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We study estimates for Hardy space norms of analytic projections. We firstfind a sufficient condition for the Bergman projection of a function in theunit disc to belong to the Hardy space $H^p$ for $1 < p < \infty$. We apply theresult to prove a converse to an extension of Ryabykh's theorem about Hardyspace regularity for Bergman space extremal functions. We also prove that the$H^q$ norm of the Szeg\"{o} projection of $F^{p/2} \overline{F}^{(p/2)-1}$cannot be too small if $F$ is analytic, for certain values of $p$ and $q$. Weapply this to show that the best analytic approximation in $L^p$ of a functionin both $L^p$ and $L^q$ will also lie in $L^q$, for certain values of $p$ and$q$.

Bergman and Szego projections, Extremal Problems, and Square Functions