Journal of the Institute of Engineering, 2016, 12(1): 210-213 210 TUTA/IOE/PCU © TUTA/IOE/PCU Printed in Nepal
Some Properties of AP Weight Function
Santosh Ghimire Department of Engineering Science and Humanities, Institute of Engineering Pulchowk Campus, Tribhuvan University, Kathmandu, Nepal Corresponding author: [email protected]
Received: June 20, 2016 Revised: July 25, 2016 Accepted: July 28, 2016
Abstract: In this paper, we briefly discuss the theory of weights and then define A1 and Ap weight functions. Finally we prove some of the properties of AP weight function.
Key words: A1 weight function, Maximal functions, Ap weight function.
1. Introduction
The theory of weights play an important role in various fields such as extrapolation theory, vector-valued inequalities and estimates for certain class of non linear differential equation. Moreover, they are very useful in the study of boundary value problems for Laplace's equation in Lipschitz domains. In 1970, Muckenhoupt characterized positive functions w for which the Hardy-Littlewood maximal operator M maps Lp(Rn, w(x)dx) to itself. Muckenhoupt's characterization actually gave the better understanding of theory of weighted inequalities which then led to the introduction of Ap class and consequently the development of weighted inequalities.
2. Definitions
n Definition: A locally integrable function on R that takes values in the interval (0,∞) almost everywhere is called a weight. So by definition a weight function can be zero or infinity only on a set whose Lebesgue measure is zero.
We use the notation to denote the w-measure of the set E and we reserve the notation Lp(Rn,w) or Lp(w) for the weighted L p spaces. We note that w(E)< ∞ for all sets E = contained in some ball since the weights are locally integrable functions. Definition: The uncentered Hardy-Littlewood maximal operators on Rn over balls B is defined as
1 Similarly the uncentered Hardy-Littlewood = sup Avg maximal|| = sup operators on| R|n over. cubes Q is defined as ∈ ∈ || Ghimire 211
Some Properties of AP Weight Function 1 In each of the definition above, the = suprema sup Avg are|| ta =ken sup over all balls| B| and. cubes Q containing the Santosh Ghimire ∈ ∈ || point x. H-L maximal functions are widely used in Harmonic Analysis. For the details about the Department of Engineering Science and Humanities, Institute of Engineering H-L maximal operators, see [2]. Pulchowk Campus, Tribhuvan University, Kathmandu, Nepal Corresponding author: [email protected] Definition: A function w(x)≥0 is called an A1 weight if there is a constant C1>0 such that
M(w)(x)≤ C1 w(x) Received: June 20, 2016 Revised: July 25, 2016 Accepted: July 28, 2016 where M(w) is uncentered Hardy-Littlewood Maximal function given by
Abstract: In this paper, we briefly discuss the theory of weights and then define 1 A1 and Ap weight functions. Finally we prove some of the properties of AP If w is an A1 weight, then the quantity (which = sup∈ is finite) given . by weight function. ||
Key words: A1 weight function, Maximal functions, Ap weight function. 1 is called the A characteristic[] = constant sup of w. || || || 1 || Definition 1. Introduction : Let 1 < p < ∞ . A weight w is said to be of class A p if is finite where is defined as The theory of weights play an important role in various fields such as extrapolation theory, [] [] vector-valued inequalities and estimates for certain class of non linear differential equation. Moreover, they are very useful in the study of boundary value problems for Laplace's equation in 1 1 We remark that[] in the= above sup definition of A| and A| one can also| use set| of all balls. in Rn instead Lipschitz domains. In 1970, Muckenhoupt characterized positive functions w for which the || 1 p || Hardy-Littlewood maximal operator M maps Lp(Rn, w(x)dx) to itself. Muckenhoupt's of all cubes in Rn. characterization actually gave the better understanding of theory of weighted inequalities which then led to the introduction of Ap class and consequently the development of weighted 3. Properties inequalities. Now we state and prove some of the properties of AP weight functions: -1 ∞ n 2. Definitions Property 1: Let k be a nonnegative measurable function such that k, k are in L (R ). Then if w is an AP for some 1≤ p < ∞, then kw is also an AP weight function. Definition: A locally integrable function on Rn that takes values in the interval (0, ) almost ∞ Proof: For kw to be an A P weight function, we must show (as described in definition) everywhere is called a weight. So by definition a weight function can be zero or infinity only on where is given by; a set whose Lebesgue measure is zero. [] < ∞, [] We use the notation to denote the w-measure of the set E and we reserve the p n p p 1 1 notation L (R ,w) or L (w) for the weighted L spaces. We note that w(E)< ∞ for all sets E -1 ∞ n | | | | = Note that k []and k are= in L sup(R ), so we have < ∞ and < ∞. Using this. together contained in some ball since the weights are locally integrable functions. -1 || || with nonnegativity of k and k , we have Definition: The uncentered Hardy-Littlewood maximal operators on Rn over balls B is defined |||| || || as 1 1 || || || || 1 Similarly the uncentered Hardy-Littlewood = sup Avg maximal|| = sup operators on| R|n over. cubes Q is defined as ∈ ∈ || 1 1 = || || 212 Some Properties Of AP Weight Function
| | | | = . ≤|| |||| || || || . 1 1 ≤ || | | || || .
