ISSN 0001-4346, Mathematical Notes, 2013, Vol. 94, No. 3, pp. 320–329. © Pleiades Publishing, Ltd., 2013. Original Russian Text © A. V. Ivanov, V. I. Ivanov, 2013, published in Matematicheskie Zametki, 2013, Vol. 94, No. 3, pp. 338–348.
Optimal Arguments in Jackson’s Inequality in the Power-Weighted Space L2(Rd ) A. V. Ivanov* and V. I. Ivanov** Tula State University, Tula, Russia Received February 10, 2013
Abstract—This paper is devoted to the determination of the optimal arguments in the exact Jackson inequality in the space L2 on the Euclidean space with power weight equal to the product of the moduli of the coordinates with nonnegative powers. The optimal arguments are studied depending on the geometry of the spectrum of the approximating entire functions and the neighborhood of zero in the definition of the modulus of continuity. The optimal arguments are obtained in the case where the first skew field is a lpd -ball for 1 ≤ p ≤ 2, and the second is a parallelepiped. DOI: 10.1134/S0001434613090034 Keywords: Jackson’s inequality, power-weighted space L2 (Rd ), modulus of continuity, skew field, Dunkl transform, Logan’s problem, Holder’s ¨ inequality.
1. INTRODUCTION Let Rd be d-dimensional real Euclidean space with inner product (x, y) = |x|2 = (x, y), let e1 , . . . , ed be a standard orthonormal basis in Rd , let |(α, x)|2k(α) vk (x) =
d
j=1 xj yj
and norm
(1)
α∈R+
be the generalized power weight defined by the positive subsystem R+ of a root system R ⊂ Rd and a function k(α) : R → R+ invariant with respect to the reflection group G(R) generated by R (see [1]), let ˆ 2 e−|x|2 /2 vk (x) dx ck = Rd
d be the Macdonald–Meta–Selberg integral, let dμk (x) = c−1 k vk (x) dx, let Lp,k (R ), 1 ≤ p ≤ 2, be the d space of complex Lebesgue measurable functions f on R with finite norm ˆ 1/p |f (x)|p dμk (x) < ∞, f p,k = Rd
and let Cb (Rd ) be the space of continuous bounded functions. Harmonic analysis in the spaces with generalized power weight was constructed in papers of Dunkl, ¨ Rosler, de Jeu, Trimesh, etc. The necessary facts can be obtained from [1]. Harmonic analysis is performed by using the Dunkl difference differential operators ∂f (x) f (x) − f (σα (x)) , j = 1, . . . , d, + k(α)(α, ej ) Dj f (x) = ∂xj (α, x) α∈R+
where σα is the orthogonal reflection in the hyperplane (α, x) = 0. * **
E-mail:
[email protected] E-mail:
[email protected]
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For y ∈ Rd , the system Dj f (x) = yj f (x),
f (0) = 1,
has a unique (in Rd ) solution of Ek (x, y) known as the Dunkl kernel, which is extended to an analytic function in Cd × Cd . The generalized exponential ek (x, y) = Ek (ix, y) possesses properties similar to those of the exponential ei(x,y) . The generalized exponential is used to define the Dunkl transform ˆ k f (y)ek (x, y) dμk (y). (2) f (x) = Rd
The inverse Dunkl transform is given by the formula ˆ f (y)ek (x, y) dμk (y) = fk (−x). fˇk (x) = Rd
For the Dunkl transform, the usual L1 and L2 theories hold, including Parseval’s equality. Thus, the Dunkl transform is a linear bounded operator from L1,k (Rd ) to Cb (Rd ) and from L2,k (Rd ) to L2,k (Rd ); therefore, by the Riesz–Torin interpolation theorem, it can be extended to a bounded operator from Lp,k (Rd ) to Lp ,k (Rd ), 1 < p < 2, 1/p + 1/p = 1. Further, let V and U be convex centrally symmetric compact skew fields, which are invariant with respect to the reflection group G(R), and let |x|V and |x|U be the norms on Rd defined by their Minkowski functionals 1/p d p |xj | , 1 ≤ p < ∞, |x|∞ = max |xj |, |x|p = j
