V5B7 ADVANCED TOPICS IN ANALYSIS: SOBOLEV SPACES

V5B7 ADVANCED TOPICS IN ANALYSIS: SOBOLEV SPACES OLLI SAARI Abstract. These are the notes about materal covered durng the lectures. The exposton ams at gvng overvew on technques needed to deal wth the subject as well as relevant phenomena, and that wll also be n focus n the exam. Sharper results and more detals can be found n the references, whch you should also consult n case you feel presence of gaps you do not know how to fll. Contents 1. Short motvaton 2 1.1. Duals of normed spaces 2 1.2. Convergence of smooth functons 2 1.3. Dstrbutons 3 1.4. Weak approxmaton by smooth functons 3 1.5. Dervatve 3 2. Sobolev spaces 4 2.1. Defnton and approxmaton 4 2.2. Basc propertes and examples 5 2.3. Poncaré s nequalty 5 3. Maxmal functons 6 4. Self-mprovng of generalzed Poncaré nequaltes 7 4.1. Weak L p spaces 7 4.2. Rearrangement 8 4.3. Controllng oscllaton 8 4.4. Truncaton argument 11 5. Hardy Sobolev spaces 13 5.1. Motvaton 13 5.2. Hardy spaces 13 5.3. Hardy Sobolev spaces 13 5.4. Characterzaton wth Calderón maxmal functon 15 5.5. Pontwse characterzatons 16 6. Embeddngs for Calderón Hardy spaces 16 6.1. Hgher order spaces 16 6.2. Fractonal ntegral 17 6.3. Inequalty for Calderón functons 17 7. Lttlewood Paley theory 18 7.1. Schwartz class 18 7.2. Fourer transform 18 7.3. Lttlewood Paley decomposton 19 7.4. Calderón Zygmund theory 19 7.5. Khntchne s nequalty 19 8. Functon spaces 2 8.1. Resz potentals 21 8.2. Potental spaces 21 8.3. Besov and Trebel Lzorkn scales 22 8.4. Fnte dfferences 22 9. Real nterpolaton 25 9.1. Prelmnares 25 9.2. K functonal 25 Date: January 31, 219. 1

9.3. J functonal 26 9.4. Equvalence of the methods 26 9.5. Basc propertes and dualty 27 9.6. Explct computatons 29 1. Complex nterpolaton 3 1.1. Basc propertes 31 1.2. Scalar and vector valued L p spaces 32 1.3. The method of retracts 32 1.4. Sneberg s stablty theorem 33 1.5. Surjectvty 33 1.6. Injectvty 33 1.7. Consstency 34 11. Drect method n the calculus of varatons 35 11.1. Problems wth boundary values 35 11.2. Applcaton I: Hölder contnuty for n-laplacan 36 11.3. Applcaton II: Very weak solutons to lnear equatons 38 11.4. Fractonal Laplacans 38 11.5. Caffarell Slvestre extenson 39 12. Traces and extensons 4 12.1. Trace theorems 4 12.2. Some explanatons 43 12.3. Other boundary values 43 12.4. Comment on domans 44 13. Capacty 44 13.1. Bascs 44 13.2. Hausdorff measure 47 13.3. Sets of capacty zero 47 13.4. Exceptonal sets for Sobolev spaces 49 13.5. Precse representatves 5 14. Modulus of a curve famly 52 14.1. Curves and condensers 54 14.2. Sobolev spaces usng curves 57 14.3. Remark on metrc spaces 58 15. Resz capacty and Frostman s lemma 58 References 6 1. Short motvaton Ths secton ponts at the theory of dstrbutons to motvate the study of weak dfferentablty. Not everythng s proved, not everythng s precse, nothng s needed n the man part of the course. More detals on dstrbutons can be found n Rudn s book [21]. 1.1. Duals of normed spaces. Let X be a normed vector space over reals. A functonal on X s a mappng f X R. Algebrac dual of X s the collecton of lnear functonals. The topologcal dual or brefly dual s the collecton of contnuous lnear functonals. We denote X = {f X R f lnear and contnuous (bounded)}. In addton to ts norm (strong) topology, X has a weak topology, whch s the topology nduced by the famly X. Next consder the dual X. It s endowed by a norm topology (operator norm as X R), but snce every x X defnes a map X R, we can regard X as a subset (X) X where (x)(f) = f(x). The coarsest or weakest topology that makes all (x) contnuous s called the weak* topology of X. In addton, one could also consder the weak topology of X nduced by the famly X. 2

1.2. Convergence of smooth functons. Every topology defnes a noton of convergence. We study functon spaces that contan C (Ω) as a subset (usually even a dense one). The strongest topology one mght want to use s the followng. We defne the convergence n D as follows: We say ϕ ϕ n D f there s K R n compact wth supp ϕ K. For all α N n, D α ϕ D α ϕ unformly n K. Ths convergence s related to the topology nduced by the famly of norms (N postve nteger and K compact set from an exhauston of R n ) ρ N K(f) = max{ D α f(x) x K, α N}. It keeps C (R n ) closed. The space D tself we let be C c (Ω). That s not closed because of the requrement on supports. 1.3. Dstrbutons. At the other end of the world les the space of dstrbutons D, the topologcal dual of D. Convergence n the sense of dstrbutons s exactly the weak* convergence of D, that s f f f f, ϕ f, ϕ as real numbers for all ϕ D. The Japanese bracket denotes the dualty parng. Examples of dstrbutons nclude All test functons n D. All locally ntegrable functons. The drac mass ϕ ϕ() and ts dervatves that are not even measures anymore. In the case of locally ntegrable f, we can defne for a test functon ϕ, f, ϕ = fϕ f L 1 ϕ. 1.4. Weak approxmaton by smooth functons. Let η = 1 B(,1) ce 1 x 2 1 be normalzed to have ntegral one. Then t s smooth, supported n B(, 1), even, and ts dlates η δ = δ n η(δ 1 ) are an approxmaton of dentty (see Secton 1.2.4 n Grafakos [8]), n partcular, for f L p (R n ) wth p [1, ) lm ɛ f η ɛ f Lp (Ω) and the convergence also takes place unformly n compact sets for contnuous functons. Hence closure of D under L p norm s L p. For ψ D defne τ h ψ(x) = ψ(x h), ψ(x) = ψ( x) h R d f ψ = f, τ ψ Proposton 1.1. Let f D (R n ). Then η δ f f as dstrbuton when ɛ. Moreover, each η δ f s smooth. Sketch of the proof. Snce the dfference quotent s a lnear operator and f s contnuous, we see that f η δ (x) s smooth. To prove the convergence, we compute for an arbtrary test functon ψ f, ψ = lm δ f, ψ η δ = f, ψ(y)η δ ( y) dy = lm δ ψ(y) f, τ y η δ dy = lm δ f η δ, ψ where we resorted to exstence result of vector valued ntegral, see Rudn [21] Theorem 6.32 for detals. Ths tells that closure of D under the convergence of dstrbutons s D. 3

1.5. Dervatve. Let D α be the dfferental operator correspondng to partal dervatves α. For any dstrbuton, we can defne the dstrbutonal (weak) dervatve through approxmaton (also wthout, but ths s to serve as ntuton). Let f be any sequence of smooth functons convergng to f n the sense of dstrbutons. Let D α f, ϕ = lm D α f, ϕ = lm ( 1) α f, D α ϕ = ( 1) α f, D α ϕ. The last expresson s clearly ndependent of the sequence, and t s the usual defnton of dstrbutonal dervatve. In general t s just a dstrbuton and as such rather useless. However, there are stll many notons of convergense we may us n order to close C c. 2. Sobolev spaces Most of ths secton can be found n Evans and Garepy [6]. 2.1. Defnton and approxmaton. For ths secton, let Ω be an open set n R n. A locally ntegrable functon f defnes a dstrbuton, and hence t makes sense to speak about ts dstrbutonal dervatve. Defnton 2.1. Let f be a dstrbuton. We wrte g = k f and call t the dstrbutonal dervatve f f, k ϕ = g, ϕ. In case all frst order dstrbutonal dervatves are locally ntegrable functons, that s f, k f L 1 loc(r n ), k {1,..., n} we call f a Sobolev functon and the defnton gets the expresson For f Sobolev and p 1, we defne the norms f k ϕ = gϕ. f W s,p = f L p = ( f p 1/p ) f W s,p = f L p + f W s,p. and the Sobolev space as the set of Sobolev functons wth fnte Sobolev norm and structure as normed space. W 1,p (R n ) = {f f W 1,p (Ω) < } Theorem 2.2. Sobolev space W 1,p (Ω) s a Banach space. Proof. We show the completeness. Let f be Cauchy n W 1,p. Then t s Cauchy n L p as s k f. By completeness of L p, there are lmt functons f and g such that f f and k f g n L p. It remans to show that g s the dstrbutonal dervatve of f. Take ϕ C c (Ω). Then f k ϕ = lm f k ϕ = lm ( k f )ϕ = gϕ where we used convergence n L p. Theorem 2.3 (Meyers-Serrn). C (Ω) s dense n W 1,p (Ω) whenver p [1, ). Lemma 2.4. Let f W 1,p (Ω), ɛ > and f ɛ = η ɛ f be the standard mollfcaton. Then f ɛ f W 1,p (Ω ) f p [1, ) and d(ω, Ω) >. Proof. Take ɛ < 1 1 d(ω, Ω). Snce supp η ɛ B(, ɛ), we have supp η ɛ ( x) Ω so that f ɛ s well defned. Then for any partal dervatve f ɛ (x) = (η ɛ )(x y)f(y) dy = (η ɛ (x y))f(y) dy = η ɛ (x y) f(y) dy = η ɛ f(x) and the rght hand sde converges to f n L p. 4

