In [5], the author has proved extended sharp extrapolation theorem on $L^p$ spaces with $\sum$-method ([1]), which asserted $\sum_{1<p<q}((p-1)^{-\alpha}L^p)^{p/q}=(L\log^{\alpha}L+L^q)^{1/q}$. In the present paper, on that result, we shall consider the case $q\approx 1$.

We consider orientation-preserving embeddings of the disk having seven or less fixed points all of which are transversal. We classify all the braid types which can be realized as the braid type of the fixed point set for such an embedding.

We give a local normal form of first order partial differential equations with singular solution up to contact diffeomorphism.