabla\_X u|^2 d\nu \leq \int\_{\Omega} \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2} |\mathcal{L}u|^2 d\nu, \end{split}$$

which shows (12.72).

Let us now prove (12.73). Inequality (12.64) can be rewritten as

$$\begin{aligned} \frac{16}{(\beta+\alpha-4)^2(\beta-\alpha)^2} \int\_{\Omega} \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2} |\mathcal{L}u|^2 d\nu - \frac{16}{(\beta+\alpha-4)^2(\beta-\alpha)^2} \mathcal{D}u \\ \geq \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2 |u|^2 d\nu, \end{aligned}$$

where

$$\begin{split} \mathcal{D} & \coloneqq \frac{(\beta + \alpha - 4)(\beta - \alpha)}{8} \int\_{\Omega} \Gamma^{\frac{\alpha - 4}{2 - \beta}} |\nabla\_{X} \Gamma^{\frac{1}{2 - \beta}}|^{2} \left( \ln \frac{R}{\Gamma^{\frac{1}{2 - \beta}}} \right)^{-2} |u|^{2} d\nu \\ & \quad + \frac{(\beta + \alpha - 4)(\beta - \alpha)}{4(\beta - 2)} \int\_{\partial\Omega} \Gamma^{\frac{\alpha - 2}{2 - \beta} - 1} \left( \ln \frac{R}{\Gamma^{\frac{1}{2 - \beta}}} \right)^{-1} |u|^{2} \langle \widetilde{\nabla} \Gamma, d\nu \rangle \\ & \quad + \frac{(\beta + \alpha - 4)^{2}(\beta - \alpha)}{4(\beta - 2)} \int\_{\partial\Omega} \Gamma^{\frac{\alpha - 2}{2 - \beta} - 1} |u|^{2} \langle \widetilde{\nabla} \Gamma, d\nu \rangle + \frac{(\beta + \alpha - 4)(\beta - \alpha)}{4} \mathcal{C}(u). \end{split}$$

Combining it with (12.76) we obtain

$$\begin{split} &\frac{1}{2}\mathcal{L}(u) + \left(\frac{(\beta+\alpha-4)(\alpha-2)}{2} + \epsilon\right) \frac{16}{(\beta+\alpha-4)^{2}(\beta-\alpha)^{2}} \mathcal{D} + \int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}}\_{2-\beta} |\nabla\_{X}u|^{2} d\nu \\ &\leq \left(\frac{16\epsilon}{(\beta+\alpha-4)^{2}(\beta-\alpha)^{2}} + \frac{8(\alpha-2)}{(\beta+\alpha-4)(\beta-\alpha)^{2}} + \frac{1}{4\epsilon}\right) \int\_{\Omega} \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|\nabla\_{X}\Gamma^{\frac{1}{2-\beta}}|^{2}} |\mathcal{L}u|^{2} d\nu. \end{split}$$

Taking = (β+α−4)(β−α) 8 we obtain

$$\begin{aligned} \frac{(\beta-\alpha)^2}{8} \mathcal{C}(u) + \frac{\beta+3\alpha-8}{2(\beta+\alpha-4)} \mathcal{D} + \frac{(\beta-\alpha)^2}{4} \int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}} |\nabla\_X u|^2 d\nu \\ \leq \int\_{\Omega} \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2} |\mathcal{L}u|^2 d\nu, \end{aligned}$$

completing the proof. -

Remark 12.7.3. Let us formulate several consequences of the described estimates for the setting of functions u ∈ C∞ 0 (Ω), so that we have C(u)=0. Thus if M = G is a stratified group of homogeneous dimension Q ≥ 3, we take β = Q, so

that Γ 1 2−*β* (x) = d(x) is the L-gauge on the group G. Then the above estimates give refinements compared to known estimates, as, e.g., in Kombe [Kom10], with respect to the inclusion of boundary terms. Thus, for all u ∈ C∞ 0 (Ω), we have that

1. Estimate (12.63) is reduced to

$$\int\_{\Omega} \frac{d^{\alpha}}{|\nabla\_{X}d|^{2}} |\mathcal{L}u|^{2} d\nu \geq \frac{(Q+\alpha-4)^{2}(Q-\alpha)^{2}}{16} \int\_{\Omega} d^{\alpha-4} |\nabla\_{X}d|^{2} |u|^{2} \, d\nu,$$

for Q>α> 4 − Q.

2. Estimate (12.64) is reduced to

$$\begin{split} \int\_{\Omega} \frac{d^{\alpha}}{|\nabla\_{X}d|^{2}} |\mathcal{L}u|^{2} d\nu &\geq \frac{(Q+\alpha-4)^{2}(Q-\alpha)^{2}}{16} \int\_{\Omega} d^{\alpha-4} |\nabla\_{X}d|^{2} |u|^{2} \, d\nu \\ &+ \frac{(Q+\alpha-4)(Q-\alpha)}{8} \int\_{\Omega} d^{\alpha-4} |\nabla\_{X}d|^{2} \left(\ln \frac{R}{d}\right)^{-2} |u|^{2} d\nu, \end{split}$$

for Q>α> 4 − Q.