Proof of the Meyers-Serrn theorem. Fx δ >. Let Ω = Ω 2 1 Ω 2 +1 so that Ω form a fntely overlappng cover of Ω. Let ψ be the assocated smooth partton of unty. For each, choose ɛ so that (ψ f) ɛ ψ f W 1,p (Ω) < 2 δ, here the subscrpt ɛ denotes convoluton by the mollfer η ɛ. For l > and x Ω l, we note that Hence f(x) = =1 l+1 u v W 1,p (Ω l ) l+1 (ψ u)(x) = =1 =1 (ψ u)(x), v(x) = l+1 ψ u (ψ f) ɛ W 1,p (Ω l ) =1 =1 l+1 (ψ f) ɛ (x) = =1 (ψ f) ɛ (x). ψ u (ψ f) ɛ W 1,p (Ω) 2 δ = δ. Sendng l, we obtan u v W 1,p (Ω) δ by monotone convergence. The functon v s obvously smooth by convoluton and fnte overlap of Ω. 2.2. Basc propertes and examples. Proposton 2.5. The followng hold:. Let p 1. If f W 1,p and g W 1,p wth p = p/(p 1), then (fg) = f g + g f L 1.. If F C 1, F L and F () =, then F (f) = F (f) f L p.. Let f + = 1 f> f and f = f + f. Then f + = 1 f> f almost everywhere. Proof. Frst and second tems follow by approxmatng wth smooth functons to whch rules of calculus apply. The thrd follows from the second by choosng F ɛ (r) = (r 2 + ɛ 2 ) 1/2 ɛ and sendng ɛ. See Evans and Garepy secton 4.2.2 for more detals. Example 2.6.. Any smooth and compactly supported s trvally Sobolev.. f R R defned va f(x) = x s not smooth, but t s Sobolev.. Let n 2 and take smooth ϕ wth 1 B(,1) ϕ 1 B(,2). Then v. f(x) = ϕ(x) x 1/2 W 1,1 (R 2 ) though t s unbounded. Let f be as prevously. Let q be an enumeraton of ratonals n B(, 1). Then g(x) = 2 f(x q ) W 1,1 (R 2 ) =1 though t s unbounded n any open subset of B(, 1), dscontnuous at every pont n B(, 1) and defntely not dfferentable there ether. v. If n = 1, then W 1,1 (R) s subset of absolutely contnuous functons. 2.3. Poncaré s nequalty. For bounded measurable E and a locally ntegrable f denote f E = f = 1 E E f. E Theorem 2.7 (Poncaré s nequalty). Let f W 1,1 loc (Rn ). Then for all balls B = B(x, r) wth center x R n and radus r > we have where 2B = B(x, 2r) s the concentrc dlate. B f f B Cr 2B f Proof. By approxmaton, we may assume f s smooth. Then B f f B B B f(x) f(y) dxdy B B Cr B 1 2B f(x) dxdtdy = Cr 2B f wth constant only dependng on the dmenson of the ambent space. 5 1 =1 x y f(x + t(x y)) dtdxdy

Defnton 2.8. For α (, 1], defne the Hölder semnorm and norm by f(x) f(y) f Cα (Ω) = sup x,y Ω x y α x y f Cα (Ω) = f L (Ω) + f Cα (Ω) Theorem 2.9 (Morrey s embeddng). Suppose f W 1,p (R n ) and p > n. Then f can be redefned n a set of measure zero to a bounded and Hölder contnuous functon (stll denoted by f) satsfyng for α = 1 n/p. f Cα (R n ) C f W 1,p (R n ) Proof. Suppose z s a Lebesgue pont and take R >. Then by telescopng, Poncaré and Hölder f B(z,R) f(z) = f B(z,2 1 R) f B(z,2 R) C f f =1 >1 B(z,2 1 B(z,2 1 R) ) C >1(2 1 R) f C B(z,2 1 ) >1(2 1 R) 1 n/p ( f p 1/p ) B(z,2 1 R) CR 1 n/p ( f p 1/p ) B(z,R) Then choose x, y Lebesgue ponts and R = x y. It follows by trangle nequalty, Poncaré and the prevous property f(x) f(y) f(x) f B(x,R) + f B(x,R) f B(y,R) + f B(y,R) f(y) CR 1 n/p f Lp (R n ). Takng supremum over x, y, we obtan the bound for f C α. To prove boundedness, take an arbtrary Lebesgue pont z. By the prevous reasonng and Hölder s nequalty f(z) f(z) f B(z,1) + f B(z,1) C f Lp (R n ) + f Lp (R n ). 3. Maxmal functons Next we am at showng that Poncaré nequalty captures the essental of what t s to be a Sobolev functon. To put ths dea nto the rght framework, we wll need several maxmal functons. The frst one s classcal. Defnton 3.1 (Hardy Lttlewood maxmal functon). For f L 1 loc, defne Mf(x) = sup f B x B where the supremum s over the Eucldean balls. The maxmal functon satsfes M f f almost everywhere (Lebesgue dfferentaton theorem), t s lower semcontnuous (exercse), Mf = f only f f =, and Mf x n whenever f n a set of postve measure. Hence Mf L 1 for any non-trval f. Theorem 3.2 (Hardy Lttlewood Wener). For f L 1 loc we have Mf L p f L p f p = 1, t holds {Mf > λ} C λ f. Proof can be found for nstance n Grafakos [8] as Theorem 2.1.6. whenever p > 1 and Defnton 3.3 (Calderón maxmal functon). For p (, ), f L p loc (Rn ), and α [, 1], we defne Np α f(x) = sup nf c R rα ( f c p 1/p ). B B x Warnng: ths s not a standard name for ths maxmal functon and s sometmes used to refer to another quantty (that s of comparable sze though). Ths operator and ts relatves are aextensvely studed n [5]. Proposton 3.4. If p > 1 and f W 1,p (R n ), then N 1 1 f L p (R n ). 6

