Our aim in this paper is to treat Hardy’s inequalities for Musielak-Orlicz-Sobolev functions on proper open subsets of $R^N$.

One of the generalizations of McShane’s identities by Tan, Wong and Zhang is an identity concerning lengths of simple closed geodesics which pass through two Weierstrass points on a hyperbolic one-holed torus. The Fuchsian groups which uniformize the surface are purely hyperbolic and free of rank two. Another type of Fuchsian groups of the same property is of type (0, 3) corresponding to hyperbolic three-holed spheres. In this paper we show a McShane-type identity which holds for all Fuchsian groups of type (0, 3).

In our previous study, the author and Tamaru proved that a left-invariant Riemannian metric on a three-dimensional simply-connected solvable Lie group is a solvsoliton if and only if the corresponding submanifold is minimal. In this paper, we study the minimality of the corresponding submanifolds to solvsolitons on fourdimensional cases. In four-dimensional nilpotent cases, we prove that a left-invariant Riemannian metric is a nilsoliton if and only if the corresponding submanifold is minimal. On the other hand, there exists a four-dimensional simply-connected solvable Lie group so that the above correspondence dose not hold. More precisely, there exists a solvsoliton whose corresponding submanifold is not minimal, and a left-invariant Riemannian metric which is not solvsoliton and whose corresponding submanifold is minimal.

We characterize (conditional) Hardy spaces of the Laplacian and the fractional Laplacian by using Hardy-Stein type identities.

Cromwell introduced the notion of an arc presentation of a knot. An arc presentation is an embedding of a knot in an open-book. In this note, we define a transformation on an arc presentation called a page move, and prove that any knot is transformed into the trivial knot by a single page move. We also study a relationship with the unknotting number.

We complete the classification of smooth 4-manifolds which admit genus-1 simplified broken Lefschetz fibrations.

For a super-polyharmonic function $u$ on the unit ball satisfying a growth condition on spherical means, we study a growth property of the Riesz measure of $u$ near the boundary.