3. Estimate (12.72) is reduced to

$$\int\_{\Omega} \frac{d^{\alpha}}{|\nabla\_X d|^2} |\mathcal{L}u|^2 d\nu \ge \frac{(Q-\alpha)^2}{4} \int\_{\Omega} d^{\alpha-2} |\nabla\_X u|^2 d\nu,$$

for Q>α> 8−Q 3 .

4. Estimate (12.73) is reduced to

$$\begin{split} \int\_{\Omega} \frac{d^{\alpha}}{|\nabla\_{X}d|^{2}} |\mathcal{L}u|^{2} d\nu &\geq \frac{(Q-\alpha)^{2}}{4} \int\_{\Omega} d^{\alpha-2} |\nabla\_{X}u|^{2} d\nu \\ &+ \frac{(Q+3\alpha-8)(Q-\alpha)}{16} \int\_{\Omega} d^{\alpha-4} |\nabla\_{X}d|^{2} \left(\ln \frac{R}{d}\right)^{-2} |u|^{2} d\nu, \end{split}$$

for Q>α> 8−Q 3 .

For unweighted versions (with α = 0) inequalities (12.63)–(12.64) work under the condition Q ≥ 5 which is usually appearing in Rellich inequalities, while (12.72)– (12.73) work for homogeneous dimensions Q ≥ 9.

# 12.8 Rellich inequalities via Picone identities

In this section we discuss weighted Rellich inequalities in the spirit of Hardy inequalities from Section 12.5, relying on an appropriate version of Picone identities.

Theorem 12.8.1 (Weighted anisotropic Rellich type inequality). *Let* Ω ⊂ M *be an admissible domain. Let* Wi(x) ∈ C2(Ω) *and* Hi(x) ∈ L1 loc(Ω) *be the non-negative weight functions. Let* v > 0*,* v ∈ C2(Ω) C1(Ω) *with*

$$X\_i^2 \left( W\_i(x) |X\_i^2 v|^{p\_i - 2} X\_i^2 v \right) \ge H\_i(x) v^{p-1} \quad \text{and} \quad -X\_i^2 v > 0,\tag{12.77}$$

*a.e. in* Ω*, for all* i = 1,...,N*. Then for every* 0 ≤ u ∈ C2(Ω) C1(Ω) *we have the following inequality*

$$\begin{split} \sum\_{i=1}^{N} \int\_{\Omega} H\_i(x) |u|^{p\_i} d\nu &\le \sum\_{i=1}^{N} \int\_{\Omega} W\_i(x) |X\_i^2 u|^{p\_i} d\nu \\ &+ \sum\_{i=1}^{N} \int\_{\partial\Omega} W\_i(x) |X\_i^2 v|^{p\_i - 2} X\_i^2 v \langle \tilde{\nabla}\_i \left( \frac{u^{p\_i}}{v^{p\_i - 1}} \right), d\nu \rangle \\ &- \sum\_{i=1}^{N} \int\_{\partial\Omega} \left( \frac{u^{p\_i}}{v^{p\_i - 1}} \right) \langle \tilde{\nabla}\_i (W\_i(x) |X\_i^2 v|^{p\_i - 2} X\_i^2 v), d\nu \rangle, \end{split} \tag{12.78}$$

*where* 1 < pi < N *for* i = 1,...,N*, and* ∇ iu = XiuXi*.*

The proof of Theorem 12.8.1 will rely on first establishing an appropriate version of the second-order Picone identity. Before giving its formulation and the proofs, let us make some remarks and also formulate several of its consequences.

#### Remark 12.8.2.

As a consequence of Theorem 12.8.1 we can obtain several Rellich type inequalities involving the sum of squares operator.

Corollary 12.8.3 (Rellich inequalities for sums of squares). *Let* Ω ⊂ M *be an admissible domain, and let the operator* L *is the sum of squares of vector fields:*

$$\mathcal{L} := \sum\_{i=1}^{N} X\_i^2.$$

*Then we have the following estimates:*

(1) *Let* β > 2*,* α ∈ R, β + α > 4 *and* β>α*. Then we have*

$$\int\_{\Omega} \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2} |\mathcal{L}u|^2 d\nu \ge \frac{(\beta+\alpha-4)^2(\beta-\alpha)^2}{16} \int\_{\Omega} \Gamma^{\frac{\alpha-4}{2-\beta}} |\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2 |u|^2 d\nu,$$

*for all* u ∈ C∞ 0 (Ω\{0})*.*

(2) *Let* 1 <p< ∞ *and* 2 − β<α< min{(β − 2)(p − 1),(β − 2)}*. Then for all* u ∈ C∞ 0 (Ω\{0}) *we have*

$$\int\_{\Omega} \frac{\Gamma^{\frac{\alpha+2p-2}{2-\beta}}}{|\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^{2p-2}} |\mathcal{L}u|^p d\nu \ge C(\beta, \alpha, p)^p \int\_{\Omega} \Gamma^{\frac{\alpha-2}{2-\beta}} |\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^2 |u|^p d\nu,\tag{12.79}$$

*where* C(β, α, p) := (β+α−2) p (β−2)(p−1)−α p *.*

*Proof of Corollary* 12.8.3*.* To prove Part (1), we take γ = −β+α−4 2 , and choose the functions W(x) and v such that

$$W(x) = \frac{\Gamma^{\frac{\alpha}{2-\beta}}}{|X\_i \Gamma^{\frac{1}{2-\beta}}|^2} \quad \text{and} \quad v = \Gamma^{\frac{\gamma}{2-\beta}},$$

and apply Theorem 12.8.1.

