The purpose of this note is to prove some boundedness/compactness results of a harmonic analysis flavor for the Bergman and Szegő projections on certain classes of planar domains using conformal mappings. In particular, we prove weighted estimates for the projections, provide quantitative
L
p
L^p
estimates and a specific example of such estimates on a domain with a sharp
p
p
range, and show that the “difference” of the Bergman and Szegő projections is compact at the endpoints
p
=
1
,
∞
p=1, \infty
for domains with sufficient smoothness. We also pose some open questions that naturally arise from our investigation.