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James constant for interpolation spaces
J. Math. Anal. Appl. 382 (2011) 127–131
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
James constant for interpolation spaces ✩ Anna Betiuk-Pilarska a , Supaluk Phothi b , Stanisław Prus a,∗ a b
Institute of Mathematics, M. Curie-Skłodowska University, 20-031 Lublin, Poland Department of Mathematics, Faculty of Science, Chiangmai University, Chiangmai, Thailand
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 31 December 2010 Available online 23 April 2011 Submitted by T.D. Benavides Keywords: Uniformly nonsquare Banach space James constant Interpolation space
Estimates for the James constant for various norms in real interpolation spaces for finite families of Banach spaces are given. As a corollary it is shown that if a family contains at least one space which is uniformly nonsquare, then the interpolation space is uniformly nonsquare. © 2011 Elsevier Inc. All rights reserved.
1. Introduction Geometric properties of interpolation spaces have been intensively studied by many authors. This in particular concerns properties which can be applied to different fields of Banach space theory or metric fixed point theory. One of the most standard of such properties is uniform convexity. For interpolation spaces this property and some of its generalizations were studied for instance in [1,4,18,16,12]. In this note we consider a property weaker than uniform convexity. It is called uniform nonsquareness. The class of uniformly nonsquare Banach spaces was introduced by R.C. James in [9]. He proved that these spaces are super-reflexive, i.e., they admit equivalent uniformly convex norms. In [6], J. Gao and K.S. Lau introduced a coefficient related to uniform nonsquareness. It is called the James constant or the Gao–Lau coefficient. In this paper we use the former name. Uniform nonsquareness turned out to be useful in metric fixed point theory. The James constant and its modifications give sufficient conditions for normal structure (see [7,5,10]) and even estimates for the normal structure coefficient (see [15]). In [8], it was proved that all uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings on bounded closed convex sets. Uniform nonsquareness for spaces obtained by complex interpolation methods was studied in [3]. In particular, it was proved that if at least one of the spaces A 0 , A 1 is uniformly nonsquare, then the Calderón interpolation space [ A 0 , A 1 ]θ has the same property. This result was generalized in [14], where an estimate for the James constant of the interpolation space was given. In this note we consider a real interpolation method for a finite family of spaces developed in [19] and [17]. It is a generalization of the well-known Lions–Peetre interpolation method for couples of spaces. We give estimates for the James constant for interpolation spaces endowed with various norms. As a corollary we observe that if at least one of spaces is uniformly nonsquare, then so is the interpolation space.
✩
*
The first and third authors were partially supported by the Polish MNiSW grant No. N N201 393737. Corresponding author. E-mail addresses: [email protected] (A. Betiuk-Pilarska), [email protected] (S. Phothi), [email protected] (S. Prus).
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.04.040
©
2011 Elsevier Inc. All rights reserved.
128
A. Betiuk-Pilarska et al. / J. Math. Anal. Appl. 382 (2011) 127–131
2. Preliminaries Throughout the paper by B X and S X we denote the closed unit ball and the unit sphere of a normed space X , respectively. The James constant of X is defined as
J ( X ) = sup min x + y , x − y : x, y ∈ S X .
√
In this definition S X can be replaced by the ball B X . In [7], J. Gao and K.S. Lau proved that, in general, 2 J ( X ) 2. A space X is uniformly nonsquare if and only if J ( X ) < 2. All uniformly convex spaces are uniformly nonsquare (see [9] or [10]). In particular, L p spaces are uniformly nonsquare for 1 < p < ∞ and we have
1
J (l p ) = J ( L p ) = max 2 p , 2
1− 1p
(see [7]). In [2], a general result on the James constant for product spaces was proved. It shows in particular that
J Lp(X) 2 −
1 2
2 − J ( X ) 2 − J (L p )
for any Banach space X and 1 < p < ∞. It follows that if X is uniformly nonsquare, then L p ( X ) has the same property. In [11], the following useful refinement of the triangle inequality was obtained. Let X be a normed space. Then
x y + x + y + 2 − x y min x, y x + y
for any nonzero elements x, y ∈ X . From the above inequality we can easily derive the next one
min x + y , x − y max x, y + (1 − λ) min x, y
(1)
for all x, y ∈ X , where λ = 2 − J ( X ). Let X 0 , . . . , X n be Banach spaces. We say that X = ( X 0 , . . . , X n ) is an interpolation (n + 1)-tuple if the spaces X 0 , . . . , X n are linearly and continuously embedded in a Hausdorff topological vector space E. In fact we can assume that E is a Banach space (see [17]). Let us recall the construction of an interpolation space given by Sparr (see [17, p. 263]). We take vectors t = (t 1 , . . . , tn ) ∈ Rn+ , θ = (θ0 , . . . , θn ) ∈ Rn++1 such that ni=0 θi = 1 and a parameter q ∈ [1, ∞). We put |t −θ | = t 1−θ1 . . . tn−θn . Additionally we set t 0 = 1. For i = 0, . . . , n by Y i we denote the space of all measurable functions u from Rn+ into X i for which the following norm is finite
−θ t t i u i (t ) q dt 1 · · · dtn X
u Y i =
i
Rn+
t1
tn
1q .
