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Bilinear multipliers of weighted Wiener amalgam spaces and variable exponent Wiener amalgam spaces

Journal of Inequalities and Applications20142014:476

https://doi.org/10.1186/1029-242X-2014-476

Received: 9 September 2014

Accepted: 12 November 2014

Published: 27 November 2014

Abstract

Let ω 1 , ω 2 be slowly increasing weight functions, and let ω 3 be any weight function on R n . Assume that m ( ξ , η ) is a bounded, measurable function on R n × R n . We define

B m ( f , g ) ( x ) = R n R n f ˆ ( ξ ) g ˆ ( η ) m ( ξ , η ) e 2 π i ξ + η , x d ξ d η

for all f , g C c ( R n ) . We say that m ( ξ , η ) is a bilinear multiplier on R n of type ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) if B m is a bounded operator from W ( L p 1 , L ω 1 q 1 ) × W ( L p 2 , L ω 2 q 2 ) to W ( L p 3 , L ω 3 q 3 ) , where 1 p 1 q 1 < , 1 p 2 q 2 < , 1 < p 3 , q 3 . We denote by BM ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) the vector space of bilinear multipliers of type ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) . In the first section of this work, we investigate some properties of this space and we give some examples of these bilinear multipliers. In the second section, by using variable exponent Wiener amalgam spaces, we define the bilinear multipliers of type ( W ( p 1 ( x ) , q 1 , ω 1 ; p 2 ( x ) , q 2 , ω 2 ; p 3 ( x ) , q 3 , ω 3 ) ) from W ( L p 1 ( x ) , L ω 1 q 1 ) × W ( L p 2 ( x ) , L ω 2 q 2 ) to W ( L p 3 ( x ) , L ω 3 q 3 ) , where p 1 , p 2 , p 3 < , p 1 ( x ) q 1 , p 2 ( x ) q 2 , 1 q 3 for all p 1 ( x ) , p 2 ( x ) , p 3 ( x ) P ( R n ) . We denote by BM ( W ( p 1 ( x ) , q 1 , ω 1 ; p 2 ( x ) , q 2 , ω 2 ; p 3 ( x ) , q 3 , ω 3 ) ) the vector space of bilinear multipliers of type ( W ( p 1 ( x ) , q 1 , ω 1 ; p 2 ( x ) , q 2 , ω 2 ; p 3 ( x ) , q 3 , ω 3 ) ) . Similarly, we discuss some properties of this space.

MSC:42A45, 42B15, 42B35.

Keywords

bilinear multipliersweighted Wiener amalgam spacevariable exponent Wiener amalgam space

1 Introduction

Throughout this paper we will work on R n with Lebesgue measure dx. We denote by C c ( R n ) , C c ( R n ) and S ( R n ) the space of infinitely differentiable complex-valued functions with compact support on R n , the space of all continuous, complex-valued functions with compact support on R n and the space of infinitely differentiable complex-valued functions on R n that rapidly decrease at infinity, respectively. Let f be a complex-valued measurable function on R n . The translation, character and dilation operators T x , M x and D s are defined by T x f ( y ) = f ( y x ) , M x f ( y ) = e 2 π i x , y f ( y ) and D t p f ( y ) = t n p f ( y t ) , respectively, for x , y R n , 0 < p , t < . With this notation out of the way, one has, for 1 p and 1 p + 1 p = 1 ,
( T x f ) g ˆ ( ξ ) = M x f ˆ ( ξ ) , ( M x f ) g ˆ ( ξ ) = T x f ˆ ( ξ ) , ( D t p f ) g ˆ ( ξ ) = D t 1 p f ˆ ( ξ ) .
For 1 p , L p ( R n ) denotes the usual Lebesgue space. A continuous function ω satisfying 1 ω ( x ) and ω ( x + y ) ω ( x ) ω ( y ) for x , y R n will be called a weight function on R n . If ω 1 ( x ) ω 2 ( x ) for all x R n , we say that ω 1 ω 2 . For 1 p , we set
L ω p ( R n ) = { f : f ω L p ( R n ) } .
It is known that L ω p ( R n ) is a Banach space under the norm
f p , ω = f ω p = { R n | f ( x ) ω ( x ) | p d x } 1 p , 1 p <
or
f , ω = f ω = ess sup x R n | f ( x ) ω ( x ) | , p = .
The dual of the space L ω p ( R n ) is the space L ω 1 q ( R n ) , where 1 p + 1 q = 1 and ω 1 ( x ) = 1 ω ( x ) . We say that a weight function υ s is of polynomial type if υ s ( x ) = ( 1 + | x | ) s for s 0 . Let f be a measurable function on R n . If there exist C > 0 and N N such that
| f ( x ) | C ( 1 + x 2 ) N
for all x R n , then f is said to be a slowly increasing function [1]. It is easy to see that polynomial-type weight functions are slowly increasing. For f L 1 ( R n ) , the Fourier transform of f is denoted by f ˆ . We know that f ˆ is a continuous function on R n which vanishes at infinity and it has the inequality f ˆ f 1 . We denote by M ( R n ) the space of bounded regular Borel measures, by M ( ω ) the space of μ in M ( R n ) such that
μ ω = R n ω d | μ | < .

