  In a recent paper, we exhibit a link between the average local growth of
Laplace eigenfunctions on surfaces and the size of their nodal set. In that
paper, the average local growth is computed using the uniform - or $L^\infty$ -
growth exponents on disks of wavelength radius. The purpose of this note is to
prove similar results for a broader class of $L^q$ growth exponents with $q \in
(1, \infty)$. More precisely, we show that the size of the nodal set is bounded
above and below by the product of the average local $L^q$ growth with the
frequency. We briefly discuss the relation between this new result and Yau's
conjecture on the size of nodal sets.
