Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 22, 1071 - 1078
Weighted Composition Operators on Sobolev - Lorentz Spaces S. C. Arora Department of Mathematics University of Delhi, Delhi - 110007, India
[email protected] Satish Verma SGTB Khalsa College University of Delhi, Delhi - 110007, India
[email protected] Abstract For Ω an open subset of the Euclidean space Rn , T : Ω → Ω a measurable non-singular transformation and u a real-valued measurable function on Rn , we study boundedness of the weighted composition operator uCT : f → u · (f ◦ T ) on the Sobolev-Lorentz space W 1,n,q (Ω), consisting of those functions of the Lorentz space L(n, q), whose distributional derivatives of the first order belong to L(n, q), 1 ≤ q ≤ n.
Mathematics Subject Classification: Primary 47B33, 47B38; Secondary 46E30, 46E35 Keywords: Lorentz space, Sobolev-Lorentz space, distribution function, non-increasing rearrangement, measurable non-singular transformation, composition operator, weighted composition operator
1
Introduction
Suppose (Ω, A, μ) is a measure space where Ω is an open subset of the Euclidean space Rn , A the σ-algebra of Lebesgue measurable subsets of Ω and μ the Lebesgue measure. Let f be a complex-valued Lebesgue measurable function defined on Ω. For s ≥ 0, define μf the distribution function of f as μf (s) = μ{x ∈ Ω : |f (x)| > s}.
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By f ∗ we mean the non-increasing rearrangement of f given as f ∗ (t) = inf{s > 0 : μf (s) ≤ t},
t ≥ 0.
For 1 ≤ q ≤ n, the Lorentz norm of f is given by f n,q =
∞ 0
dt f (t)) t
1/n ∗
(t
q
1/q
.
The Lorentz space L(n, q) is the set of equivalence classes of complex-valued Lebesgue measurable functions f on Ω with f n,q < ∞. L(n, q) is a Banach space [11] with respect to above norm. The Sobolev-Lorentz space W 1,n,q (Ω) is defined as the set of all complexvalued functions f in L(n, q) whose weak partial derivatives ∂f /∂xi (in the distributional sense) belong to L(n, q), i = 1, 2, . . . , n. It is a Banach space [25] with respect to the norm: f 1,n,q =
n ∂f f n,q + i=1
∂xi
n,q
.
On σ-finite measure space (Ω, A, μ), let T : Ω → Ω be a measurable (T (E) ∈ A for every E ∈ A) non-singular transformation (μ(T −1 (E))=0, whenever μ(E) = 0). Let the function fT = d(μ ◦ T −1 )/dμ be the RadonNikodym derivative. Suppose u is a complex-valued measurable function defined on Rn . Then T induces a well-defined linear transformation uCT on W 1,n,q (Ω) defined by −1
(uCT f )(x) = u(x)f (T (x)), x ∈ Ω, f ∈ W 1,n,q (Ω). If uCT maps W 1,n,q (Ω) into itself and is bounded, then we call uCT a weighted composition operator on W 1,n,q (Ω) induced by T with weight u. If u ≡ 1, then CT is called a composition operator induced by T . Our study here, of weighted composition operators, on Sobolev-Lorentz space W 1,n,q (Ω) is motivated by the work of Herbert Kamowitz and Dennis Wortman[12]. Other similar references include [2–7, and 15–17]. The paper contains two sections. In the first section we define the composition operator on W 1,n,q (Ω), and the second section is devoted to the study of weighted composition operator on W 1,n,q (Ω).
2
Composition operator on W 1,n,q (Ω) ∂Tk ∂xi
∞
≤ M, (M > 0) i, k = ∂xi ∞ k 1, 2, . . . , n, where T = (T1 , T2 , . . . , Tn ) and ∂T denotes the first order par∂xi tial derivative(in the classical sense). Then for each f in W 1,n,q (Ω) we have Lemma 2.1 Let fT ,
∈ L (μ) with
∂Tk
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Weighted composition operators
f ◦ T ∈ L(n, q), 1 ≤ q ≤ n, and the first order distributional derivatives of (f ◦ T ), given by, n ∂f ∂Tk ∂ (f ◦ T ) = ◦T (1) ∂xi ∂xi k=1 ∂xk for i = 1, 2, . . . , n, are in L(n, q). Proof. The Radon-Nikodym derivative fT = d(μ ◦ T −1 )/dμ ∈ L∞ (μ) implies that for each E ∈ A, (μ ◦ T −1 )(E) =
E
fT dμ ≤ fT ∞ μ(E).
