# fa.functional analysis – How to think action of orthogonal projection operator on Heaviside function of some operator?

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.