If μ M ( R n ) , the Fourier-Stieltjes transform of μ is denoted by μ ˆ [2].

The space ( L p ( R n ) ) loc consists of classes of measurable functions f on R n such that f χ K L p ( R n ) for any compact subset K R n , where χ K is the characteristic function of K. Let us fix an open set Q R n with compact closure and 1 p , q . The weighted Wiener amalgam space W ( L p , L ω q ) consists of all elements f ( L p ( R n ) ) loc such that F f ( z ) = f χ z + Q p belongs to L ω q ( R n ) ; the norm of W ( L p , L ω q ) is f W ( L p , L ω q ) = F f q , ω [35].

In this paper, P ( R n ) denotes the family of all measurable functions p : R n [ 1 , ) . We put
p = ess inf x R n p ( x ) , p = ess sup x R n p ( x ) .
We shall also use the notation
Ω = { x R n : p ( x ) = } .
The variable exponent Lebesgue spaces (or generalized Lebesgue spaces) L p ( x ) ( R n ) are defined as the set of all (equivalence classes) measurable functions f on R n such that ϱ p ( λ f ) < for some λ > 0 , equipped with the Luxemburg norm
f p ( x ) = inf { λ > 0 : ϱ p ( f λ ) 1 } .

If p < , then f L p ( x ) ( R n ) if ϱ p ( f ) < . The set L p ( x ) ( R n ) is a Banach space with the norm p ( x ) . If p ( x ) = p is a constant function, then the norm p ( x ) coincides with the usual Lebesgue norm p [6]. The spaces L p ( x ) ( R n ) and L p ( R n ) have many common properties. A crucial difference between L p ( x ) ( R n ) and the classical Lebesgue spaces L p ( R n ) is that L p ( x ) ( R n ) is not invariant under translation in general. If p < , then C c ( R n ) is dense in L p ( x ) ( R n ) . The space L p ( x ) ( R n ) is a solid space, that is, if f L p ( x ) ( R n ) is given and g L loc 1 ( R n ) satisfies | g ( x ) | | f ( x ) | a.e., then g L p ( x ) ( R n ) and g p ( x ) f p ( x ) by [6]. In this paper we will assume that p < .

The space ( L p ( x ) ( R n ) ) loc consists of classes of measurable functions f on R n such that f χ K L p ( x ) ( R n ) for any compact subset K R n . Let us fix an open set Q R n with compact closure, p ( x ) P ( R n ) and 1 q . The variable exponent amalgam space W ( L p ( x ) , L ω q ) consists of all elements f ( L p ( x ) ( R n ) ) loc such that F f ( z ) = f χ z + Q p ( x ) belongs to L ω q ( R n ) ; the norm of W ( L p ( x ) , L ω q ) is f W ( L p ( x ) , L ω q ) = F f q , ω [7].

2 The bilinear multipliers space BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ]

Lemma 2.1 Let 1 p q < and ω be a slowly increasing weight function. Then C c ( R n ) is dense in the Wiener amalgam space W ( L p , L ω q ) .

Proof Since C c ( R n ) ¯ = L ω q ( R n ) [8], we have C c ( R n ) ¯ = W ( L p , L ω q ) by a lemma in [9]. Also we have the inclusion
C c ( R n ) C c ( R n ) W ( L p , L ω q ) .
For the proof that C c ( R n ) is dense in W ( L p , L ω q ) , take any f W ( L p , L ω q ) . For given ε > 0 , there exists g C c ( R n ) such that
f g W ( L p , L ω q ) < ε 2 .
(2.1)
Also, since g C c ( R n ) L ω q ( R n ) and C c ( R n ) is dense in L ω q ( R n ) , by Lemma 2.1 in [10], there exists h C c ( R n ) such that
g h q , ω < ε 2 .
Furthermore, by using the inequality p q , we write
g h W ( L p , L ω q ) g h q , ω < ε 2
(2.2)
(see [11] and [5]). Combining (2.1) and (2.2), we obtain
f h W ( L p , L ω q ) f g W ( L p , L ω q ) + h g W ( L p , L ω q ) < ε .

This completes the proof. □

Definition 2.1 Let 1 p 1 q 1 < , 1 p 2 q 2 < , 1 < p 3 , q 3 and ω 1 , ω 2 , ω 3 be weight functions on R n . Assume that ω 1 , ω 2 are slowly increasing functions and m ( ξ , η ) is a bounded, measurable function on R n × R n . Define
B m ( f , g ) ( x ) = R n R n f ˆ ( ξ ) g ˆ ( η ) m ( ξ , η ) e 2 π i ξ + η , x d ξ d η

for all f , g C c ( R n ) .

m is said to be a bilinear multiplier on R n of type ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) if there exists C > 0 such that
B m ( f , g ) W ( L p 3 , L ω 3 q 3 ) C f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 )

for all f , g C c ( R n ) . That means that B m extends to a bounded bilinear operator from W ( L p 1 , L ω 1 q 1 ) × W ( L p 2 , L ω 2 q 2 ) to W ( L p 3 , L ω 3 q 3 ) .

