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This paper is concerned with the Helmholtz decomposition of vector fields on bounded domains and exterior domains, and provides the decompositions in Sobolev spaces and Besov spaces of order in certain interval. In particular, the decompositions in homogeneous Besov spaces $\dot{B}_{1,q}^{s}$ and $\dot{B}_{\infty,q}^{s}$ are given.

A novel molecule, TatBim-Alexa, consisting of the HIV1 Tat cell-penetrating peptide, the Bim apoptosis-inducing peptide, and Alexa Fluor 546 was synthesized for photoinducion of apoptosis. The Alexa Fluor 546 was used as a photosensitizer and covalently attached at the C-terminus of TatBim peptide by the thiol-maleimide reaction. Photo-dependent cytosolic internalization of TatBim-Alexa and photo-dependent apoptosis using TatBim-Alexa were demonstrated in several kinds of mammalian cells including human cancer cell lines.