Clearly, Y i is isometrically isomorphic to a space L q ( X i ). The interpolation space X θ q; K obtained by the K -method is defined as the space of all elements a ∈ X 0 + · · · + X n which admit a decomposition of the form
a = u 0 (t ) + · · · + un (t )
(2)
for every t ∈ R+ , where u 0 ∈ Y 0 , . . . , un ∈ Y n . Given a ∈ X θ q; K , we set n
a p = inf
n
i =0
1p p u i Y i
where the infimum is taken over all decompositions of the form (2). This formula gives us a norm and X θ q; K with this norm is a Banach space. It is clear that for any r , s ∈ [1, ∞) the norms · r and · s are equivalent. They are also equivalent to the norm
a∞ = inf max u i Y i : i = 0, . . . , n
where the infimum is taken over all decompositions of the form (2). In the case when n = 1 the space X θ q; K reduces to the well-known Lions–Peetre interpolation space [ X 0 , X 1 ]θ0 ,q (see [13]). Given a decomposition of the form (2), we have the following estimate
a∞
n i =0
θ
u i Yii .
(3)
A. Betiuk-Pilarska et al. / J. Math. Anal. Appl. 382 (2011) 127–131
129
Consequently, n
a p C p
i =0
θ
u i Yii
(4)
for every p ∈ [1, ∞), where C p = (n + 1)1/ p . Theorem 1. Let X = ( X 0 , . . . , X n ) be an interpolation (n + 1)-tuple and let t, q, θ be as above. Then
J ( Z ∞ ) min J (Y k )θk 21−θk : k = 0, . . . , n
where Z ∞ denotes the space X θ q; K endowed with the norm · ∞ .
> 0, we choose a1 , a2 ∈ X θ q; K such that a1 ∞ = 1, a2 ∞ = 1 and J ( Z ∞ ) − min a1 + a2 ∞ , a1 − a2 ∞ .
Proof. Given
j
j
Next, we find functions u 0 ∈ Y 0 , . . . , un ∈ Y n , j = 1, 2, so that j
j
a j = u 0 (t ) + · · · + un (t ) for every t ∈ Rn+ and
j max u i Y : i = 0, . . . , n 1 + i
for j = 1, 2. Let us fix k ∈ {0, 1, . . . , n}. From (3) we obtain
J ( Z ∞ ) − min a1 + a2 ∞ , a1 − a2 ∞
θk 1 2 θi u + u min uk1 + uk2 Y , uk1 − uk2 Y i k Y k
k
k
i =k
J (Y k )θk 21−θk (1 + ) and passing to the limit with
→ 0, we get
J ( Z ∞ ) J (Y k )θk 21−θk .
2
Theorem 2. Let X = ( X 0 , . . . , X n ) be an interpolation (n + 1)-tuple and let t, q, θ be as above. Given p ∈ [1, ∞), by Z p we denote the space X θ q; K endowed with the norm · p . Then
p
J ( Z p ) min 1 + 1 − 2 − J (Y k )
− θ1
(n + 1)
k
1p
: k = 0, . . . , n .
> 0. There exist a1 , a2 ∈ X θ q; K such that a1 p = 1, a2 p = 1 and J ( Z p ) − min a1 + a2 p , a1 − a2 p .
