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A numerical point of view at the Gurov-Reshetnyak inequality on the real line

A"norm"of power function in the Gurov-Reshetnyak class on the real line is computed. Moreover, a lower bound for the norm of the operator of even extension from the semi-axis to the whole real line in the Gurov-Reshetnyak class is obtained from numerical experiments.


Introduction
Let us consider functions f : R → R + where R is an interval of R. In what follows, R is the real line R or the semi-axis R + = [0, ∞). We also assume that the function f is locally summable on R, i.e. it is summable on each bounded subinterval I of R.
The mean value of the function f on a bounded interval I is defined by and the mean oscillation of this function is where | · | denotes the Lebesgue measure. For a given ε ∈ (0, 2] the Gurov-Reshetnyak class GR = GR(ε) = GR R (ε) is defined as the set of all non-negative functions f which are locally summable on R and such that the Gurov-Reshetnyak condition Ω(f ; I) ≤ ε f I is satisfied on all bounded intervals I ⊂ R (see [7]). Note that since any non-negative function f on any interval I satisfies the inequality Ω(f ; I) ≤ 2f I , the class GR R (2) is trivial and it coincides with the class of all functions locally summable on R. However, if ε ∈ (0, 2) then GR R (ε) is a non-trivial class (see [10, P. 112], [16]). If I is a subinterval of R, then the expression f I = Ω(f ; I)/f I is called the relative oscillation of the function f on the interval I. Further, the term f R = sup I⊂R f I is called the "norm" of function f in the Gurov-Reshetnyak class GR R .
One of the main results of the present work is the computation of the "norm" f α R of the function f α in the Gurov-Reshetnyak class on the real line R (see Theorem 2.1 below). In particular, this theorem implies the equation The above problem can be reformulated as follows: If a monotone function f belongs to the class GR R + (ε) for an ε ∈ (0, 2), then its even extension to R, which is also denoted by f , belongs to the Gurov-Reshetnyak class GR R (ε ) with an ε ∈ [ε, 2) (see Lemma 2.1). Therefore, one can also ask a question about the norms of the operator T of the even extension of monotone functions f ∈ GR R + (ε) to the real line R.
In the Table 1 below we report the results of numerical calculations of the values of ε R (p) for various p > 1. Comparing these results with known values of ε R + (p), one obtains lower bounds for the norms T (ε) GR and T GR . An analogous problem, concerning the norm of the operator of the even extension for the class BM O of monotone functions with bounded mean oscillation, has been considered in [9]. Some estimates for the norm of such an extension have been obtained in [17]. It is remarkable that the lower estimate presented in Remark 3.1 for the norm of the operator of the even extension T GR of the present work coincides with the lower estimate T BM O obtained in [2] for the corresponding operator of the even extension of monotone functions f ∈ BM O (see Remark 3.2).

Gurov-Reshetnyak inequality for power functions on the real line
Let us recall that the mean value f I = γ of function f on a subinterval I is uniquely defined by the condition 1 It is easily seen that According to [11], the Gurov-Reshetnyak "norm" of any monotone function f on R + can be computed by the formula and this fact will be used in what follows. Let us show that the even extension of monotone functions from a non-trivial Gurov-Reshetnyak class GR R + belong to a non-trivial class GR R . Lemma 2.1 For any ε ∈ (0, 2) there is an ε ∈ (0, 2) such that if a monotone function f belongs to GR R + (ε), then the even extension of f from R + to R belongs to GR R (ε ).
Proof. The proof of this lemma can be split into three steps.
Step 2. Let us show that the function f satisfies the Gehring inequality on R. It suffices to consider only the intervals of the form I = (−a, b), where 0 < a < b. Since f is an even function, one can apply the inequality (2) on the interval (0, b) and obtain Step 3. Now it remains to use the fact that the Gehring inequality implies the Gurov-Reshetnyak inequality with an ε ∈ (0, 2) (see [10, P. 114], [13]), which completes the proof.

Remark 2.1
The above obtained value of ε ≡ ε (ε) is not exact since at each step of the proof of Lemma 2.1, the parameters used are overestimated.
Remark 2.2 For ε < 1 there is a simpler proof of Lemma 2.1. In this case one can employ the simple inequality [2] Indeed, since then Other details of the proof are left to the reader. 2). Therefore, one also has T GR ≤ 2.
On the other hand, a lower bound for T GR , obtained in numerical experiments, is presented in Remark 3.1 below.

Consider now the expression
Let ψ be the function defined by Then and the proof is completed.
Corollary 2.2 Let p > 1 and let ψ be the function defined in Theorem 2.1. Then Recall that the "norms" on the Gurov-Reshetnyak class are denoted by ε − R (p) ≡ f 1/(p−1) R and ε + R (p) ≡ f −1/p R , p > 1. However, one has and Theorem 2.1 and Corollary 2.2 lead to the following result.

Corollary 2.3
If p > 1, then The next corollary allows us to improve Theorem 2.1 by using a better description of the set where the function ψ can attain its maximum.
Proof. Since the function ψ(α, η) is continuous on the interval [0, 1] in the variable η, it suffices to show that for any fixed α the function ψ 1 (α, η), defined in the proof of Theorem 2.1, is decreasing on the interval [η 1 , 1]. For this we compute the derivative ∂ ∂η and observe that if 0 < η < 1, then the derivative is negative. This completes the proof. Corollaries 2.2 and 2.4 imply the following result: Let us compute the derivative of the function ψ 0 (α, η). Thus Note that for η = η 1 the second factor in the second line of the formula is equal to zero. Moreover, the expression in the third line is nothing else but (1+η)Ω(f α ; I)/(2 sign α), and elementary computations lead to the following representation for the derivative of the function ψ 0 (α, η): It is easily seen that On the other hand, This leads to the following result.
Corollary 2.6 means that η max is the root of the equation However, even in the simplest case α = 1 the authors do not know an analytic solution of this equation (see Example 3.1 below).

Remark 2.4
The numerical study of the behaviour of the function ψ 0 (α, η) for different values of α shows that the derivative ∂ ∂η ψ 0 (α, η) has a unique root η max = η max (α) in the interval (0, η 1 ). Nevertheless, the authors do not know any rigorous proof of this fact.
Comments to the graphs. Figure 1. The lower graph in the left part of the Figure 1 shows the dependance of the Gurov-Reshetnyak "norms" ε R + (p) on the parameter p. These results are obtained from formula (1). Note that the data are represented in the logarithmic scale and do not include all results from Column 2. The upper graph shows the dependance of the exponents ε R (p) on p, presented in Corollary 2.3.
In the right part of Figure 1 we show the graphs of the inverse relations, i.e. these are values of those p, for which the function f 1/(p−1) belongs to the class GR R + (ε) (the lower line) or to the class GR R (ε) (the upper line) for a given ε.   In the left graph we show the quotient of the "norms" of the function f 1/(p−1) in classes GR R and GR R + with respect to the parameter p. The right graph reflects the behaviour of the parameter C ε (see (5)) with respect to ε. In both cases the variables p and ε are shown in the logarithmic scale. Table 1, the "norm" of the operator T can be estimated as follows

Remark 3.1 As is seen from
where the constant C is defined in (5). The corresponding numerical value C is shown in boldface in the last row of Table 1.  Example 3.2 Let α = 1/2, p = 3, and ε ≈ 0.296. In Table 1 the numerical results corresponding to this case are shown in the line partially written in boldface. In Figure 3 the graph of the corresponding function ψ(1/2, η) is represented by the dashed line. In this case, the equation (3) takes the form √ η = 2 3 + η .