Proof. By Poncaré s nequalty 2.7 and Hardy Lttlewood maxmal theorem 3.2 N 1 1 f Lp (R n ) M f Lp (R n ) f Lp (R n ). In order to reverse the mplcaton, we have to use a result of Haj lasz [1] (extendng earler work of Calderón): Theorem 3.5. Let f L 1 loc (Rn ) and suppose there exsts g L 1 loc (Rn ) such that B f f B r(b) 2B g, for all balls B. Then f W 1,1 loc (Rn ) and f Cg for some constant C only dependng on the dmenson n. Proof. Let ϕ be the usual smooth functon ϕ, supp ϕ B(, 1) and ϕ = 1 so that ϕ ɛ (x) = ɛ n ϕ(xɛ 1 ) wth ɛ > s an approxmaton of dentty. Now ϕ ɛ (x) only f x /ɛ 1 so that supp ϕ ɛ B(, ɛ). Also, by compact support and fundamental theorem on calculus Then Dϕ =, where D = k for any k. D(f ϕ ɛ )(x) = f Dϕ ɛ (x) = ɛ 1 f (Dϕ) ɛ (x) = ɛ 1 f(y)(dϕ) ɛ (x y) dy = ɛ 1 B(x,ɛ) (f(y) f B(x,ɛ) )(Dϕ) ɛ (x y) dy Cɛ 1 n B(x,ɛ) f(y) f B(x,ɛ) dy C B(x,2ɛ) g where the last nequalty followed by the assumpton. Then take any test functon ψ C c (R n ) wth supp ψ = K. Let K ɛ = {x d(x, K) < ɛ}. Then D(f ϕ ɛ, ψ C K ψ(y) B(y,2ɛ) g(x) dxdy = Cɛ n C K2ɛ K ψ(y) 1 K(y) 1 B(y,2ɛ )(x)g(x) dydx ψ L (R n ) K2ɛ g Takng the lmt ɛ, the we see that the dstrbuton on the left extends to a boudned lnear functonal on any C c (Ω) for Ω an open subset of R n. By the Resz representaton theorem (Secton 1.8 n Evans and Garepy [6]) t s gven as ntegraton aganst a sgned Radon measure µ. To see that µ s absolutely contnuous wth respect to the Lebsgue measure, suppose for contradcton there exsted K compact wth µ(k) > = K. Let χ (x) = (1 d(x, K)) + so that these functons are Lpschtz and χ 1 K. Then < µ(k) lm χ dµ C lm χ L (R n ) supp χ g = so that µ s absolutely contnuous wth respect to Lebesgue measure. By Radon-Nkodym theorem dµ = h(x) dx for a measurable h. By Lebesgue dfferentaton theorem we conclude h Cg. Corollary 3.6. If p 1 and N 1 1 f L p (R n ), then f L p (R n ). Proof. For any ball B, we have B f f B 2r nf z B N 1 1 f(z) r B N 1 1 f. Now N1 1 f L p (R n ) L 1 loc (Rn ). By the prevous theorem f s weakly dfferentable and f CN1 1 f L p (R n ) 7

4. Self-mprovng of generalzed Poncaré nequaltes 4.1. Weak L p spaces. For f measurable and t >, let d f (t) = {x R n f(x) > t}. We defne for p > f L p, (R n ) sup t[d f (t)] 1/p. t> Standard examples are L p functons and power functons. If f L p (R n ), then d f (t) = dx = ( f p { f >t} t ) On the other hand x n/p L p (R n ) but { x n/p > t} = B(, t p/n ) whose measure s t p. Consequently x n/p L p, (R n ). f L p, (R n ) f Lp (R n ). 4.2. Rearrangement. Infmums and supremums are to be understood as essental ones. Stll f beng measurable and t >, let f (t) = nf{λ d f (λ) < t}. Here s a lst of some useful propertes, assumng f (n Grafakos [8] Secton 1.4.1 you wll fnd more but mnd the dfference n the defnton): f s non-ncreasng. Indeed, {λ d f (λ) < t 1 } {λ d f (λ) < t 2 } whenever t 2 t 1 so that f (t 2 ) f (t 1 ). d f (f (t)) t. There s an decreasng sequence λ f (t) so that d f (λ ) < t. But d f (f (t)) = {f > λ } = lm d f (λ ) t. =1 {f f (t)} t. If the set has nfnte measure, there s nothng to prove so assume t s fnte. Then {f f (t)} = {f > f (t) 2 } = lm =1 k k {f > f (t) 2 } = lm {f > f (t) 2 k } t k =1 snce each member of the sequence s below nfmum of what would gve measure above t. f () = sup t> f (t) = f L (R n ). Frstly f () = sup nf{λ > d f (λ) < t} nf{λ > d f (λ) = } = f L (R n ) t> just by ncluson. The other drecton follows by contradcton. Suppose f L (R n ) > f (). Assume f () < snce otherwse there s nothng to prove. There s a sequence of t and f (t ) f (). If f () < f L (R n ), then < d f (f ()) = lm d f (f (t )) lm t = whch s a contradcton. The last lne follows along the same lnes as the prevous tems. nf z E f(z) (1 E f )( E ). For any < λ < nf z E f(z) we have d 1E f (λ) = E. Hence d 1E f (λ) < E only f λ nf z E f(z) and consequently f ( E ) nf z E f(z). (f + g) (t) f (θt) + g ((1 θ)t) for all θ (, 1). Let A = {λ d f (λ) < θt} and B = {λ d g (λ) < (1 θ)t}. If then a A and b B, then and d f+g (a + b) d f (a) + d g (b) < t (f + g) (t) nf(a + B) = nf A + nf B = f (θt) + g ((1 θ)t). d f (t) < s f and only f f (s) < t so d f (t) s f and only f f (s) > t. It holds f L p, (R n ) = sup f (t)t 1/p t> as can be seen from the prevous tems. 8

4.3. Controllng oscllaton. Defnton 4.1. Let Q be the famly of all cubes wth sdes parallel to coordnate axes and p >. Let a Q [) be n D r (or satsfy the condton D r ) f for all cubes Q and all collectons {Q } of ts parwse dsjont subcubes a(q ) r Q C a a(q) r Q where C a s a constant only dependng on the functon a but not on Q or the famly Q. Examples nclude The usual rght hand sde of Poncaré nequalty for f W 1,p (R n ) wth p > 1: a(q) = l(q) ( f p 1/p ) 2Q s n D r wth r = pn/(n p) The fractonal averages wth p > and αp [, n) and µ a locally fnte postve Borel measure a(q) = l(q) α ( µ(q) 1/p ) Q are n D r wth r = np/(n αp). The argument for the latter s as follows: Fx Q and Q as n the defnton a(q ) r Q = Q α/n 1/p+1 µ(q ) r/p Q α/n 1/p+1 µ(q ) r/p r/p Q α/n 1/p+1 ( µ(q )) Q α/n 1/p+1 µ(q) r/p = a(q) r Q. We used the exponent n the Lebesgue measure of the cube beng postve and r/p > 1. There s a slght complcaton when usng a(q) = l(q) ( f p 1/p ) 2Q because of the dlaton on the rght hand sde and the above argument must be refned n order to verfy the D r condton for such an functon a. We wll omt that and move on to the man theorem about generalzed Poncaré nequaltes. The theorem as well as the proof are here n the case of R n and Lebesgue measure, but the orgnal theorem n [18] s more general. It apples n spaces of homogeneous type and for weghted measures. Theorem 4.2 (Lerner and Peréz [18]). Let r > and suppose a satsfes D r. Suppose nf [1 Q(f c)] (λ Q ) C λ a(q), λ (, 1) c R holds for all cubes Q. Then nf c Q 1/r 1 Q (f c) L r, (R n ) Ca(Q) for all Q and a constant Q only dependng on r, C a and n. The strength of the assumpton s best understood through the followng Proposton 4.3. Let f be measurable, Q a cube and λ (, 1). Then for any δ >. (f1 Q ) (λ Q ) λ 1/δ ( f δ 1/δ ) Q Proof. Take s so that s < (f1 Q ) (λ Q ). Then d f1q (s) λ Q and for any δ > so that λ Q {x Q f > s} < ( f δ s ) s < ( 1 λ f δ 1/δ ). Q Takng lmt s (f1 Q ) (λ Q ) concludes the proof. 9

