Revista Matemática Iberoamericana

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Volume 30, Issue 1, 2014, pp. 349–367
DOI: 10.4171/RMI/782

Published online: 2014-03-23

On the $L^p$-differentiability of certain classes of functions

Giovanni Alberti[1], Stefano Bianchini[2] and Gianluca Crippa[3]

(1) Università di Pisa, Italy
(2) SISSA-ISAS, Trieste, Italy
(3) Universität Basel, Switzerland

We prove the $L^p$-differentiability at almost every point for convolution products on $\mathbb{R}^d$ of the form $K*\mu$, where $\mu$ is bounded measure and $K$ is a homogeneous kernel of degree $1-d$. From this result we derive the $L^p$-differentiability for vector fields on $\mathbb{R}^d$ whose curl and divergence are measures, and also for vector fields with bounded deformation.

Keywords: Approximate differentiability, Lusin property, convolution operators, singular integrals, Calderón–Zygmund decomposition, Sobolev functions, functions with bounded variation, functions with bounded deformation

Alberti Giovanni, Bianchini Stefano, Crippa Gianluca: On the $L^p$-differentiability of certain classes of functions. Rev. Mat. Iberoam. 30 (2014), 349-367. doi: 10.4171/RMI/782