$\partial_i P$ is either zero or a hhP of degree $d-1$. An important relation between the norm of a hhP and the norm of its gradient is given by the following lemma.

\[lemma\_pdp\][@HFT lemma 5.13] Let $P,\ Q:{\mathbb{R}}^n\to {\mathbb{R}}$ be two homogeneous harmonic polynomials of degree $d$. Then $$\begin{gathered}
 {\left\langleP\middle\vertQ\right\rangle} = \frac{1}{d(2d+n-2)} {\left\langle\nabla P\middle\vert\nabla Q\right\rangle} =\frac{1}{d(2d+n-2)} \sum_{i=1}^n {\left\langle\partial_i P\middle\vert\partial_i Q\right\rangle}\, .\end{gathered}$$

Moreover, it is possible to prove a simple bound on the sup norm of a hhP $P$ given its degree and its $L^2(\partial B(0,1))$ norm.

\[lemma\_dest\] Given a $P_d\in {\mathcal{P}}_d$, we have the sharp upper bound $$\begin{gathered}
 {\left\|P_d\right\|}_{C^0(B(0,1))}\leq \sqrt{\operatorname{dim}({\mathcal{H}}_d)} {\left(\fint_{\partial B(0,1)} P_d^2 \right)}^{1/2}\leq C(n) d^{\frac n 2 -1 }{\left(\fint_{\partial B(0,1)} P_d^2 \right)}^{1/2}\, .\end{gathered}$$

### Two Variables

A special case which deserves to be studied on its own is the case of hhPs defined in ${\mathbb{R}}^2$. The following is a standard but useful point:

Let $n=2$, then ${\mathcal{H}}_{d,2}$ is a $2$ dimensional space for every $d\geq 2$, and an orthonormal basis for this space is given by $$\begin{gathered}
 P_d(r,\theta) = 2r^{d} \sin{\left(d\theta\right)}\, , \quad C_d(r,\theta)= 2r^{d}\cos{\left(d\theta\right)} = P_d(r,\theta+\pi/(2d))\, .\end{gathered}$$ Moreover, by direct computation one has $$\begin{gathered}
\label{eq_dxP}
 \partial_x P_d(r,\theta) = 2d r^{d-1} \sin((d-1)\theta)\, , \quad \partial_y P_d(r,\theta) = 2d r^{d-1} 