Proof. Let us fix
j
j
Next, we find functions u 0 ∈ Y 0 , . . . , un ∈ Y n , j = 1, 2, so that j
j
a j = u 0 (t ) + · · · + un (t ) for every t ∈ Rn+ and
n
j p u i
i =0
Yi
1p 1 + .
Let us fix k ∈ {0, 1, . . . , n}. We can assume that uk1 Y k uk2 Y k . Using estimate (1), we see that
J ( Z p ) − min a1 + a2 p , a1 − a2 p
2 u
k Yk
p 1 u + u 2 p + (1 − λ)uk1 Y + i Y i Y k
i =k
i
i
1p
130
A. Betiuk-Pilarska et al. / J. Math. Anal. Appl. 382 (2011) 127–131
where λ = 2 − J (Y k ) ∈ [0, 1). Therefore
J (Z p) −
n
2p u i
1p +
Yi
i =0
1
p p 1 + u
p (1 − λ) p uk1 Y k
i
i =k
Yi
1
p p p 1p 1 1 + + (1 − λ) uk Y + ui Y . k
i
i =k
From (4) it follows that
1 θk u k Y k
1 C p (1 + )
.
Applying the inequality (1 − λ) p 1 − λ p and the above estimate, we obtain n
p
p p 1p −p u 1 u − λ p u 1 p (1 + ) p − λ p C p θk (1 + )− θk . (1 − λ) p uk1 Y + i Y i Y k Y k
i
i =k
i
i =0
k
Hence
− θp
J ( Z p ) − 1 + + (1 + ) p − λ p C p Passing to the limit with
k
− θp
(1 + )
k
1p
.
→ 0, we get
− θp
J ( Z p ) 1 + 1 − λp C p
k
1p
p − 1 1 = 1 + 1 − 2 − J (Y k ) (n + 1) θk p .
2
Corollary 3. Let X = ( X 0 , . . . , X n ) be an interpolation (n + 1)-tuple and let Z denote either Z p with some p ∈ [1, ∞) or Z ∞ . If X k is uniformly nonsquare for some k, then Z is uniformly nonsquare. In [17] and [19] another interpolation method is also studied. It is called the J -method and the interpolation space obtained by this method is denoted by X θ q; J . In this case for a ∈ X 0 + · · · + X n we consider representations of the form
a=
u (t )
dt 1
Rn+
t1
···
dtn tn
where u is a measurable function with values in the intersection such that the following norm is finite
a p = inf
n
i =0
(5)
, n
i =0
X i . The space X θ q; J consists of all a ∈ X 0 + · · · + X n
1p p u Y i
where the infimum is taken over all functions u satisfying (5). We can also consider the norm
a∞ = inf max u Y i : i = 0, . . . , n where the infimum is taken over all representations (5) and given such representation, we have the estimate
a∞
n i =0
θ
u Yii .
It is therefore easy to see that in our results the space X θ q; K can replaced by X θ q; J . References [1] [2] [3] [4] [5] [6] [7]
B. Beauzamy, Espaces d’interpolation réels: topologie et géométrie, Springer-Verlag, Berlin, Heidelberg, New York, 1978. A. Betiuk-Pilarska, S. Prus, Uniform nonsquareness of direct sums of Banach spaces, Topol. Methods Nonlinear Anal. 34 (2009) 181–186. E. Casini, M. Vignati, The uniform non-squareness for the complex interpolation spaces, J. Math. Anal. Appl. 164 (1992) 518–523. M. Cwikel, S. Reisner, Interpolation of uniformly convex Banach spaces, Proc. Amer. Math. Soc. 84 (1982) 555–559. S. Dhompongsa, A. Kaewkhao, S. Tasena, On a generalized James constant, J. Math. Anal. Appl. 285 (2003) 419–435. J. Gao, K.S. Lau, On the geometry of spheres in normed linear spaces, J. Aust. Math. Soc. Ser. A 48 (1990) 101–112. J. Gao, K.S. Lau, On two classes of Banach spaces with uniform normal structure, Studia Math. 99 (1991) 41–56.