We denote by BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] the space of all bilinear multipliers of type ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) and m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) = B m .

The following theorem is an example to a bilinear multiplier on R n of type ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) .

Theorem 2.1 Let 1 p 1 + 1 p 2 = 1 p 3 , 1 q 1 + 1 q 2 = 1 q 3 and ω 3 ω 1 . If K L ω 3 1 ( R n ) , then m ( ξ , η ) = K ˆ ( ξ η ) defines a bilinear multiplier and m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) K 1 , ω 3 .

Proof We know by Theorem 2.1 in [10] that for f , g C c ( R n ) ,
B m ( f , g ) ( t ) = R n f ( t y ) g ( t + y ) K ( y ) d y .
(2.3)

Also by Proposition 11.4.1 in [5], T y f W ( L p 1 , L ω 1 q 1 ) , T y g W ( L p 2 , L ω 2 q 2 ) . So, we write F T y f L ω 1 q 1 ( R n ) , F T y g L ω 2 q 2 ( R n ) .

Using the Minkowski inequality and the generalized Hölder inequality, we have
B m ( f , g ) W ( L p 3 , L ω 3 q 3 ) = B m ( f , g ) χ Q + x p 3 q 3 , ω 3 = { R n f ( t y ) g ( t + y ) K ( y ) d y } χ Q + x p 3 q 3 , ω 3 R n f ( t y ) g ( t + y ) χ Q + x ( t ) p 3 q 3 , ω 3 | K ( y ) | d y R n f ( t y ) χ Q + x ( t ) p 1 g ( t + y ) χ Q + x ( t ) p 2 q 3 , ω 3 | K ( y ) | d y = R n F T y f ( x ) ω 3 ( x ) F T y g ( x ) q 3 | K ( y ) | d y .
(2.4)
Again, by using Proposition 11.4.1 in [5] and the assumption ω 3 ω 1 , we write
F T y f ω 3 q 1 ω 3 ( y ) f W ( L p 1 , L ω 1 q 1 ) < .
(2.5)
From this result, we find F T y f L ω 3 q 1 ( R n ) . Hence by (2.4), (2.5) and the generalized Hölder inequality, we obtain
B m ( f , g ) W ( L p 3 , L ω 3 q 3 ) R n F T y f ( x ) ω 3 ( x ) q 1 F T y g ( x ) q 2 | K ( y ) | d y R n f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) | K ( y ) | ω 3 ( y ) d y = f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) K 1 , ω 3 = C f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) ,
(2.6)
where C = K 1 , ω 3 . Then m ( ξ , η ) = K ˆ ( ξ η ) defines a bilinear multiplier. Finally, using (2.6), we obtain
m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) = sup f W ( L p 1 , L ω 1 q 1 ) 1 , g W ( L p 2 , L ω 2 q 2 ) 1 B m ( f , g ) W ( L p 3 , L ω 3 q 3 ) f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) K 1 , ω 3 .

 □

Definition 2.2 Let 1 p 1 p 2 < , 1 q 1 q 2 < , 1 < p 3 , q 3 and ω 1 , ω 2 , ω 3 be weight functions on R n . Suppose that ω 1 , ω 2 are slowly increasing functions. We denote by M ˜ [ ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] the space of measurable functions M : R n C such that m ( ξ , η ) = M ( ξ η ) BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] , that is to say,
B M ( f , g ) ( x ) = R n R n f ˆ ( ξ ) g ˆ ( η ) M ( ξ η ) e 2 π i ξ + η , x d ξ d η

extends to a bounded bilinear map from W ( L p 1 , L ω 1 q 1 ) × W ( L p 2 , L ω 2 q 2 ) to W ( L p 3 , L ω 3 q 3 ) . We denote M ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) = B M .

Let ω be a weight function. The continuous function ω 1 cannot be a weight function. But the following lemma can be proved easily by using the technique of the proof of Lemma 2.1.

Lemma 2.2 Let 1 p q < and ω be a slowly increasing continuous weight function. Then C c ( R n ) is dense in W ( L p , L ω 1 q ) Wiener amalgam space.

Theorem 2.2 Let 1 p 3 + 1 p 3 = 1 , 1 q 3 + 1 q 3 = 1 , q 3 p 3 1 and ω 3 be a continuous, symmetric slowly increasing weight function. Then m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] if and only if there exists C > 0 such that
| R n R n f ˆ ( ξ ) g ˆ ( η ) h ˆ ( ξ + η ) m ( ξ , η ) d ξ d η | C f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) h W ( L p 3 , L ω 1 q 3 )

for all f , g , h C c ( R n ) .