n Now taking supremum over all cubes Q in R , we get This shows that kw is an A ≤ || | | [] < ∞ P weight. [] < ∞. Property 2: Suppose that w is in Ap for some and 0 < δ < 1. Then belongs to Aq where q = δp+1-δ. Moreover, . ∈ [1. ∞] Proof: We have [ ] ≤ []
1 1 = || || = || || 1 Let us denote Next we claim that f(q) is an increasing function of q. For this we note that by Holder's inequality, =|| , ≥ 0, > 0.
1 1 1 ≤ 1 , ∀ > 1, + = 1. Let q1 > q. Set One| |has = > 1. Taking q-th root both sides, we have ≤ || .
This proves the claim that f(q) is an increasing function of q. Using the claim we have, =|| ≤ || =, ∀ > .
1 1 ≤ , ∀ 0 < || || From (1) and (2), we have, < 1 2 Ghimire 213
= || || . Thus, ≤ || || ≤[] . [ ] = sup ⊂ ≤ [] . ≤|| |||| || || || . Property 3: Show that if the Ap characteristics constants of a weight w are uniformly bounded for all p >1, then w is in A1. 1 1 Proof: As given, are uniformly bounded for all p > 1. Thus there exists a constant M < ≤ || | | ∞ || || . such that for all p >1. For a fixed p >1, by definition we know that, [] n Now taking supremum over all cubes Q in R , we get This shows that kw is an A ≤ || | | [] < ∞ P [] ≤ weight. [] < ∞. 1 1 [] = sup . Property 2: Suppose that w is in Ap for some and 0 < δ < 1. Then belongs to Aq || || So, for all balls B. Specifically, when , the where q = δp+1-δ. Moreover, . ∈ [1. ∞] inequality holds. That is, Proof: We have || || ≤ →1 [ ] ≤ []
1 1 1 1 1 This→lim is true because = | | ≤ . = || || || || ||
= → → → limand notice ||the fact that for = lim || | | = lim | | || || 1 Let us denote Next we claim that f(q) is an increasing Now taking supremum over0<<≤ ∞, all balls B, we get,| | ≤ | | . function of q. For this we note that by Holder's inequality, =|| , ≥ 0, > 0. Hence w is in A by definition. [] 1 = Sup || | | ≤ <∞. There are other properties of A which can be proved using the elementary analysis tools. For 1 1 1 p ≤ 1 , ∀ > 1, + = 1. more properties, we refer [1, 2, 3]. Let q1 > q. Set One| |has = > 1. 4. Conclusion We studied A1 and Ap weight functions and we proved some important properties of these weight function using the elementary analysis tools. Taking q-th root both sides, we have ≤ || .
This proves the claim that f(q) is References an increasing function of q. Using the claim we have, =|| ≤ || =, ∀ > . [1] Ba~nelos R and Moore CN (1991), Probabilistic Behavior of Harmonic Functions, Birkhauser Verlag, USA. [2] Ghimire S (2014), Two Different Ways to Show a Function is an A Weight Function, The 1 1 1 Nepali Mathematical Sciences Report, 33(1): 53-58. ≤ , ∀ 0 < || || [3] Ghimire S (2014), Weighted Inequality, Journal of Institute of Engineering, 10(1): 121-124. From (1) and (2), we have, < 1 2 [4] Grafakos L (2009), Modern Fourier Analysis, Second Edition, Springer.