On the convergence of mappings with k-finite distortion

It is well known that the limit of a uniformly converging sequence of analytic functions is an analytic function. Reshetnyak generalized this result to mappings with bounded distortion: the limit of a locally uniformly converging sequence of mappings with bounded distortion is a mapping with bounded distortion. Reshetnyak used the weak convergence of Jacobians to prove the following theorem on the limit of a sequence of mappings with bounded distortion.

It is well known that the limit of a uniformly converging sequence of analytic functions is an analytic function. Reshetnyak generalized this result to mappings with bounded distortion: the limit of a locally uniformly converging sequence of mappings with bounded distortion is a mapping with bounded distortion.
Reshetnyak used the weak convergence of Jacobians to prove the following theorem on the limit of a sequence of mappings with bounded distortion. Theorem 1 ([1]). Let f m : Ω → R n , m = 1, 2, . . . , be an arbitrary sequence of mappings with bounded distortion locally converging in L n (Ω) to a mapping f 0 : Ω → R n . Assume that the sequence of distortion coefficients K(f m ), m = 1, 2, . . . , is bounded. Then the limit mapping f 0 is a mapping with bounded distortion and the following inequality holds: (1) The weighted mappings with bounded (p, q)-distortion, which generalize the mappings with bounded distortion, were defined and studied in [2]. It was shown in [3] that the locally uniform limit of mappings with bounded (θ, 1)-weighted (p, q)-distortion is also a mapping with bounded (θ, 1)-weighted (p, q)-distortion, and an estimate similar to (1) was obtained. The proofs of theorems in both [3] and the present paper are based on a new method developed in [4] for generalizing Reshetnyak's results to the Carnot groups.
In the present paper, we generalize these assertions to classes of mappings with k-finite distortion which naturally arise in the problem of pull-backs of differential forms of degree k (see [5]).

PRELIMINARIES
Let U be a domain in R n . We consider the Banach space L p (U, Λ k ) of differential forms ω of degree k with measurable coefficients which have the following finite norm: . * E-mail: vodopis@math.nsc.ru ** E-mail: nkudr@itam.nsc.ru A mapping f : U → R n is said to be approximate differentiable at a point x ∈ U [6] if there exists a linear mapping L : R n → R n such that for any ε > 0. It is well known that the approximate differential is unique [6] if x is a density point. In what follows, it is denoted by the symbol ap Df (x).
Let a mapping f = (f 1 , . . . , f n ): U → W of Euclidean domains U, W ⊂ R n be approximate differentiable almost everywhere in U . The differential canonically generates a linear mapping of spaces of k-vectors and the following operation f * of pull-back of k-forms. Namely, any k-form where the summation is performed over all k-dimensional ordered multi-indices β = (β 1 , . . . , β k ), 1 ≤ β 1 < · · · < β k ≤ n, and where can be pulled back to the domain U so that one obtains a k-form Here the partial derivatives are understood in an approximate sense, and M β α (x) are k × k minors of the matrix with ordered rows and columns. We denote the norm of this linear mapping by the symbol |Λ k f (x)|.
The minimal analytic and geometric properties of the mapping f generating a bounded operator of pull-back of differential forms of degree k were obtained in [5].
We shall say that an approximate differentiable In addition to the property of k-finite distortion, the mapping in (3) must also exhibit a certain behavior of a distortion characteristic containing the ratios (for details, see [5]). Here we formulate a simpler version of this result for homeomorphic mappings.
Proposition ([5]). Let f : U → W be an approximate differentiable homeomorphism. An operator is bounded if and only if the following conditions are satisfied: (1) f : U → W has a k-finite distortion; (2) the function In this case, the norm of the operator f * is comparable with K k,p ( · ) | L κ (U ) .
In our paper, we consider the mappings differentiable in the sense of Sobolev. Such mappings are unconditionally approximate differentiable.
We say that a homeomorphism f ∈ CD k (U ; W ) has a (q, p)-bounded distortion (f ∈ CD k q,p (U ; W )) if it satisfies condition (2) of the proposition stated above.