j=1
Bpd = {x ∈ Rd : |x|p ≤ 1}. d (σV ), 1 ≤ p ≤ 2, σ > 0, be the class of functions f ∈ L (Rd ) ∩ C (Rd ) for which the Let Ep,k p,k b k d support (spectrum) satisfies supp f ⊂ σV . The function f ∈ Ep,k (σV ) can be extended to Cd as follows: ˆ fk (y)ek (z, y) dμk (y), f (z) = σV
It follows from this equality that f (z) is an entire function and it satisfies the exponential estimate |f (z)| ≤ cf eσ|Im z|V ∗ ,
cf > 0,
Im z = (Im z1 , . . . , Im zd ),
where V ∗ = {x ∈ Rd : max |(x, y)| ≤ 1} y∈V
is a polar of V . The converse statement, known as the Paley–Wiener theorem, was proved for cases in which V is a Euclidean ball or k(α) takes integer values [2]. It is also valid when V is a parallelepiped with faces parallel to the coordinate planes and vk (x) =
d
|xj |2λj +1 ,
j=1
1 λj ≥ − . 2
d (σV ) we mean the quantity By the best approximation of a function f ∈ L2,k (Rd ) by the class E2,k d (σV )}. E(σV, f )2,k = inf{f − g2,k : g ∈ E2,k
By Parseval’s equality,
ˆ E 2 (σV, f )2,k =
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|y|V ≥σ
|fk (y)|2 dμk (y).
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The modulus of continuity of a function f ∈ L2,k (Rd ) is defined by the equality 1/2 ˆ 2 ω(τ U, f )2,k = sup 2 (1 − Re ek (t, y))|f (y)| dμk (y) , τ > 0. t∈τ U
Rd
If k(α) ≡ 0, then it coincides with the classical modulus of continuity ω(τ U, f )2 = sup{f (x + t) − f (x)2 : t ∈ τ U }. The Jackson constant D(σV, τ U )2,k is defined as a sharp constant in Jackson’s inequality E(σV, f )2,k ≤ Dω(τ U, f )2,k , i.e., we set D(σV, τ U )2,k
E(σV, f )2,k d = sup : f ∈ L2,k (R ) . ω(τ U, f )2,k
For all σ, τ > 0, the following inequalities hold: 1 √ ≤ D(σV, τ U )2,k < ∞, 2 where the lower bound is sharp [3]–[5]. The optimal argument in Jackson’s inequality or the Chernykh point is defined by the equality
1 τd,k (σV, U ) = inf τ > 0 : D(σV, τ U )2,k = √ . 2 The problem of the optimal argument is related to the Logan problem for a certain class of entire functions. The Logan problem for the class of real functions M consists in the calculation of the quantity Λ(M, V ) = inf{λ(f, V ) : f ∈ M, f ≡ 0}, where λ(f, V ) = sup{|x|V : f (x) > 0} is the radius of the smallest ball in the norm of the skew field V outside which the functions are nonpositive. Let W d(σU ) be the class of real even entire functions f ∈ E d (σU ) for which f (0) > 0, fk (y) ≥ 0 k
1,k
on Rd . Let us present some well-known results. Theorem A. The following relations hold: τd,k (σV, U ) = Λ(Wkd (σU ), V ) =
Λ(Wkd (U ), V ) τs,k (V, U ) = . σ σ
For k(α) ≡ 0, Theorem A was proved by Berdysheva in [6]; in the general case, it was proved in [5]. Further, we set σ = 1. Let Jλ (x) be the Bessel function of the first kind of order λ ≥ −1/2, let Jλ (x) xλ be the normalized Bessel function, and let qλ be its least positive zero. jλ (x) = 2λ Γ(λ + 1)
Theorem B. If λk =
d −1+ k(α), 2 α∈R+
then τd,k (B2d , B2d ) = Λ(Wkd (B2d ), B2d ) = 2qλk . MATHEMATICAL NOTES
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The extremal function in the Logan problem is of the form jλ2k (|x|2 /2) 1 − (|x|2 /(2qλk ))2
.