To prove Part (2), we set

$$W(x) = \frac{\Gamma^{\frac{\alpha+2p-2}{2-\beta}}}{|\nabla\_X \Gamma^{\frac{1}{2-\beta}}|^{2p-2}} \text{ and } v = \gamma^{-\frac{\beta+\alpha-2}{p(2-\beta)}},$$

and apply Theorem 12.8.1. -

Now let us prove the following anisotropic (second-order) Picone type identity, extending its horizontal version in Lemma 6.10.4. Then, as a consequence, we obtain Theorem 12.8.1.

Lemma 12.8.4 (Second-order Picone identity). *Let* Ω ⊂ G *be an open set. Let* u, v *be twice differentiable a.e. in* Ω *and satisfying the following conditions:*

$$u \ge 0, \ v > 0, \ X\_i^2 v < 0 \quad a.e. \ in \ \Omega.$$

*Let* pi > 1*,* i = 1,...,N*. Then we have*

$$L\_1(u,v) = R\_1(u,v) \ge 0,\tag{12.80}$$

*where*

$$R\_1(u,v) := \sum\_{i=1}^N |X\_i^2 u|^{p\_i} - \sum\_{i=1}^N X\_i^2 \left(\frac{u^{p\_i}}{v^{p\_i - 1}}\right) |X\_i^2 v|^{p\_i - 2} X\_i^2 v,$$

*and*

$$\begin{aligned} L\_1(u,v) &:= \sum\_{i=1}^N |X\_i^2 u|^{p\_i} - \sum\_{i=1}^N p\_i \left(\frac{u}{v}\right)^{p\_i - 1} X\_i^2 u X\_i^2 v |X\_i^2 v|^{p\_i - 2} \\ &+ \sum\_{i=1}^N (p\_i - 1) \left(\frac{u}{v}\right)^{p\_i} |X\_i^2 v|^{p\_i} \\ &- \sum\_{i=1}^N p\_i (p\_i - 1) \frac{u^{p\_i - 2}}{v^{p\_i - 1}} |X\_i^2 v|^{p\_i - 2} X\_i^2 v \left(X\_i u - \frac{u}{v} X\_i v\right)^2 .\end{aligned}$$

*Proof of Lemma* 12.8.4*.*  By Young's inequality we have

$$\frac{u^{p\_i - 1}}{v^{p\_i - 1}} X\_i^2 u X\_i^2 v |X\_i^2 v|^{p\_i - 2} \le \frac{|X\_i^2 u|^{p\_i}}{p\_i} + \frac{1}{q\_i} \frac{u^{p\_i}}{v^{p\_i}} |X\_i^2 v|^{p\_i}, \quad i = 1, \dots, N, 1$$

where pi > 1 and qi > 1 with 1 p*i* + 1 q*i* = 1. Since X2 i v < 0, i = 1,...,N, we arrive at

$$\begin{split} L\_{1}(u,v) &\geq \sum\_{i=1}^{N} |X\_{i}^{2}u|^{p\_{i}} + \sum\_{i=1}^{N} (p\_{i}-1) \frac{u^{p\_{i}}}{v^{p\_{i}}} |X\_{i}^{2}v|^{p\_{i}} - \sum\_{i=1}^{N} p\_{i} \left( \frac{|X\_{i}^{2}u|^{p\_{i}}}{p\_{i}} + \frac{1}{q\_{i}} \frac{u^{p\_{i}}}{v^{p\_{i}}} |X\_{i}^{2}v|^{p\_{i}} \right) \\ &- \sum\_{i=1}^{N} p\_{i} (p\_{i}-1) \frac{u^{p\_{i}-2}}{v^{p\_{i}-1}} |X\_{i}^{2}v|^{p\_{i}-2} X\_{i}^{2}v \left| X\_{i}u - \frac{u}{v}X\_{i}v \right|^{2} \\ &= \sum\_{i=1}^{N} \left( p\_{i}-1 - \frac{p\_{i}}{q\_{i}} \right) \frac{u^{p\_{i}}}{v^{p\_{i}}} |X\_{i}^{2}v|^{p\_{i}} \\ &- \sum\_{i=1}^{N} p\_{i} (p\_{i}-1) \frac{u^{p\_{i}-2}}{v^{p\_{i}-1}} |X\_{i}^{2}v|^{p\_{i}-2} X\_{i}^{2}v \left| X\_{i}u - \frac{u}{v}X\_{i}v \right|^{2} \geq 0. \end{split}$$

This completes the proof of Lemma 12.8.4. -

*Proof of Theorem* 12.8.1*.* Let us give a brief outline of the following proof as it is similar to the proof of Theorem 12.5.1: we start by using the Picone type identity (12.80), then we apply an analogue of Green's second formula and the hypothesis (12.77), respectively. Finally, we arrive at (12.78) by using Hi(x) ≥ 0. Summarizing, we have

0 ≤ Ω Wi(x)L1(u, v)dν = Ω Wi(x)R1(u, v)dν = Ω Wi(x)|X2 i u| p*i* dν − Ω X2 i up*i* vp*i*−1 Wi(x)|X2 i v| p*i*−2X2 i vdν = Ω Wi(x)|X2 i u| p*i* dν − Ω up*i* vp*i*−1 X2 i Wi(x)|X2 i v| p*i*−2X2 i v dν + ∂Ω Wi(x)|X2 i v| p*i*−2X2 i v∇i up*i* vp*i*−1 , dν − up*i* vp*i*−1 ∇i(Wi(x)|X2 i v| p*i*−2X2 i v), dν ≤ Ω Wi(x)|X2 i u| p*i* dν − Ω Hi(x)|u| p*i* dν + ∂Ω Wi(x)|X2 i v| p*i*−2X2 i v∇i up*i* vp*i*−1 , dν − up*i* vp*i*−1 ∇i(Wi(x)|X2 i v| p*i*−2X2 i v), dν .

