DOI: 10.15415/mjis.2013.21008
A Note on a Multiplier Class R. G. VYAS Department of Mathematics, Faculty of Science, The Maharaja Sayajirao University of Baroda, Vadodara-390002, Gujarat, India. E-mail: drrgvyas@yahoo.com Abstract Let f be a complex valued function on [0, 2π]N, Λ1 = { f : Tδ f −
}
}
{
f ∞ = O ( δ ) , Λ11 = f : Tδ f − f 1 = O ( δ ) and the multiplier class, ∞
( L , Λ1 ) = { f ∈ L ([0, 2π ] ) : f ∗ g ∈ Λ1 , ∀g ∈ L∞ ([0, 2π ]N }, 1
N
where
N
δ = (δ1 , δ2 ,..., δN ), Tδ f (x ) = f ( x + δ ), ∀x, δ ∈ [0, 2π ] . Here, we have characterized the class Λ11 as Λ11 = ( L∞ , Λ1 ) . 2010 Mathematics Subject classifcation: Primary 42B15 and secondary 42B35, 42A85. Keywords and phrases: Lipschitz condition, convolution product, multiplier class.
1. INTRODUCTION
T
he concept of generalized bounded variations, Lipschitz condition, absolute continuity, etc., are generalized to integral setting as well as from one variable to several variables to obtain various classes of functions which are of interest in Functional Analysis as well as in Fourier analysis (Avdispahic, 1986; James Caveny, 1970; De Leeuw, 1961; Ghorpade and Limaye, 2010; Vyas and Darji, 2012). Lipschitz conditions and integrated Lipschitz conditions for single variable functions as well as for several variables functions give classes which are important for the discussion of the convergence, uniform convergence and absolute convergence of single Fourier series as well as of multiple Fourier series respectively. Objective of the paper is to obtain some relationship between the Lipschitz classes and the integrated Lipschitz classes of functions of several variables. First, recall some standard notations for the function classes of interest.
Mathematical Journal of Interdisciplinary Sciences Vol. 2, No. 1, September 2013 pp. 95–98
2. NOTATIONS AND DEFINITIONS
In the sequel, f is a complex valued function of N-variables defined over 1 N q q 1≤ q ≤ ∞) [0, 2π]N, which is 2π-periodic in each variable, δ = (δ1 ,.....δN ), δ q = (∑ i=1δ©2013 i ) ( by Chitkara 1 q University. All Rights ...δN ), δ q = (∑ iN=1δi q ) (1 ≤ q ≤ ∞) and Reserved.
Vyas, R. G.
(∆h f )(x ) = (T( h ) f − f )(x ) 1
1
η1=0
η N =0
= ∑ .....∑ (−1)η1 +....+ηN f ( x1 + η1h1 , x2 + η2 h2 ,.... x N + η N hN ) , (2.1)
96
where (Thf)(x) = f(x + h) for all x = (x1, x2, .... xN), h = (h1, h2, ...., hN ) ∈ [0, 2π]N. For any h and δ define h ≤ δ if and only if hi ≤ δi for all i = 1 to N. L1 ([0, 2π ]N ), is the space of complex valued Lebesgue integrable functions over [0, 2π]N which are 2π periodic in each variables and L∞ ([0, 2π ]N ) consists of the L1 ([0, 2π ]N ) functions which are essentially bounded. . 1 and . ∞ denote the norms in the spaces L1 ([0, 2π ]N ) and L∞ ([0, 2π]N ) respectively. Λ ([ 0, 2 π ] ) = f ε L ([ 0, 2 π ] ) : 0 ≤ h ≤ δ T f − f = O ( δ ) .
{ ]) { ]) { ]) { ]) { ]) { ]) { N
1
(
Λ 1 [ 0, 2 π 1
(
N
Lip [ 0, 2 π Lip
1
([ 0, 2π
(
Λ11 [ 0, 2 π
(
Lip11 [ 0, 2 π
(
Lip11 [ 0, 2 π 1
(
h
N
N
∞
h
1
([ 0, 2π ] ) : 0 ≤ h ≤ δ T f − f = O ( δ )}. sup
N
1
∞
2
sup
N
h
1
sup
([ 0, 2π ] ) : 0 ≤ h ≤ δ ∆ N
h
([ 0, 2π ] ) : 0 ≤ h ≤ δ ∆
1
([ 0, 2π ] ) : 0 ≤ h ≤ δ ∆
1
N
} )} )}
h
f ∞ = O ( δ 1
h
f 1 = O ( δ 1
sup
N
1
f ∞ = O ( δ 2 ) .
sup
= fεL
= fεL
Lip*11 [ 0, 2 π ]
1
([ 0, 2π ] ) : 0 ≤ h ≤ δ T f − f = O ( δ )}.
∞
= fεL
N
sup
N
= fεL
N
2
([ 0, 2π ] ) : 0 ≤ h ≤ δ T f − f = O ( δ )}.
1
= fεL
N
∞
h
= fεL
N
}
sup
N
∞
.
.
) = { f ε Lip ([ 0, 2π ] ) : N
11
sup
(
hi ∈ (0, δi ] ∆( 0 , ...., 0 , hi , 0 , ....0 ) f ∞ = O (| δi |), ∀ i = 1 to N}
Lip*11 [ 0, 2 π ] 1
N
) = { f ∈ Lip ([ 0, 2π ] ) : 1
N
11
sup
hi ∈ (0, δi ] ∆( 0 , ...., 0 , hi , 0 , ....0 ) f 1= O (| δi |), ∀ i = 1 to N
For any f , g ∈ L1 ([0, 2π ]N )}their convolution product f ∗ g is the L1([0, 2π]N) function defined (Vyas, 2013) as
( f ∗ g )(x ) =
1 (2 π ) N
∫
[ 0 , 2 π ]N
f (x − t )g(t )dt, ∀x ∈ [0, 2π ]N .
