I'm learning Fourier analysis by Elias M. Stein & Rami Shakarchi and got stuck on the step in problem 3 on page 213.
Here is my attempt: $$ \begin{array}{l} \sum _{j=1}^{d}\frac{\partial u}{\partial x_{j}}\frac{\partial u}{\partial t} v_{j} \ =\frac{\partial u}{\partial t} \ \begin{bmatrix} \frac{\partial u}{\partial x_{1}}\\ ...\\ \frac{\partial u}{\partial x_{d}} \end{bmatrix} \cdot \hat{n} \ \leqslant \left| \frac{\partial u}{\partial t} \ \begin{bmatrix} \frac{\partial u}{\partial x_{1}}\\ ...\\ \frac{\partial u}{\partial x_{d}} \end{bmatrix}\right| \\ =\left(\frac{\partial u}{\partial t}^{2}\sum _{j=1}^{d}\frac{\partial u}{\partial x_{j}}^{2}\right)^{1/2} \leqslant ?\frac{1}{2}\left(\frac{\partial u}{\partial t}^{2} +\sum _{j=1}^{d}\frac{\partial u}{\partial x_{j}}^{2}\right) \end{array}$$ And I suppose I can simplify it that way, since I don't see any connection between inequality and derivatives (by the way, is that correct? I think the fact that u(x,t) is a solution to the wave equation can do the job, but I can't figure out how to). $$\left( x_{d+1}^{2}\sum _{j=1}^{d} x{_{j}}^{2}\right)^{1/2} \leqslant ?\frac{1}{2}\left( x_{d+1}^{2} +\sum _{j=1}^{d} x{_{j}}^{2}\right)$$
I think, I figured it out, maybe it will be helpful for someone: $$ \begin{array}{l} \left( x_{d+1}^{2}\sum _{j=1}^{d} x{_{j}}^{2}\right)^{1/2} \leqslant ?\frac{1}{2}\left( x_{d+1}^{2} +\sum _{j=1}^{d} x{_{j}}^{2}\right)\\ ( ab)^{1/2} \leqslant ?\frac{1}{2}( a+b)\\ 2ab\leqslant \left( a^{2} +b^{2}\right) \ -\ which\ is\ a\ known\ result\ \end{array}$$