In the last line, we have used (12.77) which leads to (12.78). -

Open Access. This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# Bibliography

M. Ruzhansky, D. Suragan, *Hardy Inequalities on Homogeneous Groups*, Progress in Mathematics 327, https://doi.org/10.1007/978-3-030-02895-4

© The Editor(s) (if applicable) and The Author(s) 2019

# Index

∇ , 53 ∇H, horizontal gradient, 47 ∇m H, 277 ∇γ, 358 A = EE∗, 38 admissible domain, 50, 503 angle function, 374, 380 anisotropic p-sub-Laplacian, 314 anisotropic H¨older spaces, 463 approximation of identity, 25 Baouendi–Grushin distance, 359 operator and vector fields, 358 Bessel kernel, 255 Bessel potential, 249 Bessel–Riesz kernels, 419 Bessel–Riesz operators, 419 generalized, 432 boundary value problem Dirichlet, 452 Kac, 464, 467, 474, 477 Neumann, 453 Robin, 454 C∗, 354 C, Coulomb potential operator, 396 Caffarelli–Kohn–Nirenberg inequality classical, 175 extended, 186 for Euclidean decomposition, 276 fractional, 203 fractional, critical, 212 horizontal, 272, 295, 302 horizontal, with boundary term, 365 Lp, 177 Lp, higher-order, 181

Lp, horizontal, higher-order, 277 new type, 184 Campanato space, 445 characteristic point, 456 characteristic set, 456 comparison principle for p-sub-Laplacian, 497 control distance, 503 convolution, 23 convolution Hardy inequality, 248 critical, 255 coordinate functions on the group, 28 Coulomb potential operator, 396 (−Δp,|·|) s, fractional p-sub-Laplacian, 191 D, double layer, 459 Δγ, 358 D´ıaz–Sa´a inequality, 499 dilations, 13 distribution function, 21 divH, horizontal divergence, 47 divergence formula, 50, 505, 511 domain admissible, 50, 503 half-space, 489 l-strip, 491 l-wedge, 486 of class C1, 456 quadrant-space, 490 strongly admissible, 503 double layer potential, 459 E, Euler operator, 36 εa,b, 65 embeddings horizontal, 296

Engel group, 379

Euler operator, 36 an identity, 171 lower bound, 398 Pythagorean relation, 398 Euler semigroup, 42 Euler–Hilbert–Sobolev spaces, 407 embeddings, 407 Euler–Sobolev spaces, 406 exponential mapping, 12 formulae for invariant derivatives, 28 fractional p-sub-Laplacian, 191 fractional Caffarelli–Kohn–Nirenberg inequality, 203 critical, 212 fractional Gagliardo–Nirenberg inequality, 202 fractional integral operator, 436 fractional Sobolev inequality, 196 fractional Sobolev spaces, 192 weighted, 203 functional spaces S1,p(Ω), ◦ S 1,p (Ω), 493 fundamental solutions for p-sub-Laplacian, 60 on H-type groups, 69 on complex affine group, 355 on Heisenberg group, 65 on stratified groups, 47 G+, half-space, 374 Γα, anisotropic H¨older spaces, 463 Gagliardo seminorm, 192 Gagliardo–Nirenberg inequality, 444 fractional, 202 Gauss' mean value formula, 54, 510 for p-harmonic functions, 57 Gesztesy and Littlejohn inequality, 84 graded group, 230, 249 Green function, 486 Green's identities, 53, 508 for p-sub-Laplacian, 56 group Abelian, 14 Carnot, homogeneous, 46 Carnot, polarizable, 60 complex affine, 354

Engel, 379 graded, 14 H-type, prototype, 68 Heisenberg, 14, 62 homogeneous, 14 nilpotent, 12 stratified, 14, 46 upper triangular, 15 H¨older spaces anisotropic, 463 for control distance, 503 Haar measure, 12 half residue, 474 half-space, G+, 374 Hardy identity L2(G), 71 L2(G), weighted, 76 Lp(G), weighted, 79 Hardy inequality anisotropic, 519 critical, 94, 98 critical, Q = 2, 111 critical, global, 229 critical, local, 229 critical, remainder estimates, 118 critical, subcritical, 109 for Baouendi–Grushin vector fields, 359 for convolution, 248, 255 for fractional powers of sub-Laplacian, 228 fractional, 193, 406 generalised weights, boundary, 363 geometric, L2, 374, 382 geometric, L2, Engel group, 379 geometric, L2, Heisenberg group, 378 geometric, Lp, 380, 385 homogeneous weights, 240 horizontal, 308 horizontal, L2, 288 horizontal, L2, refined, 289 horizontal, Lp, 273 horizontal, Lp, weighted, 279 horizontal, anisotropic, 319 horizontal, critical, 280 horizontal, weighted, 273