Mathematical Journal of Interdisciplinary Sciences, Volume 2, Number 1, September 2013
In general the multiplier class is defined as
A Note On A Multiplier Class
(L∞[0, 2π]N, E) = {f ∈ L1([0, 2π]N) : f * g ∈ E for all g ∈ L∞([0, 2π]N)},
for E = Λ1 ([0, 2π ]N ), or E = Lip ([0, 2π]N), or E = Lip11 ([0, 2π]N), or E = Lip*11([0, 2π]N ). Here, we prove the following results for functions of several variables. Theorem 2.1. Λ11 ([0, 2π ]N ) = ( L∞ ([0, 2π ]N ), Λ1 ([0, 2π ]N )).
97
Theorem 2.2. Lip1 ([0, 2π ]N ) = ( L∞ ([0, 2π ]N ), Lip ([0, 2π ]N )). N ∞ N N 1 Theorem 2.3. Lip*11 ([0, 2π ] ) = ( L ([0, 2π ] ), Lip*11 ([0, 2π ] )).
For the simplicity of proof, we will prove all these results for N = 2. Proof of the Theorem 2.1. For any f ∈ Λ11 ([0, 2π ]2 ), g ∈ L∞ ([0, 2π ]2 ) and x, y ∈ ([0, 2π ]2 ), we have
( f ∗ g )(x ) − ( f ∗ g )(y )
1 f ( x − t ) − f ( y − t ) g ( t ) dt (2π )2 ∫[ 0,2 π ]2 g ∞ f (x − t ) − f (y − t ) dt = O( x − y 2 ). ≤ (2π )2 ∫[ 0,2 π ]2 ≤
Now suppose that f ∈ ( L∞ ([0, 2π ]2 ), Λ1 ([0, 2π ]2 )) The Lipschitz class Λ1 ([0, 2π ]2 ) is a Banach space with respect to the norm
h = h (0, 0) +
h( y ) − h( x ) sup . x,y∈[0,2π ]2 y − x 2
Define the map
T : L∞ ([0, 2π ]2 ) → Λ1 ([0, 2π ]2 ) as T ( g ) = f ∗ g, g ∈ L∞ ([0, 2π ]2 )
To prove that T is continuous it is enough to show that the graph of T is closed. Consider gn → g in L∞ ([0, 2π ]N ) and T (gn) → h in Λ1 ([0, 2π ]2 ) then for any fixed x ∈ ([0, 2π ]2 ), we have
(T ( g ))(x ) − h(x ) ≤ ( f ∗ g )(x ) − ( f ∗ gn )(x ) + (T ( gn ))(x ) −h(x ) ≤ f 1 gn − g ∞ + T ( gn ) − h .
Mathematical Journal of Interdisciplinary Sciences, Volume 2, Number 1, September 2013
Hence, T (g) = h implies T is continuous and
Vyas, R. G.
T ( g ) ≤ T g ∞ , ∀g ∈ L∞ ([0, 2π ]2 ), where T is a constant
Thus, for any x, y ∈ ([0, 2π ]2 ), we have |
98
1 (2 π )2
∫
[ 0 ,2 π ]2
( f (x − t ) − f (y − t )) g(t ) dt ≤ T g ∞ x − y 2 .
{−i arg ( f ( x−t )− f ( y−t ))} , we have In the above inequality for g(t ) = e 1 f (x − t ) − f (y − t ) dt ≤ T x − y 2 . (2π )2 ∫[ 0,2 π ]2
This completes the proof. Similarly, we can prove the Theorem 2.2. By using the fact that; Lip*11 ([0, 2π ]) is a Banach space with respect to the norm h = h (0, 0) +
+
h( x1 , x2 ) − h( y1 , x2 ) − h( x1 , y2 ) + h( y1 , y2 ) sup x,y∈[0,2π ]2 y1 − x1 + y2 − x2
h( x1 , x2 ) − h( y1 , x2 ) h( x1 , x2 ) − h( x1 , y2 ) sup sup + 2 2 x,y∈[0,2π ] x,y∈[0,2π ] y1 − x1 y2 − x2
where x = (x1, x2) and y = (y1, y2). Similarly one can prove the Theorem 2.3. References Avdispahic, M. (1986) Concepts of generalized bounded variation and the theory of Fourier series, Inter. J. Math. and Math. Sci., V. 9, No. 2, 223-244. http://dx.doi.org/10.1155/S0161171286000285 Caveny James (1970) On integral Lipschitz conditions and integral bounded variation, J. London Math Soc. (2), 2, 346-352. http://dx.doi.org/10.1112/jlms/s2-2.2.346 De Leeuw, K. (1961), Banach spaces of Lipschitz functions, Studia Mathematica, 55-66. Ghorpade, S. R. and Limaye, B. V. (2010) A Course in Multivariable Calculus and Analysis, Springer. http://dx.doi.org/10.1007/978-1-4419-1621-1 Vyas, R. G. and Darji, K. N. (2012) On Banach algebra valued functions of bounded generalized variation on one and several variables, Bulletin of Mathematics Analysis and Applications, V.4, 1, 181-189. Vyas, R. G. (2013) Convolution function of several variables with generalized bounded variation, Analysis Mathematica, V. 39, 2, 153-161. Zygmund, A. (1988) Trigonometric series. Vol. I,II. Reprint of the 1979 edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge.
Mathematical Journal of Interdisciplinary Sciences, Volume 2, Number 1, September 2013