Proof of the theorem. Fx Q. We frst show for ɛ > small enough that f t (, ɛ Q /2), then (4.1) (1 Q f) (t) ca(q) ( Q 1/r t ) + (f1 Q ) (2t). The left sde s the level above whch there s t mass, the rght hand sde s the smaller level wth mass 2t above plus an oscllaton term pad for the dfference. Set-up and CZ cover. Defne E = {x Q f(x) (f1 Q ) (t)}. We frst clam that t E 2t. The lower bound s the thrd bullet n the property lst. For the upper bound, note that we may assume f (2t) < f (t) snce otherwse (4.1) s trvally true. Now E d f (f (2t)) 2t by the second bullet of the property lst. Next we let {Q } be the Calderón-Zygmund cubes of 1 E, that s, the maxmal relatvely dyadc subcubes of Q so that E Q > ɛ Q. We have the propertes ɛ Q < E Q 2 n ɛ Q. E Q Q Q Q j = unless = j. Ths s possble, snce E Q = E 2t ɛ Q by our restrcton on values of t to be studed. Indeed, we start dvdng Q nto dyadc subcubes. If a cube gves hgh densty of E, we choose t to be some Q and leave t asde from further subdvsons. The large cube Q gves densty of E small enough so that the process s well defned. Control over Q by CZ cubes. Frst, by the defnton of E, the ffth bullet n the property lst of rearrangements and the fact that f s non-ncreasng, we see (4.2) (f1 Q ) (t) nf f(x) = nf x E nf f(x) nf (f1 E Q ) ( E Q ) nf (f1 E Q ) (ɛ Q ) x E Q where the last nequalty used the lower bound from CZ decomposton. Thnk ɛ as beng small so that the value above s almost the sup. We next control t by almost the nf plus an oscllaton. Precsely, for any c R (f1 Q ) (ɛ Q ) = ((f c)1 Q ) (ɛ Q ) + c and c nf x Q ( f(x) c + f(x) ) [( f c + f )1 Q ] ( Q ) [ f c 1 Q ] (ɛ Q ) + [f1 Q ] ((1 ɛ) Q ). The frst lne s clear from defnton and the second lne used some of the propertes of rearrangements. Now altogether (f1 Q ) (ɛ Q ) 2 nf c [ f c 1 Q ] (ɛ Q ) + [f1 Q ] ((1 ɛ) Q ) and pluggng ths n (4.2), we arrve at (f1 Q ) (t) nf (2 nf [ f c 1 Q] (ɛ Q ) + [f1 Q ] ((1 ɛ) Q )) c Reducton to small oscllaton cubes. Dvde the CZ cubes n two famles. We say I f a(q ) M 1/r a(q) ( Q 1/r t ) where M s a large constant to be chosen. Snce we are estmatng an nfmum over, t wll suffce to fnd even one ndex such that I but eventually we wll need that there are many. Ths s proved as follows. For I, we have the reversed nequalty so I 1 Q Q a(q ) r (Ma(Q) r Q I t ) tc a M where the last nequalty used the D r condton. On the other hand, the CZ condton on the cubes gve Puttng the nequaltes together, we see (4.3) I Q 2 n ɛ 1 Q E = 2 n ɛ 1 E 2 n ɛ 1 t. Q = Q Q t ( 1 I 1 2 n ɛ C a M ) > 2t

by choce of M large enough only dependng on C a and ɛ and n. Hence there must be at least one ndex n I. We conclude t s legal to restrct attenton to cubes n I havng low oscllaton n comparson to the large cube Q, and (4.4) (f1 Q ) (t) nf (2 nf [ f c 1 Q] (ɛ Q ) + [f1 Q ] ((1 ɛ) Q )) Ca(Q) ( Q c t ) so that the frst term s of the desred form. 1/r +nf I [f1 Q ] ((1 ɛ) Q ) Iteraton. It remans to estmate T = nf I [f1 Q ] ((1 ɛ) Q ), whch s of the same form as the startng pont n (4.2) but now restrcted to a smaller cube. Set E = {x Q f(x) (f1 Q ) ((1 ɛ) Q )} Then usng agan the thrd bullet n the lst about rearrangements as well as the lower bound for the measure of Q cubes n the famly I (equalty (4.3)), we conclude provded M s large enough. Now T nf E = E (1 ɛ) Q > 2t nf f(x) = nf f(x) (f1 Q ) ( E ) (f1 Q ) (2t). x E x E Pluggng ths n (4.4) we have proved (4.1). Next choose a postve nteger k such that ɛ2 k 1 Q t < ɛ2 k Q and terate (4.1): (f1 Q ) (t) C ɛ,a a(q) ( Q t ) 1/r k j= 2 j/r + (f1 Q ) (2 k+1 t) C ɛ,a a(q) ( Q 1/r t ) + (f1 Q ) (ɛ Q ). Then note that the functon f = f c wth c R satsfes the hypothess of the theorem whenever f does. Hence we conclude nf ((f c)1 Q) (t) C ɛ,a a(q) ( Q 1/r c t ) + nf ((f c)1 Q) (ɛ Q ) C ɛ,a a(q) ( Q 1/r c t ). whch concludes the proof for t < ɛ Q /2. For t ɛ Q /2 the clam s trval. 4.4. Truncaton argument. The prevous theorem s an example of an argument yeldng an estmate wth weak L r norm on the left hand sde. In certan stuatons of practcal nterest, such estmates can be mproved to strong L r bounds. In partcular, f a(q) = l(q) ( Q f p ) 1/p, we can apply what s called Maz ya s truncaton argument. Presentaton follows mostly Haj lasz [12]. Take t 1 < t 2 and a measurable functon u. Let t 1, u(x) t 2 ũ t2 t 1 (x) = u(x), t 1 < u(x) < t 2 t 2, t 1 u(x) and u t2 t 1 (x) = ũ t2 t 1 t 1. Proposton 4.4. Let A k = {x Ω u(x) > 2 k } for some measurable Ω R n. Fx q > p and assume for a par of measurable functons (u, g) and all k Z 1 Ω u 2k+1 2 k L q, (R n ) C g Lp (A k A +1). Then u Lq (Ω) C g Lp (Ω) where the constants C need not be same but they only depend on q, n and p. Proof. u q 2 qk {x Ω 2 k 1 < u(x) 2 k } 2 qk {x Ω 2 k 2 < u(x) 2 k 2 } Ω k Z k Z 2 qk {x Ω 2 k 2 < u 2k 1 2 k 2} C ( g p q/p ) k Z k Z Ak A k+1 11 q/p C ( g p ) g q/p L k Z Ak A p (Ω). k+1

Proposton 4.5. If u W 1,p loc (Rn ), p 1 and a < b, then u b a W 1,p loc and (4.5) nf Q 1/p 1 Q (u b a c) L p, (R n ) Cl(Q) ( 1 1/p Q u p ) 2Q {a< u b} where p = np/(n p). Proof. We only prove the case p > 1. Recall the truncaton property u ± = 1 u± u from Proposton 2.5. We restrct the attenton to postve u and wrte M = max(u, a) = (u a) + + a and m = mn(u, b) = (u b) + b so that u b a = M + m u W 1,p loc. Hence by Poncaré s nequalty 2.7 and Hölder s nequalty u b a (u b Q a) Q Cl(Q) u b 2Q a l(q) ( (M 1 2Q u b Q a ) p 1/p ) = a(q). The rght hand sde clearly satsfes D p so by Theorem 4.2 and the maxmal functon theorem nf Q 1/p 1 Q (u b a c) L p, (R n ) Cl(Q) ( 1 1/p Q u p ). 2Q {a< u b} Theorem 4.6 (Sobolev-Poncaré). Let u W 1,p loc (Rn ) wth p 1. Then for 1/n = 1/p 1/p and all cubes Q. 1/p ( u u Q p ) Cl(Q) ( u p 1/p ) Q 2Q Proof. We only present proof for the case p > 1. We can restrct the attenton to postve part of u and hence asssume u. Take λ such that {u λ} Q Q 2 and {u λ} Q Q 2. Let v = (v λ) + so that u λ = v + (u λ). We am at estmatng Q v p. Suppose ṽ s such that {ṽ = } Q Q /2. Then for any t > and c R t follows {ṽ > t} { ṽ c > t} + {ṽ, c > t/2} = { ṽ c > t} + {ṽ } 1 { c >t/2} { ṽ c > t} + {ṽ = } 1 { c >t/2} = 2 { ṽ c > t} Ths holds n partcular for all truncatons v b a wth < a < b. Then by equaton (4.5) {va b > t} t p 2 nf { v b c tp a c > t/2} Cl(Q) p ( 1 p /p Q u p ) 2Q {a< u b} whence by Proposton 4.4 v p l(q) p ( 1 p /p Q Q u p ). 2Q Snce the same argument apples to (u λ), we have from whch the clam follows snce u λ p l(q) p ( 1 p /p Q Q u p ) 2Q Q u u Q p Q u λ p + λ u Q p. Corollary 4.7. Suppose u W 1,p (R n ) and p 1. Then Proof. The prevous nequalty s equvalent to u L p (R n ) u W 1,p (R n ). 1/p ( u u Q p ) C ( u p 1/p ) Q 2Q where we can send l(q) to obtan the clam. Note that u Q ( u p ) 1/p as l(q). 12 Q