For d = 1, the power weight is of the form vk (x) = |x|2λ+1 , λ ≥ −1/2. For λ = −1/2, the optimal argument was obtained by Chernykh (see [6]), while the Logan problem was solved by Logan [7]. For λ > −1/2, Theorem B was proved by Gorbachev [8]. For d > 1, k(α) ≡ 0, Theorem B was also proved by Gorbachev; in the general case, it was proved in [4], [5]. Euclidean balls are invariant with respect to the group of all orthogonal transformations; therefore, Theorem B holds for all weights (1). In the other cases, the optimal argument and the solution of the Logan problem depend on the reflection group G(R) and the corresponding weight (1). Consider the root system R = {±e1 , . . . , ±ed }. For it, R+ = {e1 , . . . , ed } and G(R) is the group, isomorphic to Zd2 , of diagonal matrices of order d with entries ±1 on the principal diagonal. For an invariant function, k(±ej ) = λ + 1/2, λj ≥ −1/2 and the power weight is of the form vk (x) =
d
|xj |2λj +1 .
(3)
j=1
The system of generalized exponentials has the form ek (x, y) =
d
(4)
eλj (xj yj ),
j=1
where eλj (xj yj ) = jλj (xj yj ) − ijλ j (xj yj ).
(5)
We introduce the notation jλ (x, y) =
d
jλj (xj yj ),
λ = (λ1 , . . . , λd ).
(6)
j=1
The invariant norms on Rd depend on the moduli of the coordinates. For the skew field U we shall consider the parallelepiped Πa =
d
[−aj , aj ],
a = (a1 , . . . , ad ),
aj > 0.
j=1
Theorem C. If λ = (λ1 , . . . , λd ), λj ≥ −1/2, and a = (a1 , . . . , ad ), aj > 0, the weight vk is defined by (3) and 2qλd 2qλ1 ,..., , (7) bλ,a = a1 ad then τd,k (B2d , Πa ) = Λ(Wkd (Πa ), B2d ) = |bλ,a |2 . If 1 ≤ p < 2 and qλ qλ1 = ··· = d , a1 ad then τd,k (Bpd , Πa ) = Λ(Wkd (Πa ), Bpd ) = |bλ,a |p . MATHEMATICAL NOTES
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The extremal entire function in the Logan problem is of the form F (x) = (|bλ,a |22 − |x|22 )
d
jλ2j (aj xj /2)
j=1
(1 − (aj xj /(2qλj ))2 )2
.
d , Theorem C was proved by Berdysheva in [6]; in the general case, it was For k(α) ≡ 0 and Πa = B∞ proved in [5]. The extremal entire function of the class Wkd(Πa ) was constructed by using the Yudin method [9]. Our goal is to get rid from the constraint (8) in Theorem C. To do this, it is necessary to modify the Yudin method for constructing the extremal function.
