Sobolev type inequalities in ultrasymmetric spaces with applications to Orlicz-Sobolev embeddings (Q558509)

Summary: Let \(D^kf\) be the vector composed by all partial derivatives of order \(k\) of a function \(f(x)\), \(x\in\Omega\subset\mathbb{R}^n\). Given a Banach function space \(A\), we look for a possibly small space \(B\) such that \(\|f\|_B\leq c\| |D^kf| \|_A\) for all \(f\in C_0^k(\Omega)\). The estimates obtained are applied to ultrasymmetric spaces \(A=L_{\varphi, E}\), \(B=L_{\psi,E}\), giving some optimal (or rather sharp) relations between the parameter-functions \(\varphi(t)\) and \(\psi(t)\) and new results for embeddings of Orlicz-Sobolev spaces.