The Besicovitch covering theorem and near-minimizers for the couple ( L 2 , BV )

Let Ω be a rectangle in R2. A new algorithm for the construction of a near-minimizer for the couple (L2(Ω),BV (Ω)) is presented. The algorithm is based on the Besicovitch covering theorem and analysis of local approximations of the given function f ∈ L2(Ω).

The theory of real interpolation is based on the notion of Peetre's K-functional, which is defined for the couple of Banach spaces (X 0 , X 1 ) by the formula where x ∈ X 0 + X 1 and t > 0.
It is well known that the K-functional is connected (see [1,2]) to the more general L p,q -functional, defined for 0 < p, q < ∞ by the expression L p,q (t, x; X 0 , X 1 ) = inf u∈X 1 ( x − u p X 0 + t u q X 1 ), t > 0.
Definition 1.An element x t ∈ X 1 will be called a near-minimizer for the L p,q -functional of the element x ∈ X 0 + X 1 at the point t > 0 if x − x t p X 0 + t x t q X 1 ≤ cL p,q (t, x; X 0 , X 1 ).In the case when c = 1 the element x t will be called the exact minimizer for the L p,q -functional.Let us give one example connected to inverse problems (see, for example, [3]).Let A : X → Y be a given linear bounded operator which maps the Hilbert space X to the Hilbert space Y .Suppose that we observe the quantity where η ∈ Y is a 'noise' and we are interested in the reconstruction of the exact solution x ∈ X of ( 1).
An attempt to define an approximate solution of (1) as A −1 y does not lead to success even in the case when A is injective and y belongs to the image of the operator A, since such a procedure usually increases the level of noise.For example, this happens even in the finite-dimensional case when A has rather small singular numbers.Hence more sophisticated ideas are used to solve this problem.One of the most popular methods to construct an approximate solution to (1) is the classical Tikhonov regularization strategy, which suggests that as an approximate solution we take an element x t ∈ X which minimizes the Tikhonov functional for some parameter t > 0 chosen on the basis of a priori information on the noise η and the exact solution x.
In fact, the Tikhonov method is deeply connected to the problem of constructing a minimizer for the L 2,2 -functional for the Hilbert couple (Y, A(X)), where by A(X) we denote the image of the space X with the norm Indeed, from the equality it follows that if the element x t ∈ X is the minimizer for the Tikhonov functional, then , and therefore Ax t is a minimizer for the L 2,2 -functional.Moreover, in the case when A is an injective operator the element x t can be reconstructed as A −1 u t , where u t is a minimizer for the L 2,2 -functional of the element y for the Hilbert couple (Y, A(X)).
The Tikhonov method provides regularization only for the case of Hilbert spaces, however, during the last years regularization in the case when X, Y are not Hilbert spaces have attained a great importance.For example, a ROF (Rudin-Osher-Fatemi) model, which is very popular in image processing, suggests (see the books [4,5]) reduction of the level of noise in the 'noisy' image f ∈ L 2 by taking instead of f an element f t which minimizes the L 2,1 -functional of the couple (L 2 , BV ) However, to construct the exact minimizer and even a near-minimizer for the functional (2) is not a simple task and several different approaches have been suggested (see, for example, [6][7][8][9], and the book [4]).The most popular methods are based on nonlinear elliptic partial differential equations and on wavelet theory.Below we will present a new algorithm for constructing a near-minimizer for (2).This algorithm is a limiting case of the general algorithm (see [10,11]) and is based on the theory of local approximations (see [12,13]) and the Besicovitch covering theorem [14].
The algorithm consists of three steps (see Fig. 1).In the first step we analyse the local approximations of the given function f ∈ L 2 and construct the initial family of cubes 1 {Q x }.In the second step we apply the Besicovitch covering theorem to the family {Q x } and obtain a family of cubes {K i } i∈I consisting of a finite number of cubes.In the third step we use the family of cubes {K i } i∈I to construct a partition of unity and to define a near-minimizer f t .
Let us consider the algorithm in detail.Let f ∈ L 2 = L 2 (Ω), where Ω ⊂ R 2 is a rectangle, and let t > 0 be a given number.We will construct a near-minimizer for (2), i.e. an element f t such that with the constant c > 1 independent of f ∈ L 2 and t > 0.
1 Here and below we consider only cubes with sides parallel to the coordinate axes.If t ≥ t * , where then it is possible to prove that the near-minimizer can be defined as a constant function on Ω equal to f Ω .Hence it is enough to consider the case when t < t * .
In the first step of the algorithm we construct the initial family of cubes Here and below by Q(x, r) we denote a cube Q with the centre at the point x and radius r equal to the half side length of Q.
To construct the cube Q(x, r x ) for x ∈ Ω, we consider the function , where It is clear that the function ϕ x (r) is a nondecreasing continuous function with values from zero to t * defined in (3).As t < t * , therefore for each x ∈ Ω there exists a number r = r x such that (see Fig. 2) As the initial family of cubes {Q x } x∈Ω we will take the constructed family {Q(x, r x )} x∈Ω .Note that from the condition ϕ x (r x ) = t it follows that if the subfamily {Q(x i , r x i )} i∈I of {Q(x, r x )} x∈Ω consists of pairwise disjoint cubes, then the number of cubes |I| in the subfamily satisfies the inequality and therefore is finite.Now we will apply the Besicovitch covering theorem to the initial family of cubes {Q x } x∈Ω .Let us remind the formulation of the Besicovitch theorem (see [15]).
Theorem 2. Let Ω ⊂ R n be a bounded set.Suppose that for any x ∈ Ω the cube Q x with the centre in x is given.Then there exists a subfamily {Q x i } which covers Ω, i.e.Ω ⊂ ∪Q x i , and which can be split into a finite number of subfamilies of pairwise disjoint cubes π k = {Q x i } i∈I k .Moreover, the number of subfamilies π k can be estimated only by the dimension n.
From (4) we see that each subfamily π k consists of a finite number of cubes and therefore the whole family of constructed cubes {K i = Q x i } i∈I is finite and covers Ω.
Below we will assume that cubes K i are indexed in agreement with the decrease in their volume.Now we are ready to give the formula for the near-minimizer f t for the L-functional (2).On K 1 ∩ Ω we will take f t equal to f K 1 , then we define f t on (K 2 ∩ Ω)\K 1 by the constant f K 2 on (K 3 ∩ Ω)\(K 1 ∪ K 2 ), etc.As the number of the cubes in the family {K i } is finite, the process of constructing the function f t consists of a finite number of steps.The constructed near-minimizer can be written as where the functions ψ 1 = χ K 1 ∩Ω , ψ 2 = χ K 2 ∩Ω\K 1 , ... , ψ N = χ K N ∩Ω\∪ i<N K i form a partition of unity of Ω.Note that the constructed near-minimizer is not a continuous function.
Remark 3. Construction of the family of cubes {K i } in the Besicovitch covering theorem is constructive: as the first cube K 1 we take the largest or almost the largest cube in the family {Q x } x∈Ω .Then from the family {Q x } x∈Ω we exclude cubes with centres in K 1 .As the cube K 2 we take the largest or almost the largest cube from the family of remaining cubes, etc.

Fig. 1 .
Fig. 1.The three steps of the algorithm.