For f in W 1,n,q (Ω), the distribution of f ◦ T satisfies μf ◦T (s) = μ{x ∈ Ω : |f (T (x))| > s} = (μ ◦ T −1 ){x ∈ Ω : |f (x)| > s} ≤ fT ∞ μ{x ∈ Ω : |f (x)| > s} = fT ∞ μf (s). Therefore {s > 0 : μf (s) ≤ t} ⊆ {s > 0 : μf ◦T (s) ≤ fT ∞ t}. This gives (f ◦ T )∗ (fT ∞ t) ≤ f ∗ (t), i.e., (f ◦ T )∗ (t) ≤ f ∗ (t/fT ∞ ). Now f ∈ L(n, q), 1 ≤ q ≤ n, gives f ◦
T qn,q
= ≤
∞ 0
∞ 0
(t1/n (f ◦ T )∗ (t))q (t1/n f ∗ (t/b))q
dt t
dt q = fT q/n ∞ f n,q . t
Thus f ◦ T n,q ≤ fT 1/n ∞ f n,q , and hence f ◦ T ∈ L(n, q). By the same arguments, as each weak partial derivative ∂f /∂xk ∈ L(n, q), it follows that ∂f /∂xk ◦ T ∈ L(n, q), for each k = 1, 2, . . . , n. Also ∂Tk /∂xi ∈ L∞ (μ), therefore
∂f ∂Tk ◦T ∈ L(n, q), for each i, k = 1, 2, . . . , n. ∂xk ∂xi
Hence, using triangle inequality, it follows that the function in right hand side of (1) belongs to L(n, q), for each i = 1, 2, . . . , n. Since f ∈ W 1,n,q (Ω), 1 ≤ q ≤ n, following the same computation as in Friedrich’s Theorem [14, Theorem 2.2.1, p. 57], there exists a sequence < fm >
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S. C. Arora and Satish Verma
∂f m in D(Rn ) = C0∞ (Rn ) such that fm → f in L(n, q)(Ω) and ∂f → ∂x in ∂xi i L(n, q)(Ω ) for every 1 ≤ i ≤ n and for every relatively compact set Ω in Ω Let g ∈ D(Rn ). We choose relatively compact set Ω in Ω with supp(g) ⊂ Ω . Then by the ordinary chain rule for smooth function fm , we have for each i = 1, 2, . . . , n
∂g ∂g (fm ◦ T ) dμ = (fm ◦ T ) dμ ∂xi ∂xi Ω Ω ∂ (fm ◦ T )gdμ = − Ω ∂xi n ∂fm ∂Tk ◦T gdμ. = − ∂xi Ω k=1 ∂xk
(2)
∂g ∈ X , where X is the associate ∂xi space of the Banach function space X = L(n, q)(Ω), 1 ≤ q ≤ n. Therefore by using the H¨ older’s inequality in Banach function spaces, we have
Now g ∈ D(Rn ) implies that for each i, g,
Ω
∂g | fm ◦ T − f ◦ T | dμ ≤ (fm − f ) ◦ T X ∂xi
∂g ∂xi X ∂g 1/n → 0. ≤ fT ∞ fm − f X ∂xi X
Therefore as m → ∞, for each i = 1, 2, . . . , n
∂g (fm ◦ T ) dμ → ∂xi Ω
By the similar arguments, using we obtain that in L(n, q)(Ω ), n ∂fm k=1
∂xk
◦T
∂fm ∂xk
Ω
→
(f ◦ T )
∂f ∂xk
∂g dμ. ∂xi
in L(n, q)(Ω ) and
∂Tk ∂xi
∈ L∞ (μ),
n ∂Tk ∂f ∂Tk → ◦T ∂xi ∂xi k=1 ∂xk
So by the H¨ older’s inequality in Banach function spaces again, we obtain as m → ∞, for each i = 1, 2, . . . , n
n ∂fm
Ω k=1
∂xk
◦T
∂Tk gdμ → ∂xi
n
Ω k=1
∂f ∂Tk ◦T gdμ. ∂xk ∂xi
Hence by taking limits on both the sides of (2) as m → ∞, we obtain
∂g (f ◦ T ) dμ = − ∂xi Ω = −
n
Ω k=1 n Ω k=1
∂f ∂Tk ◦T gdμ ∂xk ∂xi
