JUHA KINNUNEN. Harmonic Analysis

Transcription

1 JUHA KINNUNEN Harmonic Analysis Department of Mathematics and Systems Analysis, Aalto University 27

2 Contents Calderón-Zygmund decomposition. Dyadic subcubes of a cube Dyadic cubes of the whole space Calderón-Zygmund decomposition of a function Dyadic maximal function Marcinkiewicz interpolation theorem 9 3 Bounded mean oscillation Basic properties of BMO Completeness of BMO The John-Nirenberg inequality The sharp maximal function BMO and interpolation Muckenhoupt weights The A p condition Properties of A p weights A weak type characterization of A p The Gehring lemma Reverse Hölder inequalities and A p A strong type characterization of A p A p and BMO Characterizations of A p Characterizations of BMO

3 Calderón-Zygmund decomposition Dyadic cubes and the Calderón-Zygmund decomposition are very useful tools in harmonic analysis. The property of dyadic cubes, that either one is contained in the other or the interiors of the cubes are disjoint, is very useful in constructing coverings with pairwise disjoint cubes. The Calderón -Zygmund decomposition gives decompositions of sets and functions into good and bad parts, which can be considered separately using real variable and harmonic analysis techniques.. Dyadic subcubes of a cube A closed cube is a bounded interval in R n, whose sides are parallel to the coordinate axes and equally long, that is, = [a, b ] [a n, b n ] with b a =... = b n a n. The side length of a cube is denoted by l(). In case we want to specify the center, we write (x, l) = {y R n : y i x i l2 }, i =,..., n for a cube with center at x R n and side length l >. If = (x, l), we denote α = (x,αl) for α >. Thus α the cube with the same center as, but the side length multiplied by factor α. The integral average of f L loc (Rn ) in a cube is denoted by f = f (x) dx = f (x) dx. Let = [a, b ]... [a n, b n ] be a closed cube in R n with side length l. We decompose into subcubes recursively. Denote D = {}. Bisect each interval

4 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION 2 [a i, b i ], i =,2,..., and obtain 2 n congruent subcubes of. Denote this collection of cubes by D. Bisect every cube in D and obtain 2 n subcubes. Denote this collection of cubes by D 2. By continuing this way, we obtain generations of dyadic cubes D k, k =,,2,... The dyadic subcubes in D k are of the form [ a + m l 2 k, a + (m ] [ + )l 2 k a n + m nl 2 k, a n + (m ] n + )l 2 k, where k =,,2,... and m j =,,...,2 k, j =,..., n. The collection of all dyadic subcubes of is D = D k. k= A cube D is called a dyadic subcube of. Figure.: Collections of dyadic subcubes. Remark.. Dyadic subcubes of have the following properties: () Every D is a subcube of. (2) Cubes in D k cover and the interiors of the cubes in D k are pairwise disjoint for every k =,,2,... (3) If, D, either one is contained in the other or the interiors of the cubes are disjoint. This is called the nesting property, see Figure.2. (4) If D k and j < k, there is exactly one parent cube in D j, which contains.

5 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION 3 (5) Every cube D k is a union of exactly 2 n children cubes D k+ with = 2 n. (6) If D k, then l( ) = 2 k l() and = 2 nk. Figure.2: Nestedness property. Assume that f L loc (Rn ). By the Lebesgue differentiation theorem f (y) f (x) d y = for almost every x R n. (.) lim r B(x,r) A point x R n, at which (.) holds, is called a Lebesgue point of f. For every Lebesgue point x we have since B(x,r) f (y) d y f (x) lim f (y) d y = f (x), r B(x,r) B(x,r) f (y) f (x) d y as r. Moreover, every Lebesgue point x of f is a Lebesgue point of f, since f (y) f (x) d y f (y) f (x) d y as r. B(x,r) B(x,r) We shall need the following version of the Lebesgue differentiation theorem. Lemma.2. Assume that x R n is a Lebesgue point of f L loc (Rn ). Then lim f (y) d y = f (x) i i i

6 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION 4 whenever, 2, 3,... is any sequence of cubes containing x such that lim i i =. T H E M O R A L : The Lebesgue differentiation theorem does not only hold for balls but also for cubes and dyadic cubes. Proof. Let i = (x i, l i ), where x i R n is the center and l i = l( i ) is the side lenght of the cube i for every i =,2,... We observe that (x i, l i ) B(x, nl i ) for every i =,2,... Figure.3: (x i, l i ) B(x, nl i ) for every i =,2,... This implies f (y) d y f (x) f (y) f (x) d y (x i,l i ) (x i,l i ) B(x, nl i ) (x i, l i ) B(x, f (y) f (x) d y nl i ) = B(,) n n 2 f (y) f (x) d y as i, B(x, nl i ) since l i as i. The following Calderón-Zygmund decomposition will be extremely useful in harmonic analysis. Theorem.3 (Calderón-Zygmund decomposition of a cube (952)). Assume that f L loc (Rn ) and let be a cube in R n. Then for every t f (y) d y

7 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION 5 there are countably or finitely many dyadic subcubes i, i =,2,..., of such that () the interiors of i, i =,2,..., are pairwise disjoint, (2) t < f (y) d y 2 n t for every i =,2,... and i (3) f (x) t for almost every x \ i= i. The collection of cubes i, i =,2,..., is called the Calderón-Zygmund cubes in at level t. T H E M O R A L : A cube can be divided into good and bad parts so that in the good part (complement of the Calderón-Zygmund cubes) the function is small and in the bad part (union of the Calderón-Zygmund cubes) the integral average of a function is in control. Note that the Calderón-Zygmund cubes cover the set {x : f (x) > t}, up to a set of measure zero, and thus the bad part contains the set where the function is unbounded. Proof. The strategy of the proof is the following stopping time argument. For every x such that f (x) > t we choose the largest dyadic cube D containing x such that f (y) d y > t. Then we use the fact that for any collection of dyadic subcubes of there is a subcollection of dyadic cubes with disjoint interiors and with the same union as the original cubes. These are the desired Calderón-Zygmund cubes. Then we give a rigorous argument. Consider (possible empty) collection of dyadic subcubes D of, that satisfy f (y) d y > t. (.2) The cubes in are not necessarily pairwise disjoint, but we construct a new collection of cubes so that for every we consider all cubes such that. Since f (y) d y t, for every cube there exists a maximal cube. Let = { i } i be the collection of these maximal cubes with respect to inclusion. We show that this collection has the desired properties. () This follows immediately from maximality of the cubes in and the nestedness property of the dyadic subcubes. Indeed, if the interiors of two different cubes in intersect then one is contained in the other, and hence one of them cannot be maximal, see Figure.4. (2) By (.2), we note that. If i D k for some k, then by properties (4) and (5) of the dyadic subcubes we conclude that i is contained in some cube D k with = 2 n i, see Figure.5. Since i maximal, cube does not

