Complex Anal. Oper. Theory https://doi.org/10.1007/s11785-018-0795-4
Complex Analysis and Operator Theory
The Harmonic Dirichlet–Besov Space and the Optimal Norm for the Bergman Projection Djordjije Vujadinovi´c1
Received: 20 September 2017 / Accepted: 27 April 2018 © Springer International Publishing AG, part of Springer Nature 2018
Abstract We found the optimal operator-norm constant for the harmonic Bergman projection in the context of harmonic Dirichlet–Besov space regarding the natural semi-norm defined on it. Also, we give the two-side norm estimation when it comes to the complete norm. Keywords Harmonic Besov space · Harmonic Bergman projection · Zonal harmonics · Bochner spaces Mathematics Subject Classification Primary 46E20 · 46E22
1 Introduction and Notation Throughout this paper, B will be the open unit ball in Rn for a fixed positive integer n ≥ 2. The weighted measure dvα on B is defined by dvα (x) = cα (1 − |x|2 )α−1 dv(x) for any α > 0, where dv is the Lebesgue volume measure on B. Here, the constant cα is 2( n +α) chosen such that vα is normalized, i.e. vα (B) = 1 and it is given by cα = n|B|(2n )(α) . 2
By |B| we denote the volume of B and we let dτ (x) = (1 − |x|2 )−n dv(x). The class of all harmonic functions f on the unit ball B is denoted by H (B) and by h 2 (B) the space of all square-integrable harmonic functions f (with respect to dv) in B, i.e.,
Communicated by Daniel Aron Alpay.
B 1
Djordjije Vujadinovi´c
[email protected] Faculty of Mathematics, University of Montenegro, Dzordza Vašingtona bb, 81000 Podgorica, Montenegro
D. Vujadinovi´c
f 2L 2 (B,dv) =
| f (x)|2 dv(x) < ∞, f ∈ h 2 (B). B p
p
More generally, the weighted harmonic Bergman space bα = bα (B), 1 ≤ p ≤ ∞, is the space of all complex-valued harmonic functions f, such that the norm 1/ p
f L p (B,dvα ) =
B
| f | p dvα
is finite. We will use the following notation for the partial derivatives |∂ m f (x)| =
|∂ k f (x)|, ∂ k f (x) =
|k|=m
∂ |k| f (x) . ∂xk
n Here k = (k1 , . . . , kn ) is multi-index and |k| = i=1 ki = m. The Dirichlet–Besov spaces of harmonic functions in the unit ball represents the special case of the harmonic Besov spaces B p when p = 2. The harmonic Besov spaces have been studied by many authors in various settings of conditions and function domains. One may see the extensive study on them in [5]. Following the work of Jevti´c and Pavlovi´c (see [6]) on the harmonic Besov space B p , 1 ≤ p ≤ ∞, which is defined as a space consisting of all harmonic functions f ∈ H (B) such that the function (1 − |x|2 )k |∂ k f (x)| belongs to L p (B, dτ ) for some positive integer k, k > n−1 p . In fact, if 1 ≤ p ≤ ∞ and f ∈ H (B), then the following statements are equivalent: 2 m 0 m 0 f (x)| ∈ (a) There is a positive integer m 0 > n−1 p such that (1 − |x| ) |∂ L p (B, dτ ). 2 m m p (b) For all positive integers m > n−1 p , (1 − |x| ) |∂ f (x)| ∈ L (B, dτ ), (see Theorem 3.2 in [6]).
The definition is independent from the choice of integer m. The Besov space-norm, in this paper, is defined in the following manner: f B p =
|∂ α f (0)| +
1/ p (1 − |x|2 )mp |∂ m f (x)| p dτ (x)
, 1 ≤ p < ∞,
B
|α|
(1.1) and f B∞ =
|∂ α f (0)| + sup(1 − |x|2 )m |∂ m f (x)|, p = ∞.
(1.2)
B
|α|
The derived semi-norm (1.1) is given by f B˜ p =
1/ p (1 − |x|2 )mp |∂ m f (x)| p dτ (x) B
.
