Afr. Mat. DOI 10.1007/s13370-015-0365-0
Besov norms in terms of the S-transform Sunil Kumar Singh1
Received: 25 August 2014 / Accepted: 4 July 2015 © African Mathematical Union and Springer-Verlag Berlin Heidelberg 2015
Abstract In this paper, the S-transform of derivatives and derivatives of S-transform are obtained. The characterization of Besov and weighted Besov spaces in terms of the continuous S-transform are given. Keywords
Besov space · Fourier transform · S-Transform 65R10 · 30H25
Mathematics Subject Classification
1 Introduction The S-transform is a time-frequency localization technique that has characteristics superior to both of the Fourier transform and the wavelet transform [13]. The one-dimensional continuous S-transform of f is defined as [15] e−i2π ξ t f (t) ω(τ − t, ξ ) dt, (1.1) (Sω f )(τ, ξ ) = R
where the window ω is assumed to satisfy the following: ω(t, ξ )dt = 1 for all ξ ∈ R. R
(1.2)
The most usual window ω is the Gaussian one |ξ | − ξ 2 t 2 ω(t, ξ ) = √ e 2k 2 , ξ ∈ R\{0} , k > 0, k 2π
B 1
(1.3)
Sunil Kumar Singh
[email protected] Department of Mathematics, Rajiv Gandhi University, Doimukh 791112, Arunachal Pradesh, India
123
S. K. Singh
in which ξ is the frequency, t is the time variable, and k is a scaling factor that controls the number of oscillations in the window. The Eq. (1.1) can be rewritten in the following forms: (Sω f )(τ, ξ ) = e−i2π ξ τ ei2π ξ x f (τ − x) ω(x, ξ ) d x R = f (·)e−i2π ξ · ∗ ω(·, ξ ) (τ ).
(1.4) (1.5)
If fˆ(ξ ) and (Sω f )(τ, ξ ) are the Fourier transform and S-transform of f respectively, then there exists a direct relation between the S-transform and Fourier transform which is given as: fˆ(ξ ) =
R
(Sω f )(τ, ξ )dτ,
(1.6)
and hence f (t) = F −1
R
(Sω f )(τ, ·)dτ (t).
(1.7)
Some basic properties of S-transform can be found in [6,8–10] and certain examples of S-transform are given below the proofs of which can be found in [8, page 892]. (a) (Sω δ)(τ, ξ ) = ω(τ, ξ ), 2 2 (b) (Sω 1)(τ, ξ ) = e−i2π ξ τ e−2π k , where ω is the Gaussian window. Also, the fractional S-transform has been studied on the spaces of type S and W by Singh [7,11]. Based on the idea of the S-transform, a new integral transform has been defined by Singh [12] analogous to the continuous wavelet transform.
2 The S-transform of derivatives In this section, we derived a general formula for the S-transform of derivatives and derivatives of S-transform which will may be useful in the theory of differential equations. Theorem 2.1 Let f be a continuously n-times differentiable function such that (i) f, f , . . . f (n) ∈ L 1 (R), and (ii) lim|t|→∞ f (r ) (t) = 0 for r = 0, 1, 2, . . . , (n − 1). (n)
Let ω(t, ξ ) be a window function such that Dt ω(t, ξ ) and (S D (n−m) ω f ) exist for each m, t 0 ≤ m ≤ n, and n = 0, 1, 2, . . . , then
n n (i2πξ )m S D (n−m) ω f (τ, ξ ). Sω f (n) (τ, ξ ) = t m m=0
123
(2.1)
Besov norms in terms of the S-transform
Proof We prove the result by induction on n. For n = 1, we have by definition (Sω f )(τ, ξ ) = f (t) ω(τ − t, ξ ) e−i2π ξ t dt R ∞ − f (t)Dt ω(τ − t, ξ )e−i2π ξ t dt = f (t)ω(τ − t, ξ )e−i2π ξ t t=−∞ R −i2π ξ t =− f (t)Dt ω(τ − t, ξ )(−1)e dt R − f (t)ω(τ − t, ξ )(−i2πξ )e−i2π ξ t dt R
= (S Dt ω f )(τ, ξ ) + (i2πξ )(Sω f )(τ, ξ ).