8 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION 6 Figure.4: Collection of maximal subcubes. satisfy (.2). Thus t < i f (y) d y i i f (y) d y 2 n t. (3) Assume that x \ i= i. By the beginning of the proof, t f (y) d y for every dyadic subcube D containing point x. Thus there exist k D k such that x k for every k =,2,... Note that and i= = {x}, see i the Figure.6. If x is a Lebesgue point of f, Lemma.2 implies f (x) = lim k k k f (y) d y t..2 Dyadic cubes of the whole space Next we consider the dyadic cubes in R n and a global version of the Calderón- Zygmund decomposition. A half open dyadic interval in R is an interval of the form [m2 k,(m + )2 k ),

9 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION 7 Figure.5: i is contained in some cube D k with = 2 n i. where m, k Z. The advantage of considering half open intervals is that they are pairwise disjoint. A dyadic interval of R n is a cartesian product of one-dimensional dyadic intervals n [m j 2 k, (m j + )2 k ), j= where m,..., m n, k Z. The collection of dyadic cubes D k, k Z, consists of the dyadic cubes with the side length 2 k. The collection of all dyadic cubes in R n is D = k ZD k. Observe that D k consist of cubes whose vertices lie on the lattice 2 k Z n and whose side length is 2 k. The dyadic cubes in the kth generation can be defined as D k = 2 k ([,) n + Z n ). The cubes in D k cover the whole R n and are pairwise disjoint, see Figure.7. Moreover, these dyadic cubes have the same properties (2)-(5) in Remark. as the dyadic subcubes of a given cube. W A R N I N G : It is not true that every cube is a subcube of a dyadic cube. For example, consider [,] n. However, there is a substitute for this property: For every cube there is a dyadic cube D such that 5 (exercise). Remarks.4: () For any subcollection D of dyadic cubes whose union is a bounded set, there is a subcollection of pairwise disjoint maximal cubes with the

10 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION 8 Figure.6: k D k such that x for every k =,2,... k same union. A cube is called maximal, if there does not exist any strictly larger with, see Figure.8. A useful property is that the collection maximal cubes are always pairwise disjoint. This follows at once from nestedness property of the dyadic cubes. Indeed, if two different cubes in satisfy, then one is contained in the other, and hence one of them cannot be maximal. (2) Every nonempty open set can be represented as a union of countably many pairwise disjoint dyadic cubes. Let us consider the collection D of all dyadic cubes contained in the given open set. These cubes are not pairwise disjoint, but we may consider cubes in D that are maximal in the sense of inclusion, which means that they are not contained any of the other cubes in D. This collection of cubes satisfies the required properties (exercise). In the one-dimensional case every open set is a union of countably many disjoint open intervals. The Lebesgue measure of an open set is the sum of volumes of these intervals. (3) The Whitney decomposition of a nonempty proper open subset Ω of R n states that it can be represented as a union of countably many pairwise disjoint dyadic intervals whose side lengths are comparable to their distance to the boundary of the open set. More precisely, there are pairwise disjoint dyadic cubes i, i =,2,..., such that Ω = i= i,

11 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION 9 Figure.7: Dyadic cubes in R n. nl( i ) dist( i,r n \ Ω) 4 nl( i ), if the boundaries of i and j touch, then 4 l( i) l( j ) 4, for every i there exist at most 2 n cubes in the collection that touch it. (4) Assume that f L loc (Rn ). Then E k f (x) = D k ( ) f (y) d y χ (x) is the conditional expectation of f with respect to the increasing collection of σ-algebra generated by D k, k Z. Note that E k (x) dx = f (x) dx R n R n for every k Z and E k can be considered as a discrete analog of an approximation of the identity. Theorem.5 (Global Calderón-Zygmund decomposition (952)). Assume that f L (R n ). Then for every t > there are countably or finitely many dyadic cubes i, i =,2,..., in R n such that

12 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION Figure.8: A collection of maximal cubes. () cubes i, i =,2,..., are pairwise disjoint, (2) t < f (y) d y 2 n t for every i =,2,... and i (3) f (x) t for almost every x R n \ i= i. The collection of cubes i, i =,2,..., is called the Calderón-Zygmund cubes in R n at level t. T H E M O R A L : The difference to the Calderón-Zygmund in a cube is that we assume global integrability instead of local inregrability. With this assumption, we obtain the Calderón-Zygmund decomposition at every level t >. Note that, if the function is bounded, the Calderón-Zygmund decomposition may be empty for some values of t >. Proof. As in the proof of Theorem.3, consider the collection of maximal dyadic cubes D in R n that satisfy (.2). Note that (.2) gives l( ) n = < f (y) d y f (y) d y t t R n for every cube D that satisfy (.2). Thus the maximal cube i exists as in the proof of Theorem.3. Otherwise, the proof is similar. Example.6. Consider the Calderón-Zygmund decomposition for f : R R, f (x) = χ [,] (x) at level t >. We may assume that < t <, since f (x) dx for

13 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION every interval R. In other words, if t, there are no intervals in R for which f (x) dx > t. For < t <, choose k {,, 2,...} such that 2 k t < 2 k. We claim that the Calderón-Zygmund decomposition at level t consists only of one interval [, 2 k ). To see this, we observe that m() 2 k f (x) dx = 2 k χ [,] (x) dx = 2 k > t. On the other hand, if,, where D, then = [, 2 k+l ), l {,2,...}, and thus m( ) f (x) dx = 2 k l < 2 k t so that is the maximal cube with the property f (x) dx > t..3 Calderón-Zygmund decomposition of a function For a function f L (R n ), and any level t >, we have the decomposition f = f χ { f t} + f χ { f >t} (.3) into good part g = f χ { f t}, which is bounded, and the bad part b = f χ { f >t}. These parts can be analyzed separately using real variable techniques. For the good part we have the bounds g f and g t and for the bad part b f and {x R n : b(x) } {x R n : f (x) > t} t f. The last bound follows from Chebyshev s inequality and tells that the measure of the support of the bad part is small. This truncation method is will be useful in later in connection with interpolation, see Lemma 2., but here we consider a more refined way to decompose an arbitrary integrable function into its good and large bad parts so that not only the absolute value but also the local oscillation is in control. Theorem.7 (Calderón-Zygmund decomposition of a function (952)). Assume that f L (R n ) and let t >. Then there are functions g and b, and countably or finitely many pairwise disjoint dyadic cubes i, i =,2,..., such that () f = g + b,

14 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION 2 (2) g f, (3) g 2 n t, (4) b = b i, where b i = in R n \ i, i =,2,..., i= (5) b i (x) dx =, i =,2,..., i (6) b i (x) dx 2 n+ t and i (7) i t f. i= T H E M O R A L : Any function f L (R n ) can be represented as a sum of a good and a bad function f = g + b, where g is bounded and b = i= b i, where b i, i =, 2,..., are highly oscillating localized function with zero integral averages. Note that the bad function b contains the unbounded part of function f. Remarks.8: () It follows from () and (2) that b f g f + g 2 f (2) and thus b L (R n ).This shows that the bad function g is integrable. ( g p = g(x) p dx R n ( p g ) p = ( g p R n g(x) dx (2n t) p f (2n t) p g(x) p g(x) dx R n ) p g p p g p ) p and thus g L p (R n ) whenever p. This shows that the good function g is essentially bounded and belongs to all L p -spaces. Proof. Let i, i =,2,..., be the Calderón-Zygmund cubes for f at level t >, see Theorem.5. Define f (x), x R n \ i, g(x) = i= f (y) d y, x i, i =,2,... i and b i (x) = (f (x) f i )χ i (x), i =,2,...