(1.3)
The Harmonic Dirichlet–Besov Space and the Optimal Norm…
In the rest of the paper, the integer m is fixed in (1.1) and (1.3), and satisfies the inequality m > n−1 p . We shall underline the definition of the harmonic Dirichlet–Besov space B 2 . Definition 1.1 The harmonic Dirichlet–Besov space B 2 is the set of functions f harmonic in B such that f B˜ 2 = where m >
1/2 (1 − |x| )
2 2m
|∂ f (x)| dτ (x) m
2
< ∞,
(1.4)
B
n−1 2 .
Weighted harmonic Bergman kernel Since the Hilbert space bα2 is closed in the weighted Lebesgue space L 2 (B, dvα ) there is a unique reproducing kernel Rα (x, y) defined on B × B which is real and symmetric, such that Rα (x, ·) ∈ bα2 stands for any fixed x ∈ B and f (x) = f (y)Rα (x, y)dvα (y), f ∈ bα2 . B
Following [1] (see Theorem 8.9, page 177) the unweighted harmonic Bergman kernel is given by the sequent formula ∞
R0 (x, y) =
1 (n + 2k)Z k (x, y), x, y ∈ B, n|B|
(1.5)
k=0
where Z k are extended zonal harmonic. For more information about zonal harmonics we refer to [1] (Chapter 5). The series (1.5) converges absolutely and uniformly on K × B for every compact set K ⊂ B. The formula for the weighted harmonic Bergman kernel for α > 0 is given by Rα (x, y) = ωα
∞ (k + n/2 + α) k=0
where ωα =
(n/2) (n/2+α) .
(k + n/2)
Z k (x, y), x, y ∈ B,
(1.6)
We will denote the above coefficients with An,α k =
(k + n/2 + α) . (k + n/2)
The weighted harmonic Bergman projection Pα is presented as an integral operator f (y)Rα (x, y)dvα (y), f ∈ L 2 (B, dvα ), x ∈ B . Pα f (x) = B
It is known that Pα is bounded on L p (B, dvα ) for 1 < p < ∞.
D. Vujadinovi´c
The estimation of the norm for the harmonic Bergman projection in case we conp p sider the harmonic Bergman space bα , Pα : L p (B, dvα ) → bα was observed in [3]. The authors obtained the optimal two-side norm estimation. Namely, they proved the existence of the constant Cα > 0 for a given α > −1, such that Cα−1
p2 p2 ≤ Pα L p (B,dvα )→bαp ≤ Cα , p−1 p−1
when 1 < p < ∞. Of course, Pα L 2 (B,dvα )→bα2 = 1. The following theorem describes the connection between the harmonic Besov space and the harmonic Bergman projection (see [6]). Theorem 1.2 For the 1 ≤ p ≤ ∞, the Bergman projection Pα , α > 0, is bounded and maps L p (B, dτ ) onto the harmonic Besov space B p . The main inducement for this paper is natural continuation of the previous research in the context of the harmonic Besov spaces. Some estimations of the norm for Pα : L p (B, dτ ) → B p , when 1 < p < ∞ have already been done in [9]. The fact that the harmonic Bergman projection Pα : L 2 (B, dτ ) → B 2 is not an orthogonal projection motivates us to observe this particular case. The analogous problem in the one-dimensional analytic case for the Besov space was considered in [8].
2 Preliminaries Gamma function’s inequalities The following inequalities for the Gamma function (see Theorem 2 in [4], page 7) will be of interest. Proposition 2.1 Let m, p and k be real numbers with m, p > 0 and p > k > − m: If k( p − m − k) 0( 0)
(2.1)
( p)(m) ()( p − k)(m + k).
(2.2)
then we have Proposition 2.2 Let p > 0 and q ∈ R such that |q| < p. Then 2 ( p) < ( p − q)( p + q). According to the introduced semi-norm (1.3) the inner product associate with the harmonic Dirichlet–Besov space B 2 is f, g =
|k|=m B
and f 2B˜ 2 = f, f .