2. MAIN RESULT First, let us obtain a general lower bound in the Logan problem. Let λ ≥ −1/2, k ∈ N, and let · · · < qλ,−2 < qλ,−1 < qλ,0 < 0 < qλ,1 < qλ,2 < · · · ,
qλ,1−k = −qλ,k
be the zeros of the Bessel function jλ (x). If f is an entire function of exponential type a > 0 and f ∈ L1,k (R), vk (x) = |x|2λ+1 , then, it satisfies the quadrature formula [10], [11] ˆ ∞ 2qλ,k ; f (x)|x|2λ+1 dx = rλ (k, a)f a −∞ k∈Z
here the series on the right-hand side converges absolutely and rλ (k, a) =
22λ+3 |qλ,k |2λ > 0. a2λ+2 (Jλ (qλ,k ))2
Let λ = (λ1 , . . . , λd ), λj ≥ −1/2, let a = (a1 , . . . , ad ), aj > 0, let k = (k1 , . . . , kd ) ∈ Zd , let d 2qλ1 ,k1 2qλd ,kd kd k1 k rλj (kj , aj ), ba = (b1 , . . . , bd ) = ,..., , rλ (k, a) = a1 ad j=1
and let E d,a be the class of entire functions of d variables of exponential type aj in the jth variable. The multidimensional version of the quadrature formula for entire functions of exponential type was proved in [5]. Theorem D. If the weight vk is defined by (3) and f ∈ E d,a ∩ L1,k (Rd ), then ˆ f (x)vk (x) dx = rλ (k, a)f (bka ); Rd
k∈Zd
here the series on the right-hand side converges absolutely. If
θ = (θ1 , . . . , θd ) ∈ (0, 1]d ,
bka,θ =
f ∈ E d,a ∩ L1,k (Rd ),
fθ (x) = f (θ1 x1 , . . . , θd xd ),
then fθ ∈ E d,a ∩ L1,k (Rd ), ˆ R
2θ1 qλ1 ,k1 2θd qλd ,kd ,..., , a1 ad
fθ (x)vk (x) dx = d d
ˆ
1 2(λj +1)
j=1 θj
Rd
f (x)vk (x) dx;
therefore, Theorem D has the following corollary. MATHEMATICAL NOTES
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Corollary. If f ∈ E d,a ∩ L1,k (Rd ) and θ ∈ (0, 1]d , then ˆ Rd
f (x)vk (x) dx =
d
2(λj +1)
θj
j=1
rλ (k, a)f (bka,θ ).
k∈Zd
Let the norm |x|V = |(|x1 |, . . . , |xd |)|V be invariant with respect to the reflection group G(R) for the root system R = {±e1 , . . . , ±ed }. The function |(|x1 |, . . . , |xd |)|V does not decrease with respect to each variable |x1 |, . . . , |xd |. Lemma. If λ = (λ1 , . . . , λd ), λj ≥ −1/2, a = (a1 , . . . , ad ), aj > 0, the norm | · |V is a function even in each coordinate, the weight vk is defined by (3), and the vector bλ,a is defined by (7), then Λ(Wkd (Πa ), V ) ≥ |bλ,a |V . Proof. Assume the converse: there exists an ε > 0 and a function f ∈ Wkd (Πa ) for which |x|V ≥ (1 − ε)|bλ,a |V .
f (x) ≤ 0,
Since fk (0) ≥ 0 and, for θ ∈ [1 − ε, 1]d , k ∈ Zd , 2θ1 qλ1 ,1 2θ q d λ ,1 k d ≥ (1 − ε)|bλ,a |V , ,..., |ba,θ |V ≥ a1 ad V it follows from the corollary that ˆ d 2(λ +1) f (x)vk (x) dx = θj j rλ (k, a)f (bka,θ ) ≤ 0. 0≤ Rd
Hence
j=1
2θd qλd ,1 2θ1 qλ1 ,1 ,..., f a1 ad
k∈Zd
for 1 − ε ≤ θ1 ≤ 1, . . . , 1 − ε ≤ θd ≤ 1.