15 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION 3 T H E M O R A L : The function g is defined so that it is equal to f outside the Calderón-Zygmund cubes and in a Calderón-Zygmund cube the it is the average of the function in that cube. () g(x) = f (x) (f (x) f i )χ i (x) = f (x) b i (x) = f (x) b(x). i= i= (2) g(x) dx = R n = = = = R n \ i= i R n \ i= i R n \ i= i R n \ i= i R n \ i= i R n \ i= i g(x) dx + g(x) dx i= i f (x) dx + g(x) dx i= i f (x) dx + f i dx i= i f (x) dx + f i dx i= i f (x) dx + f (x) dx i i= i f (x) dx + f (x) dx = f (x) dx. R n i= i (3) By Theorem.5, we have f (x) t for almost every x R n \ i= i and f (y) d y f (y) d y 2 i n t, i =,2,... i This implies that g(x) 2 n t for almost every x R n. (4) See the proof of (). (5) b i (x) dx = (f (x) f i )χ i (x) dx i i = f (x) dx f i =, i =,2,... i (6) By Theorem.5, we have b i (x) dx ( f (x) + f i ) dx i i 2 f (x) dx 2 n+ t i, i =,2,... i (7) i = i i= i= f (y) d y = t i= i t f.

16 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION 4.4 Dyadic maximal function There is an interpretation of the Calderón-Zygmund decomposition in terms of maximal functions. The dyadic maximal function of f L loc (Rn ) is M d f (x) = sup f (y) d y, (.4) where the supremum is taken over all dyadic cubes containing x. By Lemma.2, for almost every x R n, we have f (x) = lim f (y) d y M d f (x), k k where k D k. Thus the dyadic maximal function is bigger than the absolute value of the function almost everywhere. This explains the name maximal function. W A R N I N G : The dyadic maximal function is not comparable to the standard Hardy-Littlewood maximal function M f (x) = sup f (y) d y, (.5) where the supremum is taken over all cubes in R n containing x. It is clear that M d f (x) M f (x) for every x R n, but the inequality in the reverse direction does not hold. For example, consider f : R n R,, x n, f (x) =, x n <. Then M d f (x) = f (x) for every x R n, but the standard Hardy-Littlewood maximal function is strictly positive everywhere. Lemma.9. Assume that f L (R n ) and let t > such that the set E t = {x R n : M d f (x) > t} has finite measure. Then E t is the union of pairwise disjoint dyadic Calderón- Zygmund cubes i, i =,2,..., given by Theorem.5. In particular, cubes i, i =,2,..., satisfy properties ()-(3) in Theorem.5. Proof. We show that E t = i= i. If x E t, then M d f (x) > t and thus there exists a dyadic cube D such that x and f (y) d y > t.

17 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION 5 The Calderón-Zygmund cubes i, i =,2,..., given by Theorem.5 is a collection of maximal dyadic cubes with this property. This implies that x i= i and thus E t i= i. On the other hand, if x i= i, then x i for some i =,2,... and by the Calderón-Zygmund decomposition M d f (x) f (y) d y > t. i This shows that x E t and thus i= i E t. This completes the proof. Figure.9: The distribution set of the dyadic maximal function. T H E M O R A L : The union of the Calderón-Zygmund cubes is the distribution set of the dyadic maximal function. This means that the Calderón-Zygmund decomposition is more closely related to f = f χ {Md f t} + f χ {Md f >t} instead of f = f χ { f t} + f χ { f >t} in (.3). Note carefully, that this is not the Calderón-Zygmund decomposition of a function constructed in the proof of Theorem.5, but Lemma.9 shows that {M d f > t} is the union of the Calderón-Zygmund cubes. This suggest another point of view to the Calderón-Zygmund decomposition, in which we analyse the distribution set of the dyadic maximal function, for example, using the Whitney covering theorem. Remarks.: () By summing up over all Calderón-Zygmund cubes, we have E t = i i= f (y) d y = f (y) d y. t i= i t E t

18 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION 6 This is a weak type estimate for the dyadic maximal function. Observe that in contrast with the standard Hardy-Littlewood maximal function, there is no dimensional constant in the estimate. This estimate holds true also for other measures than Lebesgue measure. (2) We also have an inequality to the reverse direction, since f (y) d y = f (y) d y 2 n t i = 2 n t E t. (.6) E t i i= This is a reverse weak type inequality for the dyadic maximal function. (3) If t > s, then E t E s and, by maximality of the Calderón-Zygmund cubes, each cube in the decomposition at level t is contained in a cube in the decomposition at level s. In this sense, the Calderón-Zygmund decompostions are nested. (4) Observe that the Calderón-Zygmund decomposition in Lemma.9 can be done under the assumption that {x R n : M d f (x) > t} <. By the weak type estimate in (), this is weaker than assuming f L (R n ). Next we show how we can use the Calderón-Zygmund decomposition to obtain estimates for the standard maximal function defined by (.5). Lemma.. Assume that f L (R n ) and let i, i =,2,..., be the Calderón- Zygmund cubes of f at level t > given by Theorem.5. Then () i {x R n : M f (x) > t} and i= (2) {x R n : M f (x) > 4 n t} 3 i. i= T H E M O R A L : The first claim is essentially a restatement of the fact that M d f (x) M f (x) for every x R n and thus i= {x R n : M d f (x) > t} {x R n : M f (x) > t}. The seconds claim implies the following inequality in the reverse direction {x R n : M f (x) > 4 n t} 3 i = 3 n i = 3 n {x R n : M d f (x) > t} (.7) i= for every t >. In particular, this gives the weak type estimate for the standard Hardy-Littlewood maximal function as well, since { {x R n : M f (x) > t} 3 n x R n : M d f (x) > t } 4 n 3n 4 n f (y) d y t R n for every t >. Thus the weak type estimate for the standard Hardy-Littlewood maximal function follows from the corresponding estimate for the dyadic maximal function. This shows that information on dyadic cubes can be used to obtain i= information over all cubes, see also Example 2.6 (2).