(1 − |x|2 )2m
∂m f ∂m g (x) (x)dτ (x), ∂xk ∂xk
f, g ∈ B 2 , (2.3)
The Harmonic Dirichlet–Besov Space and the Optimal Norm…
In this section we are going to find the “exact” norm for the weighted Bergman projection Pα restricted on the subspace L 2 (B, dτ ) regarding the semi-norm generated with (2.3) in B 2 . Specifically, we find the operator norm Pα g B˜ 2 . g L 2 (B,dτ ) =0 g L 2 (B,dτ )
Pα L 2 (B,dτ )→ B˜ 2 =
sup
Moreover, the completion of the semi-norm · B˜ 2 in B 2 is defined as f B2 =
|∂ k f (0)| + f B˜ 2 ,
(2.4)
k
and we will also consider the operator norm Pα L 2 (B,dτ )→B 2 =
Pα g B 2 g L 2 (B,dτ ) =0 g L 2 (B,dτ ) sup
or briefly Pα . At this point, we recall the basic definition of the Bochner space which will be significant in order to prove the main result. Namely, for the given measure space (X, , μ), and the Banach space (Y, · Y ), the Bochner space L p (X, Y ) 1 ≤ p ≤ ∞, is defined to be the space of all Bochner measurable functions f : X → Y in a way that the corresponding norm is finite f L p (X,Y ) =
p
X
1/ p
f (x)Y dμ(x)
< ∞,
(2.5)
with a usual modification for p = ∞, f L ∞ (X,Y ) = ess sup f (x)Y .
(2.6)
x∈X
We return to the initial problem. The next lemma deals with the problem of finding the image of a polynomial by harmonic Bergman projection. More precisely, we are going to observe the function ϕ(x) p(x), where p is a polynomial in Rn and n ϕ(x) = (1 − |x|2 ) 2 . The homogenous polynomial p ∈ Pm (Rn ) can be represented as a unique sum of harmonic homogenous polynomials (see Theorem 5.7 in [1], page 77), i.e., p = pm + |x|2 pm−2 + · · · + |x|2k pm−2k ,
(2.7)
where k = [ m2 ] and each p j ∈ H j (Rn ). The above decomposition reduces on p = pm + pm−2 + · · · + pm−2k , on S.
(2.8)
D. Vujadinovi´c
Here, Pm (Rn ) is a notation for the set of all homogenous polynomials of degree m ∈ N, while H j (Rn ) is a set of all harmonic homogenous polynomial of degree j. By P(Rn ) we denote the set of all polynomials in Rn . In the sequel lemma, for any p ∈ Pm (Rn ) by pi , i ≤ m, we mean the harmonic homogenous polynomial of degree i which appears in decomposition (2.7). Lemma 2.3 Let p ∈ P(Rn ). Then, Pα (ϕp)(x) = D(n, α)
p(x) =
∞
Ain,α
i=0
j≥i
( i+ 2j+n ) (n +
i+ j 2
+ α)
ψ ij (x), where
ψ j (x), ψ j ∈ P j (Rn ),
(2.9)
j≥0 ( n2
+ α) . |B|(α)
D(n, α) =
Proof Using the homogenous expansion of the polynomial p ∈ P(Rn ), we write ψ j (x), (2.10) p(x) = j≥0
where the sum in (2.10) is finite, and as we have stated above, ψ j is the homogenous polynomial of degree j. Thus, Pα (ϕp)(x) = Rα (x, y)ϕ(y) p(y)dvα (y) B
= ωα
∞
Ain,α
i=0 ∞
= ωα cα
Ain,α
i=0 ∞
= ωα cα n = ωα cα n = ωα cα n =
i=0 ∞ i=0 ∞
ωα cα n 2
i=0 ∞ i=0
Z i (x, y)ϕ(y) p(y)dvα (y)
B
(1 − |y|2 )n/2+α−1 ψ j (y)Z i (x, y)dv(y)
j≥0 B
Ain,α Ain,α
ψ ij (x) B
S
Z i (x, ξ )ψ j (ξ )dσ (ξ )
(1 − r 2 )n/2+α−1r j+i+n−1 dr
j≥i
1
0
j≥i
Ain,α
(1 − r 2 )n/2+α−1r j+n+i−1 dr
j≥0 0
j≥i
Ain,α
1
1
S
Z i (x, ξ )ψ ij (ξ )dσ (ξ )
(1 − r 2 )n/2+α−1r j+i+n−1 dr
0
j +n+i n + α, 2 2
ψ ij (x).