=0
It follows from the analyticity of f in each variable that f ≡ 0, and this contradicts the condition f (0) > 0. The lemma is proved. For λ = (−1/2, . . . , −1/2), the lemma was proved by Berdysheva [6]. Theorem. If λ = (λ1 , . . . , λd ), λj ≥ −1/2, a = (a1 , . . . , ad ), aj > 0, 1 ≤ p ≤ 2, the weight vk is defined by (3), and the vector bλ,a is defined by (7), then τd,k (Bpd , Πa ) = Λ(Wkd (Πa ), Bpd ) = |bλ,a |p . The extremal entire function in the Logan problem is of the form d d jλ2j (aj xj /2) aj 2−p 2 p xj . Fp (x) = |bλ,a |p − 2qj (1 − (aj xj /(2qj ))2 )2 j=1
j=1
Proof. Suppose that λ ≥ −1/2, a > 0, vλ (x) = |x|2λ+1 , dμλ (x) = f (x) is an even function on R, fλ (y) =
ˆ
∞ −∞
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f (x)jλ (xy) dμλ (x)
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√ is its Dunkl (Hankel) transform, cλ = Γ(λ + 1)/( π Γ(λ + 1/2)), and ˆ π t T f (x) = cλ f ( x2 + t2 − 2xt cos ϕ ) sin2λ ϕ dϕ,
t∈R
0
is the generalized shift operator. Note the following properties of the generalized shift operator [12], [13]: • if f (x) ≥ 0, then T tf (x) ≥ 0;
(10)
• the function T f (x) is even in x and t;
(11)
tf ) λ (y) = j (ty)fλ (y); • (T λ
(12)
t
• if supp f ⊂ [−δ, δ], then supp T f ⊂ [−|t| − δ, |t| + δ]. t
Consider the function
⎧ 2qλ ⎪ ⎨jλ x , a uλ (x) = ⎪ ⎩0,
Using the formula [10] ˆ a xJλ (μx)Jλ (νx) dx = 0
μ2
(13)
a , 2 a |x| > . 2 |x| ≤
(14)
a {μJλ+1 (μa)Jλ (νa) − νJλ (μa)Jλ+1 (νa)}, − ν2
we obtain u λλ (x) = −
qλ a2λ jλ (qλ ) jλ ((a/2)x) · . 23λ Γ(λ + 1) (2qλ /a)2 − x2
(15)
Let λ = (λ1 , . . . , λd ), λj ≥ −1/2, a = (a1 , . . . , ad ), aj > 0, t = (t1 , . . . , td ) ∈ Rd , and let uλ (x) =
d
d
T t uλ (x) =
uλj (xj ),
j=1
T tj uλj (xj ).
j=1
By (2), (4)–(6), (10), (12), and (14), we have u λk (x) =
d
λ
u λjj (xj ),
t u )k (x) = j (t, x) (T uλk (x), λ λ
T t uλ (x) ≥ 0.
(16)
j=1
If |tj | ≤ δj , δ = (δ1 , . . . , δd ), δj > 0, then, from (13), we obtain supp T t uλ (x) ⊂ Πa/2+δ .
(17)
According to the Yudin method, the required extremal function in [5] is obtained as the Dunkl transform of the function ˆ ∂uλ (t) vk (t) dσλ (t), T t uλ (x) (18) G(x) = − ∂n ∂Πa/2 where ∂uλ (t)/∂n is the derivative along the outward normal to the boundary ∂Πa/2 of the parallelepiped Πa/2 , vk (t) is the weight (3), and dσ(t) is an area element of the surface on ∂Πa/2 , dσλ (t) = d
dσ(t)
λj +1 Γ(λ j j=1 2
+ 1)
.
