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On the Space of Bounded Mean Oscillations

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On the Space of Bounded Mean Oscillations Journal of the Institute of Engineering Volume 16, No. 1 Published: April 2021 ISSN: 1810-3383 © TUTA j IOE j PCU Printed in Nepal On the Space of Bounded Mean Oscillations Santosh Ghimire a, Arjan Kumar Sunar b a Dept. of Applied Sciences & Chemical Engineering, Pulchowk Campus, Institute of Engineering, Tribhuvan University, Nepal b Kathford International College of Engineering and Management, Tribhuvan University, Nepal Corresponding Authors: a [email protected], b [email protected] Received: 2021-01-16 Revised: 2021-01-26 Accepted: 2021-02-19 Abstract: The space of bounded mean oscillations, abbreviated BMO, was first introduced by F. John and L. Nirenberg in 1961 in the context of partial differential equations. Later, C. Fefferman proved that the BMO is the dual space of well-known Hardy space, popularly known as H1 space and became the center of attraction for mathematicians. With the help of BMO space, many mathematical phenomenon can be characterized clearly. In this article, we discuss the connections of function of bounded mean oscillations with weight functions, sharp maximal functions and Carleson measure. Keywords: BMO, weight function, maximal function, Carleson measure n 1. Introduction L¥(R ) in harmonic analysis. Every bounded function belongs to BMO, but there exist unbounded functions The Lebesgue spaces, popularly known as Lp spaces, with bounded mean oscillations. Such a functions play an important role in Fourier analysis. However, typically blow up logarithmically as shown by the many important classes of operators are not well John-Nirenberg theorem[1]. The relevance of BMO is behaved on the Lp spaces particularly in L1 and L¥ attested by the fact that the classical singular integral spaces. Therefore, L is too large to be the domain of ¥ n ¥ n 1 operators fail to map L (R ) to L (R ), but instead such operators. Similarly, the target space of many ¥ n they map L (R ) to BMO. Moreover, BMO is the dual canonical operators exceeds L¥. Consequently, L¥ is space of the Hardy space H1. This observation was too small to be the range of such operators. These two announced by C. Fefferman in [1] and then proved in spaces are considered as dual of each other in certain [2]. We can show that every bounded function is in sense. The motivation to find substitutes for the spaces BMO. This gives that L¥ ⊂ BMO. But there exist L¥ and L1 led to the space of bounded mean functions which are in BMO but not in L¥. A famous oscillations and the Hardy space H1. The space of n example for this is f (x) = logjxj which is in BMO(R ) bounded mean oscillations is abbreviated BMO space. ¥ n but not in L (R ). But BMO is not much larger than The functions of bounded mean oscillation was L¥ space. Also BMO scales the same way as L¥: if originally introduced by F. John and L. Nirenberg in the f (x) is in BMO then so is f (lx) for any l > 0 with the context of partial differential equations [1]. This space same BMO norm. The space of BMO is much more naturally arises as the class of function whose deviation needed in the study of various situations such as from their means over cubes is bounded. These spaces boundedness of Calderon-Zygmund operators, real turned out to be the ”right” spaces to study instead of interpolation, Carleson measure, study of paraproducts L1 and L¥ respectively. In fact many operators which etc. to just name a few. BMO also plays a central role 1 are ill-behaved on L1 or L¥ are bounded on H and on in the regularity theory for non linear partial differential BMO. Thus the space of BMO is strictly including the equations. In this article, we discuss the relations of L¥ space and is a good extension of the Lebesgue functions of BMO with weight function, sharp maximal spaces Lp with 1 < p < ¥ from a point of view of functions and Carleson measure. In order to do this, we Harmonic Analysis. Thus the space of bounded mean first state some definitions. We begin with the definition oscillation turns out to be a natural substitute for of functions of bounded mean oscillations. Pages: 163 – 167 On the Space of Bounded Mean Oscillations p n Definition [Functions of bounded mean oscillations] Let f 2 L (R );1 ≤ p < ¥. A real-valued locally