(2.11)
The Harmonic Dirichlet–Besov Space and the Optimal Norm…
The important single-sum formula for the inner-product on the unit sphere ( ·, · S ) of two harmonic polynomials (see Theorem 5.14 in [1], page 84) is given by the following expression, Theorem 2.4 If p = α bα x α and q = α cα x α are harmonic polynomials on Rn , then p, q S = bα cα wα , α
where wα =
α! . n(n + 2) · · · (n + 2|α| − 2)
We conclude this section by presenting the main result of this paper and its corollaries. Theorem 2.5 For the defined semi-norm (2.3) on the harmonic Dirichlet–Besov space, the operator norm of the harmonic Bergman projection is expressed by Pα L 2 (B,dτ )→ B˜ 2 =
n2m−1 ( n2 )(2m − n + 1)(n + 2α − 1) π n/2 |B| 2 (α)
1/2 .
(2.12)
Now, we observe the problem of finding the operator norm for the Pα , where we consider the Banach norm defined by (2.4). For this purpose we define first the functional
m : L 2 (B, dτ ) → C, ∂ k Pα g(0).
m (g) =
(2.13)
k
Lemma 2.6 m L 2 (B,dτ )→C = 2
2 n (n (An,α k k!) dim Hk (R )
k
+ 2α − 1)(k + n2 )
n|B|(k +
3n 2
+ 2α − 1)
1/2 .
Proof According to the Theorem 1.2 for f ∈ B 2 there is g ∈ L 2 (B, dτ ) in a way that f = Pα g. For x ∈ B, we may assume that the vector x is of the form xe1 , x ∈ R. So, ∂ k f (0) = lim ∂ k f (x) k
x→0
= lim
x→0
k
k
∂ k Rα (x, y)g(y)dvα (y)
D. Vujadinovi´c
= lim ωα x→0
= lim ωα x→0
= lim ωα x→0
= ωα
= ωα
An,α d
∂ k Z d (x, y)g(y)dvα (y)
B
∂ k Z d (x, ξ )|y|d g(y)dvα (y)
∂k Zd (x, ξ )|y|d g(y)dvα (y) β B ∂x
|β|=k
k
An,α k k!
B
An,α d
k
B
Z k (e1 , ξ )|y|k g(y)dvα (y)
An,α k k!
k
where ξ =
An,α d
k
k
∞
B
Z k (e1 , y)g(y)dvα (y),
(2.14)
y |y| .
Now, simple duality argument and identity (2.14) implies 2 sup ∂ k Pα g(0) g L 2 (B,dτ ) ≤1 k
= (ωα cα ) =
2
k
k1 ,k2
×
k1 ,k2
n,α An,α k1 Ak2 k1 !k2 !
× =
S
n,α An,α k1 Ak2 k1 !k2 !
1
B
Z k1 (e1 , y)Z k2 (e1 , y)(1 − |y|2 )n+2α−2 dv(x)
(1 − r 2 )n+2α−2 r k1 +k2 +n−1 dr
0
Z k1 (e1 , ξ )Z k2 (e1 , ξ )dσ (ξ )
n,α n n |B|(ωα cα )2 . (Ak k!)2 dim Hk (Rn )B n + 2α − 1, k + 2 2 k
(2.15)
Theorem 2.7 max{ m , Pα L 2 (B,dτ )→ B˜ 2 } ≤ Pα ≤ m + Pα L 2 (B,dτ )→ B˜ 2 .
(2.16)
Proof Out of the definition of the norm in (2.4) and Lemma 2.6 we derive the following easy inequality (2.17) Pα ≤ m + Pα L 2 (B,dτ )→ B˜ 2 .
The Harmonic Dirichlet–Besov Space and the Optimal Norm…
On the other hand, for a positive number > 0, let g ∈ L 2 (B, dτ ) be a function such that g L 2 (B,dτ ) ≤ 1 and Pα g B˜ 2 > Pα L 2 (B,dτ )→ B˜ 2 − . Furthermore, we may choose h ∈ L 2 (B, dτ ) in a way that h L 2 (B,dτ ) ≤ 1 and Pα h B˜ 2 > m − .
