An estimate for the best constant in the L p-Wirtinger inequality with weights

We prove an estimate for the best constant C in the following Wirtinger type inequality ∫ 2π


Introduction
In this paper we consider some Wirtinger-type inequalities.More precisely, our aim is to give an estimate for the best constant C(a, b) in the following type inequality (for p > 1): (1.1) We note that when a = b = 1 and p = 2 , inequality (1.1)-(1.2) reduces to the well-known Wirtinger inequality.In this case (1.3) C(a, b) = C(1, 1) = 1.
The case b = a and p = 2 has been analyzed by Piccinini and Spagnolo in [3], who obtained the estimate .
The case b = a −1 and p = 2 has been studied in [4] (when a is a measurable function bounded from above and away from 0) and in [2] (when a is simply a function belonging to L 1 (0, 2π)), where the following estimate was obtained: The case a = b and p = 2 has been studied in [4] (when a, b are positive measurable functions on (0, 2π) bounded from above and away from 0) and in [2] (when a and b are non negative measurable functions on (0, 2π) such that a and 1 b belong to L 1 (0, 2π)), where the following estimate was obtained: In this paper we will extend the estimates (1.3)-(1.6) to the more general case p > 1.
We will denote by p the conjugate exponent to p (i.e. Furthermore, if a : (0, 2π) → R is an integrable positive function we will denote by L p 1 the set of all measurable functions u on [0, 2π] such that 2π 0 the set of all functions w ∈ L 1 (0, 2π) such that the distributional derivative w belongs to L p ( 1 a p−1 ) and w(0) = w(2π).If b : (0, 2π) → R is any nonnegative measurable function, W 1,p  per (b) is defined similarly.We will prove the following two theorems: Theorem 1.1.Let a ∈ L 1 (0, 2π) be a nonnegative function.Then the following Wirtinger inequality holds: where Note that when a is equal to 1 , Theorem 1.1 yields which is the optimal constant in the Wirtinger inequality, as proved by Croce and Dacorogna in [1].
In order to prove Theorem 1.2, we need Lemmas 3.1 and 3.2.The first one gives an estimate for C(a, a) (when a = b ) that reduces to the estimate (1.5) obtained by Piccinini and Spagnolo when p = 2 ; the second one gives an estimate for C(a, b) for arbitrary weight functions a, b belonging to L ∞ .

Proof of Theorem 1.1
First we proceed with a lemma and then with the proof of the theorem.Then W ∈ W 1,p (R) is β− periodic where β = 2π 0 adt.It easily seen that (2.1) and (2.2) may be rewritten as Therefore, applying Theorem 1.1 of [1] we conclude the proof.
Proof of the Theorem 1.1.The proof is similar in spirit to the one of [2] but it differs in technical aspects.It will be divided into 3 steps.
For h > 1 we set where h is a constant, hence w h (0) = w h (2π).
A simple continuity argument yields that for all h there exists h such that i.e., the sequence { h } is bounded, we claim that h → 0 .In fact if this were not true there would exist a subsequence, still denoted by h , such that h → with = 0. Now, we show that the value is equal to zero, thus proving the claim.The function is strictly increasing and from (1.2) (2.9) φ(0) = 0.
On the other hand (2.10) Combining (2.9) and (2.10) we have By contradiction, we have proved the claim.From Lemma 2.1 we have the following inequality for w h and a h (2.12) where Since h → 0, w h → w uniformly in (0, 2π); moreover a h → a in L 1 (0, 2π) and we have On the other hand (2.6) and (2.7) imply 2π 0 Notice that since w is bounded, the last integral is infinitesimal as Step 2. Now, still assuming that w ∈ W 1,p per (0, 2π)∩W 1,∞ (0, 2π), we make no special assumptions on a. Notice, however, that we may always assume, without loss of generality, that 2π 0 a(x)dx > 0 , otherwise (1.7) would be trivial.For δ > 0 we set where δ is a constant.
As in the previous step it is possible to prove that for any δ there exists δ such that (2.17) From the previous step we have the following inequality for w δ and a δ (2.19) As in (2.13), from (2.18) we have, (2.20) Combining (2.19) and (2.20) we obtain (1.7).
Step 3. Finally we assume that w ∈ W 1,p per ( 1 a p−1 ).For h > 0 we set where h is a constant and (2.23) When {T h (w )>0} T h (w )dx < − {T h (w )<0} T h (w )dx, σ h (x) is defined similarly to (2.24).Also in this case with similar considerations to those of the previous steps we obtain: for any h there exists h such that Moreover, for any h, w h (0) = w h (2π).
From the previous case, we have the Wirtinger inequality for a and w h (2.27) Since w h → w uniformly, we get