Proof We assume that m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] . By Theorem 2.2 in [10], we write, for all f , g , h C c ( R n ) ,
| R n R n f ˆ ( ξ ) g ˆ ( η ) h ˆ ( ξ + η ) m ( ξ , η ) d ξ d η | = | R n h ( y ) B ˜ m ( f , g ) ( y ) d y | R n | h ( y ) | | B ˜ m ( f , g ) ( y ) | d y ,
(2.7)
where B ˜ m ( f , g ) ( y ) = B m ( f , g ) ( y ) . If we set t = u , we have
B ˜ m ( f , g ) W ( L p 3 , L ω 3 q 3 ) = F B ˜ m ( f , g ) Q q 3 , ω 3 = B m ( f , g ) ( u ) χ Q + x ( u ) p 3 q 3 , ω 3 = B m ( f , g ) ( u ) χ Q x ( u ) p 3 q 3 , ω 3 = F B m ( f , g ) Q ( x ) q 3 , ω 3 .
(2.8)
Since ω 3 is a symmetric weight function, if we set x = y , we have
B ˜ m ( f , g ) W ( L p 3 , L ω 3 q 3 ) = F B m ( f , g ) Q ( y ) q 3 , ω 3 .
(2.9)
We know from [3] and [5] that the definition of W ( L p 3 , L ω 3 q 3 ) is independent of the choice of Q. Then there exists C > 0 such that
F B m ( f , g ) Q ( y ) q 3 , ω 3 C 1 F B m ( f , g ) Q ( y ) q 3 , ω 3 .
(2.10)
Hence, by (2.9) and (2.10), we have
B ˜ m ( f , g ) W ( L p 3 , L ω 3 q 3 ) C 1 F B m ( f , g ) Q ( y ) q 3 , ω 3 = C 1 B m ( f , g ) W ( L p 3 , L ω 3 q 3 ) .
(2.11)
Since from the assumption m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] the right-hand side of (2.11) is finite, thus B ˜ m ( f , g ) W ( L p 3 , L ω 3 q 3 ) . On the other hand, since m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] , there exists C 2 > 0 such that
B m ( f , g ) W ( L p 3 , L ω 3 q 3 ) C 2 f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) .
(2.12)
Combining (2.11) and (2.12), we have
B ˜ m ( f , g ) W ( L p 3 , L ω 3 q 3 ) C 1 C 2 f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) .
(2.13)
If we apply the Hölder inequality to the right-hand side of inequality (2.7) and use inequality (2.13), we obtain
| R n R n f ˆ ( ξ ) g ˆ ( η ) h ˆ ( ξ + η ) m ( ξ , η ) d ξ d η | B ˜ m ( f , g ) W ( L p 3 , L ω 3 q 3 ) h W ( L p 3 , L ω 1 q 3 ) C 1 C 2 f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) h W ( L p 3 , L ω 3 1 q 3 ) .
For the proof of converse, assume that there exists a constant C > 0 such that
| R n R n f ˆ ( ξ ) g ˆ ( η ) h ˆ ( ξ + η ) m ( ξ , η ) d ξ d η | C f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) h W ( L p 3 , L ω 3 1 q 3 )
(2.14)
for all f , g , h C c ( R n ) . From the assumption and (2.14), we write
| R n h ( y ) B ˜ m ( f , g ) ( y ) d y | C f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) h W ( L p 3 , L ω 3 1 q 3 ) .
(2.15)
Define a function from C c ( R n ) W ( L p 3 , L ω 1 q 3 ) to such that
( h ) = R n h ( y ) B ˜ m ( f , g ) ( y ) d y .
is linear and bounded by (2.15). Also, since q 3 p 3 1 , we have C c ( R n ) ¯ = W ( L p 3 , L ω 3 1 q 3 ) by Lemma 2.2. Thus extends to a bounded function from W ( L p 3 , L ω 3 1 q 3 ) to . Then ( W ( L p 3 , L ω 3 1 q 3 ) ) = W ( L p 3 , L ω 3 q 3 ) . Again, since the definition of W ( L p 3 , L ω 3 q 3 ) is independent of the choice of Q, there exists C 3 > 0 such that
B m ( f , g ) W ( L p 3 , L ω 3 q 3 ) C 3 B ˜ m ( f , g ) W ( L p 3 , L ω 3 q 3 ) .
(2.16)
Combining (2.15) and (2.16), we obtain
B m ( f , g ) W ( L p 3 , L ω 3 q 3 ) C 3 B ˜ m ( f , g ) W ( L p 3 , L ω 3 q 3 ) = C 3 = C 3 sup h W ( L p 3 , L ω 3 1 q 3 ) 1 | ( h ) | h W ( L p 3 , L ω 3 1 q 3 ) C 3 C f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) .

This completes proof. □

The following theorem is a generalization of Theorem 2.1.