Notice that under the previous notation, we have Pα (g − g1 + h) B˜ 2 > m + Pα L 2 (B,dτ )→ B˜ 2 − , where Pα g1 (x) = k
2 n (n (An,α k k!) dim Hk (R )
+ 2α − 1)(k + n2 )
n|B|(k +
+ 2α − 1) 1/2 n2m−1 ( n2 )(2m − n + 1)(n + 2α − 1) + . π n/2 |B| 2 (α) k
1/2
3n 2
(2.18)
3 The Proof of the Theorem 2.5 Proof Since the space of all the harmonic polynomials is dense in h 2 (B), we will consider the acting of the weighted Bergman projection on a set {ϕ(x)ψ(x)|ψ ∈ P(Rn )}. From the homogeneous expansion of ψ(x) = j≥0 ψ j (x) and the Lemma 2.3 we get
Pα (ϕψ)(x) =
( n2
∞ + α)
|B|(α)
n,α
Ai
i=0
If we denote by ψ ij (x) =
j≥i
|s|=i
i+ 2j+n
ψ ij (x). n + i+2 j + α
j
as x s ,
(3.1)
(3.2)
D. Vujadinovi´c
then, the Theorem 2.4 implies ϕψ2L 2 (B,dτ )
=
=
ψi (x)ψ j (x)dv(x)
B
i, j
n|B|
r
=
i, j
=
i, j
=
i, j
= =
i+ j+n−1
0
i, j
1
n|B| i + j +n n|B| i + j +n
S
dr
S
ψi (ξ )ψ j (ξ )dσ (ξ )
⎛ ⎞⎛ ⎞ ⎝ ψil (ξ )⎠ ⎝ ψ sj (ξ )⎠ dσ (ξ ) l≤i
s≥0 S
s≤ j
ψis (ξ )ψ sj (ξ )dσ (ξ )
j d!adi ad 2π n/2 ( n2 )(i + j + n) n(n + 2) · · · (n + 2s − 2) s≥0 |d|=s
π n/2
i, j
i + j +n
s≥0 |d|=s
s≥0 |d|=s
j
d!adi ad 2s−1 ( n2 + s)
d!a i a j π n/2 d d . 2s−1 ( n2 + s) i + j +n i, j≥s
Moreover, ∂ m Pα (ϕψ)(x)
∞ i+ 2j+n n2 + α
= Ain,α ∂ k ψ ij (x) i+ j |B|(α) i=0 j≥i n + 2 + α |k|=m
∞ i+ 2j+n j,k n2 + α
= Ain,α as x s−k i+ j |B|(α) i=0 j≥i n + 2 + α |s|=i |k|=m =
∞
A˜ in,α
i=0
j,k
where as
j
= as
j≥i
n
Ci,n,α j
|s|=i |k|=m
(sd +1) d=1 (sd −kd +1)
A˜ in,α =
j,k
as x s−k ,
and s − k = (s1 − k1 , . . . , sn − kn ),
1 An,α , Ci,n,α j = |B|(α) i
i+ j+n + α 2 2
. i+ j n+ 2 +α
n
(3.3)
The Harmonic Dirichlet–Besov Space and the Optimal Norm…
Therefore, taking into account the definition of wα from the Theorem 2.4 and using the polar coordinates x = r ξ, ξ ∈ S, we obtain Pα (ϕψ)2B˜ 2 =
∞
( A˜ in,α )2
i=0
B
⎛ ⎞2 n,α j,k (1 − |x|2 )2m−n ⎝ Ci, j as x s−k ⎠ dv(x) |s|=i |k|=m
j≥i
1 ∞ n,α 2 ˜ = n |B| ( Ai ) (1 − r 2 )2m−n r 2(i−m)+n−1 dr 0
i=0
⎛
×
⎝
S
⎛ ⎝
|s|=i |k|=m
⎞
⎞2
j,k ⎠ s−k ⎠ Ci,n,α ξ j as
dσ (ξ )
j≥i
n,α n n |B| ( A˜ i )2 B 2m − n + 1, i − m + 2 2 i=0 ⎛ ⎞2 n,α j,k ⎝ × Ci, j as ⎠ ws−k ∞
=
|s|=i |k|=m
j≥i
n ( A˜ in,α )2 B 2m − n + 1, i − m + 2 i=0 ⎛ ⎞2 n,α j ws−k ((s)!)2 ⎝ × Ci, j as ⎠ . ((s − k)!)2
=
n |B| 2
∞
|s|=i |k|=m
(3.4)
j≥i
Now, let us consider the set Nn0 of all non-negative n−tulpes and discrete measure μ defined on Nn0 as 1 μ(s1 , s2 , . . . , sn ) = |s|−1 n 2 2 + |s| and the Hilbert space L 2 ([0, 1], dvn ), dvn (x) = x n−1 d x. Our intention is to transfer the obtained identities in the certain norm relations to the context of the Bochner space L 2 (Nn0 , L 2 ([0, 1], dvn )). Let us first introduce the mapping : L 2 (S) ∩ P(Rn ) → L 2 (Nn0 , L 2 ([0, 1], dvn (x))) given by (ψ) = f, ψ(ξ ) = f (s, x) = where ψ j (ξ ) =
ψ j (ξ ), ψ j ∈ H j (S)
j≥0
π n/2 s!