(2.2)
Assume that result is true for n − 1, that is, n−1 n−1 (n−1) (τ, ξ ) = Sω f (i2πξ )m S D (n−1−m) ω f (τ, ξ ). t m
(2.3)
m=0
By putting f = u in the above equation and using (2.2), we get Sω u
(n)
(τ, ξ ) =
n−1 n−1 m
m=0
=
t
n−1 n−1 m
m=0
(i2πξ )m S D (n−1−m) ω u (τ, ξ ) (i2πξ )m
S D (n−m) ω u (τ, ξ ) t
+(i2πξ ) S D (n−1−m) ω u (τ, ξ )
t
=
n−1
m=0
+
n−1 (i2πξ )m S D (n−m) ω u (τ, ξ ) t m
n−1 n−1
m=0
m
(i2πξ )m+1 S D (n−1−m) ω u (τ, ξ ) t
n−1
= S D (n) ω u (τ, ξ ) + t
+
n−1 n−1
m=1
n−1 (i2πξ )m S D (n−m) ω u (τ, ξ ) t m
(i2πξ )r S D (n−r) ω u (τ, ξ ) + (i2πξ )n (Sω u)(τ, ξ )
t r −1 = S D (n) ω u (τ, ξ ) t n−1 n−1 n − 1 n−1 + (i2πξ )m S D (n−m) ω u (τ, ξ ) + t m m−1
r =1
m=1
m=1
+(i2πξ )n (Sω u)(τ, ξ ) = S D (n) ω u (τ, ξ ) t
+
n−1 n−1
m=1
m
+
n−1 m−1
(i2πξ )m S D (n−m) ω u (τ, ξ ) t
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S. K. Singh
+(i2πξ )n (Sω u)(τ, ξ ) = (S D (n) ω u)(τ, ξ ) t
+
n−1 n
m=1
m
(i2πξ )m S D (n−m) ω u (τ, ξ ) t
+(i2πξ ) (Sω u)(τ, ξ ) n n (i2πξ )m S D (n−m) ω u (τ, ξ ) = S D (n) ω u (τ, ξ ) + t t m m=1 n n = (i2πξ )m S D (n−m) ω u (τ, ξ ). t m n
m=0
Remark 2.2 For the Gaussian window case (1.3), |ξ | d n − ξ 2 t22 Dtn ω(t, ξ ) = √ e 2k k 2π dt n n ξ ξt |ξ | − ξ 2 t 2 = √ e 2k 2 (−1)n H √ √ n k 2π k 2 k 2 n ξ ξt ω(t, ξ )Hn = (−1)n √ √ , k 2 k 2
(2.4)
where Hn is the Hermite polynomial of order n. Using (2.4) in (2.1) we deduce
n n Sω f (n) (τ, ξ ) = (i2πξ )m e−i2π ξ t f (t) Dtn−m ω(τ − t, ξ ) dt m R m=0 n n e−i2π ξ t f (t) = (i2πξ )m m R m=0 ξ(τ − t) −ξ n−m dt ω(τ − t, ξ )Hn−m × √ √ k 2 k 2 n n −1 n−m m (i2π) = √ m k 2 m=0 ξ(τ − t) × e−i2π ξ t f (t) ω(τ − t, ξ )Hn−m dt. √ k 2 R
(2.5)
Theorem 2.3 If ω(t, ξ ) is continuously n-times differentiable with respect to ξ and (S D (n−m) ω t m f ) exists for each m, 0 ≤ m ≤ n, and n = 0, 1, 2, . . . , then ξ
Dξn (Sω f ) (τ, ξ ) =
n n m=0
123
m
(−i2π)m S D (n−m) ω t m f (τ, ξ ). ξ
(2.6)
Besov norms in terms of the S-transform
Proof We prove the result by induction on n. For n = 1, we have by definition Dξ (Sω f ) (τ, ξ ) =
R
f (t)Dξ ω(τ − t, ξ )e−i2π ξ t dt
f (t) Dξ ω(τ − t, ξ ) e−i2π ξ t dt + f (t) ω(τ − t, ξ ) Dξ e−i2π ξ t dt R R = (S Dξ ω f )(τ, ξ ) + (−i2π) t f (t) ω(τ − t, ξ ) e−i2π ξ t dt =
R
= (S Dξ ω f )(τ, ξ ) + (−i2π)(Sω t f )(τ, ξ ).