19 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION 7 Proof. () If x i= i, then x i for some i =,2,... By Theorem.5 M f (x) f (y) d y > t i and thus i= i {x R n : M f (x) > t}. (2) Assume that x R n \ i= 3 i and let any closed cube in R n containing x. Choose k Z such that 2 k < l() 2 k. Then there exists at most 2 n such dyadic cubes R,..., R m D k, which intersect the interior of. We note that Figure.: At most 2 n dyadic cubes R,..., R m D k intersect the interior of. 3R j for every j =,..., m. Each cube R j, j =,..., m, cannot be a subset of any of the cubes i, i =,2,..., since otherwise x 3 i for some i =,2,..., which is not possible, since x R n \ i= 3 i. Since R j is not contained in the union of the Calderón-Zygmund cubes, by the proof of Theorem.5 and Theorem.3, we have R j R j f (y) d y t, i =,..., m. On the other hand, R j = 2 kn = 2 n 2 kn n 2 n l() n 2 n and m 2 n, thus f (y) d y = m f (y) d y j= R j m R j f (y) d y m2 n t 4 n t. R j R j j=

20 CHAPTER. CALDERÓN-ZYGMUND DECOMPOSITION 8 Since this holds true for every cube containing x, we have M f (x) 4 n t for every x R n \ i= 3 i. In other words, R n \ 3 i {x R n : M f (x) 4 n t}, i= from which the claim follows.

21 2 Marcinkiewicz interpolation theorem Interpolation of operator is an important tool in harmonic analysis. Consider an operator, which maps Lebesgue measurable functions to functions. A typical example is the maximal operator. The rough idea of interpolation is that if we know that the operator is a bounded in two different function spaces, then it is bounded in the intermediate function spaces. We are mainly interested in L p (R n ) spaces with p and we begin with a useful decomposition of an L p (R n ) function into two parts. To this end, we define L p (R n ) + L p 2 (R n ), p < p 2, to be the space of all functions of the form f = f + f 2, where f L p (R n ) and f 2 L p 2 (R n ). Lemma 2.. Let p < p 2 and p p p 2. Then L p (R n ) L p (R n ) + L p 2 (R n ). T H E M O R A L : Every L p (R n ) function can be written as a sum of an L p (R n ) function and an L p 2 (R n ) function whenever p p p 2. Proof. If p = p or p = p 2, there is nothing to prove, since f = f +. Thus we assume that p < p < p 2. Assume that f L p (R n ) and let t >. Define f (x), if f (x) > t, f (x) = f (x)χ { f >t} (x) =, if f (x) t, and Clearly f (x), if f (x) t, f 2 (x) = f (x)χ { f t} (x) =, if f (x) > t. f (x) = f (x)χ { f >t} (x) + f (x)χ { f t} (x) = f (x) + f 2 (x), 9

22 CHAPTER 2. MARCINKIEWICZ INTERPOLATION THEOREM 2 see (.3). First we show that f L p (R n ). Since p < p, we obtain f (x) p dx = f (x) p dx = f (x) p p f (x) p dx R n { f >t} { f >t} t p p f (x) p dx t p p f p R n p <. Then we show that f 2 L p 2 (R n ). Since p 2 > p, we have f 2 (x) p 2 dx = f (x) p 2 dx = f (x) p2 p f (x) p dx R n { f t} { f t} t p 2 p f (x) p dx t p2 p f p R n p <. Thus f = f + f 2 with f L p (R n ) and f 2 L p 2 (R n ), as required. Definition 2.2. Let T be an operator from L p (R n ) to Lebesgue measurable functions on R n. () T is sublinear, if for every f, g L p (R n ), T(f + g)(x) T f (x) + T g(x) and T(af )(x) = a T f (x), a R, for almost every x R n. (2) T is of strong type (p, p), p, if there exists a constant c, independent of the function f, such that T f p c f p for every f L p (R n ). (3) T is of weak type (p, p), p <, if there exists a constant c, independent of the function f, such that ( c {x R n : T f (x) > t} t f p for every t > and f L p (R n ). T H E M O R A L : Operator is of strong type (p, p) if and only if it is a bounded operator from L p (R n ) to L p (R n ). The corresponding weak type condition is a substitute for this for several operators in harmonic analysis which fail to be bounded in certain L p (R n ) spaces. For example, the Hardy-Littlewood maximal operator is a sublinear operator which is not of strong type (,) but it is of weak type (,). ) p

23 CHAPTER 2. MARCINKIEWICZ INTERPOLATION THEOREM 2 Remarks 2.3: () Every linear operator T is sublinear, since T(f + g)(x) = T f (x) + T g(x) T f (x) + T g(x) and T(af )(x) = at f (x) = a T f (x). (2) The notion of strong type (p, p) is stronger than weak type (p, p). If T f p c f p for every f L p (R n ), by Chebyshev s inequality {x R n : T f (x) > t} t p R n T f (x) p dx t p T f p p ( c t f p) p. Theorem 2.4 (Marcinkiewicz interpolation theorem (939)). Let p < p 2 and assume that T is a sublinear operator from L p (R n ) + L p 2 (R n ) to Lebesgue measurable functions on R n, which is simultaneously of weak type (p, p ) and (p 2, p 2 ). Then T is of strong type (p, p) whenever p < p < p 2. T H E M O R A L : Weak type estimates at the endpoint spaces imply strong type estimates spaces between. Proof. p 2 < Assume that if there exist constant c and c 2, independent of the function f, such that for every f L p (R n ) and {x R n : T f (x) > t} {x R n : T f (x) > t} for every f L p 2 (R n ). Consider the decomposition ( c t f p ) p, t >, ( c2 t f p 2 ) p2, t >. f = f + f 2 = f χ { f >t} + f χ { f t}, where f L p (R n ) and f 2 L p 2 (R n ), given by Lemma 2.. Sublinearity T f (x) T f (x) + T f 2 (x) implies that for almost every x for which T f (x) > t, either T f (x) > t 2 or T f 2(x) > t 2. Thus { {x R n : T f (x) > t } x R n : T f (x) > t } { x R n : T f 2 (x) > t } 2 2 { x R n : T f (x) > t } { + 2 x R n : T f 2 (x) > t } 2 ( c t 2 ( 2c t + f p ) p + ) p ( 2c2 t ( c2 ) p2 f t 2 p2 2 f (x) p dx {x R n : f (x) >t} ) p2 f (x) p 2 dx. {x R n : f (x) t}

24 CHAPTER 2. MARCINKIEWICZ INTERPOLATION THEOREM 22 By Cavalieri s principle T f (x) p dx = p t p {x R n : T f (x) > t} dx R n (2c ) p p t p p f (x) p dx dt + (2c 2 ) p 2 p {x R n : f (x) >t} t p p 2 {x R n : f (x) t} f (x) p 2 dx dt, where the integrals on the right-hand side are computed by Fubini s theorem as t p p {x R n : f (x) >t} f (x) f (x) p dx dt = f (x) p t p p dt dx R n = f (x) p p f (x) p dx p p R n = f (x) p dx p p R n and t p p 2 {x R n : f (x) t} f (x) p 2 dx dt = f (x) p 2 t p p2 dt dx R n f (x) = f (x) p 2 f (x) p p 2 dx p 2 p R n = f (x) p dx. p 2 p R n Thus we arrive at T f p p = T f (x) p dx R n (2c ) p p f (x) p dx + (2c 2 ) p p 2 f (x) p dx p p R n p 2 p R ( n (2c ) p = p + (2c ) 2) p2 f p p. p p p 2 p p 2 = Assume that T f c 2 f for every f L (R n ) and write f = f + f 2 = f χ { f > t 2c 2 } + f χ { f t 2c 2 }. Then f L p (R n ) as in Lemma 2. and f 2 L (R n ), since f 2 strong (, ) estimate for f 2 and obtain T f 2 (x) T f 2 c 2 f 2 c 2 t 2c 2 = t 2 for almost every x R n and, consequently, { x R n : T f 2 (x) > t } =. 2 t 2c 2. We apply

