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 ?
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$\begingroup$ If you assume that new neasure $L'$ (or $\sigma$) is doubling, you can use harmonic analysis on homogenuous spaces. But I do not know if this is the case of interest to you. $\endgroup$– Giorgio MetafuneCommented Mar 15, 2020 at 15:43
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$\begingroup$ Thank you for this hint. Indeed, the doubling condition came up at some point. I am interested in Hölderian type regularity result. $\endgroup$– user153086Commented Mar 15, 2020 at 17:34
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