The value p = 1 can be ncluded n the Sobolev Poncaré nequalty but ths requres ether more careful proof of the Poncaré nequalty wth no dlaton on the rght hand sde or a more general verson of the Lerner Pérez theorem, none of whch were done. Note also that for mere applcaton to Sobolev Poncaré nequalty the tools n ths secton are unnecessarly complcated. However, they wll fnd use also elsewhere. 5. Hardy Sobolev spaces 5.1. Motvaton. We have been tryng to understand smoothness as somethng that allows to control mean oscllatons n small scales or approxmate by constant wth good error bound. When p 1, ths property of Sobolev functons was made precse by Poncaré s nequalty. When p < 1, the L p norm of the dervatve s not enough to control mean oscllaton. Indeed, let φ C 1 (R) satsfy, x 1 φ(x) = 1, x 1 Take the dlate ψ ɛ = ɛ 1 φ( ɛ 1 ). Then for arbtrarly small ɛ > but 1 1 nf c ( 1 1 (φ ɛ ) (x) p = ɛ 1 p 1/p φ ɛ c ) p 1 1 1 φ (x) dx ɛ 1 p as ɛ. Hence a Poncaré nequalty wth a dervatve on the rght hand sde s not possble unless p 1. On the other hand, we saw n connecton wth the Calderón maxmal functon that a maxmal functon control s always true. It wll turn out, that N 1 p f beng n L p s equvalent wth dstrbuton (or actually functon) f L p wth p > n/(n + 1) havng dervatves n Hardy space. 5.2. Hardy spaces. Let C N = {ψ C supp ψ B(, 1), sup α ψ(x) 1, for all α N, ψ = 1}. x For ϕ C N wth N 2, let a > and defne the maxmal functons actng on dstrbutons f M ϕ f(x) = sup ϕ t f(x) t> Mϕf(x) a = sup ϕ t f(y), x y <at M N f(x) = sup{m 1 ϕf ϕ C N }. Theorem 5.1. Fx p > and a >. Let ϕ, ψ C N wth N large enough dependng only on a and p. Then M ϕ f Lp (R n ) a M a ψf Lp (R n ) a,n M N f Lp (R n ). For proof, see Grafakos [9] Theorem 6.4.4. Defnton 5.2. Let η = c1 B(,1) e 1/(1 x 2). For f a dstrbuton, defne and H p = {f f Hp (R n ) < }. The few propertes and observatons: f Hp (R n ) = M η f Lp (R n ) H p s a quas-normed space. Contnuous functons are dense n H p. For ϕ ɛ a smooth approxmaton of dentty, the lmt lm ɛ ϕ ɛ f(x) exsts almost everyhwere ndependently of the exact choce of ϕ. For p > 1, H p = L p. For p 1, ths s not the case. A compactly supported and bounded f fals to be n H p unless f =. In the case of mean zero f H p wth p > n/(n + 1). Membershp n lower ndex Hardy spaces comes wth more vanshng moments. 13

5.3. Hardy Sobolev spaces. What follows s mostly a smple case of more general treatment n Myach [2]. He deals wth domans more general that the full space and classes of functons dfferentable more than once. Defnton 5.3. A dstrbuton f s sad to be n the homogeneous Hardy Sobolev space Ḣ1,p (R n ) f k f H p (R n ) for all k = 1,..., n. We defne a semnorm as n f Ḣ1,p (R n ) = k=1 k f Hp (R n ). The semnorm does not see dfference by constant. There are ways to defne a proper norm, but the choce of the non-homogeneous part dependes n the applcaton one has n mnd. For our purposes, the most natural defnton wll be L p Ḣ1,p. Once could also thnk about H p Ḣ1,p, but ths choce mposes a zero-cancellaton condton on the functon tself. In the study of dfferentablty, such a condton would just be an unnecessary complcaton. Lemma 5.4. Let Q be a cube and ψ smooth wth supp ψ Q and ψ =. Then there are v k smooth and supported n Q such that ψ = l(q) n k=1 k ψ. Moreover for all α m and some m. If α ψ l(q) n α, then also α v k Cl(Q) n α Proof. We normalze Q = [ 1, 1] n. The general clam follows by change of varables. Let g C, supported n [ 1, 1] and wth g = 1. For f wth f = and support n [ 1, 1], let T1 1 f = t f(s) ds. Hence we have the clam for n = 1 and can assume t for dmenson n 1 as we prove t for hgher dmensons by nducton. Suppose we have defned operators Tk n 1 for all k = 1,..., n 1. For f Cc ([ 1, 1] n ) havng mean zero. Set f(x ) = R f(x, t) dt x R n 1, T n k f(x, t) = g(t) T n 1 k T n n f(x, t) = t f(x ), for k = 1,..., n 1 (f(x, s) g(s) f(x )) ds. We take v k = T n k ψ. Gven the constructon, checkng the propertes s easy.suppose we already have the clam for dmenson n 1. Then n k=1 Tk n ψ(x, x n ) = ψ(x, x n ) g(x n )ψ(x n 1 ) + g(x n ) by nducton. Dervatve bounds are deduced smlarly. Proposton 5.5. Let f Ḣ1,p. Then k=1 T n k ψ = ψ(x, x n ) For any sequence ɛ and smooth, compactly supported ϕ wth ϕ = 1, the lmt lm ϕ ϕ ɛ f(x) exsts almost everywhere and s ndependent of the precse choce of ϕ. For ϕ as above and < ɛ < δ, t holds for every x. f (ϕ ɛ ϕ δ )(x) δm k f(x) Proof. We start wth the second tem. Fx k to be an nteger such that 2 k δ ɛ < 2 k+1 δ and let ϕ = ϕ 2 δ for all {,..., k 1}. Wrte ϕ ɛ ϕ δ = ϕ ɛ ϕ k + (ϕ ϕ 1 ) = k =1 k Φ = where Φ = ϕ ϕ 1 for < k and Φ k = ϕ ɛ ϕ k. Now each Φ satsfes the assumptons of the prevous lemma so there are functons v j, wth n Φ = 2 δ j v j,. 14 j=1

Hence (ϕ ɛ ϕ δ ) f(x) δ k = 2 n j=1 j v j, f(x) δ M N k f(x). k Ths concludes the proof of the second tem. It also shows that the sequence ϕ ɛ (x) s Cauchy whenever M N k f are fnte. Hence there s a lmt. Smlarly, t can be used to show that the lmt does not depend on the approxmate dentty n queston. Lemma 5.6. Let f Ḣ1,p (R n ) L p (R n ) wth p >. Then for Q a cube, c(q) ts center, ϕ C N and almost every x Q f(x) ϕ l(q) f(c(q)) l(q) M N k f(x). Proof. We may assume Q = 1. Choose a pont x such that the approxmatons ϕ ɛ f(x) f(x) converge as ɛ. Take a sequence of nested cubes Q Q 1 such that Q = Q, l(q ) = 2 and x c where c = c(q ) s always mnmzed so that from some ndex on c = x. where f(x) c = (f ϕ 2 1(c 1 ) f ϕ 2 (c )) = f Φ (c ) = n k=1 Φ (y) = ϕ 2 1(y + c 1 c ) ϕ 2 (y). By the prevous lemma, there are v k such that Φ = 2 n k=1 k v k wth v k satsfyng good bounds. Consequently = f Φ (c ) 2 n = k=1 5.4. Characterzaton wth Calderón maxmal functon. = ( k v k f)(c ) M N k f. Proposton 5.7. Let f Ḣ1,p (R n ) L p (R n ). Then N 1 p f Lp (R n ) f Ḣ1,p (R n ) Proof. Take q < p so that p > nq/(n q). By the prevous lemma, nf c 1 Q ( 1/n f c q 1/q ) Q n k=1 n k=1 ( M N k f q 1/q ). Q After multpcaton by Q 1/n, the rght hand sde satsfes the summablty condton D q wth q = nq/(n q). Hence we can apply Theorem 4.2 to conclude Denote f = 1 Q (f c). For Q f p = p nf c 1 Q (f c) L q, (R n n ) Q 1/q d f (t)t p 1 dt = p Q T p q + f one may optmze the rght hand sde to see so that nf c k=1 L q, (R n ) T ( M N k f q 1/q ). Q 1 Q ( 1/n f c p 1/p 1 Q (f c) L q ) nf, (R n ) Q c Q 1/q n Np 1 f M q M N k f. k=1 t p q 1 dt Q T p q + f T p q, L q, (R n ) Here M q f = (M f q ) 1/q and now the clam follows form the maxmal functon theorem as q < p. As useful pece of the prevous proof s the followng Corollary 5.8 (of the proof.). Let N α p f L p (R n ) and q < pn/(n αp) (5.1) N α q f(x) M p N α p f(x) for all x. Proposton 5.9. Suppose p > n/(n + 1) and N 1 p f L p (R n ). Then k f H p (R n ) for all k = 1,... n. 15