Index 569

integral form, 237, 244 L-gauge, boundary term, 366 L2(G), weighted, 77 local, for sums of squares, 513 logarithmic, 100 Lp, classical, 71 Lp, remainder estimate, 348 Lp, with L-gauge, 332 Lp(G), weighted, 79, 104 many-particle, 325 maximal, 266 metric measure spaces, 239 on complex affine group, 355 on homogeneous group, 71 on open sets, 408 on quasi-balls, 110 one-dimensional, 225 original, 237 radial, weighted, 92 remainder estimates, 103, 118 remainder estimates, Q = 2, 116 stability, 121, 123 stability, critical, 125 super weights, 80 two-weight, 85 two-weight, Lp, 341 two-weight, Lp, remainder estimate, 346 two-weight, polarizable Carnot group, 343 two-weight, special cases, 343 with L-gauge, polarizable Carnot groups, 335 with boundary terms, 481 with exponential weights, 328 with multiple singularities, 321 with superweights, 80, 83 Hardy–Littlewood maximal operator, 422 Hardy–Littlewood–Sobolev inequality, 262 reversed, 264 Hardy–Rellich identities Lp-weighted, higher-order, 153 Hardy–Rellich inequality, 144 higher-order, 146 horizontal, 292 horizontal, p-sub-Laplacian, 304

two-parameter, 283 weighted Lp, higher-order, 154 weighted Lp, uncertainty principle, 155 Hardy–Riesz inequalities, xiv Hardy–Sobolev inequality, 228 on graded groups, 230 Hβ(G), Euler–Hilbert–Sobolev, 407 heat kernel, 249 Heisenberg group, 63 quaternionic, 66 Heisenberg–Pauli–Weyl uncertainty principle, 397, 402 homogeneous degree, 26 dimension, 15 quasi-norm, 48 homogeneous Carnot group, 46 homogeneous functions and operators, 19 horizontal divergence, 47 embeddings, 296 gradient, 47 p-Laplacian, 47 horizontal Picone identity, 314, 317 hypoellipticity, 47 Iα,γ, Bessel–Riesz operators, 419 inner quasi-radius, 214 integral Hardy inequality, 237, 244 Iρ,γ - , generalized Bessel–Riesz operator, 432 Kac boundary value problem, 463 Kac's principle of not feeling the boundary, 463 Kα,γ, Bessel–Riesz kernels, 419 Kohn-Laplacian, 65 extended, 472 Komatsu-non-negative operator, 39 L-gauge, 48 La,b, 64 Lp, p-sub-Laplacian, 304 L1,p(Ω), 408 Landau–Kolmogorov inequality, 233

Lk,p(G), Euler–Sobolev space, 406

LMp,φ, generalized Morrey spaces, 421

LMp,φ, Campanato space, 445 LMp,q, Morrey spaces, 421 Lorentz spaces, 409 Lorentz–Zygmund spaces, 409 Lp, p-sub-Laplacian, 47 Lp,q,λ(G), Lorentz–Zygmund, 409 Lp,q(G), Lorentz type spaces, 409 Lp,q,λ, Lp,q,λ1,λ2 (G), Lorentz–Zygmund type spaces, 410 Lp,ρ, weighted p-sub-Laplacian, 306 L|·|,Q,p,q(G), Lorentz type spaces, 409 Lyapunov inequality Euclidean, 213 for Riesz potential, 224 for weighted Riesz potentials, 222 fractional p-sub-Laplacian, 215 fractional p-sub-Laplacian system, 218 p-Laplacian, 213 *M*, Hardy–Littlewood maximal operator, 422 M, momentum operator, 390, 392 maximum principle for p-sub-Laplacian, 496 Minkowski inequality, 242 momentum operator, 390, 392 Morrey spaces, 421 generalized, 421 Newton potential for sub-Laplacian, 464 generalized, 466 trace, 464, 467, 474, 477 nilpotent Lie group and algebra, 12 Olsen inequality, 435 for Bessel–Riesz operators, 440 for fractional integral operators, 439 P, position operator, 390, 392 p-sub-Laplacian anisotropic, 314 fractional, 191 Peetre inequality, 20 perimeter measure, 511 Picone identity, 314, 494 anisotropic, 521 second-order, 317, 541

Picone inequality, 496

Plemelj jump relations, 459 Poincar´e inequality, 408 horizontal, critical, 283 polar decomposition, 22 special case, 21 polynomials, 27 Taylor, 30 position operator, 390, 392 position-momentum identities, 394 Pythagorean inequality, 395, 398 quasi-norm, 16, 17, 48 quaternion Heisenberg group, 66 quaternions, 65 R, radial derivative, 35 R˜, Rellich operator, 130 radial derivative R, 35 rearrangement, 21 Rellich identity Lp, 137 Rellich identity for R˜ critical, higher-order, 160 Rellich inequalities for sums of squares, 540 Rellich inequality classical, 129 for sub-Laplacian with drift, 349 for sums of squares, 531 for sums of squares, refined, 536 horizontal, 279, 292 horizontal, anisotropic, 320 horizontal, for sub-Laplacian with drift, 310 identity, L2, 131 identity, Lp, 137 L2, generalized weights, boundary, 368 L2, weighted, with L-gauge, 339 Lp, generalized weights, boundary, 369 Lp, higher-order, weighted, 155 Lp, weighted, with L-gauge, 337 on homogeneous groups, L2, 131 on homogeneous groups, L2-weighted, 134 on homogeneous groups, Lp, 138 on homogeneous groups, Lp-weighted, 139

