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

Degenerations of log Hodge de Rham spectral sequences, log Kodaira  vanishing theorem in characteristic $p>0$ and log weak Lefschetz conjecture  for log crystalline cohomologies