Geometric intuition: from Hölder spaces to the Calderón-Zygmund estimate

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1 Geometric intuition: from Hölder spaces to the Calderón-Zygmund estimate A survey of Lihe Wang s paper Michael Snarski December 5, 22 Contents Hölder spaces. Control on functions Control on sets Regularity results 4 3 Calderòn-Zygmund decomposition 4 4 Vitali covering lemma 5 5 The Hardy-Littlewood Maximal function 7 6 Proof of theorem 9 Abstract The following report is based on paper by Wang [?]. We include Wang s discussion of Hölder norms and supplement it with results from class and assignments. Then, we introduce the Hardy- Littlewood maximal function, the Calderòn-Zygmund decomposition, and the Vitali covering lemma. Finally, we include the proof of the Calderòn-Zygmund estimate ( ) D 2 u p c f p + u p. B B B Hölder spaces. Control on functions Let us first discuss the geometry of Hölder spaces. We are motivated by the discussion in Wang. Hölder norms given us intuition on the density of a function and what can be said about sets { f > λ} = {x : f(x) > λ} and their measure. Recall that the definition of Hölder norm C,α for α (, ] is u C,α u(x) u(y) = sup u(x) + sup x x,y x y α. () Note that, from homework assignment 4, question 5. (a), the space for which (??) holds for α = + ε, for any ε >, is only the space of constants. Moreover, from (b), this space is much larger than C ; continuously differentiable functions are not dense in C,α. Thus, there is an advantage at looking at this class of functions. Consider what the control on these functions looks like. For > x y > and α (, ), we have x y < x y α, and so f(x) f(y) x y α < f(x) f(y). x y

2 Figure : Graph of f(x) = x α for α = (left) and α = 2 (right). In particular, this makes it clear that C C,α. Taylor s theorem gives us the error on first order approximation, so one can interpret the Hölder norm as a sort of error estimate. However, it gives much more local information about the function than the L p norm. Indeed, because integrals do not detect sets of measure, and the L p norm does not tell us about any local behaviour of a function, we would not be able to bound D 2 u without additional information. Now consider the definition of density. Provided the limit exists, the metric density of a measurable set E at a point x is defined as lim r m(b(x, r) E). m(b(x, r)) With this definition in mind, consider the geometry of Hölder space. In his paper, Wang uses the following statements for motivation: Lemma. If u is a solution of (4) and h is a continuous harmonic function with h = u on B, then u(x) h(x) 2n x 2 ) f. (2) Corollary. For any < α <, there are positive universal contsants r < and ε > such that for a solution u of (4) in B, with u and f ε, there is a constant A such that u(x) A r α for all x B r. Taking B = B r, we can iterate this to obtain u(x) A k r kα for x B r k. In other words, we can get very precise local precision of u. This will be the guiding idea for the measure-theoretic bounds on the sizes of sets..2 Control on sets Equation (2) tells us that the function u is almost harmonic. In other words, we should be able to get some control over it, possibly by looking at averages over balls (we will see this when we introduce the maximal function). Let s look at how we can translate Lemma into set-theoretic language. We first include the following theorem. Theorem (Rudin 8.6). If f : Ω [, ) is a measurable function, µ is a σ finite positive measure on the Borel sigma algebra of Ω R d, ϕ : [, ] [, ] is monotonic, absolutely continuous on [, T ] for every T <, ϕ() = and ϕ(t) ϕ( ) as t, then (ϕ f)dµ = µ{x : f(x) > t}ϕ (t)dt. Ω Proof. Let E be the set of all (x, t) Ω [, ) such that f(x) > t. E is clearly a measurable set whenever f is a simple function, and its measurability follows from the approximation of any 2