#### Index 571

stability, 141 weighted, anisotropic, 540 weighted, higher-order, 150 weighted, iterated, 149 Rellich inequality for R˜ critical, 158 critical, higher-order, 162 Rellich operator R˜, 130 representation formulae for functions polarizable Carnot groups, 61 stratified groups, 55 Rg, generator of dilations, 398 Riesz kernel, 226 Riesz potential, 221, 249 first eigenvalue, 223 Rockland operator, 249 Schur test, 230, 254 simple boundary, 503 single layer potential, 457 continuity, 458 Sj , S0, single layer, 457 Sobolev inequality classical, 164 fractional, 196 Sobolev spaces fractional, 192 fractional, weighted, 203 Sobolev type inequality, 164 horizontal, 299 horizontal, L2, 300 weighted, 170 Sobolev type spaces horizontal weights, 295 Sobolev–Lorentz–Zygmund spaces, 409 Stein–Weiss inequality, 262 graded groups, 263 step of a nilpotent group, 12 of a stratified group, 46 stratified group, 46 free, 49 representation formulae for functions, 55 Stubbe type remainder estimate, 172

sub-Laplacian, 46 with drift, 57 super weights, 80 Taylor formula, 32, 33 polynomials, 30 Tρ, fractional integral operator, 436 triangle inequality, 18 reverse, 20 Trudinger–Moser inequality, 229 uncertainty principle a logarithmic version, 231 critical, 97 for sums of squares, 526 for sums of squares, refined, 528 Hardy–Rellich, weighted Lp, higher-order, 155 Heisenberg–Kennard, 395 Heisenberg–Pauli–Weyl, 179, 397, 402 Heisenberg–Pauli–Weyl, horizontal, 274, 297 Heisenberg–Pauli–Weyl, weighted, with boundary term, 367 logarithmic, 101 on complex affine group, 357 on homogeneous groups, 75 two-weight, polarizable Carnot group, 345 two-weight, special cases, 345 with boundary terms, 484 with multiple singularities, 323 uniqueness of positive solutions, 500 W1Lp,q,λ(G), W1 0 Lp,q,λ(G), Sobolev–Lorentz–Zygmund spaces, 410

Young's inequality, 23 for weak-Lp spaces, 23


**5. Conclusions**

In this population-based study of men with breast cancer, treatment with chemotherapy was significantly associated with elevated risk of CVD mortality, with the highest risk observed among Hispanic men. These findings have important implications for cardiooncology care as well as extending research in the context of noncancerous outcomes in men with breast cancer. With the proportion of cancer patients receiving radiation therapy and chemotherapy increasing over the past few decades [16], future studies on cardiovascular outcomes due to cancer treatment regimens among racially and ethnically diverse MBC patients are warranted to enhance the clinical management of breast cancer in men.

**Author Contributions:** Conceptualization, D.A., M.M. and K.P.; methodology, D.A., M.M. and K.P.; formal analysis, D.A.; investigation, D.A.; resources, D.A., M.M. and K.P.; data curation, D.A.; writing—original draft preparation, D.A.; writing—review and editing, D.A., M.M. and K.P.; supervision, D.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding. **Institutional Review Board Statement:** Institutional review board approval was not required for this study as the SEER registry is a de-identified publicly available database.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data used for this study are publicly available from the National Cancer Institute at https://seer.cancer.gov/, accessed on 20 November 2022.

**Conflicts of Interest:** The authors declare no conflict of interest.


**2.1 Patient positioning and the uterine manipulator insertion**

The uterine manipulator allows to move the uterus up and down or in a lateral position. To facilitate the insertion of a uterine manipulator, the patient must be placed in a lithotomy position with legs in stirrups and spread apart in slight ventral flexion. The buttocks must be placed slightly over the edge of the operating table, a position that allows movements of the uterine manipulator.

A bladder catheter is inserted before the uterine manipulator placement.

The patient arms must be placed along the body to avoid injury to the brachial plexus.

We use the Koh manipulator with three different cups according to the dimensions of the cervix and vagina. For robotic surgery, we need a prominent Trendelenburg position (25–30°). The shoulder braces are mandatory to avoid slipping.

#### **2.2 Abdominal access. First port insertion. Pneumoperitoneum**

In most cases, the access for the vision port is done transumbilical. The umbilical area is the thinnest abdominal wall area, where aponeurosis detaches difficult from the peritoneum.

We anchor the umbilicus with two forceps, and a small incision of the skin is made to allow the insertion of the Veress needle. The correct positioning of the Veress needle inside the peritoneum is checked using known tests: suction, drop, or pressure tests. For abdominal placement of the vision port, the surgical table must be placed in a neutral position to avoid the risk of vascular injury of great vessels.

The pneumoperitoneum, for gynecological surgeries, is done to a pressure of 12–15 mmHg. To allow the insertion of an 8 mm trochar, the umbilical incision will be extended. The trochar is inserted perpendicular to the abdominal wall with the shutter mounted. When removed the shutter, a gas jet under pressure is heard, a sign that the instrument is correctly placed. Connect the cannula tap to the CO2 insufflation system and insert the camera [11, 12].