Theorem 2.3 Let 1 p 1 + 1 p 2 = 1 p 3 , 1 q 1 + 1 q 2 = 1 q 3 , ω 3 ω 1 and υ ( x ) = C ( 1 + x 2 ) N , C 0 , N N be a weight function. If μ M ( υ ) and m ( ξ , η ) = μ ˆ ( α ξ + β η ) for α , β R , then m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] . Moreover,
m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) μ υ if  | α | 1 , m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) | α | 2 N μ υ if  | α | > 1 .
Proof Let f , g C c ( R n ) . By Theorem 2.3 in [10], we have
B m ( f , g ) ( t ) = R n f ( t α y ) g ( t β y ) d μ ( y ) .
(2.17)
Also by [5] we write the inequalities
T α y f W ( L p 1 , L ω 1 q 1 ) ω 1 ( α y ) f W ( L p 1 , L ω 1 q 1 )
(2.18)
and
T β y g W ( L p 2 , L ω 2 q 2 ) ω 2 ( α y ) g W ( L p 2 , L ω 2 q 2 ) .
(2.19)
From these inequalities, we have F T α y f L ω 1 q 1 ( R n ) and F T β y g L ω 2 q 2 ( R n ) . If we use the inequality ω 3 ω 1 and set x α t = u , we obtain
F T α y f ω 3 q 1 F T α y f ω 1 q 1 ω 1 ( α y ) f W ( L p 1 , L ω 1 q 1 ) ,
(2.20)
and hence F T α y f ω 3 L q 1 ( R n ) . Then by (2.17), (2.18), (2.19), (2.20) and the Hölder inequality, we have
B m ( f , g ) W ( L p 3 , L ω 3 q 3 ) R n f ( t α y ) g ( t β y ) χ Q + x ( t ) p 3 d | μ | ( y ) q 3 , ω 3 R n f ( t α y ) χ Q + x ( t ) p 1 g ( t β y ) χ Q + x ( t ) p 2 d | μ | ( y ) q 3 , ω 3 R n F T α y f ( x ) F T β y g ( x ) q 3 , ω 3 d | μ | ( y ) R n F T α y f ( x ) ω 3 ( x ) q 1 F T β y g ( x ) q 2 d | μ | ( y ) R n ω 1 ( α y ) f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L q 2 ) d | μ | ( y ) f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) R n ω 1 ( α y ) d | μ | ( y ) .
(2.21)
Now, suppose that α 1 . Since ω 1 is a slowly increasing weight function, there exist C 0 and N N such that
ω 1 ( x ) C ( 1 + x 2 ) N = υ ( x ) .
Then
R n ω 1 ( α y ) d | μ | ( y ) R n C ( 1 + α 2 y 2 ) N d | μ | ( y ) R n C ( 1 + y 2 ) N d | μ | ( y ) = μ υ .
Hence by (2.21)
B m ( f , g ) W ( L p 3 , L ω 3 q 3 ) f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) μ υ .
(2.22)
Thus m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] , and by (2.22) we obtain
m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) = sup f W ( L p 1 , L ω 1 q 1 ) 1 , g W ( L p 2 , L ω 2 q 2 ) 1 B m ( f , g ) W ( L p 3 , L ω 3 q 3 ) f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) μ υ .
Similarly, if α > 1 , then
R n ω 1 ( α y ) d | μ | ( y ) R n C ( α 2 + α 2 y 2 ) N d | μ | ( y ) = α 2 N R n υ ( y ) d | μ | ( y ) = α 2 N μ υ .
Therefore by (2.21) we have
B m ( f , g ) W ( L p 3 , L ω 3 q 3 ) α 2 N f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) μ υ .
(2.23)
Hence, we obtain m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] , and by (2.23)
m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) = sup f W ( L p 1 , L ω 1 q 1 ) 1 , g W ( L p 2 , L ω 2 q 2 ) 1 B m ( f , g ) W ( L p 3 , L ω 3 q 3 ) f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) α 2 N μ υ .

 □

Now, we will give some properties of the space BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] .

Theorem 2.4 Let m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] .
  1. (a)
    T ( ξ 0 , η 0 ) m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] for each ( ξ 0 , η 0 ) R 2 n and
    T ( ξ 0 , η 0 ) m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) = m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) .
     
  2. (b)
    M ( ξ 0 , η 0 ) m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] for each ( ξ 0 , η 0 ) R 2 n and
    M ( ξ 0 , η 0 ) m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) ω 1 ( ξ 0 ) ω 2 ( η 0 ) m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) .
     
