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A homeomorphism U of the unit disk D ⊂ R 2 , U = ( u 1 , u 2 ) : D ⟶ onto D is a quasiharmonic map, if u i ∈ W l o c 1 , 1 , i = 1 , 2 are solutions to the system (0.1) { div B ( y ) ∇ u 1 = 0 a.e. in D div B ( y ) ∇ u 2 = 0 a.e. in D for a symmetric degenerate elliptic conductivity B = B ( y ) i.e. (0.2) ∣ ξ ∣ 2 H ( y ) ≤ 〈 B ( y ) ξ , ξ 〉 ≤ H ( y ) ∣ ξ ∣ 2 a.e. in y ∈ D ∀ ξ ∈ R 2 where H : D ⟶ onto [ 1 , ∞ [ is measurable. A sufficient condition that the Sobolev homeomorphism U ∈ W l o c 1 , 1 is a quasiharmonic map is that U − 1 ∈ W loc 1 , 1 . This condition is not necessary because we construct a quasiharmonic map U such that U − 1 ∈ BV ∖ W loc 1 , 1 (see Theorem 1.1 ).

In the recent literature certain BMO-type seminorms provide characterizations of Sobolev functions. In the same order of ideas, we obtain the norm of the gradient of a function in L p ( Ω ) , where Ω ⊂ R n , n > 1 and p > 1 , as limit of BMO-type seminorms involving families of pairwise disjoint sets with all orientations, the sets being not necessarily cubes or tessellation cells. An analogous result is obtained when rotations are not allowed.

The purpose of this paper is to present the relation between certain BMO-type seminorms and the total variation of SBV functions. Following some ideas of a recent paper by L. Ambrosio and G.E. Comi, we give a representation formula of the total variation of SBV functions which does not make use of the distributional derivatives. We consider an anisotropic variant of the BMO-type seminorm introduced in 2015 in a paper by J. Bourgain, H. Brezis and P. Mironescu, by using, instead of cubes, covering families made by translations of a given open bounded set with Lipschitz boundary.

Buckley (1993) [3] proved the linear dependence ‖ M ‖ L 2 ( R n , w ) ≤ c ( n ) A of the L 2 ( R n , w ) -norm for the Hardy–Littlewood maximal operator M on the classical A 2 -constant A = A 2 ( w ) = sup Q ⨏ Q w ⨏ Q w − 1 where the supremum is taken over all cubes with sides parallel to the axes. We prove in the case n = 1 that, for p 0 = 1 + A − 1 A p ≤ 2 , the dependence on the constant A is precisely preserved ‖ M ‖ L p ( R , w ) ≤ c ( p ) [ A 1 − p ( 2 − p ) A ] 1 p − 1 and it is impossible to decrease the value of p 0 . Similar exact continuation theorems hold for the L 2 -norm inequalities of weighted maximal operators.