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Multiplication and Composition Operators on Weak Lp Spaces

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Multiplication and Composition Operators on Weak Lp Spaces Multiplication and Composition Operators on Weak Lp spaces Ren´eErlin Castillo1, Fabio Andr´esVallejo Narvaez2 and Julio C. Ramos Fern´andez3 Abstract In a self-contained presentation, we discuss the Weak Lp spaces. Invertible and compact multiplication operators on Weak Lp are char- acterized. Boundedness of the composition operator on Weak Lp is also characterized. Keywords: Compact operator, multiplication and composition oper- ator, distribution function, Weak Lp spaces. Contents 1 Introduction 2 2 Weak Lp spaces 2 3 Convergence in measure 7 4 An interpolation result 13 5 Normability of Weak Lp for p > 1 25 6 Multiplication Operators 39 7 Composition Operator 49 A Appendix 52 02010 Mathematics subjects classification primary 47B33, 47B38; secondary 46E30 1 1 Introduction One of the real attraction of Weak Lp space is that the subject is sufficiently concrete and yet the spaces have fine structure of importance for applications. WeakLp spaces are function spaces which are closely related to Lp spaces. We do not know the exact origin of Weak Lp spaces, which is a apparently part of the folklore. The Book by Colin Benett and Robert Sharpley[4] contains a good presentation of Weak Lp but from the point of view of rearrangement function. In the present paper we study the Weak Lp space from the point of view of distribution function. This circumstance motivated us to undertake a preparation of the present paper containing a detailed exposition of these function spaces. In section 6 of the present paper we first prove a charac- terization of the boundedness of Mu in terms of u, and show that the set of multiplication operators on Weak Lp is a maximal abelian subalgebra of B WeakLp , the Banach algebra of all bounded linear operators on WeakLp. For the systemic study of the multiplication operator on different spaces we refereed to ([1], [2], [5] [3], [10], [18], [21]). We use it to characterize the invertibility of Mu on Weak Lp. The compact multiplication operators are also characterized in this section. In section 7 a necessary and sufficient condition for the boundedness of com- position operator CT is given. For the study of composition operator on different function spaces we refereed to ([6], [11], [12], [16], [18], [19]). 2 Weak Lp spaces Definition 2.1. For f a measurable function on X, the distribution function of f is the function Df defined on [0; 1) as follows: Df (λ) := µ fx 2 X : jf(x)j > λg : (1) The distribution function Df provides information about the size of f but not about the behavior of f itself near any given point. For instance, a func- tion on Rn and each of its translates have the same distribution function. It follows from definition 2.1 that Df is a decreasing function of λ (not neces- sarily strictly). Let (X; µ) be a measurable space and f and g be a measurable functions on (X; µ) then Df enjoy the following properties: For all λ1, λ2 > 0: 2 1. jgj ≤ jfj µ-a.e. implies that Dg ≤ Df ; λ 2. D (λ ) = D 1 for all c 2 =f0g; cf 1 f jcj C 3. Df+g(λ1 + λ2) ≤ Df (λ1) + Dg(λ2); 4. Dfg(λ1λ2) ≤ Df (λ1) + Dg(λ2). For more details on distribution function see ([7] and [15]). Next, Let (X; µ) be a measurable space, for 0 < p < 1, we consider C p Weak L := f : µ(fx 2 X : jf(x)j > λg) ≤ ; p λ for some C > 0. Observe that Weak L1 = L1. Weak Lp as a space of functions is denoted by L(p;1). Proposition 2.1. Let f 2 Weak Lp with 0 < p < 1. Then C p kfk = inf C > 0 : D (λ) ≤ L(p;1) f λ 1=p p = sup λ Df (λ) λ>0 1=p = sup λ fDf (λ)g : λ>0 