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131
[8] J. García-Falset, E. Llorens-Fuster, E.M. Mazcuñan-Navarro, Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings, J. Funct. Anal. 233 (2006) 494–514. [9] R.C. James, Uniformly nonsquare Banach spaces, Ann. of Math. 80 (1964) 542–550. [10] M. Kato, L. Maligranda, Y. Takahashi, On James and Jordan–von Neumann constants and the normal structure and the normal structure coefficient of Banach spaces, Studia Math. 144 (2001) 275–295. [11] L. Maligranda, Simple norm inequalities, Amer. Math. Monthly 113 (2006) 256–260. [12] D. Kutzarova, L. Nikolova, S. Prus, Infinite-dimensional geometric properties of real interpolation spaces, Math. Nachr. 191 (1998) 215–228. [13] J.L. Lions, J. Peetre, Sur une classe d’espaces d’interpolation, Inst. Hautes Etudes Sci. Publ. Math. 19 (1964) 5–68. [14] L. Nikolova, T. Zachariades, On Edmunds–Triebel spaces Banach, J. Math. Anal. 4 (1) (2010) 146–158. [15] S. Prus, Some estimates for the normal structure coefficient in Banach spaces, Rend. Circ. Math. Palermo 40 (1991) 128–135. [16] M. Salvatori, M. Vignati, On the stability of multi-dimensional convexity under interpolation, Math. Nachr. 164 (1993) 299–308. [17] G. Sparr, Interpolation of several Banach spaces, Ann. Mat. Pura Appl. 99 (1974) 247–316. [18] M. Vignati, Complex interpolation and convexity, Proc. Amer. Math. Soc. 99 (4) (1987) 705–711. [19] A. Yoshikawa, Sur la théorie d’espaces d’interpolation — les espaces de moyenne de plusieurs espaces de Banach, J. Fac. Sci. Univ. Tokyo 14 (1970) 407–468.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
James constant for interpolation spaces ✩ Anna Betiuk-Pilarska a , Supaluk Phothi b , Stanisław Prus a,∗ a b
Institute of Mathematics, M. Curie-Skłodowska University, 20-031 Lublin, Poland Department of Mathematics, Faculty of Science, Chiangmai University, Chiangmai, Thailand
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 31 December 2010 Available online 23 April 2011 Submitted by T.D. Benavides Keywords: Uniformly nonsquare Banach space James constant Interpolation space
Estimates for the James constant for various norms in real interpolation spaces for finite families of Banach spaces are given. As a corollary it is shown that if a family contains at least one space which is uniformly nonsquare, then the interpolation space is uniformly nonsquare. © 2011 Elsevier Inc. All rights reserved.
1. Introduction Geometric properties of interpolation spaces have been intensively studied by many authors. This in particular concerns properties which can be applied to different fields of Banach space theory or metric fixed point theory. One of the most standard of such properties is uniform convexity. For interpolation spaces this property and some of its generalizations were studied for instance in [1,4,18,16,12]. In this note we consider a property weaker than uniform convexity. It is called uniform nonsquareness. The class of uniformly nonsquare Banach spaces was introduced by R.C. James in [9]. He proved that these spaces are super-reflexive, i.e., they admit equivalent uniformly convex norms. In [6], J. Gao and K.S. Lau introduced a coefficient related to uniform nonsquareness. It is called the James constant or the Gao–Lau coefficient. In this paper we use the former name. Uniform nonsquareness turned out to be useful in metric fixed point theory. The James constant and its modifications give sufficient conditions for normal structure (see [7,5,10]) and even estimates for the normal structure coefficient (see [15]). In [8], it was proved that all uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings on bounded closed convex sets. Uniform nonsquareness for spaces obtained by complex interpolation methods was studied in [3]. In particular, it was proved that if at least one of the spaces A 0 , A 1 is uniformly nonsquare, then the Calderón interpolation space [ A 0 , A 1 ]θ has the same property. This result was generalized in [14], where an estimate for the James constant of the interpolation space was given. In this note we consider a real interpolation method for a finite family of spaces developed in [19] and [17]. It is a generalization of the well-known Lions–Peetre interpolation method for couples of spaces. We give estimates for the James constant for interpolation spaces endowed with various norms. As a corollary we observe that if at least one of spaces is uniformly nonsquare, then so is the interpolation space.
✩
*
The first and third authors were partially supported by the Polish MNiSW grant No. N N201 393737. Corresponding author. E-mail addresses: [email protected] (A. Betiuk-Pilarska), [email protected] (S. Phothi), [email protected] (S. Prus).