(2.7)
Assume that result is true for n − 1, that is, Dξn−1 (Sω f ) (τ, ξ ) =
n−1 n−1 m
m=0
(−i2π)m S D (n−1−m) ω t m f (τ, ξ ). ξ
(2.8)
Differentiate above equation with respect to ξ and using (2.7), we get Dξn (Sω f ) (τ, ξ ) =
n−1 n−1 m
k=0
=
n−1 n−1 m
m=0
(−i2π)m Dξ
S D (n−1−m) ω t m f ξ
(−i2π)
m
(τ, ξ )
m
S D (n−m) ω t f ξ
(τ, ξ )
+(−i2π) S D (n−1−m) ω t m+1 f (τ, ξ )
=
n−1
m=0
+
n−1 n−1 m
m=0
=
ξ
n−1 (−i2π)m S D (n−m) ω t m f (τ, ξ ) ξ m
(−i2π)m+1 S D (n−1−m) ω t m+1 f (τ, ξ ) ξ
S D (n) ω f ξ
n−1
(τ, ξ ) +
n−1 n−1 m=1
m
(−i2π)m S D (n−m) ω t m f (τ, ξ ) ξ
n−1 (−i2π)m S D (n−m) ω t m f (τ, ξ ) ξ m−1 m=1 +(−i2π)n Sω t n f (τ, ξ ) n−1 n−1 = S D (n) ω f (τ, ξ ) + (−i2π)m S D (n−m) ω t m f (τ, ξ ) ξ ξ m +
m=1
n−1 (−i2π)m S D (n−m) ω t m f (τ, ξ ) ξ m−1 m=1 + (−i2π)n Sω t n f (τ, ξ ) = S D (n) ω f (τ, ξ ) +
n−1
ξ
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S. K. Singh
n−1 n−1
n−1 + + m m−1 m=1 +(−i2π)n Sω t n f (τ, ξ ) = S D (n) ω f (τ, ξ )
(−i2π)
m
m
S D (n−m) ω t f ξ
(τ, ξ )
ξ
n (−i2π)m S D (n−m) ω t m f (τ, ξ )+(−i2π)n Sω t n f (τ, ξ ) ξ m m=1 n n = (−i2π)m S D (n−m) ω t m f (τ, ξ ). ξ m +
n−1
m=0
2.1 Applications of the S-transform to boundary value problems We consider the initial value problem ∂u ∂ 2u = , −∞ < x < ∞, t > 0, ∂t ∂x2
(2.9)
u(x, 0) = f (x)e−i2π x , −∞ < x < ∞.
(2.10)
with the initial condition
Solution of this problem using the Fourier transform is ∞ (x−α)2 1 u(x, t) = √ f (α)e−i2π α e− 4t dα. 2 πt −∞ By putting t =
k2 2
in the right hand side of the above equation we get ∞ (x−α)2 1 −i2π α − 2k 2 √ u(x, t) = √ f (α)e e dα √ = [(Sω f )(x, 1)]k= 2t , k 2π −∞ k= 2t
where (Sω f ) is the S-transform of f with respect to Gaussian window(1.3). Example 2.4 (i) If f (x) = 1 in the Eq. (2.10), then 2 2 u(x, t) = [(Sω 1)(x, 1)]k=√2t = e−i2π x e−2π k
√ k= 2t
= e−i2π x e−4π t . 2
(ii) If f (x) = δ(x) in the Eq. (2.10), then 1 2 u(x, t) = [(Sω δ)(x, 1)]k=√2t = [ω(x, 1)]k=√2t = √ e−x /4t . 2 πt The S-transform has been studied on spaces of distributions (see [8–10]). Sections 3 and 4 are devoted to study the S-transform on Besov spaces.