Other methods of making pneumoperitoneum, the Henson trochar approach, the visiport trochar, or the Veress optical needle, are rarely used [12].

#### **2.3 Ancillary trochars placement**

The working trochars are inserted under visual control. The incision in the skin will be about 8 mm to allow the tight insertion of the trocar. We use, for gynecological surgery, 3 or 4 Da Vinci trochars and one assistant's trochar placed in line. In the case of a simple robotic hysterectomy, we put four trochars in the umbilical cord. The distance between the trochars should be about 8–10 cm to avoid crossing the instruments during surgery. For fragile patients, the holes can be placed at distances of up to 4 cm without exceeding this value. The Da Vinci system trochars, marked with three black lines, are inserted so that the midline is placed at the level of the peritoneum.

The assistant surgeon uses a 12 mm trochar to couple the Air Seal system. It is placed on the right or left side, either on the umbilical line or triangular and cranial, between the optical and Da Vinci 2 or 3 ports.

The distance between the assistant and Da Vinci ports must exceed 7 cm.

Place the patient in the Trendelenburg position at 25– 30°. In the case of the adjoining syndrome, under visual control, adhesiolysis is performed. Remove the small intestine and epiploon from the visual field with an atraumatic pen. Place the surgical table in the lowest position possible. Dock arm 2 of the Da Vinci System at the endoscopic port. Insert the 8 mm 30 degrees endoscope and target the pubic symphysis. Arms 2 and 3 are docked.

The work tools are inserted as follows: on arm 2, the bipolar or vessel sealer extends is mounted, and on arm three is mounted monopolar curved scissors. Arm 4 will remain stowed (**Figure 11**).

## **3. The robotic hysterectomy**

Hysterectomy is one of the most common non-pregnancy-related gynecological surgeries. There are three main types of hysterectomy [13]: *Total hysterectomy, Supracervical* hysterectomy, and *Radical hysterectomy.* The uni- or bilateral adnexectomy or salpingectomy can be associated with any type of hysterectomy.


**3. Effects of Ornithinephenylacetate on glutaminase activity in bile duct rats: Inflammation and ammonia**

In cirrhotic patients it has been shown that the effects of hyperammonemia are synergistic with inflammation (Shawcross, Davies et al. 2004). The effects on cell swelling by cytokines in ammonia-sensitized cultured astrocytes has also been shown (Rama Rao, Jayakumar et al. 2010). However, the mechanisms by which ammonia produces brain swelling are still subject of much investigation. Although the effects on inflammatory processes have been found to contribute to the formation of cerebral edema, it is not clear whether ammonia promotes inflammation or both are independent factors. Inflammatory pathways identified as contributing to the edema include cyclo-oxygenase, nitric oxide (NO)/cyclic guanosine monophosphate (cGMP) signaling and cytokine release (Andrade, Lucena et al. 2005; Montoliu, Piedrafita et al. 2009; Montoliu, Rodrigo et al. 2010).

transcriptional activity. In Figure 1 is shown that patients with long microsatellite showed

0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 **Follow-up (months)**

Fig. 1. Actuarial curve showing patients free of bouts of overt hepatic encephalopathy according to the microsatellite in the promoter region. Patients with long microsatellite

**3. Effects of Ornithinephenylacetate on glutaminase activity in bile duct rats:**

In cirrhotic patients it has been shown that the effects of hyperammonemia are synergistic with inflammation (Shawcross, Davies et al. 2004). The effects on cell swelling by cytokines in ammonia-sensitized cultured astrocytes has also been shown (Rama Rao, Jayakumar et al. 2010). However, the mechanisms by which ammonia produces brain swelling are still subject of much investigation. Although the effects on inflammatory processes have been found to contribute to the formation of cerebral edema, it is not clear whether ammonia promotes inflammation or both are independent factors. Inflammatory pathways identified as contributing to the edema include cyclo-oxygenase, nitric oxide (NO)/cyclic guanosine monophosphate (cGMP) signaling and cytokine release (Andrade, Lucena et al. 2005;

showed higher risk of overt hepatic encephalopathy (log-rank: 7.74; p<0.01)

Montoliu, Piedrafita et al. 2009; Montoliu, Rodrigo et al. 2010).

higher risk of overt hepatic encephalopathy.

