Journal of the Institute of Engineering Volume 16, No. 1 Published: April 2021 ISSN: 1810-3383 © TUTA | IOE | 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 , 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 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 . 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 spaces. Therefore, L is too large to be the domain of ∞ n ∞ n 1 operators fail to 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) = log|x| 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 (λx) for any λ > 0 with the context of partial differential equations [1]. This space same BMO . 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 ∈ 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 | f (y)| dy mean oscillation, if for any measurable set R , we r>0 |B(x,r)| B(x,r) have: 1 Z Then M f is called the Hardy-Littlewood maximal sup | f (x) − fQ| dx < ∞ Q |Q| 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, |Q| 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 | f (y)| dy fQ = f (x) dx r>0,x∈B |B| B |Q| 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 (0,∞) almost everywhere is called a weight. k f kBMO = sup | f (x) − fQ| dx Q |Q| 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 is zero. We use n the notation w(E) = R w(x) dx to denote the w-measure functions in BMO(R ) and λ1,λ2 ∈ R, then one can E of the set E and we reserve the notation Lp( n,w) or easily show that the linear combination λ1 f + λ2g 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) kλ1fkBMO ≤ |λ1|kfkBMO where M(w) is uncentered Hardy-Littlewood maximal function. If w is an A weight, then the quantity (which If kfk = 0, then R | f (x) − f | 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 |w(t)| dt kw k ∞ in Q(N + 1) = [−(N + 1),N + 1] . This forces that A1 L (Q) Q cubes in Rn |Q| 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

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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 ∈ 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 . This relationship was proved by α > 0, we have BMOw BMO Muckenhoupt and Wheeden. Please refer [4] for the −Aαk f kBMO |{x : | f (x) − AvgQ f | > α}| ≤ e|Q|e detailed proof of these relations. with A = (2ne)−1. 3. BMO and Sharp Function Operator For the proof the theorem, please refer, page 124 of [3]. An immediate consequence of the John-Nirenberg In this section, we discuss the relation of BMO and sharp inequality is the p-invariance property for any 1 < p < function operator. We first begin with the definition of ∞, sharp function operator.

1  Z  p Definition: 1 p k f kBMO ≈ sup | f (x) − fQ| dx Let f be a locally integrable function defined on n.A |Q| Q R Q sharp function operator, also known as Fefferman-Stein Moreover, using the John-Nirenberg inequality, for any sharp maximal operator is denoted by f # or M# f and is f ∈ BMO, the function e| f (x)|/ρ is locally integrable for defined as: some appropriate constant ρ > 0. Now we note the Z # # 1 following result which can be easily proved: Let V(t) f (x) = M f (x) = sup | f (y) − fQ| dy n x∈Q,Q⊂ n |Q| Q be a real valued locally integrable function on R and R V(t) let 1 < p < ∞. Then the function e ∈ A if and only n n p where ∈ R . In the definition, Q is a cube in R with if two conditions are satisfied for some constant c < ∞. center at x and sides parallel to the coordinate axes and Z 1 V(t)−V fQ denotes the average of f over Q given by a. sup e Q dt ≤ C R Q |Q| Q fQ = 1/|Q| Q f (x) dx. The difference of the sharp maximal operator compared with the Hardy-Littlewood 1 1 Z p−1 b. sup e−[V(t)−VQ] dt ≤ C Maximal function is that instead of the integral Q |Q| Q averages we maximize the mean oscillation over cubes.

