Pointwise symmetrization inequalities for Sobolev functions and applications (Q986076)

The authors have developed a connection between the isoperimetric inequality and some new symmetrization inequalities. In previous papers, they considered the Euclidean space (classical\ Sobolev inequalities) and \(\mathbb R^{n}\) with Gaussian measure (logarithmic Sobolev inequalities) [\textit{J. Martin, M. Milman} and \textit{E. Pustylnik}, J. Funct. Anal. 252, No.~2, 677--695 (2007; Zbl 1136.46031); J. Funct. Anal. 256, No.~1, 149--178 (2009; Zbl 1173.26010)]. Here they considerably extend their earlier work by means of considering fairly general metric probability spaces. Let \((\Omega,d,\mu)\) be a connected, separable, metric probability space. Let \(\mu^{+}\) be the Minkowski content, and let \(I=I_{(\Omega,d,\mu)}\) be the corresponding isoperimetric profile. A standing assumption on \((\Omega,d,\mu)\) is that \(I\) is continuous, concave and symmetric about \(1/2\). In this context, the authors show several equivalent ways to formulate the isoperimetric inequality in terms of rearrangement inequalities involving a Lip function and its generalized gradient. For example, they show that the isoperimetric inequality \[ I(\mu(A))\leq\mu^{+}(A)\text{ for all Borel sets,} \] is equivalent to \[ f^{\ast\ast}(t)-f^{\ast}(t)\leq\frac{t}{I(t)}\left| \nabla f\right| ^{\ast\ast}(t)\text{ for Lip functions,}\tag{1} \] where \(f^{\ast}\) is the nonincreasing rearrangement, \(f^{\ast\ast} (t)=\frac{1}{t}\int_{0}^{t}f^{\ast}(s)\,ds\), and the modulus of the gradient is defined by \[ \left| \nabla f(x)\right| =\sup_{d(x,y)\rightarrow0}\frac{\left| f(y)-f(x)\right| }{d(x,y)}. \] The inequality (1) gives new generalized Poincaré inequalities and, in particular, it provides a unification between the classical Sobolev inequalities and the so-called logarithmic Sobolev inequalities. An important feature that allows to achieve this unification is that, given that the inequalities obtained by the authors are pointwise, one does not have to specify beforehand what norm one is using in order to compute the gain of integrability for the corresponding Sobolev inequality. Indeed, in the authors' theory, the gain of integrability is predicted by the isoperimetric profile itself. The authors' methods also allow for considerable extensions of the classical Pólya-Szegő principle (even in the classical Euclidean case, their methods provide a considerable simplification [cf. \textit{G. Leoni}, A first course in Sobolev spaces. Graduate Studies in Mathematics 105. Providence, RI: American Mathematical Society (AMS) (2009; Zbl 1180.46001)]). The paper also contains new symmetrization techniques to prove sharp integrability results for solutions of nonlinear elliptic equations. The results are sharper than known results in the sense that for borderline cases the results obtained are stronger. For details, we must refer to this interesting paper.