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In a recent paper, we exhibit a link between the average local growth ofLaplace eigenfunctions on surfaces and the size of their nodal set. In thatpaper, 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 toprove 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 boundedabove and below by the product of the average local $L^q$ growth with thefrequency. We briefly discuss the relation between this new result and Yau'sconjecture on the size of nodal sets.

Average $L^q$ growth and nodal sets of eigenfunctions of the Laplacian on surfaces