3 The S-transform on Besov spaces In this section we recall the definitions of Besov spaces due to Peetre [4] and obtain the boundedness results for S-transform on Besov spaces analogous to the boundedness results for wavelet transform on Besov spaces [1,5].
123
Besov norms in terms of the S-transform
Definition 3.1 For f ∈ L p (R) and for any fixed η ∈ R, we write M p ( f ; h, η) := f (· + h)e−i2π hη − f (·) p , for h ∈ R. L
Obviously, M p ( f ; h, η) ≤ 2 f L p and hence it is well defined. Moreover, for 1 ≤ p < ∞, M p ( f ; h, η) → 0 as h → 0. α,q
Definition 3.2 For 0 < α < 1, 1 ≤ p, q < ∞, the Besov space B p is defined by q dh α,q < ∞ for q < ∞ M p ( f ; h, η) B p = f ∈ L p (R) : |h|1+αq R and
B α,∞ = f ∈ L p (R) : |h|−α M p ( f ; h, η) ∈ L ∞ (R\{0}) . p α,q
The space B p (R) is a Banach space with norm: q f B α,q M p ( f ; h, η) = f + p L p R
and
dh |h|1+αq
1/q for q < ∞
f B α,∞ = f L p + |h|−α M p ( f ; h, η) L ∞ . p
Theorem 3.3 Suppose ω(·, ξ ) ∈ L 1 (R), then for any fixed ξ ∈ R\{0}, the operator Sω : α,q α,q B p → B p is continuous. Moreover, (Sω f )(·, ξ ) B α,q ≤ ω(·, ξ ) L 1 f B α,q . p p Proof By using Young’s inequality we have (Sω f )(·, ξ ) L p = f (·)e−i2π ξ · ∗ ω(·, ξ ) p L −i2π ξ · ω(·, ξ ) L 1 ≤ f (·)e p L
= f L p ω(·, ξ ) L 1 ,
(3.1)
and by using the Minkowski inequality we get M p ((Sω f )(·, ξ ); h, η) = (Sω f )(· + h, ξ ) e−i2π hη − (Sω f )(·, ξ ) p L p 1/ p −i2π hη = f )(· + h, ξ ) e − (S f )(·, ξ ) dτ (Sω ω R −i2π hη −i2π ξ(τ +h) e = e ei2π ξ x f (τ + h − x) ω(x, ξ ) d x R
−e−i2π ξ τ
R
R
p ei2π ξ x f (τ − x) ω(x, ξ ) d x dτ
1/ p
p 1/ p −i2π ξ τ i2π ξ x −i2π h(η+ξ ) ≤ e ω(x, ξ ) f (τ + h − x) − f (τ − x) d x e e dτ R R |ω(x, ξ )| d x M p ( f ; h, η + ξ ) = R
= ω(·, ξ ) L 1 M p ( f ; h, ξ + η).