Using the above relation, one can establish a deep From the definition of sharp function operator, one can connection between the weight functions and BMO show that a function f is a function of bounded mean # ∞ n functions. Precisely, logarithm of any A function is a oscillation, f ∈ BMO if and only if f ∈ L (R ). In 2 # BMO function and every BMO function is equal to a other words, we have k f kBMO = k f kL∞ . Moreover, # we note that k f kL∞ = 0 if f ≡ constant. Due to this constant multiple of the logarithm of an A2 weight relation ,we see that an element in the space of BMO is function. Moreover, we show that logarithm of any Ap weight function for 1 < p < ∞ is a BMO function. For an equivalence class and two function f and g are equal more details, please refer [4]. In addition to this in BMO if and only if f − g = constant. Under this n connection, there is another relation between weight equality, we see that the space BMO(R ) is a normed functions and functions in BMO. For this, we recall the space. Moreover, we can replace the average fQ by any weighted BMO space. Similar to weighted Lebesgue other constant, say CQ which depends on Q. With this n spaces, one can define weighted BMO space, denoted we have, a function f is in BMO(R ) if and only if for n any cube Q ⊂ , there exists a constant CQ depending by BMOw, as the collections of w locally integrable R functions f such that on Q such that : 1 Z 1 Z sup | f (x) −CQ(x)| dx < ∞ k f kBMOw = sup | f (x) − fw,Q|w(x) dx < ∞ n x∈Q,Q⊂ n |Q| Q Q⊂R w(Q) Q R R n In the above definition, we have w(Q) = Q w(x) dx is Also we have “ f ∈ BMO(R ) if and only if R n the w-measure of Q and the fw,Q is the weighted Q | f (x) − fQ(x)| dx < ∞” implies that “ f ∈ BMO(R )

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R p if and only if Q | f (x)− fQ(x)| dx < ∞ for 1 ≤ p < ∞”. i.e., We have , p n p n ∞ n n T : L (R ) → L (R ) and T : L (R ) → BMO(R ) 1 Z 1 Z | f (x) − fQ(x)| dx ≤ | f (x)| dx + fQ q n q n |Q| Q |Q| Q continuously. Then T : L (R ) → L (R ) continuously 1 Z for all p < q < ∞. ≤ 2 | f (x)| dx |Q| Q For this and more about the BMO and sharp maximal function, please refer [7]. n for every cube Q ⊂ R . Then by the definition of the Hardy-Littlewood maximal function f #(x) ≤ 2M f (x) n for every x ∈ R . Moreover, by the maximal function 4. BMO and Carleson Measure theorem, for every p with 1 < p ≤ ∞, there exists To connect the space of BMO with Carleson measure, constant C, depending only on n and p such that: we first recall the definition of cone and non-tangential # k f kp = 2kM f kp ≤ Ck f kp. maximal functions. n Definition: Let x ∈ R . Then a cone over x is given This shows that the sharp maximal operator is a n+1 n+1 p n by Γ(x) = (y,t) ∈ : |x − y| < t. Let F : → C. bounded operator on L (R ) whenever 1 < p ≤ ∞. R+ R+ With the help of constant functions, we see that there is Then the non-tangential maximal function associated to no point wise inequality to the reverse direction f is defined as: # M f (x) ≤ c f (x). However, using good lambda ∗ p M F(x) = sup |F(y,t)| ∈ [0,∞] inequalities, we can compare the L norms of the sharp (y,t)∈Γ(x) maximal function and the Hardy Littlewood maximal n function under certain assumptions. Even though sharp . For a cube ⊂ R , the set given by maximal operator and Hardy-Littlewood maximal T(Q) = Q × (0,l(Q)] is called the tent over the cube Q. function are not comparable point wise, they are While solving the famous corona problem, Swedish comparable on Lp level. This is given by the following mathematician L. Carleson [8] obtained the theorem due C. Fefferman and E.M. Stein[5]. characterization of all non-negative measures µ defined on 2 satisfying: p n R+ Theorem: Let 1 < p < ∞. For every f ∈ L (R ), there Z exists a constant Cp independent of f , depending only 2 2 |Pt ∗ f (x)| dµ(x,t) ≤ Ck f kL2( ) on n and p, such that: Q R

−1 # p p 2 Cp k f kL ≤ k f kLp(Rn) ≤ Cpk f kL for all f ∈ L (R). Here Pt ∗ f is the Poisson integral of f and C is the universal constant. p n p n and thus f ∈ L (R ) if and only if f # ∈ L (R ). Now we are state the definition of Carleson measure. Using the above theorem, we can show that if Let us take a particular case of characteristic function f ∈ BMO ∩ L1 f ∈ Lp < p < , then we have , for 1 ∞. given by f = χ2Q. It is easy to see that (Pt ∗ f )(x) ≥ C1 p n C There exist many operators T are bounded from L (R ) for some positive constant 1. Using this in the above to itself for 1 < p < ∞. But this fails for the case p = integral inequality, for all (x,t) ∈ T(Q) we have:

