The Lebesgue measure on $\mathbb{R}^d$
admits the following polar decomposition:
$$
L(dx) = r^{d-1} dr \lambda(dy),
$$
where 𝜆 is the uniform measure on the Euclidean unit sphere of $\mathbb{R}^d$ and where $x=ry$.
Now, change the uniform measure on the Euclidean unit sphere by another positive finite measure (non-degenerate) on the Euclidean unit sphere. Namely,
$$
L'(dx) = r^{d-1}dr\sigma(dy).
$$
where $\sigma$ is a positive finite measure (non-degenerate) on the Euclidean unit sphere $\mathbb{R}^d$ and where $x=ry$. Where can I find harmonic analysis results based on this new reference measure ?