Proof of the Theorem 1.2
In order to prove the Theorem 1.2 we need the following two lemmas.The first lemma extends an analogous one proved in Piccinini-Spagnolo (Lemma 1. of [3]).Here we will follow closely their proof by adapting to the case p = 2. Lemma 3.1.Let a(t) be a periodic measurable function, with period 2π , such that 1 ≤ a(t) ≤ L ; let w(t) be a periodic function, belonging to W 1,p loc (R), with period 2π , such that 2π 0 a|w| p−2 w = 0. Then the following inequality holds: Proof.Consider for any a(t) such that 1 ≤ a(t) ≤ L , the eigenvalue problem When p = 2 , (3.5) becomes the problem (13) in [3].
It is easy to prove that the values of λ for which this problem has not constant solutions form a sequence λ n with 0 < λ 1 < λ 2 < ..., and that for any function w(t), periodic of period 2π , such that 2π 0 a|w| p−2 w = 0, the following estimate holds Therefore, in order to prove (3.1) it is sufficient to show that, if λ = 0 and w = 0 satisfy (3.5), then necessarily It is easily seen that a solution of (3.5) in each period has at least two zeros, and that between any pair of zeros of the function there is one and only one zero of its derivative.Let t 0 , t 2 , t 4 be three consecutive zeros of w and let t 1 and t 3 be two zeros of w in such a way that t 0 < t 1 < t 2 < t 3 < t 4 .Without loss of generality we may suppose that w(t 1 ) > 0 and w(t 3 ) < 0. It is obvious that We define, for t 0 < t ≤ t 1 , the function According to (3.5) this function satisfies the following first order differential equation: We remark that, since f < 0, f is strictly decreasing.Furthermore lim t→t + 0 f (t) = +∞, f (t 1 ) = 0 .Hence, there is one and only one point, say τ , in the interval (t 0 , t 1 ) such that f (τ )= . Since f is decreasing, the following inequalities hold: Thus, calling f 0 (t) the function such that (3.10) Let us set (3.12) The solution of (3.10) is given by where .
Therefore f 0 (t) tends to infinity for t converging to and vanishes for t equal to τ + P (λ,L) λL F f0(τ ) P (λ,L) .It follows that In a similar way we can prove that hence by adding the relations above we get that . So recalling (3.6), we can state Since f0(τ ) P (λ,L) = α(L) and f0(τ ) P (λ) = β(L) the proof is complete.(3.17) Proof.Under the change of variable where k is defined by where C is defined as in (3.Proof of Theorem 1.2.The proof of this Theorem is analogous to the one of Theorem 3.1 in [2].
We divide the proof into 3 steps.
Let a h and w h be defined by (2.6) and (2.7) respectively, and for h > 1 From Lemma 3.2 we deduce the Wirtinger inequality for a h , b h and w h where S h = sup p a p−1 h b h and I h = inf p a p−1 h b h .In view of (2.7) and (3.22) we have Step 2. Let us assume now, w ∈ W 1,p per (0, 2π) ∩ W 1,∞ (0, 2π), and a ≥ 0 b < ∞.
For δ > 0, let a δ and w δ be defined by (2.15) and (2.16) respectively, and From the step 1, we deduce the Wirtinger inequality for a δ , b δ and w δ where w ∈ W 1,p (0, 2π) is a 2π− periodic function satisfying the constraint (1.2) 2π 0 a|w| p−2 w = 0, and a and b are two measurable and non negative functions belonging to suitable spaces.
Let us still denote by a the 2π− periodic extension of the function a originally defined only in [0, 2π] and set y(x) = x 0 a(t)dt and W (y) = w(x(y)).

Lemma 3 . 2 .
Suppose a and b ∈ L ∞ (R) and inf a > 0 , inf b > 0 .Assume that S = sup p √ a p−1 b , I = inf p √ a p−1 b > 0 and set A(x, y) =