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S. K. Singh
Therefore, for q < ∞ we have the following estimate q (S (Sω f )(·, ξ ) B α,q = f )(·, ξ ) + M p ((Sω f ); h, η) p ω L p R
dh |h|1+αq
1/q
1/q q dh ≤ ω(·, ξ ) L 1 f L p + ω(·, ξ ) L 1 M p ( f ; h, η+ξ ) |h|1+αq R 1/q q dh M p ( f ; h, η + ξ ) = ω(·, ξ ) L 1 f L p + |h|1+αq R = ω(·, ξ ) L 1 f B α,q . p
Corollary 3.4 If ω1 (·, ξ ), ω2 (·, ξ ) ∈ L 1 (R) and f, g ∈
R\{0}, the following estimate holds
α,q B p (R),
then for any fixed ξ ∈
(Sω f )(·, ξ ) − (Sω g)(·, ξ ) α,q ≤ (ω1 − ω2 )(·, ξ ) L 1 f α,q 1 2 Bp B p
+ ω2 (·, ξ ) L 1 f − g B α,q . p Proof
(Sω f )(·, ξ ) − (Sω g)(·, ξ ) α,q ≤ (Sω f )(·, ξ ) − (Sω f )(·, ξ ) α,q 1 2 1 2 Bp Bp + (Sω2 f )(·, ξ ) − (Sω2 g)(·, ξ ) B α,q . p
Since
(Sω f )(·, ξ ) − (Sω f )(·, ξ ) p = f (·)e−i2π ξ · ∗ (ω1 − ω2 )(·, ξ ) 1 2 L ≤ (ω1 − ω2 )(·, ξ ) L 1 f L p ,
(3.2)
Lp
using the techniques of theorem 3.3 we get (Sω f )(·, ξ ) − (Sω f )(·, ξ ) α,q ≤ (ω1 − ω2 )(·, ξ ) L 1 f α,q . 1 2 Bp B p
Furthermore,
(Sω ( f − g))(·, ξ ) α,q ≤ ω2 (·, ξ ) L 1 f − g α,q . 2 Bp B p
(3.3)
(3.4)
Therefore by using (3.3) and (3.4) in (3.2) we get (Sω f )(·, ξ ) − (Sω g)(·, ξ ) α,q ≤ (ω1 − ω2 )(·, ξ ) L 1 f α,q 1 2 Bp B p
+ ω2 (·, ξ ) L 1 f − g B α,q . p
4 The S-transform on weighted Besov spaces In this section we recall the definition of the temperate weight function which was defined by Hörmander [2, p. 4]. We give the definitions of weighted Besov spaces [1] associated with a temperate weight function and obtain boundedness results for Sω .
123
Besov norms in terms of the S-transform
Definition 4.1 A positive function κ defined on R is called a temperate weight function if there exist positive constants C and N such that κ(ξ + η) (1 + C|ξ |) N κ(η);
ξ, η ∈ R,
(4.1)
the set of all such functions κ is denoted by K . Certain examples and basic properties of weight function κ can be found in [2]. p
Definition 4.2 For 1 ≤ p < ∞, the weighted Lebesgue space L κ (R) is defined as the space of all measurable function on R such that 1/ p p f L κp = | f (x)| κ(x)d x < ∞, R
and for f ∈
p L κ (R)
and for any fixed η ∈ R, we define M p,κ ( f ; h, η) by M p,κ ( f ; h, η) := f (· + h)e−i2π hη − f (·) p , for h ∈ R. Lκ
α,q
Definition 4.3 For 0 < α < 1, 1 ≤ p, q < ∞, the weighted Besov space B p,κ (R) is defined by q dh α,q M p,κ ( f ; h, η) B p,κ = f ∈ L κp (R) : < ∞ for q < ∞ |h|1+αq R and
p −α M p,κ ( f ; h, η) ∈ L ∞ (R\{0}) . B α,∞ p,κ = f ∈ L κ (R) : |h| α,q
