Sections of Functions and Sobolev-Type Inequalities

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