Let $p\in {\mathbb{R}}[x_1,\ldots, x_n]$ be a homogenous polynomial of even degree $2d$. If $p$ is positive on the unit sphere $S\subset {\mathbb{R}}^n$, then is $|x|^{2m}p$ a sum of squares of $L^2$ functions on the sphere for some $m$? That is, can we find $g_1,\ldots, g_k\in L^2(S)$ such that $|x|^{2m} p=g_1^2+\cdots g_k^2$? Here $|x|^2=x_1^2+\cdots x_n^2$ is the standard Euclidean norm, and we endow $S$ with the round metric.

The space of $L^2$ functions on the unit sphere $S\subset {\mathbb{R}}^n$ with the round metric is a direct sum $\oplus_{m=0}^\infty H_m(S)$ where $H_m(S)$ is the vector space of harmonic homogeneous polynomials of degree $m$ restricted to the sphere. This implies that the $g_i$ should be sums of homogeneous harmonic polynomials of degree $m+d$.

A theorem of Reznick states that for large $m$, homogenous $g_i$ can always to found, but these are not necessarily harmonic.

The geometric interpretation is that the map $G=(g_1,\ldots, g_k):S\to {\mathbb{R}}^k$ is an $L^2$ isometry in the sense that the pullback $G^* |\bullet |=p$.

REMARK ON EDIT: In the original question, the given $p$ was assumed to be harmonic. After Andrew answer, I removed this assumption and rewrote the question.

Let $p\in {\mathbb{R}}[x_1,\ldots, x_n]$ be a homogenous polynomial of even degree $2d$. If $p$ is positive on the unit sphere $S\subset {\mathbb{R}}^n$, then does there exist some $m>0$ and homogeneous harmonic polynomials $g_1,\ldots, g_k$ of degree $m+d$ such that $|x|^{2m}p=g_1^2+\cdots g_k^2$ is a sum of squares. Here $|x|^2=x_1^2+\cdots x_n^2$ is the standard Euclidean metric.

A theorem of Reznick states that for large $m$, homogenous $g_i$ can always to found, but these are not necessarily harmonic.

Since harmonic polynoimals are $L^2$ on the sphere with the round metric, we have the following geometric interpretation. The map $G=(g_1,\ldots, g_k):S\to {\mathbb{R}}^k$ is an algebraic $L^2$ isometric map in the sense that the pullback $G^* |\bullet |=p$.

REMARK ON EDIT: In the original question, the given $p$ was assumed to be harmonic. After Andrew answer, I removed this assumption and rewrote the question.

REMARK ON DOUBLE EDIT: I edited the question again after Gerry's comment.

The space of $L^2$ functions on the unit sphere $S\subset {\mathbb{R}}^n$ with the round metric is a direct sum $\oplus_{m=0}^\infty H_m(S)$ where $H_m(S)$ is the vector space of harmonic homogeneous polynomials of degree $m$ restricted to the sphere.

Let $p$ be a harmonic homogenous polynomial of even degree $2d$. If $p$ is positive on the sphere, is $p$ a sum of squares of $L^2$ functions on the sphere? That is, can we find $g_1,\ldots, g_k\in L^2(S)$ such that $p=g_1^2+\cdots g_k^2$?

A theorem of Reznick states that for a homogeneous polynomial $p\in {\mathbb{R}}[x_1,\cdots, x_n]$ that is positive on the sphere $S$, then there exists large $N$ such that $|x|^{2N}p$ is a sum of squares of polynomials. Here $|x|^2=x_1^2+\cdots x_n^2$ is the standard Euclidean norm. Since $|x|\equiv 1$ on $S$, therefore my question amounts to asking whether the sums of squares can be chosen to be from $L^2(S)$.

The geometric interpretation is that the map $G=(g_1,\ldots, g_k):S\to {\mathbb{R}}^k$ is an $L^2$ isometry in the sense that $G^* |\bullet |=p$.

Let $p\in {\mathbb{R}}[x_1,\ldots, x_n]$ be a homogenous polynomial of even degree $2d$. If $p$ is positive on the unit sphere $S\subset {\mathbb{R}}^n$, then is $|x|^{2m}p$ a sum of squares of $L^2$ functions on the sphere for some $m$? That is, can we find $g_1,\ldots, g_k\in L^2(S)$ such that $|x|^{2m} p=g_1^2+\cdots g_k^2$? Here $|x|^2=x_1^2+\cdots x_n^2$ is the standard Euclidean norm, and we endow $S$ with the round metric.

The space of $L^2$ functions on the unit sphere $S\subset {\mathbb{R}}^n$ with the round metric is a direct sum $\oplus_{m=0}^\infty H_m(S)$ where $H_m(S)$ is the vector space of harmonic homogeneous polynomials of degree $m$ restricted to the sphere. This implies that the $g_i$ should be sums of homogeneous harmonic polynomials of degree $m+d$.

A theorem of Reznick states that for large $m$, homogenous $g_i$ can always to found, but these are not necessarily harmonic.

The geometric interpretation is that the map $G=(g_1,\ldots, g_k):S\to {\mathbb{R}}^k$ is an $L^2$ isometry in the sense that the pullback $G^* |\bullet |=p$.

REMARK ON EDIT: In the original question, the given $p$ was assumed to be harmonic. After Andrew answer, I removed this assumption and rewrote the question.