The space B p,κ (R) is a Banach space with norm : q α,q p f B p,κ = f L κ + M p,κ ( f ; h, η) R
and
dh |h|1+αq
1/q for q < ∞
f B α,∞ = f L κp + |h|−α M p,κ ( f ; h, η) L ∞ . p,κ
Theorem 4.4 If ω is a window function such that for any fixed ξ ∈ R\{0} |ω(x, ξ )| (1 + C|x|) N d x ≤ A < ∞,
(4.2)
R
α,q
α,q
where A, C and N are positive constants. Then the operator Sω : B p,κ → B p,κ is continuous. Moreover, (Sω f )(·, ξ ) B α,q ≤ A f B α,q . p,κ p,κ Proof By using the Minkowski inequality and (4.2), we obtain p 1/ p −i2π ξ τ i2π ξ x (Sω f )(·, ξ ) L κp = f (τ − x)e ω(x, ξ )d x κ(τ )dτ e R
≤
R
R
|ω(x, ξ )| d x
R
1/ p
| f (τ − x)| p κ(τ )dτ
1/ p
|ω(x, ξ )| d x | f (t)| p κ(x + t)dt R R p |ω(x, ξ )| (1 + C|x|) N / p d x ≤ f L κ ≤
R
≤ A f L κp
(4.3)
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S. K. Singh
and M p,κ ((Sω f )(·, ξ ); h, η) = (Sω f )(· + h, ξ )e−i2π hη − (Sω f )(·, ξ )
p
Lκ
1/ p p = (Sω f )(τ + h, ξ )e−i2π hη − (Sω f )(τ, ξ ) κ(τ )dτ R −i2π hη −i2πξ(τ +h) e = e ei2πξ x f (τ + h − x) ω(x, ξ ) d x R
−e
−i2πξ τ
≤
R
= ≤
R
R
=
R
=
R
R
R
e
i2πξ x
p f (τ − x) ω(x, ξ ) d x κ(τ )dτ
i2πξ x e ω(x, ξ )d x ω(x, ξ )d x ω(x, ξ )d x
R
R
R
1/ p
p −i2π h(η+ξ ) e f (τ + h − x) − f (τ − x) κ(τ )dτ
1/ p
−i2π h(η+ξ ) p e f (τ + h − x) − f (τ − x) κ(τ − x + x)dτ
1/ p
−i2π h(η+ξ ) p e f (τ + h − x) − f (τ − x) (1 + C|x|) N κ(τ − x)dτ
ω(x, ξ )(1 + C|x|) N / p d x
R
1/ p
−i2π h(η+ξ ) p e f (τ + h − x) − f (τ − x) κ(τ − x)dτ
1/ p
ω(x, ξ )(1 + C|x|) N / p d x M p,κ ( f ; h, η + ξ )
≤ A M p,κ ( f ; h, η + ξ ).
Consequently, for q < ∞ we have the following estimate q α,q p (Sω f )(·, ξ ) B p,κ = (Sω f )(·, ξ ) L κ + Mκ, p ((Sω f ); h, η)
R
dh |h|1+αq 1/q
1/q
q dh ≤ A f L κp + A M p,κ ( f ; h, η + ξ ) |h|1+αq R 1/q q dh = A f L κp + M p ( f ; h, η + ξ ) |h|1+αq R
. = A f B α,q p,κ
Corollary 4.5 Let ω1 and ω2 be window functions which satisfy the condition (4.2). For α,q f, g ∈ B p,κ (R) and for any fixed ξ ∈ R\{0}, the following estimate holds: (Sω f )(·, ξ ) − (Sω g)(·, ξ ) α,q ≤ A 2 f α,q + f − g α,q . 1 2 B p,κ B p,κ B p,κ
Proof We have (Sω f )(·, ξ ) − (Sω g)(·, ξ ) α,q ≤ (Sω f )(·, ξ ) − (Sω f )(·, ξ ) α,q 1 2 1 2 B p,κ B p,κ + (Sω2 f )(·, ξ ) − (Sω2 g)(·, ξ ) B α,q . p,κ
(4.4)
Since (Sω f )(·, ξ ) − (Sω f )(·, ξ ) p ≤ (Sω f )(·, ξ ) p + (Sω f )(·, ξ ) p ≤ 2 A f p , 1 2 1 2 Lκ L L L κ
123
κ
κ
Besov norms in terms of the S-transform
hence
(Sω f )(·, ξ ) − (Sω f )(·, ξ ) α,q ≤ 2 A f α,q . 1 2 B p,κ B
(4.5)
p,κ
Furthermore,