Proof (a) Let f , g C c ( R n ) . From Theorem 2.4 in [10], we write the equality
B T ( ξ 0 , η 0 ) m ( f , g ) ( x ) = e 2 π i ξ 0 + η 0 , x B m ( M ξ 0 f , M η 0 g ) ( x ) .
(2.24)
Also the equalities M ξ 0 f W ( L p 1 , L ω 1 q 1 ) = f W ( L p 1 , L ω 1 q 1 ) and M η 0 g W ( L p 2 , L ω 2 q 2 ) = g W ( L p 2 , L ω 2 q 2 ) are satisfied. Then, using equality (2.24) and the assumption m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] , we have
B T ( ξ 0 , η 0 ) m ( f , g ) W ( L p 3 , L ω 3 q 3 ) = e 2 π i ξ 0 + η 0 , x B m ( M ξ 0 f , M η 0 g ) W ( L p 3 , L ω 3 q 3 ) = B m ( M ξ 0 f , M η 0 g ) W ( L p 3 , L ω 3 q 3 ) C f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 )
for some C > 0 . Thus T ( ξ 0 , η 0 ) m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] . Moreover, we obtain
T ( ξ 0 , η 0 ) m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) = B T ( ξ 0 , η 0 ) m = sup f W ( L p 1 , L ω 1 q 1 ) 1 , g W ( L p 2 , L ω 2 q 2 ) 1 B T ( ξ 0 , η 0 ) m ( f , g ) W ( L p 3 , L ω 3 q 3 ) f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) = sup M ξ 0 f W ( L p 1 , L ω 1 q 1 ) 1 , M η 0 g W ( L p 2 , L ω 2 q 2 ) 1 B T ( ξ 0 , η 0 ) m ( M ξ 0 f , M η 0 g ) W ( L p 3 , L ω 3 q 3 ) M ξ 0 f W ( L p 1 , L ω 1 q 1 ) M η 0 g W ( L p 2 , L ω 2 q 2 ) = B m = m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) .
(b) For any f , g C c ( R n ) , we write
B M ( ξ 0 , η 0 ) m ( f , g ) ( x ) = B m ( T ξ 0 f , T η 0 g ) ( x )
(2.25)
by Theorem 2.4 in [10]. Also, the inequalities T ξ 0 f W ( L p 1 , L ω 1 q 1 ) ω 1 ( ξ 0 ) f W ( L p 1 , L ω 1 q 1 ) and T η 0 g W ( L p 2 , L ω 2 q 2 ) ω 2 ( η 0 ) g W ( L p 2 , L ω 2 q 2 ) are satisfied [5]. Since m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] , by (2.25) we have
B M ( ξ 0 , η 0 ) m ( f , g ) W ( L p 3 , L ω 3 q 3 ) = B m ( T ξ 0 f , T η 0 g ) W ( L p 3 , L ω 3 q 3 ) B m T ξ 0 f W ( L p 1 , L ω 1 q 1 ) T η 0 g W ( L p 2 , L ω 2 q 2 ) ω 1 ( ξ 0 ) ω 2 ( η 0 ) B m f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 )
(2.26)
and hence M ( ξ 0 , η 0 ) m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] . So, by (2.26) we obtain
M ( ξ 0 , η 0 ) m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) = sup f W ( L p 1 , L ω 1 q 1 ) 1 , g W ( L p 2 , L ω 2 q 2 ) 1 B M ( ξ 0 , η 0 ) m ( f , g ) W ( L p 3 , L ω 3 q 3 ) f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) ω 1 ( ξ 0 ) ω 2 ( η 0 ) B m = ω 1 ( ξ 0 ) ω 2 ( η 0 ) m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) .

 □

Lemma 2.3 If ω is a slowly increasing weight function such that ω ( x ) C 1 ( 1 + x 2 ) N = υ ( x ) and f W ( L p , L ω q ) , then D y p f W ( L p , L υ q ) . Moreover,
D y p f W ( L p , L ω q ) C f W ( L p , L υ q ) if  y 1 , D y p f W ( L p , L ω q ) < C y n q + 2 N f W ( L p , L υ q ) if  y > 1

for some C > 0 .

Proof Take any f W ( L p , L ω q ) . If we get t y = u , we obtain
D y p f W ( L p , L ω q ) = { R n | D y p f ( t ) χ Q + x ( t ) | p d t } 1 p q , ω = { R n | y n p f ( t y ) | p χ Q + x ( t ) d t } 1 p q , ω = { R n | f ( u ) | p χ y 1 Q + y 1 x ( u ) d u } 1 p q , ω = F f y 1 Q ( y 1 x ) q , ω .
(2.27)
Again, if we say y 1 x = s and use ω to be slowly increasing, then there exist C 1 > 0 and N N such that
D y p f W ( L p , L ω q ) = { R n | F f y 1 Q ( y 1 x ) | q ω ( x ) q d x } 1 q = y n q { R n | F f y 1 Q ( s ) | q ω ( y s ) q d s } 1 q y n q { R n | F f y 1 Q ( s ) | q ( C 1 ( 1 + y 2 s 2 ) N ) q d s } 1 q
(2.28)

by equation (2.27).