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.04.040
©
2011 Elsevier Inc. All rights reserved.
128
A. Betiuk-Pilarska et al. / J. Math. Anal. Appl. 382 (2011) 127–131
2. Preliminaries Throughout the paper by B X and S X we denote the closed unit ball and the unit sphere of a normed space X , respectively. The James constant of X is defined as
J ( X ) = sup min x + y , x − y : x, y ∈ S X .
√
In this definition S X can be replaced by the ball B X . In [7], J. Gao and K.S. Lau proved that, in general, 2 J ( X ) 2. A space X is uniformly nonsquare if and only if J ( X ) < 2. All uniformly convex spaces are uniformly nonsquare (see [9] or [10]). In particular, L p spaces are uniformly nonsquare for 1 < p < ∞ and we have
1
J (l p ) = J ( L p ) = max 2 p , 2
1− 1p
(see [7]). In [2], a general result on the James constant for product spaces was proved. It shows in particular that
J Lp(X) 2 −
1 2
2 − J ( X ) 2 − J (L p )
for any Banach space X and 1 < p < ∞. It follows that if X is uniformly nonsquare, then L p ( X ) has the same property. In [11], the following useful refinement of the triangle inequality was obtained. Let X be a normed space. Then
x y + x + y + 2 − x y min x, y x + y
for any nonzero elements x, y ∈ X . From the above inequality we can easily derive the next one
min x + y , x − y max x, y + (1 − λ) min x, y
(1)
for all x, y ∈ X , where λ = 2 − J ( X ). Let X 0 , . . . , X n be Banach spaces. We say that X = ( X 0 , . . . , X n ) is an interpolation (n + 1)-tuple if the spaces X 0 , . . . , X n are linearly and continuously embedded in a Hausdorff topological vector space E. In fact we can assume that E is a Banach space (see [17]). Let us recall the construction of an interpolation space given by Sparr (see [17, p. 263]). We take vectors t = (t 1 , . . . , tn ) ∈ Rn+ , θ = (θ0 , . . . , θn ) ∈ Rn++1 such that ni=0 θi = 1 and a parameter q ∈ [1, ∞). We put |t −θ | = t 1−θ1 . . . tn−θn . Additionally we set t 0 = 1. For i = 0, . . . , n by Y i we denote the space of all measurable functions u from Rn+ into X i for which the following norm is finite
−θ t t i u i (t ) q dt 1 · · · dtn X
u Y i =
i
Rn+
t1
tn
1q .
Clearly, Y i is isometrically isomorphic to a space L q ( X i ). The interpolation space X θ q; K obtained by the K -method is defined as the space of all elements a ∈ X 0 + · · · + X n which admit a decomposition of the form
a = u 0 (t ) + · · · + un (t )
(2)
for every t ∈ R+ , where u 0 ∈ Y 0 , . . . , un ∈ Y n . Given a ∈ X θ q; K , we set n
a p = inf
n
i =0
1p p u i Y i
where the infimum is taken over all decompositions of the form (2). This formula gives us a norm and X θ q; K with this norm is a Banach space. It is clear that for any r , s ∈ [1, ∞) the norms · r and · s are equivalent. They are also equivalent to the norm
a∞ = inf max u i Y i : i = 0, . . . , n
where the infimum is taken over all decompositions of the form (2). In the case when n = 1 the space X θ q; K reduces to the well-known Lions–Peetre interpolation space [ X 0 , X 1 ]θ0 ,q (see [13]). Given a decomposition of the form (2), we have the following estimate
a∞
n i =0
θ
u i Yii .
(3)
A. Betiuk-Pilarska et al. / J. Math. Anal. Appl. 382 (2011) 127–131
129
Consequently, n
a p C p
i =0
θ
u i Yii
(4)
for every p ∈ [1, ∞), where C p = (n + 1)1/ p . Theorem 1. Let X = ( X 0 , . . . , X n ) be an interpolation (n + 1)-tuple and let t, q, θ be as above. Then
J ( Z ∞ ) min J (Y k )θk 21−θk : k = 0, . . . , n
where Z ∞ denotes the space X θ q; K endowed with the norm · ∞ .
> 0, we choose a1 , a2 ∈ X θ q; K such that a1 ∞ = 1, a2 ∞ = 1 and J ( Z ∞ ) − min a1 + a2 ∞ , a1 − a2 ∞ .