(Sω ( f − g))(·, ξ ) α,q ≤ A f − g α,q . 2 B p,κ B p,κ Therefore by using (4.5) and (4.6) in (4.4) we get the required result (Sω f )(·, ξ ) − (Sω g)(·, ξ ) α,q ≤ A 2 f α,q + f − g α,q . 1 2 B B B p,κ p,κ
(4.6)
p,κ
Theorem 4.6 Suppose that ω(·, ξ ) ∈ L 1 (R) is compactly supported in time domain and for any fixed ξ ∈ R\{0} the support of ω(·, ξ ) contained in the interval [−r, r ], where α,q α,q r is a positive constant. Then the operator Sω : B p,κ → B p,κ is continuous and N / p (Sω f )(·, ξ ) B α,q ω(·, ξ ) L 1 f B α,q ≤ (1 + C|r |) . p,κ p,κ Proof Since
(Sω f )(·, ξ )
≤ f
p Lκ
p Lκ
R
|ω(x, ξ )| (1 + C|x|) N / p d x
|ω(x, ξ )| (1 + C|x|) N / p d x N/p p |ω(x, ξ )| d x ≤ f L κ (1 + C|r |)
= f L κp
|x|≤r
|x|≤r
= (1 + C|r |) N / p ω(·, ξ ) L 1 f L κp ,
and
M p,κ ((Sω f )(·, ξ ); h, η) = (Sω f )(· + h, ξ )e−i2π hη − (Sω f )(·, ξ )
p
Lκ
1/ p p −i2π hη = f )(τ + h, ξ )e − (S f )(τ, ξ ) κ(τ )dτ (Sω ω R −i2π hη −i2πξ(τ +h) e = e ei2πξ x f (τ + h − x) ω(x, ξ ) d x R
−e−i2πξ τ ≤
R
= ≤
R
R
=
R
=
R
R
R
p ei2πξ x f (τ − x) ω(x, ξ ) d x κ(τ )dτ
−i2πξ τ i2πξ x e e ω(x, ξ )d x ω(x, ξ )d x ω(x, ξ )d x
R
R
R
1/ p
−i2π h(η+ξ ) p e f (τ + h − x) − f (τ − x) κ(τ )dτ
p −i2π h(η+ξ ) e f (τ + h − x) − f (τ − x) κ(τ − x + x)dτ
1/ p
1/ p
−i2π h(η+ξ ) p e f (τ + h − x) − f (τ − x) (1 + C|x|) N κ(τ − x)dτ
ω(x, ξ )(1 + C|x|) N / p d x
R
1/ p
−i2π h(η+ξ ) p e f (τ +h − x)− f (τ − x) κ(τ − x)dτ
1/ p
ω(x, ξ )(1 + C|x|) N / p d x M p,κ ( f ; h, η + ξ )
≤ (1 + C|r |) N / p ω(·, ξ ) L 1 M p,κ ( f ; h, η + ξ ).
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S. K. Singh
Therefore, for q < ∞ we have the following estimate q α,q p (Sω f )(·, ξ ) B p,κ = (Sω f )(·, ξ ) L κ + M p,κ ((Sω f ); h, η) R
dh |h|1+αq
1/q
≤ (1 + C|r |) N / p ω(·, ξ ) L 1 f L κp q +(1 + C|r |) N / p ω(·, ξ ) L 1 M p,κ ( f ; h, η + ξ ) R
= (1 + C|r |) N / p ω(·, ξ ) L 1 q M p,κ ( f ; h, η + ξ ) × f L κp + R
= (1 + C|r |)
N/p
dh |h|1+αq
dh |h|1+αq
1/q
1/q
ω(·, ξ ) L 1 f B α,q . p,κ
Corollary 4.7 Suppose that ω1 (·, ξ ), ω2 (·, ξ ) ∈ L 1 (R) are compactly supported in time domain and for any fixed ξ ∈ R\{0} the support of ω1 (·, ξ ) and ω2 (·, ξ ) contained in α,q the interval [−r, r ], where r is a positive constant. If f, g ∈ B p,κ (R) and ξ ∈ R\{0}, then (Sω f )(·, ξ ) − (Sω g)(·, ξ ) α,q ≤ (1 + C|r |) N / p (ω1 − ω2 )(·, ξ ) L 1 f α,q 1 2 B p,κ B p,κ . + ω2 (·, ξ ) L 1 f − g B α,q p,κ Acknowledgments agement.
The author expresses his sincere thanks to Prof. R.S. Pathak for his help and encour-
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