i≥0 |s|=i
ast x t , s ∈ Nn0 , x ∈ [0, 1],
t≥|s| j
as ξ s .
(3.5)
D. Vujadinovi´c
Furthermore, let us denote by L 2 (S) ∩ P(Rn ) = M and the unit ball B = { f ∈ L 2 (Nn0 , L 2 ([0, 1], dvn ))| f L 2 (Nn ,L 2 ([0,1],dvn )) ≤ 1}. 0
Taking into account the definition of the Bochner norm (2.5) let us note that the obtained expression for the (3.3) can be rewritten as ϕψ2L 2 (B,dτ ) = f 2L 2 (Nn ,L 2 ([0,1],dv )) , f = ψ. 0
(3.6)
n
On the other hand, we define the new function g ∈ L 2 (Nn0 , L 2 ([0, 1], dvn )) as g(s, x)
⎛
= Q n,s,α (1 − x 2 )
n 2 +α−1
x |s| ⎝ ⎛
= Q n,s,α (1 − x 2 )
n 2 +α−1
x |s| ⎝
|k|=m
|k|=m
where Q n,s,α
⎞1/2 ws−k s! ⎠ ((s − k)!)2 ⎞1/2 ( n2 )s! ⎠ 2|s|−m ( n2 + |s| − m)(s − k)! ,
n 1/2 ˜ n,α 1
n
= n/4 n2|s| |B|(|s| + )B 2m − n + 1, |s| − m + A|s| . π 2 2
Then, Pα (ϕψ)2B˜ 2
=
Nn0
f (s), g(s) 2L 2 ([0,1],dv ) dμ(s).
(3.7)
n
Thus, we need to find the minimal constant C in a way that Pα (ϕψ)2B˜ 2 ≤ C 2 ϕψ2L 2 (B,dτ ) , i.e.,
(3.8)
Nn0
f (s), g(s) 2L 2 ([0,1],dv ) dμ(s) ≤ C 2 f 2L 2 (Nn ,L 2 ([0,1],dv )) . n
n
0
(3.9)
Using the Cauchy-Schwarz inequality, we have Nn0
f (s), g(s) 2L 2 ([0,1],dv ) dμ(s)
≤
Nn0
n
(3.10) f (s)2L 2 ([0,1],dv ) g(s)2L 2 ([0,1],dv ) dμ(s). n
n
The Harmonic Dirichlet–Besov Space and the Optimal Norm…
On the other hand, the duality argument and inequality Nn0
f (s)2L 2 ([0,1],dv ) g(s)2L 2 ([0,1],dv ) dμ(s) n
≤C implies
n
(3.11)
2 Nn0
f (s)2L 2 ([0,1],dv ) dμ(s) n
sup
f ∈M∩B Nn0
f (s), g(s) 2L 2 ([0,1],dv ) dμ(s) ≤ C 2 ,
(3.12)
n
where C = sup g(s) L 2 ([0,1],dvn ) . s∈Nn0
We aim to prove that the obtained constant C is indeed the desired constant from the (3.8). For this purpose, let us first notice that for the fixed multi-index s0 ∈ Nn0 the given function g(s0 , x) can be uniformly approximated by polynomials p N ,s0 (x) = N n +α−1 2 C(s0 ) d=0 (−1)d x 2d+|s0 | , N ∈ N, where d ⎛ ⎞1/2 ( n2 )s! ⎠ . C(s) = Q n,α,s ⎝ 2|s|−m ( n2 + |s| − m)(s − k)! |k|=m
On the other hand, it can easily be checked that we can find the χ ∈ L 2 (S) ∩ P(Rn ) in a way that (χ )(s0 , ·) = 2|s0 |−1 (|s0 | + n2 ) p N ,s0 . Denoting by (χ )(s0 , ·) = f, for some > 0 and N big enough, we have f (s, ·), g(s, ·) 2 dμ(s) Nn0