Proof. Given
j
j
Next, we find functions u 0 ∈ Y 0 , . . . , un ∈ Y n , j = 1, 2, so that j
j
a j = u 0 (t ) + · · · + un (t ) for every t ∈ Rn+ and
j max u i Y : i = 0, . . . , n 1 + i
for j = 1, 2. Let us fix k ∈ {0, 1, . . . , n}. From (3) we obtain
J ( Z ∞ ) − min a1 + a2 ∞ , a1 − a2 ∞
θk 1 2 θi u + u min uk1 + uk2 Y , uk1 − uk2 Y i k Y k
k
k
i =k
J (Y k )θk 21−θk (1 + ) and passing to the limit with
→ 0, we get
J ( Z ∞ ) J (Y k )θk 21−θk .
2
Theorem 2. Let X = ( X 0 , . . . , X n ) be an interpolation (n + 1)-tuple and let t, q, θ be as above. Given p ∈ [1, ∞), by Z p we denote the space X θ q; K endowed with the norm · p . Then
p
J ( Z p ) min 1 + 1 − 2 − J (Y k )
− θ1
(n + 1)
k
1p
: k = 0, . . . , n .
> 0. There exist a1 , a2 ∈ X θ q; K such that a1 p = 1, a2 p = 1 and J ( Z p ) − min a1 + a2 p , a1 − a2 p .
Proof. Let us fix
j
j
Next, we find functions u 0 ∈ Y 0 , . . . , un ∈ Y n , j = 1, 2, so that j
j
a j = u 0 (t ) + · · · + un (t ) for every t ∈ Rn+ and
n
j p u i
i =0
Yi
1p 1 + .
Let us fix k ∈ {0, 1, . . . , n}. We can assume that uk1 Y k uk2 Y k . Using estimate (1), we see that
J ( Z p ) − min a1 + a2 p , a1 − a2 p
2 u
k Yk
p 1 u + u 2 p + (1 − λ)uk1 Y + i Y i Y k
i =k
i
i
1p
130
A. Betiuk-Pilarska et al. / J. Math. Anal. Appl. 382 (2011) 127–131
where λ = 2 − J (Y k ) ∈ [0, 1). Therefore
J (Z p) −
n
2p u i
1p +
Yi
i =0
1
p p 1 + u
p (1 − λ) p uk1 Y k
i
i =k
Yi
1
p p p 1p 1 1 + + (1 − λ) uk Y + ui Y . k
i
i =k
From (4) it follows that
1 θk u k Y k
1 C p (1 + )
.
Applying the inequality (1 − λ) p 1 − λ p and the above estimate, we obtain n
p
p p 1p −p u 1 u − λ p u 1 p (1 + ) p − λ p C p θk (1 + )− θk . (1 − λ) p uk1 Y + i Y i Y k Y k
i
i =k
i
i =0
k
Hence
− θp
J ( Z p ) − 1 + + (1 + ) p − λ p C p Passing to the limit with
k
− θp
(1 + )
k
1p
.
→ 0, we get
− θp
J ( Z p ) 1 + 1 − λp C p
k
1p
p − 1 1 = 1 + 1 − 2 − J (Y k ) (n + 1) θk p .
2
Corollary 3. Let X = ( X 0 , . . . , X n ) be an interpolation (n + 1)-tuple and let Z denote either Z p with some p ∈ [1, ∞) or Z ∞ . If X k is uniformly nonsquare for some k, then Z is uniformly nonsquare. In [17] and [19] another interpolation method is also studied. It is called the J -method and the interpolation space obtained by this method is denoted by X θ q; J . In this case for a ∈ X 0 + · · · + X n we consider representations of the form
a=
u (t )
dt 1
Rn+
t1
···
dtn tn
where u is a measurable function with values in the intersection such that the following norm is finite
a p = inf
n
i =0
(5)
, n
i =0
X i . The space X θ q; J consists of all a ∈ X 0 + · · · + X n
1p p u Y i
where the infimum is taken over all functions u satisfying (5). We can also consider the norm
a∞ = inf max u Y i : i = 0, . . . , n where the infimum is taken over all representations (5) and given such representation, we have the estimate
a∞
n i =0
θ
u Yii .
It is therefore easy to see that in our results the space X θ q; K can replaced by X θ q; J . References [1] [2] [3] [4] [5] [6] [7]
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