Under the integral (18), let us introduce a specially chosen piecewise constant positive function. MATHEMATICAL NOTES
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For t ∈ ∂Πa/2 , we set tj = ±
h(t) = μj > 0, and consider the function
ˆ g(x) = −
T t uλ (x) ∂Πa/2
aj , 2
(19)
j = 1, . . . , d,
∂uλ (t) h(t)vk (t) dσλ (t). ∂n
In view of (11), the function g(x) is even in each variable. Since Eq. (14) implies 2qλj ∂uλ (t) = j (q ) uλi (ti ) ≤ 0, λ ∂n aj λj j tj =±aj /2
(20)
(21)
i=j
it follows from (16), (19) that g(x) ≥ 0. In view of (17), supp g ⊂ Πa . Further, using (16), we obtain ˆ ∂uλ (t) k t u )k (y) dσ (t) h(t)vk (t)(T g (y) = − λ λ ∂n ∂Πa/2 ˆ ∂uλ (t) h(t)jλ (t, y)vk (t) dσλ (t) u k (y). (22) =− ∂n ∂Πa/2 If Πja/2 = i=j [−ai /2, ai /2], then it follows from (19), (22) that ˆ ∂uλ (t) h(t)jλ (t, y)vk (t) dσλ (t) g1 (y) = ∂n ∂Πa/2 2λj +1 d qλj aj aj 1 yj λ +1 jλj (qλj )μj jλj = −4 j a 2 2 Γ(λj + 1) 2 j=1 j ˆ × uλi (ti )jλj (ti yi ) dμλi (ti ) Πja/2 i=j
= −4
d qλj
aj
j=1
i=j
jλ j (qλj )μj jλj
2λj +1 aj aj 1 yj λ +1 j 2 2 Γ(λj + 1) 2
d qλj
j (qλ )μj aj λj j j=1 i=j 2λj +1 aj 1 aj 1 . yj λ +1 × jλj λ j j 2 2 Γ(λj + 1) 2 u (yj ) ×
u λλii (yi ) = −4 uλk (y)
λj
λ
Substituting the expressions (14) for u λjj (yj ), we obtain g1 (y) =
u λk (y)
d
μj
j=1
2qλj aj
2
−
yj2
.
Combining this with (22), we find that g (y) = k
d
μj
j=1
2qλj aj
2
Suppose that 1 ≤ p ≤ 2, bλ,a = (b1 , . . . , bd ), 2qλj , bj = aj MATHEMATICAL NOTES
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−
μj =
yj2
|bλ,a |p bj
( uλk (y))2 .
2−p .
(23)
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In that case, d
μj b2j = |bλ,a |2−p p
d
j=1
bpj = |bλ,a |2p .
j=1
If d
μj (b2j − yj2 ) ≥ 0,
j=1
then d
μj yj2 ≤ |bλ,a |2p
j=1
¨ and, in view of Holder’s inequality, we can write p/2 1−p/2 d d d p/2 −p/(2−p) p −p/2 2 μj |yj | μj ≤ μj y j μj |y|p = j=1
j=1
≤ |bλ,a |pp |bλ,a |−p p
d
j=1
1−p/2 bpj
= |bλ,a |pp .
j=1
Thus, it follows from (23) that, for 1 ≤ p ≤ 2, λ( g k , Bpd ) ≤ |bλ,a |p . Up to normalization, the function Fp (9) coincides with the function gk . Hence, for it, supp Fpk ⊂ Πa ,
Fpk (y) ≥ 0 (y ∈ Rd ). Since [12] jλj (aj xj /2) c(λj , aj ) 1 − (aj xj /(2qj ))2 ≤ (1 + |x |)λj +5/2 , j we have
c(λ, a)(1 + |x|22 ) c1 (λ, a) ≤ d |Fp (x)|vk (x) ≤ d 4 2 j=1 (1 + |xj |) j=1 (1 + |xj |) and Fp ∈ L1,k (Rd ). Thus, Fp ∈ Wkd (Πa ) and, for it, λ(Fp , Bpd ) ≤ |bλ,a |p ; therefore, Λ(Wkd (Πa ), Bpd ) ≤ |bλ,a |p . Combining this with the lemma, we obtain the proof of the theorem. Note that, for 1 ≤ p < 2, the extremal entire function (9) is also new in the “weightless” case. For p > 2, optimization with respect to the functions (19) is not productive. We believe that, for p > 2, the following relation holds: τd,k (Bpd , Πa ) = Λ(Wkd (Πa ), Bpd ) = |bλ,a |2 . ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (grants no. no. 10-01-00564 and no. 13-01-00045). MATHEMATICAL NOTES
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