3 measurable function by a sequence of increasing simple functions φ φ 2... f. t [, ), let E t be the slice E t = {x : (x, t) E}. We therefore have µ(e t ) = χ {x:f(x)>t} (x, t)dµ(x). Hence, µ{x : f(x) > t}ϕ (t)dt = = Ω = = Ω Ω f(x) Ω Ω χ {f>t} (x, t)ϕ (t)dµ(x)dt χ {f>t} (x, t)ϕ (t)dtdµ(x) ϕ (t)dtdµ(x) ϕ(f(x))dµ(x). For every Corollary 2. We have the relationship u p dx = p Ω t p m{x Ω : u(x) > t}dt. Proof. Apply the previous theorem with ϕ(t) = t p and f(x) = u(x). We obtain u(x) p dx = p t p µ{f > t}dt. Ω Thus, if Ω u p dx =, we should have that { u > λ} λ p, so that { u > λ} is small for λ large. In order to prove that any u L p, then, we should start by showing the decay of { u > λ}. We might hope to prove { u > λ } ε { u > } for some fixed ε > and λ >, and use the scaling to get u(rx) = r 2 f(rx) { u > λ λ} ε { u > λ}. Now compare with (). We might hope to use the above inequality to get some inductive estimate. Based on the fact that D 2 u L p if u is, we might guess that { D 2 u > λ λ} ε( { D 2 u > λ} + { f δ λ} }. However, the above is not true; the failure is due to the fact that the condition D 2 u(x ) is unstable. In order to get the desired bounds, we will have to use the maximal function. 3

4 2 Regularity results We have already seen the following. Theorem 2 (Evans, Thm. 4). For u H (Ω) a solution to (4), we have We use this to obtain the following. u H 2 (Ω) C f L 2 (Ω). Proposition. If { u = f B u = B, then D 2 u 2 C f 2. B B Evans motivates this bound as follows: if u = f, we have f 2 = ( u) 2 = = = n u xix i u xjx j i,j= n u xix ix j u xj i,j= n u xix j u xix j i,j= (u ) ( 2 ) u =... u n i j.. u n 3 Calderòn-Zygmund decomposition Though it will not be used here, we briefly present the decomposition. We first present the covering lemma. Lemma 2 (Calderón-Zygmund). Let f L (R n ) with f. There is an open set Ω which is a disjoint union of open cubes and its complement F a closed set such that: i. f(x) α a.e. on F ii. Ω = k Q k, where the cubes Q k have disjoint interior and α < f 2 n α (3) Q k Q k iii. m(ω) f α. Proof. Divide R n using a dyadic grid. Here, we see that there are only two cases { Q 2 k Q 2 n. Q 2 k Q 2n = bdry Start with a Q sufficiently large so that Q Q f(x)dx α for any x, where we are integrating with respect to Lebesgue measure. Let n be the dimension of the space. We get 2 n cubes at each stage of the decomposition. Stage I: two cases for each new cube 4

5 . m(q) Q f α 2. m(q) Q f > α. In the second case, we keep the cube and put it into our collection {Q k }, which will eventually become Ω. Note that since each new cube has half the length of the one of the previous generation (say Q ), we have f 2n f 2 n α, m(q) Q m(q ) Q so that (3) holds. In the first case, we keep on dividing and get a countable collection. At each stage, we look again, since almost every point of an L function is a Lebesgue point, we find that f α almost everywhere. Clearly, then, i. and ii. hold. It remains to see that m(ω) = k k m(q k ) f(x)dx α Q k = α f L (Ω) α f. We can now show the decomposition of f into good and bad parts, f = g + b. We do so as follows: { f(x) x F g(x) = m(q k ) Q k f(x)dx x Q k Ω. We notice that g(x) 2 α almost everywhere. Then b = f g and { x F b(x) = f(x) f x Q k Ω. Thus, we have that for every cube, b =. 4 Vitali covering lemma The Vitali covering lemma is another simple combinatorial-geometric proof about sets which allows one to consider disjoint unions of balls. It is often used in the proof of the Lebesgue-Radon-Nikodym theorem. We present the proof with a figure to give intuition, and afterwards we give a modified version of Vitali which Wang uses for his proof. The following proof is from Rudin s Real & Complex analysis. Theorem 3 (Vitali covering lemma). Suppose that Ω = n B(x i, r i ) is a finite collection of balls. There is then a set S = {i,..., i k } {,..., n} such that 5