25 CHAPTER 2. MARCINKIEWICZ INTERPOLATION THEOREM 23 Thus { {x R n : T f (x) > t} x R n : T f (x) > t } { + 2 x R n : T f 2 (x) > t } 2 { x R n : T f (x) > t } 2 ( c t 2 ( 2c t f p ) p ) p By Cavalieri s principle T f p p = T f (x) p dx R n Remarks 2.5: f (x) p dx {x R n : f (x) > t 2c } 2 = p t p {x R n : T f (x) > t} dx (2c ) p p t p p = (2c ) p p f (x) p R n = (2c ) p p p p = (2c ) p c p p 2 {x R n : f (x) > t 2c } 2 2c2 f (x) t p p dt dx 2c 2 f (x) p p f (x) p dx R n p f (x) p dx p p R n f (x) p dx dt = p (2c ) p c p p 2 f p p. p p () In particular, if T is a linear operator which is strong type (p, p ) and (p 2, p 2 ), then it is strong type (p, p) whenever p p p 2. For linear operators there are better Riesz-Thorin type interpolation results, see [] and [4]. (2) If T is a linear operator instead of sublinear, then it is enough to assume weak type (p, p ) and (p 2, p 2 ) estimates for simple functions. (3) Note that the constant in strong type (p, p) estimate blows up as p p and p p 2, when p 2 <. (4) By considering f = f + f 2 = f χ { f >γt} + f χ { f γt}, where γ > is chosen appropriately, we obtain strong type bound T f p c f p, p < p < p 2, with and ( p c = 2 + p p p p 2 p ( c = 2 p p p ) p p 2 p p c p 2 c ) p c p p c p p p p p p 2 2, if p 2 <, 2, if p 2 =.

26 CHAPTER 2. MARCINKIEWICZ INTERPOLATION THEOREM 24 (5) The proof is based only on Cavalieri s principle, from which we conclude that the Marcinkewicz interpolation theorem holds for other measures than the Lebesgue measure as well. In this case we assume that the weak type estimates hold with respect to the given measure. Moreover, the Marcinkewicz interpolation theorem holds in more general spaces than the Euclidean spaces. (6) There are more general versions of the Marcinkiewicz interpolation theorem in Lorenz spaces, see [] and [4]. (7) There is a general theory of interpolation of operators in Banach spaces Examples 2.6: and more general topological spaces. The Marcinkiewicz interpolation theorem has lead to real method of interpolation and Thorin s method has lead to complex method of interpolation, see [] and references therein. () The dyadic maximal function defined in (.4) is of strong type (, ), since for every x R n, we have M d f (x) = sup f (y) d y f, where the supremum is taken over all dyadic cubes containing x. Thus M d f f for every f L (R n ). On the other hand, by Remark. () we have {x R n : M d f (x) > t} t R n f (y) d y, t >, for every f L (R n ). Since the dyadic maximal function is sublinear, by the Marcinkiewicz interpolation theorem we conclude that it is of strong type (p, p) for every < p < and M d f p p2p p f p, < p <. When p =, the bound holds with constant one. (2) Note that by Lemma., we obtain similar bounds for the standard Hardy-Littlewood maximal function defined by (.5) as well. If {x R n : M d f (x) > t} < for every t >, then by (.7), we have {x R n : M f (x) > 4 n t} 3 n {x R n : M d f (x) > t}

27 CHAPTER 2. MARCINKIEWICZ INTERPOLATION THEOREM 25 for every t > and M f p p = p = 4 n p 4 n 3 n p t p {x R n : M f (x) > t} dt = 2 n M d f p p. In particular, by () it follows that t p {x R n : M f (x) > 4 n t} dt t p {x R n : M d f (x) > t} dt M f p c f p whenever < p. This gives a dyadic proof for the Hardy-Littlewood-Wiener maximal function theorem. The Marcinkiewicz interpolation theorem shows that if a sublinear operator satisfies weak type conditions at the end points of exponents, then it is satisfies the strong type condition for all exponents in between. The next result describes the difference between the weak type and strong type conditions. This result will be used in the proof of Theorem 5. later. Theorem 2.7 (Kolmogorov). Let p <. operator from L p (R n ) to Lebesgue measurable functions on R n. T H E Assume that T is a sublinear () If T is of weak type (p, p), then for all < q < p and A R n with < A <, there exists c < such that T f (x) q dx c A q p f q p. (2.) A (2) If there exists < q < p and constant c such that (2.) holds for every A R n with < A <, then T is of weak type (p, p). M O R A L : This shows that the weak type condition of a sublinear operator at a given exponent is essentially equivalent to the strong type condition for all strictly smaller exponents. Proof. () Since T is of weak type (p, p), we have {x A : T f (x) > t} {x R n : T f (x) > t} c t p f p p, t >. Thus for every < q < p, we obtain T f (x) q dx = q t q {x A : T f (x) > t} dt A = q ( t q min A, c c f p A /p c A q p f q p. t p f p p ) dt t q A dt + cq c f p A /p t q p f p p dt

28 CHAPTER 2. MARCINKIEWICZ INTERPOLATION THEOREM 26 (2) Let t > and A = {x R n : T f (x) > t}. Then A is a measurable set and A <. If A =, there are sets A k A with A k = k for k =,2,..., such that t q k = t q A k T f (x) q dx c A k q p f q p = ck q p f q p. A k This is impossible. Thus by (2.), we obtain t q A T f (x) q dx c A q p f q p. A It follows that Thus T is of weak type (p, p). {x R n : T f (x) > t} = A c t p f p p.

