Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm

Let E be a Banach space. Let ${\displaystyle L{¹}_{\left(1\right)}({ℝ}^d,E)}$ be the Sobolev space of E-valued functions on ${\displaystyle {ℝ}^d}$ with the norm ${\displaystyle {ʃ}_{{ℝ}^d}\parallel f{\parallel }_{E}dx+{ʃ}_{{ℝ}^d}\parallel \nabla f{\parallel }_{E}dx=\parallel f\parallel ₁+\parallel \nabla f\parallel ₁}$. It is proved that if ${\displaystyle f\in L{¹}_{\left(1\right)}({ℝ}^d,E)}$ then there exists a sequence ${\displaystyle \left(g_m\right)\subset L_{\left(1\right)}¹({ℝ}^d,E)}$ such that ${\displaystyle f={\sum }_m{g}_m}$; ${\displaystyle {\sum }_m(\parallel g_m\parallel ₁+\parallel \nabla g_m\parallel ₁)<\infty }$; and ${\displaystyle \parallel g_m{\parallel }_{\infty }^{\frac{1}{d}}\parallel g_m\parallel {₁}^{\frac{d-1}{d}}\le b\parallel \nabla g_m\parallel ₁}$ for m = 1, 2,..., where b is an absolute constant independent of f and E. The result is applied to prove various refinements of the Sobolev type embedding ${\displaystyle L_{\left(1\right)}¹({ℝ}^d,E)\hookrightarrow L²({ℝ}^d,E)}$. In particular, the embedding into Besov spaces ${\displaystyle L{¹}_{\left(1\right)}({ℝ}^d,E)\hookrightarrow B_{p,1}^{\theta (p,d)}({ℝ}^d,E)}$ is proved, where ${\displaystyle \theta (p,d)=d(p^{-1}+d^{-1}-1)}$ for 1 < p ≤ d/(d-1), d=1,2,... The latter embedding in the scalar case is due to Bourgain and Kolyada.