Proof. Frst we use (5.1) to see N 1 1 f M q N 1 q M q N 1 p for any q (n/n + 1, p). By the assumpton, the rght hand sde s n L p (R n ) and hence almost everywhere fnte. The same goes for the left hand sde an f L 1 loc (Rn ) as a consequence. Take ϕ δ to be the usual approxmaton of dentty. Then Now ϕ δ k f(x) = nf c R δ n 1 (f(y) c)( k ϕ)( y x ) dy nf δ c δ 1 f c N1 1 f(x). B(x,δ) M ϕ k f N 1 1 f M q N 1 p L p (R n ). At ths pont we gve a name to the functon space defned through N α p. Defnton 5.1. Let α > and let P α be the space of all polynomals of degree at most k < α. Defne We set Np α f(x) = sup nf r α ( f P ) P P α B B x Ths s sometmes called Calderón Hardy space C α p = {f L p N α p f L p }. We wll be mostly nterested n the case α 1 so that the polynomals are of order zero. The prevous propostons can be summarzed now as Corollary 5.11. whenever p > n/(n + 1). L p (R n ) Ḣ1,p (R n ) = C 1 p 5.5. Pontwse characterzatons. Next we wrte a pontwse formula n the sprt of the Morrey embeddng theorem to characterze the spaces C α p. In the case of usual Sobolev spaces W 1,p (R n ) wth p > 1, the formula below was used gven n Haj lasz n [1] and later used as a defnton for Sobolev spaces n more general metrc measure spaces. The meanng of the formula for p = 1 remaned open for a whle untl t was connected to Hardy Sobolev space by Koskela and Saksman n [16] Theorem 5.12. Let α (, 1] and p > n/(n + α). Suppose f L p (R n ). Then N α p f L p (R n ) for all k f and only f there s g L p (R n ) such that for almost all x, y R n t holds 1/p f(x) f(y) C x y α (g(x) + g(y)). Proof. Assume frst N α p f L p. By (5.1) N α 1 f L p. For any Lebesgue pont z (almost every pont by local ntegrablty) and any R >, we wrte f(x) f B(x,R) k=1 B(x,2 k R Then for R = x y and almost every x and y f f B(x,2 k R) R α 2 kα N1 α f(x). k=1 f(x) f ( y) f(x) f B(x,R) + f B(y,R) f B(x,R) + f B(y,R) f(y) x y α (g(x) + g(y)) where g = N α 1 f. To prove the converse, take q < p and note that for any cube Q wth center c Q α/n ( f f(c) q ) ( g g(c) q 1/q ) M q g(c). Q If qn/(n qα) > p as t can be chosen to be, we conclude N α p f M q M q g L p (R n ).. 16

6. Embeddngs for Calderón Hardy spaces 6.1. Hgher order spaces. Ths s just a bref comment on values of the smoothness parameter α > 1 we have been neglectng. As ndcated n defnton 5.1, there s an extenson of Np α operator for these values usng polynomal approxmaton n stead of mean oscllaton. It s routne to verfy that Sobolev spaces W k,p can be defned here, ther members satsfy a Poncaré nequalty of type nf f P r k β f, P P k 1 B 2B where P k 1 are the polynomals of degree at most k 1, and there s an analogue of Haj lasz lemma 3.5. The concdence wth the correspondng Calderón Hardy spaces C k p s also obvous as p > 1 and eventually doable as p > n/(n kp) though not as a verbatm repetton as the the case p > 1. From pont of vew of C α p, we do not study these hgher order spaces n detal, but f need be, the book [5] as well as the paper [2] contan a lot of addtonal nformaton. 6.2. Fractonal ntegral. Defnton 6.1. Let s >. Defne k s (x) = x n+s. For f C c a pror, we defne β k I s f = k s f. k s s called Resz kernel and I s Resz potental or fractonal ntegral. If s < n and f L p wth p > 1, we can also defne I s f drectly. Proposton 6.2. Let p > 1 and sp (, n), p = np/(n sp). Then for all f L p I s f p f L p. Proof. Let R >. Then I s f(x) = I s (1 B(x,R)f + E. Let B = B(x, 2 +1 R). Then I s (1 B(x,R)f and k=1 Bk B k+1 f(y) x y dy k=1 (2 k R) n s f(y) dy R s 2 ks Mf(x) = R s Mf(x) Bk B k+1 E f L p ( x y >R x y p(s n) p 1 ) 1 1 p f L pr s n/p. Choosng R such that R s Mf(x) f L pr s n/p and applyng the Hardy Lttlewood maxmal functon theorem we conclude the proof. Defnton 6.3. Let s [, n). For f L 1 loc, we set M s f(x) = sup r s f. B x B The maxmal functon shares the L p mappng propertes of the fractonal ntegral. Proposton 6.4. M s f I s f Proof. Frst not that the supremum B x can be replaced by supremum over r > for B = B(x, r) n the defnton of the fractonal maxmal functon at cost of a dmensonal constant. Take B(x, r) and let B k = B(x, 2 k+1 r). Then k=1 r s f = Cr s n f(y) dy B k=1 Bk B k+1 whence the clam follows by takng a supremum over rad. k=1 Bk B k+1 f(y) x y n s dy I s f Corollary 6.5. M s L p L p whenever sp (, n) and p = np/(n ps). Proof. Follows from the same bound for the fractonal ntegral. 17

6.3. Inequalty for Calderón functons. Theorem 6.6. Let p >, α β > and q > p such that α β n = 1 p 1 q. Then N β 1 f L q N α 1 f L p Proof. Let B = B(x, r). Then for any δ > nf c R r β f c r α β nf N 1 α f(y) r α β (N1 α f δ ) 1/δ (M γ (N1 α f δ )) 1/δ B y B where γ = δ(α β). Now choose δ < p so that M γ L p/δ L q/δ as γ/n = δ(α β)/n = δ(1/p 1/q). Then N β 1 f L q M γ (N α 1 f δ ) 1/δ L q/δ N α 1 f L p. We can lst some nterestng specal cases easest usng the homogeneous spaces defned usng only the semnorms takng nto account the dervatves of the functon. Corollary 6.7. The followng contnuous embeddngs hold: For k > l ntegers and q > p > 1 wth (k l)/n = 1/p 1/q we have Ẇ k,p Ẇ l,q. For k/n > 1/p, denote l = max{j Z j < k/n 1/p} and σ = k/n 1/p l. Then Ẇ k,p Ċl,σ (l tmes contnuously dfferentable wth σ-hölder contnuous lth dervatves). For 1/n = 1/p 1/q wth p > n/(n + 1) we have Ḣ1,p L q. 7. Lttlewood Paley theory 7.1. Schwartz class. Here s a lst of most mportant propertes to be recalled. A good source for everythng n ths secton s Grafakos [8], chapters 2 and 5 n partcular. For the purposes of ths secton, we start assumng our functons are complex valued n contrast to the real valued consderatons n the prevous sectons. For α, β N n and f C, we let ρ α,β (f) = sup x R n x α β f(x) where x α = x α. If ρ α,β (f) < for all α and β, then we wrte f S and call t a Schwartz functon. A sequence f j f n the sense of Schwartz class f ρ α,β (f f j ) for all α and β. Clearly S s complete under ths noton of convergence. Also, n topology consstent wth ths convergence, multplcaton by polynomals as well as dfferentaton are contnuous operatons. Fnally Cc S and n partcular t s dense n all L p spaces. When dealng wth lnear or sublnear operators, t suffces to work wth Schwartz functons. If not explctly otherwse stated, all functons are supposed to be Schwartz. Together wth the Schwartz class comes the class of tempered dstrbutons that has the same role n ths theory as the dstrbtuons have for compactly supported functons. The crucal dfference s that tempered dstrbuton cannot grow too fast. For nstance, e t defnes a dstrbuton on real lne but not a tempered one. 7.2. Fourer transform. For f L 1, we wrte Ff(ξ) = f(ξ) = e 2πx ξ dx for ξ R n. Basc propertes that are repeatedly used are for f, g S, F(f g) = fĝ. for α N n, F( α f)(ξ) = (2πξ) α f(ξ) and F(( 2πx) α f) = ( α f)(ξ). Defne for p 1 and δ > the dlaton (Dl p δ f)(x) = δ n f(xδ 1 ). Then F(Dl p δ f)(ξ) = f Dl1/p δ 1 where p = p/(p 1). For translatons and modulatons we have that for a R n t holds e 2πa ξ f(ξ) = F(f( a))(ξ). fĝ = fg There s nverse F 1 f(ξ) = ˇf(ξ) = f( ξ). Fourer transform s a homeomorphsm of the S. fg = fĝ and f L 2 = f L 2 so that t s sometry n L 2 (a pror ths s norm nequalty for Schwartz functons whch allows to extend to L 2 ). 18

