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Let \(u=(u^1,\ldots , u^n)\) be a \(p\)-harmonic mapping in a domain \(\Omega \subset \mathbb {R}^n\) for \(n\ge 2\). We investigate level sets for compositions of coordinate functions \(u^i\) with convex functions satisfying growth conditions and derive the de Giorgi-type estimates. Our main result is the arithmetic three-spheres theorem for coordinate functions of mapping \(u\). The discussion is illustrated by radial \(p\)-harmonics.

Three-spheres theorem for $$p$$ p -harmonic mappings