Sobolev inequalities in the setting of the rearrangement invariant spaces of integrable functions (r.i.) have been studied largely in the last twenty years. In particular, the analysis of when two r.i. norms \(\| \cdot\|_X\) and \(\| \cdot\|_Y\) – the spaces \(X\) and \(Y\) being r.i. spaces over \([0,1]\) – satisfy that the inequality \[ \|u\|_X \leq C \| | \nabla u | \|_Y \] holds for the adequate functions \(u\), is nowadays well known. Refinements of this inequality, affecting both the r.i. norms involved and the expression appearing in the inequality, are also known, and the rather sophisticated procedures for showing them have been proved to be very productive. In this paper, the authors go further in this direction, analysing the optimal version of the Sobolev type inequality \[ \|u\|_X \leq C \Big\| t \mapsto \int_{\Omega(u,t)} | \nabla u (x) | \alpha(u,x)^{-1/n'} dx \Big\|_X, \] where \(n'\) is the conjugate number of \(n\), \(\Omega(u,t):= \{z \in \Omega: 0 < |u(z)| \leq u^*(t)\}\) and the function \(\alpha\) is defined to be \(m_n( \{ z \in \Omega: |u(z)| \geq |u(x)| \})\) if \(u(x) \neq 0\) and as \(0\) otherwise. Here, \(\Omega \subset \mathbb{R}^n\), and \(m_n\) is Lebesgue measure, which is supposed to satisfy \(m_n(\Omega)=1\). In Theorem 2, the authors prove that this inequality holds for every r.i. space \(X\) on \([0,1]\). This version of the inequality comes from the formula that appears in the optimal r.i. version of the Sobolev inequality written above for the case of \(X=Y \subseteq L^{n',1}([0,1])\), where the last space is the Lorentz space of indices \(n'\) and \(1\) on the Lebesgue measure space \([0,1]\).
For finding optimal inequalities associated to the one above, the authors center their attention on the function \[ t \in [0,1] \mapsto u_\nabla(t):=\int_{\{x \in \Omega: |u(x)| > u^*(t) \}} | \nabla u(x)| dx, \] obtaining in Theorem 7 a Sobolev type inequality in which the derivative of this function appears. For instance, as an application of this inequality in Proposition 9 it is proved that for the kernel operator \(T\) given by \(T(f)(t):= \int_t^1 f(s) s^{-1/n'} ds\), if \(X\) is any r.i. space and \([T,X]\) is the optimal domain for \(T\), then there exists a constant \(K>0\) such that \[ \|u\|_{[T,X]} \leq K \Big\| \frac{d}{dt} u_\nabla \Big\|_{[T,X]}, \quad u \in C^1_0(\Omega). \] In Section 5, comparison with other known Sobolev type inequalities is also provided.