Let $mathbb P_l$ stands for the orthogonal projection in $L^{2}(S^{n-1})$ on the space of $mathcal H_{n,l}$ of harmonic homogenous polynomial with degree $l$ in $n$ variable.

By spectral decomposition theorem of Laplacian on sphere we have

$$mathrm{Id}_{L^{2}left(mathbb{S}^{n-1}right)}=sum_{l geq 0} mathbb{P}_{l}, quad-Delta_{mathbb{S}^{n-1}}=sum_{l geq 0} l(l+n-2) mathbb{P}_{l}.$$

By writing Laplacian in polar coordinate we get

$$ r^{2} Delta_{mathbb{R}^{n}}=left(r partial_{r}right)^{2}+(n-2)left(r partial_{r}right)+Delta_{mathbb{S}^{n-1}}$$

Define $Lambda=sum_{k geq 1} k mathbb{P}_{k} $. By orthogonality we have

$$|x|^{2} Delta_{mathbb{R}^{n}}=left(r partial_{r}right)^{2}+(n-2)left(r partial_{r}right)-Lambda(Lambda+n-2).$$

Now take $r=e^t$ we get

$$|x|^{2} Delta_{mathbb{R}^{n}}=partial_{t}^{2}+(n-2) partial_{t}-Lambda(Lambda+n-2)=left(partial_{t}+Lambda+n-2right)left(partial_{t}-Lambdaright)=mathcal{L}$$

We note that for $lambda geq 1$

begin{equation}

mathcal{L}=|x|^{2} Delta_{mathbb{R}^{n}}, quad mathcal{L}_{lambda}=e^{-lambda t} mathcal{L} e^{lambda t}=mathcal{L}_{-, lambda} mathcal{L}_{+, lambda}=mathcal{L}_{+, lambda} mathcal{L}_{-, lambda}

end{equation}

Where begin{equation}

mathcal{L}_{+, lambda}=partial_{t}+lambda+Lambda+n-2, quad mathcal{L}_{-, lambda}=partial_{t}+lambda-Lambda.

end{equation}

Fundamental solution of differential operator $mathcal{L}_{+, lambda} mathcal{L}_{-, lambda}$ is given by

begin{align}

E=& H(lambda-Lambda)(2 Lambda+n-2)^{-1}left(e^{-(lambda-Lambda) t}-e^{-(Lambda+lambda+n-2) t}right) H(t)notag \ &-H(Lambda-lambda)(2 Lambda+n-2)^{-1}left(e^{-(lambda-Lambda) t} H(-t)+e^{-(Lambda+lambda+n-2) t} H(t)right)

end{align}

where $H$ is Heaviside function. i.e. begin{equation}

v=E *left(mathcal{L}_{+, lambda} mathcal{L}_{-, lambda} vright)

end{equation}

Now we wanted to estimate $left|mathbb{P}_{k} v(t)right|_{L^{p^{prime}}left(mathbb{S}^{n-1}right)}$.

For that purpose we write $F=mathcal{L}_{+, lambda} mathcal{L}_{-, lambda} v$.

By property of fundamental solution we have

begin{align*}

v(t)&=frac{H(lambda-Lambda)}{2 Lambda+n-2} int_{-infty}^{t}left(e^{-(lambda-Lambda)(t-s)}-e^{-(lambda+Lambda+n-2)(t-s)}right) F(s) d s\

&quad-frac{H(Lambda-lambda)}{2 Lambda+n-2}left(int_{t}^{+infty} e^{-(Lambda-lambda)(s-t)} F(s) d sright.\

&quadleft.quad+int_{-infty}^{t} e^{-(lambda+Lambda+n-2)(t-s)} F(s) d sright)

end{align*}

Now we apply $mathbb P_k$ to both side.

I can think that

$e^{-(lambda-Lambda)(t-s)}$ can be think as series of exponential in $Lambda$. so after applying $P_k$ to that we get

$e^{-(lambda-k)(t-s)}mathbb P_k$ because of orthogonality.

But I have doubt how to apply $mathbb P_k$ on $frac{H(lambda-Lambda)}{2 Lambda+n-2}$.

I thought about Fourier series of Heaviside function but there is problem about convergence and Heaviside on whole real line is not periodic function as such so I can not get Fourier series. Also how to apply on its denominator $2 Lambda+n-2$. Final expression is given as

begin{align*}

mathbb{P}_{k} v(t)&=frac{H(lambda-k)}{2 k+n-2} int_{-infty}^{t}left(e^{-(lambda-k)(t-s)}-e^{-(lambda+k+n-2)(t-s)}right) mathbb{P}_{k} F(s) d s\

&quad-frac{H(k-lambda)}{2 k+n-2}left(int_{t}^{+infty} e^{-(k-lambda)(s-t)} mathbb{P}_{k} F(s) d sright.\

&quadleft.quad+int_{-infty}^{t} e^{-(lambda+k+n-2)(t-s)} mathbb{P}_{k} F(s) d sright)

end{align*}

Any help or hint will be appreciated.