Let y 1 . Using inequality (2.28), we have
D y p f W ( L p , L ω q ) { R n | F f y 1 Q ( s ) | q ( C 1 ( 1 + s 2 ) N ) q d s } 1 q = F f y 1 Q q , υ .
(2.29)
Since y 1 Q is a compact set and the definition of W ( L p , L υ q ) is independent of the choice of a compact set Q, then there exists C > 0 such that
F f y 1 Q q , υ C F f Q q , υ
(2.30)
by [3, 5]. Then by (2.29) we write
D y p f W ( L p , L ω q ) F f y 1 Q q , υ C F f Q q , υ = C f W ( L p , L υ q ) .

Thus we have D y p f W ( L p , L υ q ) .

Now, assume that y > 1 . Similarly, by (2.28) and (2.30), we get
D y p f W ( L p , L ω q ) < y n q { R n | F f y 1 Q ( s ) | q ( C 1 ( y 2 + y 2 s 2 ) N ) q d s } 1 q = y n q + 2 N { R n | F f y 1 Q ( s ) | q υ ( s ) q d s } 1 q y n q + 2 N f W ( L p , L υ q ) .

Hence D y p f W ( L p , L υ q ) . □

Theorem 2.5 Let υ i ( x ) = C i ( 1 + x 2 ) N i , C i > 0 , N i > 0 for i = 1 , 2 , 3 , and let ω 3 be a slowly increasing weight function. If 2 q = 1 p 1 + 1 p 2 1 p 3 , 0 < y < and m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , υ 3 ) ] , then D y q m BM [ W ( p 1 , q 1 , υ 1 ; p 2 , q 2 , υ 2 ; p 3 , q 3 , ω 3 ) ] . Moreover, then
D y q m ( W ( p 1 , q 1 , υ 1 ; p 2 , q 2 , υ 2 ; p 3 , q 3 , ω 3 ) ) C y n q 3 2 N 3 m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , υ 3 ) ) if  y 1 , D y q m ( W ( p 1 , q 1 , υ 1 ; p 2 , q 2 , υ 2 ; p 3 , q 3 , ω 3 ) ) < C y n q 1 + n q 2 + 2 N 1 + 2 N 2 m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , υ 3 ) ) if  y > 1 .
Proof Let f W ( L p 1 , L ω 1 q 1 ) and g W ( L p 2 , L ω 2 q 2 ) be given. From Lemma 2.3, we have D y p 1 f W ( L p 1 , L ω 1 q 1 ) and D y p 2 g W ( L p 2 , L ω 2 q 2 ) . Also it is known by Theorem 2.5 in [10] that
B D y q m ( f , g ) ( y ) = D y 1 p 3 B m ( D y p 1 f , D y p 2 g ) ( y ) .
If we use this equality, we write
B D y q m ( f , g ) W ( L p 3 , L ω 3 q 3 ) = { R n | D y 1 p 3 B m ( D y p 1 f , D y p 2 g ) ( t ) χ Q + x ( t ) | p 3 d t } 1 p 3 q 3 , ω 3 = { R n | y n p 3 B m ( D y p 1 f , D y p 2 g ) ( t y 1 ) χ Q + x ( t ) | p 3 d t } 1 p 3 q 3 , ω 3 = { R n y n | B m ( D y p 1 f , D y p 2 g ) ( t y ) χ Q + x ( t ) | p 3 d t } 1 p 3 q 3 , ω 3 .
If we say y t = u in the last equality, we have
B D y q m ( f , g ) W ( L p 3 , L ω 3 q 3 ) = { R n | B m ( D y p 1 f , D y p 2 g ) ( u ) χ y Q + y x ( u ) | p 3 d t } 1 p 3 q 3 , ω 3 = F B m ( D y p 1 f , D y p 2 g ) y Q ( y x ) q 3 , ω 3 .
(2.31)
On the other hand, since ω 3 is a slowly increasing weight function, there exist C 3 > 0 , N 3 > 0 such that ω 3 ( x ) C 3 ( 1 + x 2 ) N 3 = υ 3 ( x ) . If we make the substitution y x = s in equality (2.31), we obtain
B D y q m ( f , g ) W ( L p 3 , L ω 3 q 3 ) = F B m ( D y p 1 f , D y p 2 g ) y Q ( y x ) q 3 , ω 3 = { R n | F B m ( D y p 1 f , D y p 2 g ) y Q ( s ) | q 3 ω 3 ( y 1 s ) q 3 y n d s } 1 q 3 = y n q 3 { R n | F B m ( D y p 1 f , D y p 2 g ) y Q ( s ) | q 3 ω 3 ( y 1 s ) q 3 d s } 1 q 3 y n q 3 { R n | F B m ( D y p 1 f , D y p 2 g ) y Q ( s ) | q 3 ( C 3 ( 1 + y 2 s 2 ) N 3 ) q 3 d s } 1 q 3 .
We assume that y 1 . Then
B D y q m ( f , g ) W ( L p 3 , L ω 3 q 3 ) y n q 3 { R n | F B m ( D y p 1 f , D y p 2 g ) y Q ( s ) | q 3 ( C 3 ( y 2 + y 2 s 2 ) N 3 ) q 3 d s } 1 q 3 = y n q 3 2 N 3 F B m ( D y p 1 f , D y p 2 g ) y Q q 3 , υ 3 .
Also, since yQ is a compact set, we have
B D y q m ( f , g ) W ( L p 3 , L ω 3 q 3 ) C y n q 3 2 N 3 F B m ( D y p 1 f , D y p 2 g ) Q q 3 , υ 3 = C y n q 3 2 N 3 B m ( D y p 1 f , D y p 2 g ) W ( L p 3 , L υ 3 q 3 ) .
(2.32)
Since m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , υ 3 ) ] , by Lemma 2.3 and inequality (2.32), we obtain
B D y q m ( f , g ) W ( L p 3 , L ω 3 q 3 ) C y n q 3 2 N 3 m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , υ 3 ) ) f W ( L p 1 , L υ 1 q 1 ) g W ( L p 2 , L υ 2 q 2 ) .
(2.33)
Then D y q m BM [ W ( p 1 , q 1 , υ 1 ; p 2 , q 2 , υ 2 ; p 3 , q 3 , ω 3 ) ] , and by (2.33) we have
D y q m ( W ( p 1 , q 1 , υ 1 ; p 2 , q 2 , υ 2 ; p 3 , q 3 , ω 3 ) ) C y n q 3 2 N 3 m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , υ 3 ) ) .
Now let y > 1 . Again, since m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , υ 3 ) ] , by Lemma 2.3 and inequality (2.32), we obtain
B D y q m ( f , g ) W ( L p 3 , L ω 3 q 3 ) < C B m ( D y p 1 f , D y p 2 g ) W ( L p 3 , L υ 3 q 3 ) < C y n q 1 + n q 2 + 2 N 1 + 2 N 2 m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , υ 3 ) ) f W ( L p 1 , L υ 1 q 1 ) g W ( L p 2 , L υ 2 q 2 ) .
(2.34)
Thus D y q m BM [ W ( p 1 , q 1 , υ 1 ; p 2 , q 2 , υ 2 ; p 3 , q 3 , ω 3 ) ] , and by (2.34) we have
D y q m ( W ( p 1 , q 1 , υ 1 ; p 2 , q 2 , υ 2 ; p 3 , q 3 , ω 3 ) ) < C y n q 1 + n q 2 + 2 N 1 + 2 N 2 m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , υ 3 ) ) .