29 3 Bounded mean oscillation The space of functions of bounded mean oscillation (BMO) turns out to be a natural substitute for L (R n ) in harmonic analysis. It consists of functions, whose mean oscillation over cubes is uniformly bounded. Every bounded function belongs to BMO, but there exist unbounded functions with bounded mean oscillation. Such a functions typically blow up logarithmically as shown by the John-Nirenberg theorem. The relevance of BMO is attested by the fact that classical singular integral operators fail to map L (R n ) to L (R n ), but instead they map L (R n ) to BMO. Moreover, BMO is the dual space of the Hardy space H. BMO also plays a central role in the regularity theory for nonlinear partial differential equations. 3. Basic properties of BMO The mean oscillation of a function f L loc (Rn ) in a cube R n is f (x) f dx = f (x) f dx. The mean oscillation tells that how much function differs in average from its integral average in. Definition 3.. Assume that f L loc (Rn ) and let f = sup f (x) f dx, where the supremum is taken over all cubes in R n. If f <, we say that f has bounded mean oscillation. The number f is called the BMO norm of f. The class of functions of bounded mean oscillation is denoted as BMO = { f L loc (Rn ) : f < }. 27

30 CHAPTER 3. BOUNDED MEAN OSCILLATION 28 T H E M O R A L : A function f L loc (Rn ) belongs to BMO, if there exists constant M <, independent of the cube R n, such that inequality f (x) f dx M holds for every cube R n. The BMO norm is the smallest constant M for which this is true. If f, g BMO, then f + g = sup sup f (x) + g(x) f g dx f (x) f dx + sup g(x) g dx = f + g <. Thus f + g BMO and f + g f + g. Moreover, if a R, then af = sup af (x) af dx = a sup f (x) f dx = a f <. Thus af BMO and af = a f. This shows that BMO is a vector space. However, the BMO norm is not a norm, it is only a seminorm. The reason is that f = does not imply that f =. In fact, we have f = f = c for almost every x R n, where c R is a constant. To see this, for every constant function f = c, we have f = sup f (x) f dx = sup c c dx =. Conversely, if f =, then for every cube R n, we have f (x) f dx =. This implies f (x) f = and thus f (x) = f for almost every x. This implies that f (x) = c for almost every x R n. To see this, observe that f (x) = f (,) for almost every x (,). Moreover, f (x) = f (,k) for almost every x (, k), k =,2,... Thus f (,k) = f (,), k =,2,..., and f (x) = f (,) for almost every x (, k), k =,2,... Since R n = k= (, k), this implies f (x) = f (,) = c for almost every x R n. T H E M O R A L : The previous discussion shows that functions f and f + c, c R, have the same BMO norm. We may overcame this by identifying all BMO functions, whose difference is constant by considering the equivalence relation f g f g = c, c R.

31 CHAPTER 3. BOUNDED MEAN OSCILLATION 29 The space of corresponding equivalence classes f is a normed space with norm f = f. Instead of considering the equivalence classes, we identify functions whose difference is constant almost everywhere. In this sense a function in BMO is defined only up to an additive constant. Next we study basic properties of BMO functions. The following lemma will be a useful tool in showing that certain functions belong to BMO. Lemma 3.2. Assume that for every cube R n, there is constant c, which may depend on, such that f (x) c dx M, where M < is a constant that does not depend on. f 2M. Moreover, 2 f sup inf c R T H E f (x) c dx f. Then f BMO and M O R A L : The integral average in the mean oscillation can be replaced with any other number. Proof. For every cube, we have f (x) f dx f c dx + f c dx M + f c = M + (f (x) c ) dx M + f (x) c dx 2M. First by taking infimum over c R for a fixed cube and then taking supremum over cubes, we obtain f 2sup inf c R On the other hand, it is clear that inf f (x) c dx c R f (x) c dx. f (x) f dx. This shows that sup inf c R f (x) c dx f. Remark 3.3. For a cube the constants c, for which inf c R f (x) c dx is attained, satisfy {x : f (x) > c } 2 and {x : f (x) < c } 2.

32 CHAPTER 3. BOUNDED MEAN OSCILLATION 3 Even if it is not unique, we call such a c the median of f in. On the other hand, the constant c for which inf c R f (x) c 2 dx is attained is f (exercise). T H E M O R A L : The median minimizes the L mean oscillation and the integral mean value minimizes the L 2 mean oscillation. Remark 3.4. We use Lemma 3.2 to show that f BMO implies f BMO. Indeed, by the triangle inequality f (x) f dx f (x) f dx f for every cube in R n. Lemma 3.2 with c = f implies f BMO and f 2 f. Examples 3.5: () We note that L (R n ) BMO and f 2 f. This follows, since for every cube. T H E f (x) f dx ( f (x) + f ) dx f (x) dx + f 2 M O R A L : Every bounded function is in BMO. f (x) dx + f f (x) dx 2 f (2) Let f : R n [, ], f (x) = log x. We show that f BMO. By Lemma 3.2 it is enough to show that for every cube (x, l), with x R n ja l >, there exists constant c (x,l) R so that We consider two cases. sup log y c (x,l) d y <. (x,l) (x,l) Case : First assume that x < nl. In this case, we choose c (x,l) = log l and by change of variables y = lz, d y = l n dz, we obtain l n log y c (x,l) d y = log(l z ) log l dz (x,l) ( x l,) = log z dz ( x l,) Claim: log z dz ( x l,) Reason. B(,2 n) log z dz < whenever x < nl. z x n l + 2 < n n n for every z ( x l,). Thus ( x l,) B(,2 n).

33 CHAPTER 3. BOUNDED MEAN OSCILLATION 3 Case 2: Assume then that x nl. In this case, we choose c (x,l) = log x, do the same change of variables as above, and obtain l n log y c (x,l) d y = log l z log x dz (x,l) ( x l,) = l z log ( x l,) x dz Claim: l z log x dz log2 < whenever x nl. ( x l,) Reason. We note that z x l n for every z ( x l,) and z x n l + 2 for every z ( x l,). Thus 2 l z = x l n x 2 2 = 2 l z = x + l n x = 3 2 log 2 log l z x log 3 2 for every z ( x l,). This implies l z log x log 2 = log2 for every z ( x l,). We conclude that in both cases the mean oscillation is uniformly bounded and thus log x BMO. T H E M O R A L : log x BMO, but log x L (R n ). This is an example of an unbounded BMO function. Thus inclusion L (R n ) is a proper subset of BMO. (3) Let f : R R, Since f is an odd function, we have log x, x, f (x) = log x, x >. 2a a a f (x) dx = for every interval [ a, a] R. For < a < we have f 2a a ε + ε a f (x) dx = log x dx a a = a/ a lim (x xlog x) = (a alog a) a = log a, as a +.