It holds f L f L 1 and by nterpolaton f L p f Lp whenever p [1, 2]. No nontrval compactly supported functon has ts Fourer transform compactly supported. Proposton 7.1 (Bernsten). Let R > and suppose f s supported n B(, R). Then for all q > p 1 and a dmensonal constant C For any mult-ndex α N n f L q CR n(1/p 1/q) f L p. α f Lp CR α f Lp. Proof. Let ϕ C be such that 1 B(,1) ϕ 1 B(,2). Let η = Dl 2R ϕ. Then by Young s convoluton nequalty where 1 + 1/q = 1/r + 1/p. Now f L q = ˇη f L q ˇη L r f L p η L r = Dl 1 (2R) 1 ˇϕ L r = Rn(1 1/r) C ϕ whch proves the frst clam as 1 1/r = 1/p 1/q. The second clam s done smlarly and the proof s omtted. 7.3. Lttlewood Paley decomposton. Let ϕ be smooth wth 1 B(,1) ϕ 1 B(,2) and set ϕ (x) = ϕ(2x) ϕ(x) and for j Z set ϕ j = Dl 2 j ϕ. Then supp ϕ j {2 j 1 x 2 j+1 } and ϕ j (x) = 1 whenever x. j We also denote K j = ˇϕ j and P j f = K j f s the Lttlewood Paley operator projectng f to frequences aroung 2 j. Note that each K j s an L 1 normalzed kernel lvng at scale 2 j and havng mean value zero (even arbtrarly many vanshng moments). The formula P j f = f j s very smlar to the telescopng sum of mean values argument we have been usng many tmes before. 7.4. Calderón Zygmund theory. There are certan nstances of Calderón Zygmund theory that are needed. Here are the man theorems. The proofs are not repeated here though they were dscussed n lectures. The books [8] and [9] contan all. Another good source s the lecture notes for Harmonc Analyss here https://stes.google.com/ste/oannsparsss/teachng. Theorem 7.2. Let T L 2 L 2 be lnear and bounded. Suppose T f g = f(x)k(x, y)g(y) dxdy, whenever supp f supp g = where K(x, y) L 1 loc (Rn R n {(x, x)}) and there are constants and α (, 1] wth K(x, y) x y n, for x y K(x, y) K(x, y) x x α x y n+α, for x x 1 x y 2 and the same propertes holds for the adjont T (that s, T s Calderón Zygmund operator). T H p H p for all p > and T L 1 L 1, Then We wll not prove ths but just quote t. The regon p > 1 and the weak endpont are explaned n Theorem 8.2.1 n [9]. The H p bounds are, for nstance, Theorem 6.4.14 n the same book. There are two specal cases we wll need frequently. Corollary 7.3 (Mkhln-Hörmander theorem). Let m L C n/2+1 be such that α m(ξ) ξ α for all mult-ndces α n/2 + 1. Then f F 1 (m f) s bounded L p L p whenever p > 1. Corollary 7.4. Let a l. Then f j a j P j s bounded L p L p for p > 1. 19

7.5. Khntchne s nequalty. We want to understand a characterzaton of the L p norm through the followng. There s a contnuous embeddng of the sequence spaces l p l q whenever q > p. Introducng random sgns results n un-dong the sze dfference between l 1 and l 2 sums. The l 2 way of sayng ths s Khntchne s nequalty. Proposton 7.5 (Khntchne). Let (a ) N =1 be complex numbers and ω ndependent random varables takng values { 1, 1} wth equal probablty. Let p >. Then 1/2 (E ω a p ) 1/p p ( a 2 ) where the hdden constant s n partcular ndependent of N. Proof. For smplcty, assume a are real. We start by estmatng the expectaton. Let f = ω a. For any t >, by ndependence and zero expectaton of the random varables Ee tf 1 = E e tωa = Ee tωa = 2 (eta + e ta ) e 1 2 t2 a 2 1 = e 2 t2 a 2. Also, P(f > λ) = P( f > λ) so by Tschebyschev and the precedng remark P( f > λ) = 2P(f > λ) = P(tf tλ > ) = P(e tf tλ > 1) 2e tλ Ee tf 2e tλ e 1 2 t2 a 2 = e 1 2 λ2 a 2 when we choose t = λ( a 2 ) 1. Then by Cavaler prncple p/2 E f p = p λ p 1 P( f > λ) dλ 2p λ p 1 e λ2 /( a 2 ) dλ = C p ( a 2 ) whch concludes the estmate. For the other drecton, we note that for p = 2 the comparablty holds as an equalty. Also, by Hölder s nequalty for probabltty spaces the case p > 2 s clear. For p (, 2) take s > 2 and θ (, 1) such that 1/2 = θ/s + (1 θ)/p. Then (the followng are wth respect to the probablty measure f L 2 f 1 θ L p f θ L s Cθ s f 1 θ L p f θ L 2 where the last step used the frst part of the theorem. Now hdng the L 2 norm on the left concludes the proof. Theorem 7.6 (Lttlewood Paley theorem). Let P j be the Lttlewood Paley peces. If f L p wth p > 1, then P j f 2 f L p. j Z 1/2 Lp If f s a tempered dstrbuton and the left hand sde s fnte, then there s a polynomal P such that P j f 2 j Z Proof. By Fatou, Khntchne, and Corollary 7.4, P j f 2 j Z 1/2 Lp lm nf N P j f 2 j N 1/2 Lp 1/2 Lp f P L p. lm nf (E N j N p ω P j f ) 1/p lm nf (E N f p L p)1/p L p = f L p. Conversely, we know that P j = (P j 1 + P j + P j+1 )P j so set F j = P j 1 + P j + P j+1. For f S we know that supp f(1 j ϕ) = {}. Such and expresson must be a polynomal P (ths would requre a proof, not too complcated, but we skp t, see [8] Secton 2.4.1). Let be the dualty parng of Schwartz class and let g be a Schwartz functon. The Lttlewood Paley decomposton converges as tempered dstrbuton so the followng can be justfed f P, g = F j P j f, g = lm P j f, F j g j N j N P j f 2 j Z P j f 2 j Z 1/2 Lp F j g 2 j Z 2 1/2 Lp 1/2 P j f 2 j Z F j g 2 j Z 1/2 Lp 1/2 g L p