6 Figure 2: Top: original set Ω, a disjoint union of balls. Bottom left: the final result, S = i, i 2. Bottom right: any y that was in a discarded ball will be within three times the radius of the ball that superceded it. (i) the balls B(x i, r i ), i S are disjoint, (ii) Ω i S B(x i, 3r i ), (iii) m(ω) 3 k i S m(b(x i, r i )). Proof. The proof is simple. Without loss of generality (re-label if necessary), assume the balls are ordered from largest to smallest radii, i.e. r r 2 r n. Let i =, and discard all balls which intersect B i. Let i 2 be the largest radius of the remaining balls, and repeat the same procedure. Since there is only a finite numbers of balls, this procedure ends at some i k. It is obvious that the balls are disjoint, so that (i) holds. Now, suppose that y Ω, so y B(x j, r j ) for some i. If j = i j for some i j S, we are done, so suppose the ball B(x j, r j ) was not chosen. This means it must have intersected some B(x i, r i ) before it got discarded. Since r i r j (otherwise B(x i, r i ) would ve been discarded), we must have y B(x i, 3r i ), which proves (ii). The last statement then follows easily, as by the basic properties of Lebesgue measure. (m(b(x, 3r i ))) = 3 k m(b(x, r i )) The modified Vitali is a little more complicated. Theorem 4 (Modified Vitali). Let < ε < and C D B be two measurable sets with C < ε B and satisfying the following property: for evrey x B with C B r (x) ε B r, B r (x) B D. Then D (2 n ε) C. Proof. First, C < ε B, so for almost all x C, we can find r x such that C B rx (x) = ε B rx. To see this, note that if we choose any x in the interior of C, we can certainly choose r x small enough so that B rx C and then C B rx (x) = B rx > ε B rx. Moreover, we can choose r x so big that C B rx (x) < ε B rx. Assuming C B rx (x) varies continuously with r x, we can invoke the intermediate value theorem to see there is an r x such that we have equality. Let then r x < 2 be such that C B rx (X) = ε B rx with C B r (x) < ε B r 6

7 Figure 3: Left: too small; center: too big; right: just right! for all r x < r < 2. The set C can clearly be written as a union of balls, so we can apply the Vitali covering lemma to obtain x, x 2,..., x n C which satisfy the above condition and such that B rxi B rxj =, i j, C n B 5rxi (x i ), and since C B, we also have C i B 5rxi (x i ) B. Moreover, since each B rxi (x i ) satisfies we have ε B rxi (x i ) = C B rxi (x i ), C B 5rxi (x i ) < ε B 5rxi (x i ) (5r i > r i ) = 5 n ε B rxi (x i ) = 5 n C B rxi (x i ). Moreover, since r xi < 2, we can say that B rxi (x i ) 4 n B rxi B. Thus, n C = B 5rxi (x i ) C n B 5rxi (x i ) C n 5 n 2 n n ε B rxi (x i ) ε B rxi (x i ) B n = ε2 n B rxi (x i ) B = 2 n D. 5 The Hardy-Littlewood Maximal function The Hardy-Littlewood function is one of the most important tools in analysis (or so I ve been read and been told). In some sense, it gives the worst behiaviour of a function. For f L p (R d ), the maximal function is defined as (Mf)(x) = sup r> where dm denotes the d-dimensional Lebesgue measure. f dm, m(b r ) B(x,r) 7

8 Remark For f L, Mf is never in L. Here is a simple example that is easily generalized. Take d = and w.l.o.g. suppose f > and let x 2. By choosing a different radius, we will do worse than the supremum over all radii, and so in particular choosing r = x we have Mf(x) = sup f(x) dx <r< m(b r (x)) B r(x) f(x) dx 2 x 2 x B x (x) f(x) dx = C x. Even if we take the maximal function to be on x 2, the function still diverges due to the factor of x and therefore cannot be in L. There exist other types of maximal functions. For instance, if µ is any complex Borel measure and m denotes the d-dimensional Lebesgue measure, we first set (P r µ)(x) = and we define the maximal function Mµ for µ by µ(b(x, r)) m(b(x, r)), (Mµ)(x) = sup(p r µ)(x). r> It is easy to show that the function Mµ is lower semicontinuous, hence measurable. We now use the Vitali covering lemma to prove the following lemma: Theorem 5. For a complex Borel measure µ on R d and λ >, we have {x R d : (Mµ)(x) > λ} 3 d λ u, where u = µ (R d ) < is the total variation of µ. Proof. Let µ, λ be given. There is a compact subset K of the open set {Mµ > λ} (open by the previous claims). For every x K and open ball B, we have µ (B) > λm(b). Since K is compact, we can cover K by finitely many balls B, and by the Vitali covering lemma there are balls B,..., B n such that m(k) 3 d n m(b i ) 3 d λ n µ (B i ) 3 d λ µ where the last inequality follows by disjointness of the balls. The result now follows by taking the supremum over all compact subset K {Mµ > λ}. Definition The space Weak L is the space of measurable f for which λm{ f > λ} < for < λ <. Remark Any function f L is in weak L. To see this, let λ >, and let E = { f > λ}. We then have λm(e) = λdm f dm f dm = f. E E R d Taking λ to the other side yields the result. Corollary 3. For every f L (R d ) and λ >, {Mf > λ} 3 d λ f. 8