34 CHAPTER 3. BOUNDED MEAN OSCILLATION 32 Thus f BMO, even though by (2) and Remark 3.4, we have f (x) = log x BMO. T H E M O R A L : f BMO does not imply that f BMO. Since f = f sgn f with f BMO and sgn f L (R n ) BMO, this also shows that product of two functions in BMO does not necessarily belong to BMO. (4) Consider dyadic BMO, where the supremum of the mean oscillation is taken only over dyadic cubes. It is clear that BMO is a subset of dyadic BMO, but the converse inclusion is not true. For example, the function in (3) belongs to dyadic BMO, but it does not belong to BMO (exercise). T H E (5) Let g : R R, M O R A L : BMO is a proper subset of dyadic BMO. log x, x >, g(x) =, x. Let f be as in (3). Since log x BMO by (2) and f BMO by (3), the difference g(x) = 2 (log x f (x)) = χ (, ) log x BMO. T H E M O R A L : f BMO does not imply that f χ A BMO, A R n. Next we consider two examples of the connection of BMO to partial differential equations. Examples 3.6: () We show that W,n (R n ) BMO, where W,n (R n ) is the Sobolev space of functions u L n (R n ) with Du L n (R n ). Assume that u W,n (R n ). The Poincaré inequality implies that (x,l) ( ) u(y) u (x,l) d y u(y) u (x,l) n n d y (x,l) ( ) cl Du(y) n n d y c Du n < (x,l) for every cube with constant c depending only on n. Thus if u W,n (R n ), then u BMO and u c Du n, where c depends only on n. (2) This remark is related to the course Nonlinear Partial Differential Equations. Consider weak solutions of the partial differential equation div(a(x, Du)) = n i, j= D j (a i j (x)d i u) = in R n (3.) with bounded measurable coeffients a i j L (R n ), i, j =,..., n, satisfying ellipticity assumption λ ξ 2 n a i j (x)ξ i ξ j Λ ξ 2, < λ Λ, i, j=

35 CHAPTER 3. BOUNDED MEAN OSCILLATION 33 for almost every x R n and all ξ R n. If u W,2 loc (Rn ), u >, is a weak solution to (3.), there exists c, depending only on n, λ and Λ, such that ϕ 2 D log u 2 dx c Dϕ 2 dx R n R n for every ϕ C (Rn ). Let (x,2l) be a cube in R n and take a cutoff function ϕ C ((x,2l)), ϕ such that ϕ = on (x, l) and Dϕ c l. By the logarithmic estimate above, we have D log u 2 d y ϕ 2 D log u 2 d y c Dϕ 2 d y (x,l) (x,l) Ω c l 2 (x,2l) d y = cl n 2. Together with the Poincaré inequality this gives log u (log u)(x,l) 2 d y cl 2 D log u 2 d y c Theorem 3.7. and by Hölder s (or Jensen s) inequality ( log u (log u) d y (x,l) ) 2 log u (log u) 2 d y c < for every cube in R n with c depending only on n, λ and Λ. This shows that if u > is a weak solution, then log u BMO with log u c, where c depends only on n, λ and Λ. Observe, that the bound for the BMO norm is independent of solution u. Moreover, we have () If f, g BMO, then max(f, g) BMO and min(f, g) BMO. max(f, g) 3 2 f g and min(f, g) 3 2 f g. (2) If f BMO and h R n, then function τ h f BMO, where τ h f (x) = f (x + h). Moreover, we have τ h f = f. (3) If f BMO and a R, a, then function δ a f BMO, where δ a f (x) = f (ax). Moreover, we have δ a f = f. Proof. () Since max(f, g) = 2 (f + g + f g ), by Remark 3.4, we obtain Ω max(f, g) 2 f g + 2 f + 2 g f g + 2 f + 2 g 3 2 f g. On the other hand, min(f, g) = 2 (f + g f g ) and a similar argument as above shows that min(f, g) BMO.

36 CHAPTER 3. BOUNDED MEAN OSCILLATION 34 (2) Change of variables y = x + h, dx = d y, gives (τ h f ) = τ h f (x) dx = f (x + h) dx = + h for every cube in R n. Thus τ h f = sup = sup = sup = sup +h τ h f (x) (τ h f ) dx f (x + h) f +h dx f (y) f +h d y + h +h f (y) f d y = f. (3) Change of variables x = a y, dx = a n d y gives (δ a f ) = δ a f (x) dx = f (ax) dx = a n f (y) d y = f (y) d y = f a, a a a f (y) d y = f +h where a in this connection means {az : z }. Thus δ a f = sup δ a f (x) (δ a f ) dx = sup f (ax) f a dx = sup f (y) f a d y a a = sup f (y) f d y = f. Remark 3.8. Every function f BMO can be approximated pointwise by an increasing sequence of bounded BMO functions, since truncations k, f (x) > k, f k (x) = min(max(f (x), k), k) = f (x), k f (x) k, k, f (x) < k, k =,2,..., belong to BMO with f k 9 4 f for every k =,2,..., f k f pointwise and f k f in L loc (Rn ) as k (exercise). 3.2 Completeness of BMO. We show that BMO is a Banach space using the argument from []. The proof is based on the Riesz-Fischer theorem, which asserts that L p spaces are complete.

37 CHAPTER 3. BOUNDED MEAN OSCILLATION 35 Theorem 3.9. BMO is a Banach space with the norm. Here we consider BMO as a space of equivalence classes of functions up to an additive constant, Proof. Let (f i ) i be a Cauchy sequence in BMO. Fix cube and consider the representative f i (f i ) of equivalence class f i. Sequence (f i (f i ) ) i is a Cauchy sequence in L (), since ( ) ( ) f i (f i ) f j ) (f j ) L () = = = ( ) ( ) f i (x) (f i ) f j (x) (f j ) dx ( ) ( ) f i (x) (f i ) f j (x) (f j ) dx ( f i (x) f j (x) ) (f i f j ) dx f i f j as i, j. Since L () a complete space, we conclude that sequence (f i (f i ) ) i converges in L () to function g L () and thus ( ) f i (f i ) g L () as i. (3.2) Let be a cube containing. As above, sequence (f i (f i ) ) i converges in L ( ) to function g and thus ( f i (f i ) ) g L ( ) as i. (3.3) Then consider sequence ((f i ) (f i ) ) i of real numbers, whose terms can be also interpreted as constant functions on cube. It follows from (3.2) and (3.3) that ( (f i ) (f i ) ) (g g )) L () as i. This implies that the limit g g L () is a constant function in. Denote C(, ) = g (x) g (x) for almost every x. (3.4) On the other hand, by (3.3) we have ( ) (f i ) (f i ) = f i (x) (f i ) dx g (x) dx as i and thus C(, ) = g (x) dx. (3.5) We define function f as follows. Denote k = (, k), k =,2,... and set f (x) = g k (x) C(, k ), x k. (3.6) In principle, this definition makes sense, since every x R n belongs to k for k large enough, but we have to show that f is well defined, that is, if < k < k, then g k (x) C(, k ) = g k (x) C(, k ) for almost every x k.