 □

Theorem 2.6 Let m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] .
  1. (a)
    If Φ L 1 ( R 2 n ) , then Φ m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] and
    Φ m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) Φ 1 m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) .
     
  2. (b)
    If Φ L ω 1 ( R 2 n ) such that ω ( u , υ ) = ω 1 ( u ) ω 2 ( υ ) , then Φ ˆ m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] and
    Φ ˆ m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) Φ 1 , ω m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) .
     
Proof (a) Let f , g C c ( R n ) be given. By Proposition 2.5 in [12]
B Φ m ( f , g ) ( y ) = R n R n Φ ( u , v ) B T ( u , v ) m ( f , g ) ( y ) d u d v .
If we use Theorem 2.4 and the assumption m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] , we have
B Φ m ( f , g ) W ( L p 3 , L ω 3 q 3 ) R n R n Φ ( u , v ) B T ( u , v ) m ( f , g ) W ( L p 3 , L ω 3 q 3 ) d u d v R n R n | Φ ( u , v ) | T ( u , v ) m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) d u d v = m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) Φ 1 f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) < .
(2.35)
Hence Φ m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] , and by (2.35) we obtain
Φ m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) Φ 1 m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) .
(b) Let Φ L ω 1 ( R 2 n ) . Take any f , g C c ( R n ) . It is known by Proposition 2.5 in [12] that
B Φ ˆ m ( f , g ) ( x ) = R n R n Φ ( u , v ) B M ( u , v ) m ( f , g ) ( x ) d u d v .
Since m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] , we have M ( u , v ) m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] and
M ( u , v ) m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) ω 1 ( u ) ω 2 ( v ) m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) )
by Theorem 2.4. Then
B Φ ˆ m ( f , g ) W ( L p 3 , L ω 3 q 3 ) R n R n Φ ( u , v ) B M ( u , v ) m ( f , g ) W ( L p 3 , L ω 3 q 3 ) d u d v R n R n | Φ ( u , v ) | M ( u , v ) m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) f p 1 , ω 1 g p 2 , ω 2 d u d v R n R n | Φ ( u , v ) | ω 1 ( u ) ω 2 ( υ ) m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) × f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) d u d v = m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) f W ( L p 1 , L ω 1 q 1 ) g W ( L p 2 , L ω 2 q 2 ) Φ 1 , ω .
(2.36)
Thus from (2.36) we obtain Φ ˆ m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ] and
Φ ˆ m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) Φ 1 , ω m ( W ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ) .

 □

Theorem 2.7 Let m BM [ W ( p 1 , q 1 , ω 1 ; p 2 , q 2 <