  In this article we prove that the log Hodge de Rham spectral sequences of
certain proper log smooth schemes of Cartier type in characteristic $p>0$
degenerate at $E_1$. We also prove that the log Kodaira vanishings for them
hold when they are projective. We formulate the log weak Lefschetz conjecture
for log crystalline cohomologies and prove that it is true in certain cases.