f (s, ·), g(s, ·) 2 2 L ([0,1],dvn ) . (3.13) > g(s0 )2L 2 ([0,1],dv ) − + n 2|s|−1 (|s| + n2 ) s =s0
Finally, C = A(n, m, α) sup s∈N0n
= A(n, m, α) × sup
2 (|s| +
+ α)
n 2
( 3n 2 + |s| + 2α 2 (|s| +
n 2
1/2
s! |k|=m (s−k)! − 1)(m + |s| − n2
+ α)(|s| + 1)
+ 1)
,1/2 n ( 3n + |s|+2α−1)(m +|s|− + 1)(|s| − m + 1) 2 2 1/2 n m−1 n2 ( 2 )(2m − n + 1)(n + 2α − 1) A(n, m, α) = . (3.14) π n/2 |B| 2 (α) |s|≥0
D. Vujadinovi´c
Furthermore, we consider the function (x), x ≥ m, which is defined as (x) =
3n 2
2 x + n2 + α (x + 1) . + x + 2α − 1 m + x − n2 + 1 (x − m + 1)
The Gamma function’s inequalities (1.3) and (1.4) for m ≥
n 2
gives
n n m + x − + 1 (x − m + 1) ≥ x − + 1 (x + 1) 2 2
3n n n 2 + x + 2α − 1 x − + 1 , x + +α ≤ 2 2 2 (3.15) i.e., (x) ≤ 1. On the other hand, Stirling’s asymptotic formula implies that lim (x) = 1.
x→∞
So, C = A(n, m, α).
(3.16)
Remark 3.1 Although in the inequalities (3.15) we supposed that m ≥ n2 , the fact that m is a nonnegative integer, produces no effect on the range of the initial condition m > n−1 2 .
References 1. Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory. Springer, New York (2000) 2. Choe, B.R., Koo, H., Lee, H.J.: Positive Schatten class Toeplitz operators on the ball. Studia Math. 189(1), 65–90 (2008) 3. Choe, B.R., Koo, H., Nam, K.: Optimal norm estimate of operators related to the harmonic Bergman projection on the ball. Tohoku Math. J. 62, 357–374 (2010) 4. Dragomir, S.S., Agarwal, R.P., Barnett, N.S.: Inequalities for Beta and Gamma functions via some classical and new integral inequalities. J. Inequal. Appl. 5(2), 103–165 (2000) 5. Gergün, S., Kaptano˘glu, H.T., Üreyen, A.E.: Harmonic Besov spaces on the ball. Int. J. Math. 27(9), 1650070 (2016) 6. Jevti´c, M., Pavlovi´c, M.: Harmonic Besov space on the unit ball in Rn . Rocky Mount. J. Math. 31(3), 1305–1316 (2001) 7. Jevti´c, M., Pavlovi´c, M.: Harmonic Bergman functions on the unit ball in Rn . Acta Math. Hung. 85(1–2), 91–96 (1999) 8. Vujadinovi´c, Dj: Some estimates for the norm of the Bergman projection on Besov spaces. Integr. Equ. Oper. Theory 76, 213–224 (2013) 9. Vujadinovi´c, Dj.: The norm of the Harmonic Bergman Projection and Besov space. arxiv:1709.04335