38 CHAPTER 3. BOUNDED MEAN OSCILLATION 36 By (3.4) this is equivalent to showing that C(, k ) C(, k ) = C( k, k ) whenever < k < k. However, this follows from (3.5) and (3.4), since C(, k ) C(, k ) = g k (x) dx g k (x) dx ( = g k (x) g k (x) ) dx = C( k, k ) dx = C( k, k ) This shows that f is well defined. We claim that sequence (f i ) i converges to f defined by (3.6) in BMO. It follows from (3.6) that f L loc (Rn ). Then we show that f is the required limit function. Let ε >. Since (f i ) i is a Cauchy sequence in BMO, there exists i ε such that f i f j < ε, if i, j i ε, or, equivalently, for every cube in R n, we have ( ) ( ) f i (x) (f i ) f j (x) (f j ) dx < ε, if i, j iε. By letting j and using (3.2), for every cube in R n, we have ( ) f i (x) (f i ) g (x) dx ε, if i iε. Every cube in R n is contained in k for k large enough, so that by (3.6), (3.5) and (3.4), we obtain ( f i (x) f (x) ) ( f i f ) dx = = = fi (x) g k (x) + C(, k ) }{{} =(g k ) (f i ) + f dx fi (x) g k (x) + (g k ) (f i ) + f dx fi (x) (f i ) g (x) + [ (g k ) ( g k (x) g (x) ) ] + f dx }{{} = = fi (x) (f i ) g (x) dx ε if i iε. Here we used the fact that (g k ) ( g k (x) g (x) ) + f = C(, k ) C(, k ) + f (x) dx ( = C(, k ) C(, k ) + g k (x) C(, k ) ) dx = C(, k ) C(, k ) + (g k ) C(, k ) = C(, k ) C(, k ) + C(, k ) C(, k ) =.

39 CHAPTER 3. BOUNDED MEAN OSCILLATION 37 This shows that f BMO and f i f = sup ( f i (x) f (x) ) ( f i f ) dx as i. 3.3 The John-Nirenberg inequality We begin with an example. Example 3.. Consider f : R R, f (x) = log x, which is an example of an unbounded function in BMO, see Examples 3.5 (2). Then f [ a,a] = a log x dx = 2a a a a log x dx = a a/ (xlog x x) = log a, a >, and thus for x [ a, a] and t >, we have f (x) f [ a,a] > t log x (log a ) > t e x log a > t x < a e e t. This implies {x [ a, a] : f (x) f [ a,a] > t} = 2ae t, t >, T H E M O R A L : The distribution function decays exponentially. The John-Nirenberg inequality gives a similar exponential estimate for the distribution function of oscillation of an arbitrary BMO function. The proof that we present here is based on a recursive use of the Calderón-Zygmund decomposition. Theorem 3. (The John-Nirenberg lemma (96)). There exists constants c and c 2, depending only on dimension n, such that if f BMO, then for every cube R n. T H E {x : f (x) f > t} c e c 2 t f, t >, M O R A L : The John-Nirenberg inequality tells that logarithmic blowup, as for f (x) = log x, is the worst possible behaviour for a general BMO function. In this sense the John-Nirenberg inequality is the best possible result we can hope for. Proof. Let R n be a cube and let s > f to be a parameter, which is to be chosen later. Then f (y) f d y f < s.

40 CHAPTER 3. BOUNDED MEAN OSCILLATION 38 Apply the Calderón-Zygmund decomposition for function f f at level s in, see Theorem.3. We obtain pairwise disjoint dyadic subcubes i, i =,2,..., of such that and s < f (y) f d y 2 n s, i =,2,..., i f (x) f s for almost every x \ i. (3.7) This implies f i f = (f (y) f ) d y i f (y) f d y 2 n s, i i =,2,... (3.8) Since cubes i are pairwise disjoint subcubes of and f BMO, we obtain i = i s i = i f (y) f d y s We proceed recursively. Since f BMO, we have i = f (y) f d y f. (3.9) s i f (y) f i d y f < s for every i =,2,... We take the Calderón-Zygmund decomposition for function f f i at level s in every cube i, i =,2,... We obtain pairwise disjoint dyadic subcubes i,i 2, i 2 =,2,..., of i such that s < f (y) f i d y 2 n s, i 2 =,2,... i,i 2 and This gives f (x) f i s for almost every x i \ i,i 2. (3.) i 2 = i,i 2 i 2 = f (y) f i d y (3.) s i 2 = i,i 2 f (y) f i d y f i, i =,2,... s i s By (3.8) and (3.), for every i =,2,..., we have f (x) f f (x) f i + f i f By (3.7), we obtain s + 2 n s 2 2 n s for almost every x i \ f (x) f 2 2 n s for almost every x \ i,i 2. i 2 = i,i 2 i,i 2 =

41 CHAPTER 3. BOUNDED MEAN OSCILLATION 39 an by (3.9) and (3.), we have i,i 2 = i,i 2 = i = i 2 = At kth step we observe that i,i 2 i = ( f f i s s ) 2. i,...,i k f (y) f i,...,i k d y f < s for every i,..., i k =,2,... We take the Calderón-Zygmund decomposition for function f f i,...,i k at level s in every cube i,...,i k, i,..., i k =,2,... We obtain pairwise disjoint dyadic subcubes i,...,i k, i k =,2,..., of i,...,i k such that s < f (y) f i d y 2 n s, i,...,i k k =,2,... i,...,i k and f (x) f i s for almost every x,...,i i k,...,i k \ i,...,i k. i,...,i k = As above, this gives and i,...,i k = ( ) f k i,...,i k, s f (x) f k2 n s for almost every x \ i,...,i k = i,...,i k. In other words, almost every point of the set {x : f (x) f > k2 n s} belongs to some of the cubes i,...,i k, i,..., i k =,2,... Let us then complete the proof of the exponential estimate for the distribution function. To this end, assume first that t 2 n s. Then we choose k {,2,...} such that By the beginning of the proof, we have k2 n s t < (k + )2 n s. {x : f (x) f > t} {x : f (x) f > k2 n s} i,...,i k = i,...,i k, where the last inclusion holds up to a set of measure zero. Thus {x : f (x) f > t} i,...,i k = i,...,i k i,...,i k = i,...,i k = ( ) f k s k log e f. s Since t < (k + )2 n s = k2 n s + 2 n s 2k2 n s = k2 n+ s,

42 CHAPTER 3. BOUNDED MEAN OSCILLATION 4 we have from which it follows that k > t 2 n+ s, e k log s f e t 2 n+ log s s f. Recall that s > f is a free parameter and it can be chosen as we want. By choosing s = 2 f, we obtain {x : f (x) f > t} e t 2 n+ 2 f log 2 f f = e c 2 t f, where c 2 = log2 2 n+2. This proves the claim in the case t 2 n s. Assume then that t < 2 n s = 2 n+ f. In this case and thus Thus, in both cases, we have t 2 n+ f <, {x : f (x) f > t} e e e e 2 n+ f. {x : f (x) f > t}) c e c 2 t f, with c = e and c 2 = log2 2 n+2. Remarks 3.2: The John-Nirenberg lemma gives a characterization of BMO. Assume that there exist constants c and c 2, independent of cube, such that {x : f (x) f > t} c e c 2t, t >, for every cube in R n. Then f (x) f dx = c {x : f (x) f > t} dt e c 2t dt = c c 2 for every cube. Thus f BMO with f c c 2. / t e c 2t = c c 2 Next we consider two consequences of the John-Nirenberg inequality. Assume that f L loc (Rn ) and let p <. We define ( ) f,p = sup f (x) f p p dx, where the supremum is taken over all cubes in R n, and the corresponding function space BMO p = { f L loc (Rn ) : f,p < }.