integrable function f (x) defined on n and R is said to in BMO, the space of functions of bounded 1 Z Q ⊂ n M f (x) = sup j f (y)j dy mean oscillation, if for any measurable set R , we r>0 jB(x;r)j B(x;r) have: 1 Z Then M f is called the Hardy-Littlewood maximal sup j f (x) − fQj dx < ¥ Q jQj Q function of f . In the definition of maximal function, all the balls are centered at the point x Therefore, this is where the supremum is taken over all cubes Q in n R centered Hardy-Littlewood maximal function. with sides of the cube parallel to coordinate axes, jQj Similarly, we can define uncentered Hardy-Littlewood denotes the measure of Q, and where f is the average Q maximal as: value of f on the cube Q and is defined as: 1 Z 1 Z Mu f (x) = sup j f (y)j dy fQ = f (x) dx r>0;x2B jBj B jQj Q BMO norm a function f, denoted by k f kBMO, is defined Definition [Weight] n as: A locally integrable function on R that takes values in 1 Z the interval (0;¥) almost everywhere is called a weight. k f kBMO = sup j f (x) − fQj dx Q jQj Q So by definition a weight function can be zero or infinity From the definition, we note that if f,g are two only on a set whose Lebesgue measure is zero. We use n the notation w(E) = R w(x) dx to denote the w-measure functions in BMO(R ) and l1;l2 2 R, then one can E of the set E and we reserve the notation Lp( n;w) or easily show that the linear combination l1 f + l2g is R n Lp(w) for the weighted Lp spaces. We note that w(E) < also in BMO(R ). This shows that space BMO is a linear space. Moreover, we also note that ¥ for all sets E contained in some ball since the weights are locally integrable functions. kf + gkBMO ≤ kfkBMO + kgkBMO Definition: A function w(x) > 0 is called an A1 weight and if there is a constant C1 > 0 such that M(w)(x) ≤C1w(x) kl1fkBMO ≤ jl1jkfkBMO where M(w) is uncentered Hardy-Littlewood maximal function. If w is an A weight, then the quantity (which If kfk = 0, then R j f (x) − f j dx = 0. Then f has 1 BMO Q Q is finite) given by: to be equal to its average fQ on every cube, say n QN = [−N;N] We note that the cube QN is contained Z 1 −1 n [w] = sup jw(t)j dt kw k ¥ in Q(N + 1) = [−(N + 1);N + 1] . This forces that A1 L (Q) Q cubes in Rn jQj Q f(QN) = f(Q(N + 1)). Thus we have if kfkBMO = 0, then f is a.e. equal to a constant. Consequently, we have is called the A1 Muckenhoupt characteristic constant the k:kBMO is not a norm. Note that BMO norm of of w or simply A1 characteristic constant of w. We constant is zero. This gives f and f +C have the same first begin with the relation between the functions of BMO norm with C being a constant. In the discussion BMO and weight function. For more about the weight that follows after this we take k:kBMO as a norm even functions, please refer [3]. though it is a semi norm when there is no possibility of confusion. 2. BMO and Weight Function With the introduction of maximal functions one can better understand the concept of averages of functions The theory of weights play an important role in various in analysis. Maximal functions are widely used in fields such as extrapolation theory, vector-valued differentiation theory in analysis. Roughly the maximal inequalities and estimates for certain class of non linear function is defined as the largest value of the averages differential equation. Moreover, they are very useful in of functions over all possible balls which contain a the study of boundary value problem for Laplace’s fixed point. Maximal functions appear in many forms. equation in Lipschitz domains. Some of the theory of The most important of these is the Hardy–Littlewood Muckenhoupt’s Ap weights and weighted norm maximal function. We now define Hardy-Littlewood inequalities results can be used to give characterizations maximal function: of BMO function. One of the very important property 164 Journal of the Institute of Engineering R of BMO functions is the John-Nirenberg Inequality average given by fw;Q = 1=w(Q) Q f (x)w(x) dx. The given by the theorem of John-Nirenberg stated as relation between the weights and BMO is that a follows: function f is in BMO if and only if f is bounded mean oscillation with respect to w for all w 2 A . Theorem [John-Nirenberg Theorem] For all functions f ¥ Symbolically, we write BMO = BMO and the norms in the space of BMO defined on n; for all cubes Q and w R k f k ≈ k f k .
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