hence f P extends to a bounded lnear functonal on L p whch proves the clam. A fact: One can use the Lttlewood-Paley square functon when p (, 1]. Then 1/2 Lp P j f 2 nf j Z f P H p. P We won t prove ths, but the argument contans few addtons to what we have seen. However, the theory of sngular ntegrals on Hardy spaces should be used nstead of that for L p spaces. 8. Functon spaces The Lttlewood Paley theorem and Hörmander Mkhln theorem gve a systematc way to generate functon spaces wth varyng smoothness requrements. We mostly lst dfferent defntons and then study n detal what turns out to be most commonly used fractonal Sobolev space or Sobolev Slobodeckj space. 8.1. Resz potentals. Recall the fractonal ntegrals defned through I s f = k s f where k s (x) = x s n. Suppose s (, n/2) so that k s s locally n L 1 and decays at nfnty faster than x n/2. Then k s L 1 +L 2 and we can tell ts Fourer transform s n L +L 2. Then recall (or check) that Fourer transform of radal dstrbuton s radal and Fourer transform of dstrbuton homogeneous of degree d s homogeneous of degree n d. These symmetres nal down the form C x s for the Fourer transform of k s. By Fourer nverson we can extend ths to s (n/2, n). More generally, we can defne the I s as multpler wth symbol ξ s when t acts on Schwartz functons f such that ξ s f(ξ) s locally ntegrable or wth a lenghter computaton we can dentfy t wth a convoluton by C x s n that s well defned for all Schwartz functons. F(I s f) = c ξ s f(ξ) as was dscussed. Ths s the Resz potental. See above for ssues wth the sngularty at orgn. F(( ) s/2 f) = c ξ s f(ξ) as can be seen to be consstent wth the case s an nteger. Fractonal powers of Laplacan can be nverted up to polynomals by Resz potental. F(R j f) = cξ j ξ 1 f(ξ). Ths operator s called the Resz transform and t s Calderón Zygmund operator n the sense of the Theorem 7.2. F(J s f) = (1 + ξ 2 ) s/2 f(ξ) s called the Bessel potental. It has the effect of ncreasng smoothness by s dervatves but stll beng bounded L p L p unlke I s. 8.2. Potental spaces. We start wth the homogeneous spaces. Denote by Ẇ k,p the space of tempered dstrbutons modulo polynomals so that the kth order dervatves are n L p. Proposton 8.1. Let k > be nteger and p > 1. Then Ẇ k,p = {I k f f L p }. Proof. Take frst f L p. Consder the tempered dstrbuton I k f. Let α be a mult-ndex wth α = k. Then α I k f = R α1 1 Rαn n f so that α I k f L p = R α1 1 Rαn n f L p f L p. Conversely, take f Ẇ k,p. Then there s a polynomal P of degree at most k 1 so that f = f P has Fourer transform zero at zero. Then ( f) k/2 f = ( f) k/2 f = c( f) k/2 ( Rj 2 ) k f L p where we used the dentty for Resz transforms clear from the Fourer symbol as well as boundedness of the Resz transforms. More generally, we can gve a defnton of potental spaces of non-nteger order of smoothness. Defnton 8.2. We let Ḣs,p be the collecton of tempered dstrbutons that are of the form I s f for f L p. Ths space s normed through f Ḣs,p = ( ) s/2 f L p for p >. Ths s the homogeneous potental space We defne the nhomogeneous potental space as mage of J s when p > 1: f H 1,p f f = J s g for some g L p. We norm ths through f H 1,p = (1 + ( ) 1/2 ) s/2 f L p The nhomogeneous space can be proved to be Ḣs,p L p. We delberately omt the defnton of nhomogeneous potental space when p 1 as there are at least two reasonable canddates for defnton and not so clear reason to favor one of them. 21 n j=1

Theorem 8.3. Let p > 1 and s R and f Ḣs,p. Then Let f H s,p, then f Ḣs,p j Z(2 js P j f ) 2 1/2 Lp f H s,p f L p + j (2 js P j f ) 2. 1/2 Lp Proof. Ths s very smlar to the characterzaton of L p. The upper bound for the square functon s obtaned through replacng the Lttlewood Paley pece ϕ (ξ) by ϕ (ξ) ξ s. The lower bound follows the lnes of the lower bound for L p.. Theorem 8.4. Let s n/p = σ n/q wth < s σ < n and p, q (, ). Then Ḣs,p Ḣσ,q Proof. Ths follows from H p H q bounds for the fractonal ntegrals. We only deal wth the case p > 1 for whch we have proved the relevant bounds for fractonal ntegral ( ) σ/2 L q = ( ) (σ s)/2 ( ) s/2 f L q = I s σ ( ) s/2 f L q ( ) s/2 f L p = f Ḣs,p. 8.3. Besov and Trebel Lzorkn scales. The characterzaton of L p was an L p norm of l 2 norm of the Lttlewood Paley peces. Replacng the Khntchne gven 2 by somethng else, we get some other spaces, whch go by the name of Trebel Lzorkn spaces: F F = p,q s j Z(2 js P j f ) q 1/q Lp, s R, p, q (, ). The most mportant case s q = 2 whch gave Hardy spaces and L p spaces F p,2 and homogeneous potental spaces F p,2. s One could also defne the non-homogeneous verson by cuttng the sum and addng the L p norm on the rght hand sde. However, we wll not use that n future. Whenever p = q one may change the order of sum and ntegral. Ths brngs us to Besov scale whch can be defned n ts own rght for more general values. Here we prefer the non-homogeneous verson f B s p,q = f L p + (2 js P j f L p) q j 1 The most mportand value of q s equal to p. Embeddngs for Besov spaces are easly proved. 1/q, s R, p, q (, ). Proposton 8.5. Let s n/p = σ n/q, s > σ and p, q [1, ). Then B s p,r B σ q,r. Proof. For each pece P j f L q 2 jn(1/p 1/q) P j f L p = 2 j(s σ) P j f L p by Bernsten s nequalty. Hence j 1(2 jσ P j f L q) r 1/r (2 js P j f L p) r j 1 For the other part, we frst wrte ψ = 1 j 1 ϕ j. Then we decompose f = f 1 + f 2 where f 1 = ˇψ f. For f 1, we have by Bernsten and Young f 1 L q f L p. For the other term, we take M > max(r, p, q, 2) and use the Lttlewood Paley theorem as well as Hölder s nequalty and Mnkowsk to obtan f 2 L q P j f 2 j 1 1/2 Lq (2 j(ɛ+s σ) P j f L p) M j 1 j 1(2 jɛ P j f ) M 1/M f B s p,r. 1/M Lq 1/r. (2 jɛ P j f L q) M j 1 1/M 22

8.4. Fnte dfferences. For f a measurable functon y R n and m > 1 nteger, let the dfference operators be y f(x) = 1 yf(x) = f(x) f(x + y), m y f(x) = y ( m 1 y f)(x) and for p > and t > defne the modulus of smoothness ωp m (t, f) = sup m y f L p. y <t Theorem 8.6. Assume s >, m, N N wth m+n > s and N < s. Then for p [1, ] and q [1, ] f B s p,q f L p + ( n j=1 (t N s ω m p (t, N j f)) q dt t ) 1/q. Proof. We wll prove the theorem only n the case s (, 1) and p, q (1, ). The more general case follows along the same lnes but wth an addtonal layer of techncaltes. However, before startng, let us notce the dervatves n the rght hand sde exst under the assumpton of the left hand sde beng fnte. Indeed, we have the embeddng B s p,q F N p,2 = H N,p because of the condton s > N: Let f j = P j f be the Lttlewood Paley projectons. Suppose frst p < 2, then by the contnuous embeddng l p l 2 of sequences ( k= 1/2 2 2Nk f k 2 ) If p > 2, then by Mnkowsk s nequalty ( k= 1/2 2 2Nk f k 2 ) L p ( k= 1/p 2 pnk f k p ) L p = ( L p = 2 2Nk f k 2 1/2 ( L p/2 k= k= k= 1/p 2 pnk f k p L ). p 2 2Nk f k 2 L p) 1/2. Fx then r = max(q/p, q/2, 2) > 1 so that 2r q and pr 2. For the case p > 2, we use Hölder as well as l q l 2r to see 1/2 ( 2 2Nk f k 2 1/2 L p) = ( 2 2k(N s) 2 2sk f k 2 )1/r L p) ( 2 2k(N s)r ( k= k= k= C r,n s ( k= k= 2 2rsk f k 2r L p) 1/(2r) 1/(2r) 2 qsk f k q L ) <. p The case p 2 s done exactly the same way but pr 2 beng used nstead of 2r q. Usng these two estmates we see 1/2 f F N p,2 = f L p + ( 2 2Nk f k 2 ) L p < whch proves that f F N p,2 = H N,p so that t has weak Nth dervatves n L p. To start the actual proof, we frst not that t ω(t, f) s ncreasng. Hence (t s ω p (t, f)) q dt t = 2 k+1 (t s ω p (t, f)) q dt k Z 2 k t (2 sk ω p (2 k+1, f)) q 2 k+1 dt k Z 2 k t k= (2 s(k+1) ω p (2 k+1, f)) q. k Z The same computaton can be done the other way round so we fnd (8.1) (t s ω p (t, f)) q dt t (2 ks ω p (2 k, f)) q ) k Z so that t suffces to deal wth the dscrete sum nstead of the ntegral on the left. Left controlls rght. Let P k f = ˇϕ k f be the Lttlewood Paley peces and ψ a smooth functon wth 1 B(,1) ψ 2 B(,2). To prove an L p bound for y P k f, t suffces to assume f s Schwartz as the operators are lnear, and we can take lmts to upgrade the a pror bound for nce functons to a general result L p functons. Now the Fourer transform can be wrtten as so that F( y P k f)(ξ) = ϕ k (ξ) 1 e 2πξ y f(ξ) ξ y ξ y y P k f = T y (y P k f), where F(T y g) = 1 e 2πξ y ĝ(ξ) ξ y 23