We prove necessary and sufficient conditions for the validity of the classical chain rule in the Sobolev space $W_{\text{loc}}^{1,1}({ℝ}^N;{ℝ}^d)$ and in the space $B{V}_{\text{loc}}({ℝ}^N;{ℝ}^d)$ of functions of bounded variation.

We give a new Calderón-Zygmund decomposition for Sobolev spaces on a doubling Riemannian manifold. Our hypotheses are weaker than those of the already known decomposition which used classical Poincaré inequalities.

We establish new estimates for the Laplacian, the div-curl system, and more general Hodge systems in arbitrary dimension $n$, with data in $L^1$. We also present related results concerning differential forms with coefficients in the limiting Sobolev space $W^{1,n}$.

We obtain Hardy type inequalities $${\int }_0^{\infty }M\left(\omega \left(r\right)\left|u\left(r\right)\right|\right)\rho \left(r\right)dr⩽C_1{\int }_0^{\infty }M\left(\left|u\left(r\right)\right|\right)\rho \left(r\right)dr+C_2{\int }_0^{\infty }M\left(\left|u^{\text{'}}\left(r\right)\right|\right)\rho \left(r\right)dr,$$ and their Orlicz-norm counterparts $${∥\omega u∥}_{L^M({ℝ}_{+},\rho )}⩽{\tilde{C}}_1{∥u∥}_{L^M({ℝ}_{+},\rho )}+{\tilde{C}}_2{∥u^{\text{'}}∥}_{L^M({ℝ}_{+},\rho )},$$ with an N-function M, power, power-logarithmic and power-exponential weights ω, ρ, holding on suitable dilation invariant supersets of C 0∞(ℝ+). Maximal sets of admissible functions u are described. This paper is based on authors’ earlier abstract results and applies them to particular classes of weights.

This paper studies a possible definition of Sobolev spaces in abstract metric spaces, and answers in the affirmative the question whether this definition yields a Banach space. The paper also explores the relationship between this definition and the Hajlasz spaces. For specialized metric spaces the Sobolev embedding theorems are proven. Different versions of capacities are also explored, and these various definitions are compared. The main tool used in this paper is the concept of moduli of path...

We prove that the fractional BMO space on a one-dimensional manifold is an interpolation space between C and $C^1$. We also prove that $BM{O}^1$ is an interpolation space between C and $C^2$. The proof is based on some nonclassical interpolation constructions. The results obtained cannot be transferred to spaces of functions defined on manifolds of higher dimension. The interpolation description of fractional BMO spaces is used at the end of the paper for the proof of the boundedness of commutators of the Hilbert...

Our main result shows that for a large class of nonlinear local mappings between Besov and Sobolev space, interpolation is an exceptional low dimensional phenomenon. This extends previous results by Kumlin [13] from the case of analytic mappings to Lipschitz and Hölder continuous maps (Corollaries 1 and 2), and which go back to ideas of the late B.E.J. Dahlberg [8].