Proof. Let us define C p λ = inf C > 0 : D (α) ≤ ; f α and 1=p p B = sup α Df (α) : α>0 Since f 2 Weak Lp, then C p D (α) ≤ ; f α for some C > 0, then 3 C p C > 0 : D (α) ≤ 8 α > 0 6= ;: f α On the other hand p p α Df (α) ≤ B ; p p thus fα Df (α): α > 0g is bounded above by B and so B 2 R. Therefore C p λ = inf C > 0 : D (α) ≤ α > 0 ≤ B: (2) f α Now, let > 0, then there exists C such that λ ≤ C < λ + , and thus Cp (λ + )p D (λ) ≤ < ; f λp λp then p p sup λ Df (λ) < (λ + ) λ>0 1=p p sup λ Df (λ) ≤ λ λ>0 B < λ, (3) by (2) and (3) B = λ. Definition 2.2. For 0 < p < 1 the space L(p;1) is defined as the set of all µ−measurable functions f such that C p kfk = inf C > 0 : D (λ) ≤ 8 λ > 0 L(p;1) f λ 1=p p = sup λ Df (λ) λ>0 1=p = sup λ fDf (λ)g ; λ>0 is finite. Two functions in L(p;1) will be considered equal if they are equal µ−a.e. 4 The Weak Lp = L(p;1) are larger than the Lp spaces, we have the following. Proposition 2.2. For any 0 < p < 1 and any f 2 Lp we have Lp ⊂ L(p;1); and hence f ≤ f : L(p;1) Lp (This is just a restatement of the Chebyshev inequality). Proof. If f 2 Lp, then Z Z λpµ fx 2 X : jf(x)j > λg ≤ jfjp du ≤ jfjp du = f p ; Lp fjfj>λg X therefore kfk p µ fx 2 X : jf(x)j > λg ≤ Lp : (4) λ Hence f 2 Weak Lp = L(p;1), which means that Lp ⊂ L(p;1): (5) Next, from (4) we have 1=p p sup fλ Df (λ)g ≤ f L λ>0 p f ≤ f : L(p;1) Lp Remark 2.1. The inclusion (5) is strict, indeed, let f(x) = x−1=p on (0; 1) (with the Lebesgue measure). Note 1 1 m x 2 (0; 1): > λ = m x 2 (0; 1): jxj < = 2λ−p: jxj1=p λp Thus f 2 Weak Lp(0; 1), but Z 1 1 p Z 1 dx 1=p dx = ! 1; 0 x 0 x then f2 = Lp(0; 1). 5 Proposition 2.3. Let f; g 2 L(p;1). Then 1. cf = jcj f for any constant c, L(p;1) L(p;1) p p 1=p 2. f + g ≤ 2 f + g . L(p;1) L(p;1) L(p;1) Proof. (1) For c > 0 we have λ µ fx 2 X : jcf(x)j > λg = µ x 2 X : jf(x)j > ; c thus λ D (λ) = D : cf f c And thus 1=p p kcfkL(p;1) = sup λ Dcf (λ) λ>0 1=p p λ = sup λ Df λ>0 c 1=p 1=p p p p = sup c w Df (w) = c sup w Df (w) ; cw>0 cw>0 then kcfkL(p;1) = ckfkL(p;1) : (2) Note that n o λ [ λ x 2 X : jf(x)+g(x)j > λ ⊆ x 2 X : jf(x)j > x 2 X : jg(x)j > : 2 2 Hence µ fx 2 X : jf(x) + g(x)j > λg λ λ ≤ µ x 2 X : jf(x)j > + µ x 2 X : jg(x)j > ; 2 2 then λ λ λpD (λ) ≤λpD + λpD f+g f 2 g 2 p p p p λ Df+g(λ) ≤2 sup λ Df (λ) + sup λ Dg(λ) ; λ>0 λ>0 6 therefore 1=p 1=p p p p sup λ Df+g(λ) ≤2 f L + g L λ>0 (p;1) (p;1) p p 1=p f + g ≤2 f + g : L(p;1) L(p;1) L(p;1) Remark 2.2. Proposition 2.3 (2) tell us that k:kL(p;1) define a quasi-norm on L(p;1). Definition 2.3. A quasi-norm is a functional that is like a norm except that it does only satisfy the triangle inequality with a constant C ≥ 1, that is kf + gk ≤ C kfk + kgk : 3 Convergence in measure Next, we discus some convergence notions. The following notion is of impor- tance in probability theory. Definition 3.1. Let f; fn (n = 1; 2; 3; :::) be a measurable functions on the measurable space (X; µ). The sequence ffngn2N is said to converge in measure µ to f (fn ! f) if for all > 0 there exists an n0 2 N such that µ fx 2 X : jfn(x) − f(x)j > g < for all n ≥ n0: (6) Remark 3.1. The preceding definition is equivalent to the following state- ment. For all > 0, lim µ fx 2 X : jfn(x) − f(x)j > g = 0: (7) n!1 Cleary (7) implies (6). To see the convergence given > 0, pick 0 < δ < and apply (6) for this δ. There exists an n0 2 N such that µ fx 2 X : jfn(x) − f(x)j > δg < δ; 7 holds for n ≥ n0. Since µ fx 2 X : jfn(x) − f(x)j > g ≤ µ fx 2 X : jfn(x) − f(x)j > δg : We concluded that µ fx 2 X : jfn(x) − f(x)j > g < δ; for all n ≥ n0. Let n ! 1 to deduce that lim sup µ fx 2 X : jfn(x) − f(x)j > g ≤ δ: (8) n!1 Since (8) holds for all 0 < δ < (7) follows by letting δ ! 0.
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