∂f ∂Tk ◦T gdμ. ∂xk ∂xi
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Weighted composition operators
Therefore for all i = 1, 2, . . . , n −
Ω
n ∂ ∂f ∂Tk (f ◦ T )gdμ = − ◦T gdμ. ∂xi ∂xi Ω k=1 ∂xk
As g was chosen arbitrarily, the equation (1) follows. Theorem 2.2 Let Ω ⊂ Rn be an open set and T : Ω → Ω a measurable nonsingular transformation with Radon-Nikodym derivative fT = d(μ ◦ T −1 )/dμ, ∂Tk ∂Tk ≤ M, M > 0, for i, k = 1, 2, . . . , n, where T = in L∞ (μ), and ∂xi ∂xi ∞ ∂Tk denotes the first order partial derivatives (in the clas(T1 , T2 , . . . , Tn ) and ∂xi sical sense). Then the mapping CT defined by CT (f ) = f ◦ T is a composition operator on the Sobolev-Lorentz space W 1,n,q (Ω), 1 ≤ q ≤ n. Proof. By the Lemma 2.1, we have f ◦ T ∈ W 1,n,q (Ω) and its norm satisfies the following: f ◦ T 1,n,q = f ◦ T n,q + = f ≤
n i=1
∂ (f ◦ T ) ∂xi n,q
n n ◦ T n,q +
fT 1/n ∞
i=1 k=1 n n
f n,q +
≤
fT 1/n ∞
≤
fT 1/n ∞ (1
f n,q +
∂f ∂Tk ◦T ∂xk ∂xi n,q
i=1 k=1
fT 1/n ∞
fT 1/n ∞ Mn
n
k=1
∂f ∂Tk ∂xk n,q ∂xi ∞
∂f ∂xk n,q
+ nM)f 1,n,q
The result follows.
3
Weighted composition operator on W 1,n,q (Ω)
Suppose u is a real-valued measurable function defined on Rn . Also suppose that T : Ω → Ω is a measurable non-singular transformation and (Ω, A, μ) is the σ-finite measure space, where Ω an open subset of Rn . On the same lines as in Lemma 2.1, we have the following, for i = 1, 2, . . . , n.
n ∂ ∂f ∂Tk ∂u (u · (f ◦ T )) = (f ◦ T ) + u ◦T . ∂xi ∂xi ∂xi k=1 ∂xk
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Theorem 3.1 If all the conditions stated in the Theorem 2.2 are satisfied and, in addition, u ∈ L∞ (μ) such that the first order classical partial derivatives ∂u/∂xi satisfy ∂u/∂xi ∞ ≤ M1 , M1 > 0, for i = 1, 2, . . . , n, then the mapping uCT defined by (uCT )f = u·(f ◦T ) is a weighted composition operator on the Sobolev-Lorentz space W 1,n,q (Ω), 1 ≤ q ≤ n. Proof. By the same arguments as in Lemma 2.1, we find u · (f ◦ T )n,q ≤ u∞ fT 1/n ∞ f n,q ,
∂u (f
∂xi
and
n u k=1
◦ T )
n,q
≤ M1 fT 1/n ∞ f n,q ,
n ∂f ∂Tk ∂f 1/n ◦T ≤ u Mf ∞ T . ∞ ∂xk ∂xi n,q k=1 ∂xk n,q
Hence it follows that (uCT )f 1,n,q = u · (f ◦ T )1,n,q = u · (f ≤
n ◦ T )n,q +
n ∂u 1/n u∞ fT ∞ f n,q + (f i=1
≤ u∞ fT ∞ f 1,n,q +
∂xi
i=1
◦ T )
n,q
∂ (u · (f ◦ T )) ∂xi n,q
n n + u
nM1 fT 1/n ∞ f 1,n,q
i=1
k=1
∂f ∂Tk ◦T ∂xk ∂xi n,q
+ nMu∞ fT 1/n ∞ f 1,n,q
Thus (uCT )f 1,n,q ≤ Kf 1,n,q , for some K > 0.
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