I don't want to write precisely the formulation of the Calderon-Zygmund theorem for singular integrals. The details are not so important here. 


So I consider the operator $T$ given by the following formula:
\begin{equation}
 Tf(x) = \int\limits\_{R^n}\dfrac{Y\_{k}^m(\frac{x-y}{|x-y|})}{|x-y|^n}f(y)\,
dy, \, f\in L^2(R^n),
\end{equation}
where $Y\_k^m$ is a spherical harmonic of degree $k$ ($m$ denotes the index in the basis of the related subspace of harmonics on $S^{n-1}$).


It is known that operator $T$ is well-defined in $L^2(R^n)$ (and some other spaces as well). In particular, the following estimate is usually given in the proofs of the Calderon-Zygmund theorem:
\begin{equation}
\|Tf\|\_{L^2} \leq C(n)\|\mathcal{F}[\frac{Y\_{k}^m}{r^n}]\|\_{L^{\infty}}
\|f\|\_{L^2},
\end{equation}
where $\mathcal{F}$ denotes the Fourier transform (maybe in the mean-value sense), $C(n)$ is just some constant depending only on dimension of space.


1) Is the latter estimate is precise in the sense that there is only $L^{\infty}$-norm of the Fourier transform of the kernel (of this particular kernel with spherical harmonics)? Can this norm, for example, be improved somehow (any $L^p$ norm)?


2) What if, for example, $f$ belongs to a better space, i.e., $f\in L^{\infty}(R^n)$, $\mathrm{supp}\, f\subset D$, where $D$ is some compact?


P.S. The point is that, for example, in $n=3$ (and in any dimension as well) the Fourier transform of such kernel is the spherical harmonic $Y\_k^m$ (it is known fact), but the maximum of a spherical harmonic on a sphere grows as $\sim \sqrt{2k+1}$, which is ofcourse more than, for example, as an $L^2$-norm: $\|Y\_{k}^m\|\_{L^2(S^2)} =1$. 



Calderon-Zygmund theorem for the kernel of spherical harmonics
Your operator $T$ is a Fourier multiplier with symbol $m = \mathcal{F}[Y^m\_k/r^n]$; that is, $\mathcal{F}[Tu]=m \mathcal{F}[u]$.


It is a relatively simple exercise to show that the norm of $T$ on $L^2$ is *equal* to $\|m\|\_\infty$, the essential supremum of $|m|$. In other words, $C(n)=1$. Therefore, the supremum norm of $m$ cannot be change to any other norm.


Calderón–Zygmund theory provides estimates of the norm of $T$ on other $L^p$ spaces (and much more).


