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On $\mathbb{R}^n$, we have the concept of spherically decreasing rearragement of a function, which means, given a function $f$, one can design a radial and decreasing function $f^*$ such that $\Vert f^*\Vert_{L^p} = \Vert f\Vert_{L^p}$, and $\Vert \nabla f^*\Vert_{L^2} \leq \Vert \nabla f\Vert_{L^2}$. This is described in detail in the book titled "Analysis" by Lieb and Loss (see https://books.google.de/books/about/Analysis.html?id=Eb_7oRorXJgC&redir_esc=y).

My question is: suppose we have a function $f$ on $S^n$ such that $f = 0$ at the south pole $(0,0,...,-1)$. Can we define a function $f^*$ on $S^n$ such that $\Vert f^*\Vert_{L^p} = \Vert f\Vert_{L^p}$, and $\Vert \nabla f^*\Vert_{L^2} \leq \Vert \nabla f\Vert_{L^2}$? By analogy, I would expect $f^*$ to be defined such that $f^*$ attains its maximum value at the north pole $(0,0,..,1)$ and is a function of $x_{n + 1}$ alone, where $(x_1, x_2,..., x_{n + 1})$ are the coordinates on $S^{n + 1}$ obtained from $\mathbb{R}^{n + 1}$. A reference would be highly appreciated.

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The answer is yes, but a ready reference that I have concerns spheres in dimension 2. This is called "circular symmetrization". I suppose all proofs can be extended to any dimension without much change. The reference is Hayman, Multivalent functions. Albert Baernstein II was writing a book on symmetrization and rearrangements, but he died and the fate of the book is unclear. But one can look at his papers, for example,

Baernstein, Albert, II; Taylor, B. A. Spherical rearrangements, subharmonic functions, and ∗-functions in n-space. Duke Math. J. 43 (1976), no. 2, 245–268.

which deals with the question is arbitrary dimension.

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