9 Proof. We can use the previous theorem since for any measurable set E R d, the mapping E fdm defines a complex Borel measure µ. Writing µ = fdm, we see that the definition of the maximal function for a measure agrees precisely with the Hardy-Littlewood maximal function. The above are called weak (,) estimates and give à-priori control over Mf in L. On the other hand, we have strong (p,p) type estimates, which tell us that if f L p for p >, then Mf L p as well. Proof. If Mf > λ, then by definition, there is a ball of radius r centered at x such that f dm > λ B(x, r). B(x,r) By the Vitali covering lemma, there are disjoint balls B (x, r ),..., B n (x n, r n ) such that Hence, E n B(x i, 5r i ) {x : Mf(x) > λ}. {Mf > λ} 3 d n 3 d λ B i f dm. Note that this is the same weak estimate we obtained in the previous corollary. We now apply it to obtain strong estimates. Define { f(x) f(x) > λ β(x) = 2 otherwise. Clearly then, β(x) L p for f L p, and we can apply the above weak estimate to β: {Mf > λ} 2 3d f dm. λ f >λ/2 Hence, Mf p p = = p = p Mf(x) R d Mf(x) Mf(x) 3 d 2p pλ p dλdm λ p dmdλ R d λ p m({mf > λ})dλ R d λ p f dmdλ f >λ/2 = C p f p p <. (Fubini) 6 Proof of theorem We are now in a position to prove the estimate. 9

10 Theorem 6 (Calderón-Zygmund). For u satisfying we have the following estimate for p (, ): D 2 u p C B where D denotes the total derivative operator. We first prove the following lemma Lemma 3. If u is a solution of u(x) = f(x), x < 2, (4) ( ) f p + u p, B 2 B 2 u = f in a domain Ω which contains B 4, the balls of radius 4, and {M( f 2 ) δ 2 } {M D 2 u 2 } B, (5) then there is a constant N so that for any ε >, there is a δ > depending only on ε such that {M D 2 u 2 > N 2 } B < ε B. (6) Proof. Since the set in (5) is not empty, there is x B such that, for any r >, we have D 2 u 2 2 n, f 2 2 n δ 2. (7) B(r,x ) B(r,x ) In particular, this allows us to recenter and obtain D 2 u 2, f 2 δ 2. B 4 B 4 Hence (why?) u u B4 2 C. B 4 Letting v be a solution to the corresponding system { v = on B 4 v = u ( u) B4 x u B4 on B 4, we have by minimality of harmonic functions for the Dirichlet energy, v 2 u u B4 2 C. B 4 B 4 Now, by typical regularity results, we have that v C, so we can bound the supremum of D 2 v: Meanwhile, D 2 v 2 L (B 3) N 2. D 2 (u v) 2 C f 2 Cδ 2. B 3 B 4 Using the weak L estimate proved in Corollary 3, we have the following bound: λ {x B 3 : M D 2 (u v) > λ} C N 2 D 2 (u v) 2 B 3 C N 2 f 2 Cδ 2 B 4