MAIN RESULT
almost everywhere in U . Then the limit mapping f is also a mapping with k-finite distortion and has a (q, p)-bounded distortion, and the inequality K k,p (f )(x) ≤ M (x) holds for its distortion coefficient.
Proof. It follows from the conditions of the theorem that f ∈ W 1 l,loc (U ). First, we show that the limit mapping f is also a mapping with k-finite distortion and has a (q, p)-bounded distortion. For this, we first show that the mapping f induces a bounded operator Since each mapping f m is a mapping with k-finite and (q, p)-bounded distortion, it follows from the assumption that the homeomorphism f m : U → W induces a bounded operator Moreover, the norms of the operators f * m are totally bounded Take a k-form ω ∈ L p (W, Λ k ) ∩ C (W, Λ k ) and set u m = f * m (ω). Since f * m ≤ M , the sequence of forms u m is bounded in L q (U, Λ k ). Therefore, we can extract a weakly converging subsequence. We assume that the sequence u m weakly converges in L q (U, Λ k ) to a form u 0 . The weak convergence of forms means that the coefficients of the forms u m weakly converge in L q (U ) to the corresponding coefficients of the form u 0 .
Since the sequence u m weakly converges to u 0 in L q (U, Λ k ), we have The following lemma is proved in the book [1, Chap. 2, Sec. 4].
Since the homeomorphisms f m locally uniformly converge to f and the form ω has continuous coefficients, the functions ω β (f m (x)) converge to ω β (f (x)) locally uniformly. The lemma implies that the minors of the matrices Df m weakly converge in L l/k,loc to minors of the matrix Df . Therefore, the forms u m weakly converge to f * (ω) and hence the mapping f induces a bounded operator Now we estimate the distortion coefficient of the limit mapping f . Let θ ∈ C ∞ 0 (U ) be a test function. Let Z m be the set of zeros of the Jacobian of the mapping f m . Since the rank of the matrix Df m on the set Z m is less than k, it follows that all kth-order minors are equal to zero on the set Z m . First, we consider the case q < p. We apply H¨older's inequality to derivê H¨older's inequality can be used, because q/κ + q/p = 1.
It follows from the lemma that the elements of the matrix Λ k (f m )(x), i.e., the kth-order minors of the mapping f m , weakly converge in L l/k to the elements of the matrix Λ k (f )(x). Since the norm is semicontinuous in the Banach space L l/k , the left-hand side of the inequality can be estimated aŝ Let x 0 ∈ U , and let r ≤ R be such that the ball B(x 0 , R) lies in the domain U . We consider the following family of functions θ r,ε,y ∈ C ∞ 0 (U ): As the test function θ(x), we take the function θ r,ε,x 0 (x). Then we have the inequalitŷ Since |θ r,ε,x 0 (x)| ≤ 1, it follows from Lebesgue's dominated convergence theorem that an arbitrary function g ∈ L 1,loc (Ω) satisfies the relation Since the mapping f m is differentiable, there is a set Σ m of measure zero [6], [7] such that Therefore,ˆB Let us show that |f m (B(x 0 , r)| → |f (B(x 0 , r)| as m → ∞ for almost all r.
Since |f m (B(x 0 , r))| < ∞, the mappings f m are homeomorphisms, so that the images of the spheres f m (S(x 0 , r)) do not intersect, it follows that the measure of the image of any sphere is zero for almost all r, |f m (S(x 0 , r))| = 0.
We fix r so that Thus, passing to the limit as m → ∞ and then as ε → 0 in inequality (4), we obtain Dividing the result by the measure of the ball B(x 0 , r), we obtain the inequality Since the homeomorphism f is differentiable in the Sobolev sense, by formula (2.5)  For a function g ∈ L 1,loc (U ), we use the Lebesgue fundamental theorem to derive (for details, see [9]) g(x 0 ) = lim r→0 1 |B(x 0 , r)|ˆB (x 0 ,r) f (x) dx for a.a. x 0 ∈ U.
Let us use this property, and let r tend to 0. Almost everywhere in U , we obtain the inequality Therefore, the distortion coefficient of the limit mapping f satisfies the estimate for almost all x ∈ U .
In the case q = p, we havê Further, proceeding as in the case q < p, we obtain the desired estimate