One-dimensional differential Hardy inequality

We establish necessary and sufficient conditions for the one-dimensional differential Hardy inequality to hold, including the overdetermined case. The solution is given in terms different from those of the known results. Moreover, the least constant for this inequality is estimated.


Introduction
Let I = (a, b), -∞ ≤ a < b ≤ ∞,  ≤ p, q ≤ ∞,  p +  p = , and  q +  q = . Let u ≥  and ρ >  be weight functions such that u ∈ L loc q and ρ - ≡  ρ ∈ L loc p , where L p ≡ L p (I) stands for the space of measurable functions f on I with finite norm We consider the following Hardy inequality in the differential form In [] it is shown that if then inequality () is satisfied for all functions f ∈ AC(I) such that f (a) =  or f (b) = , respectively. For example, in case (), it is equivalent to the weighted integral Hardy in- which has been studied for all values of the parameters  < p, q ≤ ∞ (see [, ], and []).
In [] it is also shown that in the last case inequality () is satisfied for all functions f ∈ AC(I) such that f (a) =  and f (b) = , that is, it is an overdetermined case. This case is studied in [, ], and [].
In the present work, for  ≤ p ≤ q ≤ ∞, we establish a criterion for the validity of inequality () with an estimate of the type for the least constant C in (), where B(u, ρ) is some functional depending on u and ρ. Moreover, we present a calculation formula for the least value of C  and its two-sided estimate.
We suppose that only condition () does not hold. In case () or (), our criterion coincides with the well-known Muckenhoupt result. However, our upper estimate in () is worse than the known one (see [-], and Remark . further). In case (), our criterion is given in terms different from those in [, ], and []. The terms in [] are close to ours, but the comparison analysis shows that our results and an estimate of type () are better than in [] (see Remark .).
At the end of the paper, we find a criterion for the compactness of the set M = {uf : f ∈ • AC(I), ρf p ≤ } in L q (I).

Proof
Since λ q+ -λ q +  , the sign of the function ϕ is defined by the value of the function d(λ) = λ q+ -λ q + .
This gives that Let us find an extremum of the function d for λ > . We have that q+ . Therefore, the function d(λ) decreases for  < λ ≤ q q+ and increases for λ > q q+ . Moreover, it has a minimum at q Thus, In view of the continuity of ϕ for λ > , from () and () there follows the existence of a point that satisfies () and ().
The last statement of Lemma . follows from the intersection of graphs of two decreasing and concave upward functions λ q λ q - and  λ- at the point λ = λ  . The proof of Lemma . is complete.
The function has a minimum at the point λ = q q+ . Therefore, By Lemma . we have that q q+ < λ  < min{q, }. Hence, Let us estimate f ( q q+ ): From (), (), (), and (), taking into account that f () < , we have (). The proof of Lemma . is complete.
Proof Necessity. Let inequality () hold with the least constant C > . Suppose We introduce the following function: It is obvious that f c,h ∈ • AC(I). Substituting f c,h into (), we get From the last inequality, taking into account that its left-hand side does not depend on α, β such that a < α < β < b, we have for all c ∈ I and  < h < d.
In the case p = , we construct f in the following way. Let numbers c and h be defined as before, δ > , and Substituting f c,h into (), we get Taking the limit in this inequality as δ → , we get for almost all x : (a < x ≤ ch) and almost all y : (c + h ≤ y < b).
For α > , there exist points x : (a < x ≤ ch) and y : (c + h ≤ y < b) such that Then This, together with u q,(x,y) ≥ u q,(c-h,c+h) , yields that Since the left-hand side of this inequality does not depend on α > , letting α → , we get () for p = . Thus, for all  ≤ p ≤ q < ∞, we have that Sufficiency. Let A p,q < ∞ be correct. Let f be a nontrivial function from • AC(I). Without loss of generality, we assume that f ≥ . Let λ > . For any integer k, we assume that T k = {t ∈ I : f (t) > λ k }, T k = T k \ T k+ . Due to the boundedness of the function f , there exists an integer n = n(f ) such that T n = ∅ and T n+ = ∅. It is obvious that I = k≤n T k = k≤n T k . The set T k is open. Therefore, there exists a family of mutually disjoint intervals In view of the continuity of the function f , we get that From () by Hölder's inequality we have that In view of f (t) < λ k+ for t ∈ T k and λ qk = (λ -q ) i≤k λ qi , we have that which, as before, means that C ≤ A p,q .
Let p = q = ∞. Sufficiency. From () we have Using these relations instead of () and (), we have This gives that uf q ≤ A p,q ρf p and C ≤ A p,q .