11 by the above. Now, taking N 2 = max(4n 2, 2 n ), we have that {x B : M D 2 u 2 > N 2 } {x B : M B3 D 2 (u v) 2 > N 2 }. How to see this? Well, if y B 3, then certainly D 2 u(y) 2 = D 2 u(y) 2 2 D 2 v(y) D 2 v(y) 2 2 D 2 u(y) D 2 v(y) 2 + 2N 2. For any x {x B : M B3 D 2 (u v) 2 > N 2 }, r 2, we have B 2 (x) B 3 and therefore D 2 u 2 2M B3 ( D 2 (u v) 2 )(x) + 2N 2 4N 2. sup r 2 B r(x) On the other hand, for r > 2, we have D 2 u 2 B r since B(x,r) B(x,r) by assumption. Finally, Wang shows that B(x,2r) D 2 u 2 2 n, D 2 u 2 2 n, f 2 2 n δ 2 B(x,r) {x B : M( D 2 u 2 ) > N 2 } {x B : M( D 2 (u v) 2 > N 2 } C N 2 f 2 Cδ 2 /N 2 < Cδ 2 = ε B where δ 2 = ε B C by choice. Corollary 4. Suppose u is a solution of (4) in a domain Ω which contains the ball B(x, 4r) for some x, r. Let B = (x, r). If {M( D 2 u 2 ) > N 2 } B ε B then Proof. By the contrapositive, if then the assumption B {M( D 2 u 2 ) > } {M( D 2 u 2 > δ 2 }. {M D 2 u 2 > N 2 } B ε B {M( f 2 ) δ 2 } {M D 2 u 2 } B must be false. In other words, the intersection is empty, and so the original set is entirely contained in the set where M( f 2 ) > δ 2 and/or M( D 2 u 2 ) >. From Wang: The moral of Corollary 2 is that the set {x : M( D 2 u 2 ) > } is bigger than the set {x : M( D 2 u 2 ) > N 2 } modulo {M(f 2 ) > δ 2 } if {x : M( D 2 u 2 ) > N 2 } B = ε B. As in the Vitali lemma, we will cover a good portion of the set {x : M( D 2 u 2 ) > N 2 } by disjoint balls so that in each of the balls, the density is ε. Corollary 5. Assuming that u is a solution of (4) in Ω B 4, with then for ε = 2 n ε, we have i. {x B : M( D 2 u 2 ) > N 2 } ε B, {x B : M( D 2 u 2 ) > N 2 } ε ( {x B : M( D 2 u 2 ) > } + {x B : M( f 2 ) > δ 2 } ).

12 ii. {x B : M( D 2 u 2 ) > N 2 } ε ( {x B : M( D 2 u 2 ) > λ 2 } + {x B : M( f 2 ) > λ 2 δ 2 } ) iii. {x B : M( D 2 u 2 ) > (N 2 ) i } k ε i {x B : M( f 2 ) > δ 2 (N 2 ) k i } +ε k {x B : M( f 2 ) > }. Proof. The important thing is to notice the resemblance between this and how we dealt with the geometry of functions. The proof is straightforward from what we have proven. First, set C = {x B : M( D 2 u 2 ) > N 2 } D = {x B : M( D 2 u 2 ) > } {x B : M( f 2 ) > δ 2 }. Now, we have that C ε B by assumption. To see that the second condition for the modified Vitali lemma is satisfied, we need to show that B r C ε B r implies B r C D. This follows immediately from the main lemma 3 that B r C ε B r D c B r =. To see ii., we need only iterate the above result on the equation (λ u) = λ f. Here, we are using the scaling property of the Laplacian: u(rx) = r 2 f(rx). Finally, repeating the same thing over and over on λ = N, N 2,... we get further local information on the geometry of these sets. We are now in a position to prove the estimate. We do so for p > 2. Theorem 7. If u = f in B 4, then D 2 u p C f p + u p. B B 4 Proof. The first step of the proof is to assume that f p is small so that we can obtain {x B : M D 2 u 2 > N 2 } ε B. Indeed, this is the crucial bound that makes everything work. We can always obtain this, however, by multiplying f by a small constant and then pulling it out into the larger one. The trick is to write D 2 u p = ( D 2 u 2 ) p/2 2

13 and show that M( D 2 u 2 ) L p/2 (B ), so that D 2 u L p (B ). Suppose then that f p = δ. We then have that (why?) (N ) ip {M( f 2 ) > δ 2 (N ) 2i pn p } δ p (N ) f p p C. Hence, D 2 u p (M( D 2 u ) ) p/2 dx B B = p λ p {x B : M D 2 u 2 λ 2 } dλ ( ) p B + (N ) kp {x B : M D 2 u 2 > (N ) 2k } p ( B + k= N kp k (N ) ip {M( f 2 ) > δ 2 (N ) 2i } C. ε i {x B : M f 2 δ 2 N 2(k i) } + Taking ε so that N p ε <, the sum converges and thus the theorem is proved. References k= N kp εk {x : M( D 2 u 2 ) } [] Lihe Wang, A geometric approach to the Calderón-Zygmund estimates. University of Iowa, lwang/cccalderon.pdf. ) 3