  The aim of the present paper is to obtain a Sj\"{o}lin-type maximal estimate
for pseudo-differential operators with homogeneous symbols. The crux of the
proof is to obtain a phase decomposition formula which does not involve the
time traslation. The proof is somehow parallel to the paper by Pramanik and
Terwilleger (P. Malabika and E. Terwilleger, A weak $L^2$ estimate for a
maximal dyadic sum operator on $R^n$, Illinois J. Math, {\bf 47} (2003), no. 3,
775--813). In the present paper, we mainly concentrate on our new phase
decomposition formula and the results in the Cotlar type estimate, which are
different from the ones by Pramanik and Terwilleger.
