I was reading chapter 5 of the book HDP(Roman Vershynin). There I find theorem 5.1.4 extremely fascinating.
I am curious to discover does this theorem hold to the inner product of two Lipschitz functions.
To prove this I was thinking to start with a product of two simple linear function
$(x,y) -> (Bx)^T . (Dy)$ where $B$ and $D$ are fixed matrices of dimensions say $q×d_1$ and $q×d_2$.
How should I proceed? I have been thinking for quite a long time but could not able to crack it. If you kindly give some suggestions will be of great help!
More Info
Suppose f is a vector valued function on say $\sqrt{d_1}S^{d_1-1}$ and g is a vector valued function on say $\sqrt{d_2}S^{d_2-1}.$
Now consider $|{f(s_1)^T g(p_1) - f(s_2)^T g(p_2)}|$ for 2 pairs of points in the domain $(s_1,p_1)$ and $(s_2,p_2)$
This is a more relevant LHS. How to process further?
If your functions are bounded over the sphere, you can do something like the following.
Let $f, g$ be your functions with $\sup_{x \in \sqrt{n}S^{n-1}} f(x) \leq M_f$, $\sup_{x \in \sqrt{n}S^{n-1}} g(x) \leq M_{g}$, and Lipschitz constants $L_f$ and $L_g$ respectively. Then $$ \begin{aligned} | f(x)g(x) - f(y)g(y) | &\leq | f(x)g(x) - f(x)g(y) | + | f(x)g(y) - f(y)g(y) | \\ &\leq M_{f} |g(x) - g(y)| + M_{g} |f(x) - f(y) \\ &\leq M_{f} L_{g} \|x - y\| + M_{g}L_{f} \|x - y\|. \end{aligned} $$ This implies your function is Lipschitz on the sphere with constant $M_{f}L_{g} + M_{g}L_{f}$.