On functions with derivatives in a Lorentz space.(English)Zbl 0976.26004

The authors of this deep and interesting paper pursue the following question: what are “minimal” assumptions on a function $$f$$ of several variables which guarantee that $$f$$ has properties such as continuity, differentiability, absolute continuity or the Lusin N-property ($$f(E)$$ is of $$n$$-dimensional Hausdorff measure zero whenever the same is true for $$E$$). The main result is that one such sharp sufficient condition guaranteeing the N-condition is that $$f$$ belongs to the Sobolev space $$W^{1,1}_{\text{loc}}$$ and its gradient $$\nabla f$$ belongs to the Lorentz space $$L^{n,1}$$. Interestingly, the same condition is also sufficient for other important properties such as continuity, $$n$$-absolute continuity or differentiability a.e. In an earlier work of A. Cianchi and the reviewer [Ark. Mat. 36, No. 2, 317-340 (1998)] it was shown that the above-mentioned condition is sufficient (and, in a certain sense, also necessary) for $$f$$ to belong to $$L^\infty$$. It is therefore interesting and even somewhat surprising that the same condition is equivalent to the $$n$$-absolute continuity of $$f$$. The paper contains plenty of nice and original thoughts. A useful by-product of the argument is a new characterization of the Lorentz space $$L^{p,q}$$ which can turn out to be quite handy.

MSC:

 26B05 Continuity and differentiation questions 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 26B35 Special properties of functions of several variables, Hölder conditions, etc. 74B20 Nonlinear elasticity
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