Stein与Shakarchi《傅里叶分析》第213页问题3推导步骤疑问求助

Hey there! Let's break down your confusion step by step—great job laying out your work so far, that makes it way easier to target the issue.

First, let's resolve the core algebraic inequality you're stuck on, since this is the immediate hurdle:

$$\left( x_{d+1}^{2}\sum {j=1}^{d} x{{j}}^{2}\right){1/2} \leqslant \frac{1}{2}\left( x_{d+1}^{2} +\sum {j=1}^{d} x{{j}}^{2}\right)$$

This is 100% correct, and it doesn't depend on any properties of the wave equation at all—it's a direct application of the classic Arithmetic Mean-Geometric Mean (AM-GM) inequality for non-negative real numbers. Let's simplify the notation to make this obvious:
Let $A = x_{d+1}^2$ (non-negative, since it's a square) and $B = \sum_{j=1}^d x_j^2$ (also non-negative). The inequality reduces to:
$$\sqrt{AB} \leq \frac{A + B}{2}$$
This is the basic two-variable AM-GM result, where equality holds exactly when $A = B$. So your step from $\left(\left(\frac{\partial u}{\partial t}\right)^2\sum_{j=1}d \left(\frac{\partial u}{\partial x_j}\right)^2\right){1/2}$ to $\frac{1}{2}\left(\left(\frac{\partial u}{\partial t}\right)^2 + \sum_{j=1}^d \left(\frac{\partial u}{\partial x_j}\right)^2\right)$ is totally valid, no differential equation required here.

Now, about your question of whether the wave equation solution property comes into play: that will almost certainly matter for the rest of the problem, not this specific inequality step. For example, if the problem is asking you to derive an energy estimate for the wave equation, you'll later use the fact that $u$ satisfies $\frac{\partial^2 u}{\partial t^2} = \sum_{j=1}^d \frac{\partial^2 u}{\partial x_j^2}$ to relate the time derivative of the energy to these cross terms you're bounding right now. But this algebraic inequality is a standalone, purely mathematical step.

Let me double-check your initial derivation to be thorough:

1. The first inequality $\left|\frac{\partial u}{\partial t} \nabla u \cdot \mathbf{v}\right| \leq \left| \frac{\partial u}{\partial t} \nabla u \right|$ is correct (you missed a pair of absolute values on the left, but the logic holds) because $|\mathbf{a} \cdot \mathbf{b}| \leq |\mathbf{a}||\mathbf{b}|$, and if $\mathbf{v}$ is a unit vector (which it looks like you're assuming here), then $|\nabla u \cdot \mathbf{v}| \leq |\nabla u|$.
2. The step $\left| \frac{\partial u}{\partial t} \nabla u \right| = \sqrt{\left(\frac{\partial u}{\partial t}\right)^2 |\nabla u|^2}$ is just the definition of the magnitude of a scalar multiple of a vector.
3. Applying AM-GM to that gives your final bound—this all checks out perfectly.

To summarize: your simplification here is completely correct, and the inequality is just basic algebraic manipulation. The wave equation's properties will be used later in the problem, not for this specific step.

备注：内容来源于stack exchange，提问作者bio grisha