1.0

0.8

0.6

0.4

**% Patients free of overt HE**

0.2

0.0

**Inflammation and ammonia**

Hyperammonemia could increase blood-brain-barrier permeability to systemic cytokines. It is also possible that several factors associated with the systemic inflammatory response syndrome could modulate brain dysfunction induced by hyperammonaemia. These processes, together with a genetic factor, may help to explain the differences that sometimes exist between lower ammonia levels and observed brain impairment in some patients. It has been shown that the presence of HE grade 3/4 correlates better with inflammation than with ammonia plasma levels (Shawcross, Sharifi et al. 2010), though extracellular brain ammonia levels may be significantly higher. One recent study showed that in a cirrhosis animal model in which plasma and brain cytokines were markedly elevated following administration of lipopolysaccharide (LPS), pre-treatment with OP prevented increased levels of TNFα and IL-6 (trend) in plasma and in brain observed in the control group (Wright G 2010). Moreover, OP reduced LPS induced development of pre-coma/coma and worsening of brain edema. It is well-known that the transcription of NFkB directly increases proinflammatory cytokines and leads to induction of nitric oxide synthase. (Li and Verma 2002). OP reduced iNOS and NFkB expressions in the cortical brain region of cirrhotic animals, indicating that ammonia reduction may modulate neuroinflammation. In cirrhosis a paradox exists between reduced intrahepatic NO generation and excess NO in the splanchnic circulation. Splanchnic vasodilatation leads to vasoconstriction of numerous vascular beds, including the liver, kidneys, and has significant effects on the brain. Asymmetric dimethylarginine (ADMA) is an endogenous inhibitor of eNOS (endothelial nitric oxide synthase), the levels of which are increased in liver failure (Leiper, Nandi et al. 2007; Mookerjee, Malaki et al. 2007). It has been shown that treatment of cirrhotic rats with OP resulted in restoration of the NOS pathways which may have a direct effect on cerebral perfusion (Balasubramaniyan, Wright et al. 2011).

Physical symptoms of HE and minimal HE have been detected by motor-evoked potentials (MEP) which examines the function of signal transmission along the nerve, which is perturbed by low grade brain edema. Similar disturbances have been found in patients with cirrhosis using magnetic resonance (MR), with signs compatible with low-grade edema along the corticospinal tract. These abnormalities were related to functional impairment detected by transcranial magnetic stimulation and were found to be reversed after liver transplantation. Recently the assessment of MEP in awake rats has been validated to monitor HE in animal models of liver failure (induced by portocaval anastomosis-PCA) and precipitated HE (simulated gastrointestinal bleed-GB). These models have been utilized to test the efficacy of OP, demonstrating that OP treatment prevents the neurophysiological abnormalities induced by a GB insult in PCA animals. Administration of OP over differing time periods (between 3 hours and 3 days) as a pretreatment prevents the decrease in the amplitude and increase in MEP latency at 6 hours post GB (Oria M, Romero-Gimenez J et al. 2011).

In preliminary studies, it has been shown that the combination of ornithine with phenylacetate to treat hyperammonaemia in cirrhosis is effective in animal models. Administration of OP results in increased conversion of glutamate to glutamine by stimulation of GS activity in the muscle with the subsequent excretion of phenylacetylglutamine in the urine. GA has been found to contribute to hyperammonaemia in cirrhosis and in hyperammonaemia animal models (Romero-Gomez, Grande et al. 2004; Romero-Gomez, Jover et al. 2006).

In this novel approach to target the altered interorgan ammonia metabolism in liver failure, OP utilizes the activity of GS to trap ammonia as glutamine, then phenylacetate facilitates its excretion as phenylacetylglutamine (Davies, Wright et al. 2009; Ytrebo, Kristiansen et al. 2009). The effectiveness of this approach with OP has been confirmed in animal models of cirrhosis and ALF. The reduction (≈50%) of plasma ammonia was associated with (a) an improvement in grade of HE in cirrhotic patients and (b) a reduction in ICP in acute liver failure. OP treatment significantly reduced ammonia concentrations, which was associated with a reduction in brain water and an increase in myoinositol levels, indicating an improvement in brain metabolism (Jalan, Wright et al. 2007; Davies 2009; Ytrebo LM 2009).

In a devascularised pig model of ALF, the rise in arterial ammonia was attenuated with OP which was accompanied by a significant decrease in extracellular brain ammonia and prevention of intracranial hypertension (Ytrebo, Sen et al. 2006).

In our studies we included twenty-five male Sprague-Dawley rats: 4 sham operated, and 21 BDL. 5 BDL's received OP (5 days, IP 0.6 g/kg), 5 BDL's received ornithine (5 days, IP 0.6 g/kg), 5 BDL's received phenylacetate (5 days, IP 0.6 g/kg) and 6 received saline (IP). We measured plasma levels for: ammonia and standard biochemical markers. Expressions of GS, GA and ornithine amino transferase (OAT) were determined by Western-blot (expressed as a % of sham values) and enzyme activity was determined by end-point methods in liver, kidney, gut, muscle and lung. We found that plasma ammonia was decreased in BDL-OP rats vs. BDL-saline (58.97±6.02 vs. 106.2±20.56 µmol/L;P<0.05). BDL-OP rats showed increased GS expression in liver (66% BDL-OP vs. 55% BDL-saline; P < 0.01) and showed further increased levels in the muscle (153% BDL-OP vs.142% BDL-saline). OP ameliorates the BDL related increases in glutaminase expression (124%vs.163%; P<0.05) and activity (0.45±0.16mIU/mg protein BDL-OP vs. 1.14±0.046mIU/mg protein BDL-saline; P<0.01) in gut. We demonstrated that this prevention is due to effect of ornithine in glutaminase activity (0.46±0.17mIU/mg protein BDL-O vs. BDL-saline; P<0.05) and not to phenylacetate. OP treatment in BDL rats increased the conversion of glutamate to glutamine by stimulation of GS in the muscle and also resulted in normalization of glutaminase expression and activity in the gut, indicating that OP effectively restricts the production of *in vivo* ammonia in a cirrhotic. Mechanism of action of OP on the metabolism of ammonia is shown in the Figure 2.