  This paper determines the sharp asymptotic order of the following reverse
H\"older inequality for spherical harmonics $Y_n$ of degree $n$ on the unit
sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$ as $n\to \infty$:
  \[\|Y_n\|_{L^q(\mathbb{S}^{d-1})}\leq C
n^{\alpha(p,q)}\|Y_n\|_{L^p(\mathbb{S}^{d-1})},\quad 0<p<q\leq \infty.\] In
many cases, these sharp estimates turn out to be significantly better than the
corresponding estimates in the Nilkolskii inequality for spherical polynomials.
Furthermore, they allow us to improve two recent results on the restriction
conjecture and the sharp Pitt inequalities for the Fourier transform on
$\mathbb{R}^d$.
