# dg.differential geometry – Comparison inequality between Sobolev-seminorm w.r.t spherical uniform distribution and gaussian distribution

Let $$d$$ be a large positive integer. Let $$f:mathbb R^d to mathbb R$$ be a continuously differentiable function. Let $$X$$ be uniformly distributed on the unit-sphere $$S_{d-1} := {x in mathbb R^d |x| = 1}$$ and let $$Z$$ be a random vector in $$mathbb R^d$$ with iid coordinates from $$N(0,1/d)$$. Let $$nabla_{S_{d-1}}f:S_{d-1} to mathbb R^d$$ be the spherical gradient of $$f$$, defined by $$nabla_{S_{d-1}} f(x) :=(I_d-xx^top)nabla f(x)$$, where $$nabla f(x)$$ is the usual / euclidean gradient of $$f$$ at $$x$$.

Question. Is there any comparison inequality between $$mathbb E|nabla_{S_{d-1}} f(X)|^2$$ and $$mathbb E|nabla f(Z)|^2$$, meaning the existence of abolute constants $$c,C>0$$ independent of $$f$$ and $$d$$, such that $$c mathbb E|nabla_{S_{d-1}} f(X)|^2 le mathbb E|nabla f(Z)|^2 le C mathbb E|nabla_{S_{d-1}} f(X)|^2$$?

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