1,∞. For counter examples please refer Stein-Weiss [6]. 0 Z 0 1 2 The substitute results for the case 1 and ∞ are T : L → C1µ(T(Q)) ≤ |Pt ∗ f (x)| dµ(x,t) ≤ C |Q| Q L1,∞ and T : L∞,∞ → L∞. The substitute result for the action of T on the space L∞ is given by T : L∞ → BMO, This gives µ(T(Q)) ≤ C|Q|. This motivates the since BMO plays the similar role as that of L∞. Finally, following defintion of Carleson measure which was we state a theorem which interpolates Lp and BMO introduced by Carleson in 1960. Please refer [8] for the and is the extension of the Marcinkiewicz interpolation details. The Carleson measure is closely related with theorem for the endpoint (∞,∞). For the proof, please the space of BMO. refer [5]. Definition: A Carleson measure is a positive measure n+1 Theorem: Let 1 < p < ∞ and T be a linear operator, µ on R+ such that there exists a constant C < ∞ for p n continuous from L into itself and from L∞ into BMO which µ(T(Q)) ≤ C|Q| for all cubes Q in R . Here C

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n is a universal constant and the smallest of such constant h ∈ BMO(R ) and the norm of h satisfies: satisfying the inequality µ(T(Q)) ≤ C|Q| is called the norm of the Carleson measure and it is given by ∞ khkBMO ≤ C ∑ khikL∞ µ(T(Q)) i=1 kµk = sup Q |Q| Finally the readers are suggested to refer [6] for more about the relation between BMO and Carleson measure In the above definition, we can use ball B(x,r) instead and various characterization of the space of functions of of cube Q and we can easily show that the these two bounded mean oscillations. definitions are equivalent. We now state the relation between the space of BMO and Carleson measure. This relation was established by C. Fefferman. We state the References theorem: Theorem: The function g ∈ BMO( n) if and only if [1] F. John and L. Nirenberg, “On functions of bounded R mean oscillation,” Comm. Pure Appl. Math, vol.14, pp. the associated measure defined on n+1 by R+ 415–426, 1961. 2 dµg(x,t) = y|∇u| (x,t) dx dt [2] C. Feffereman, “Characterization of bounded mean where u(x,t) = Pt ∗ g(x) is the Poisson integral of g, is a oscillations,” Bulletin of American Mathematical Carleson measure. Society, vol.77, 1971.

There is a decomposition theorem given by Carleson in [3] L. Grafakos, “Modern Fourier Analysis, Graduate texts 1976. Please see [5] for details. We revisit the theorem. in mathematics”, 3rd ed. New York, 2014. Let ϕ ∈ C1( n) be a radial function which satisfies: R [4] J. Gracia-Cuerva and J.-L. Rubio de Francia, “Weighted |ϕ(x)| + |∇ϕ(x)| Z norm inequalities and related topics”, North-Holland ≤ C and ϕ(x) dx = 1 Mathematics Studies, 116, Notes de Matematica, vol. ( + |x|)n+1 n 1 R 104, 1985.

If h(x) is a compactly supported BMO function, then [5] C. Fefferman and E.M. Stein, “Hp spaces of several there exists a sequence of functions, say variables”, Acta. Math, vol. 129, pp. 137–588, 1972. ∞ {hi} such that ∑i=1 khikL∞ ≤ CkhkBMO with C is a [6] E. M. Stein and G. Weiss, ‘Fourier Analysis on Euclidean constant depending only on n. In addition to this, there Spaces. Princeton University Press, New Jersey, 1971. exists a sequence {βi} depending only on x such that: [7] D. Chang and C. Sadosky,“Functions of Bounded Mean ∞ Z Oscillation,” Taiwanese Journal of Mathematics, vol. 10, h(x) = h1(x) + ϕβi(x − y)hi(y) dy +C1 no. 3, pp 573–601, 2006. ∑ n i=2 R [8] L. Carleson, “Interpolation of bounded functions and where C1 is a constant. Conversely, if a function h has the corona problem”, Ann. Mathematics, vol. 76, pp. the above decomposition, then the function 547-559, 1962.

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