Sections of functions and Sobolev-type inequalities

V. I. Kolyada

Informatics and Automation, Function spaces and related problems of analysis, Tome 284 (2014), pp. 200-211

Résumé

We study functions of two variables whose sections by the lines parallel to the coordinate axis satisfy the Lipschitz condition of order $0\alpha\le1$. We prove that if for a function $f$ the $\operatorname{Lip}\alpha $-norms of these sections belong to the Lorentz space $L^{p,1}(\mathbb R)$ ($p=1/\alpha$), then $f$ can be modified on a set of measure zero so as to become bounded and uniformly continuous on $\mathbb R^2$. For $\alpha=1$ this gives an extension of Sobolev's theorem on continuity of functions of the space $W_1^{2,2}(\mathbb R^2)$. We show that the exterior $L^{p,1}$-norm cannot be replaced by a weaker Lorentz $L^{p,q}$-norm with $q>1$.