LECTURE NOTES ON SOBOLEV SPACES FOR CCA

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1 LECTURE NOTES ON SOBOLEV SPACES FOR CCA WILLIE WAI-YEUNG WONG.. References. Before we start, some references: D. Gilbarg an N. S. Truinger, Elliptic partial ifferential equations of secon orer, Springer. Ch. 7. L. Evans, Partial ifferential equations, American Math. Soc. Ch. 5. M. E. Taylor, Partial ifferential equations I, Springer. Ch. 4. (Note: this presentation is base on heavy oses of Fourier analysis an functional analysis.) H. Triebel, Theory of function spaces, Birkhauser. Ch. 2. R. Aams, Sobolev Spaces, Acaemic Press..2. Notations. We will work in R. p : given p a real number, we efine p to be the positive real number satisfying p + (p ) = ; p is calle the Höler conjugate of p : open set in R : the bounary of, \ α : partial erivative of multi-inex α. α = (α,...,α ) (N ), with norm α = α i. α = α α 2 : there exists a compact set K such that K 2 D α : weak erivative (see.3) of multi-inex α suppf : for a function f, this enotes the support set, i.e. the set on which f C(): continuous functions taking value in the reals efine on (though most of what we say will be vali for functions taking value in a Hilbert space) C( ): the subset of C() consisting of functions that exten continuously to C (): the subset of C( ) consisting of functions which vanish on C k (): functions f such that α f C() for every α k. k is allowe to be (in which case f is smooth) or ω (in which case f is analytic). Analogously we efine C k ( ) an C k() (note that the set Cω () contains only the zero functions) Cc k (): subset of C k () such that suppf L p (),L p loc (): Lebesgue spaces (see.) W k,p (),W k,p s,p loc (),W (): Sobolev spaces (see.4) p : L p norm (see.) p,k : W k,p norm (see.4) Version as of October 27, 2.

2 2 W. W. WONG. Basic Definitions In this first part can be taken to be any open subset of R. Throughout x will be the stanar Lebesgue measure. By a measurable function we ll mean a representative of an equivalence class of measurable functions which iffer on in a set of measure. Thus sup an inf shoul be mentally replace by esssup an essinf when appropriate... Lebesgue spaces. For > p, L p () enotes the set of p-integrable measurable functions, with norm /p () u p; = u p x. If u takes values in some norme linear space, then will be the Banach space norm. For p =, L () enotes the essentially boune functions (2) u ; = sup u. For p, the spaces L p () are Banach spaces. The space L 2 () is a Hilbert space, with inner-prouct (3) u,v = u,v ; = uv x. If u,v take values in a Hilbert space H with norm, H, then the integran shoul be replace by u,v H (x). In particular, if they are complex value, the integran shoul be ūv. For the sake of completeness, let me recall the efinitions of Banach an Hilbert spaces. Let V be a linear space over R. With the obvious substitutions, you can also o over C A norm on V assigns to elements of V nonnegative real numbers, such that for v,w V : () v, with equality iff v = ; (2) sv = s v, for any scalar s R; (3) v + w v + w (triangle ineq.) The pair (V, ) is calle a norme linear space. We can efine a metric on (V, ) by (v,w) = v w. Exercise. Check that (v,w) is inee a metric. Then we can use the usual notion of convergence in metric spaces. If the norme linear space (V, ) is complete as a metric space, that is, all Cauchy sequences converge, then we say (V, ) is a Banach space. That the L p norms are in fact

3 SOBOLEV SPACES 3 norms follows easily from property of the Eucliean absolute value, an Höler s inequality (6) below. Exercise 2. Prove that L p () is a Banach space. That is, show that if u i L p () are a sequence of functions satisfying u i u j p; as i,j, then there exists u L p () such that u i u. Now let V be an R-linear space again. An inner prouct on V is a map V V R enote by,, satisfying () v,w = w,v ; (2) av + bw,x = a v,x + b w,x for all a,b R; (3) v, v, with equality iff v =. We call the pair (V,, ) an inner prouct space or pre-hilbert space. If V were over C, the symmetric property nees to be replace by Hermitian, an the linearity nees to become sesquilinearity. Exercise 3. Writing v = v,v on an inner prouct space (V,, ), show that v, w v w (Schwarz) v + w v + w (Triangle) v + w 2 + v w 2 = 2 v w 2 (Parallelogram ientity) This implies that every inner prouct space is a norme linear space. If the inuce norm is complete, we say an inner prouct space is a Hilbert space. In some texts a Hilbert space is also require to be separable: that there exists a countable subset of vectors {v,v 2,...} whose linear space is ense in V ; this will in particular imply the Hilbert space has a countable orthonormal basis. While the spaces we encounter in this note are all, in fact, separable, we will not make this assumption. Exercise 4. Check that (L 2 (),, ; ) is an inner prouct space. Check that the inner prouct efine in (3) inee leas to the norm efine in (). Therefore by Exercise 2, L 2 is a Hilbert space. Recall Young s inequality: let a,b be positive real numbers, an p real, then (4) ab ap p + bp p a p + b p. The special case p = p = 2 is known as Cauchy s inequality. By replacing a ɛa an b ɛ b, we also get the interpolate version of Young s inequality (5) ab ɛ p a p + ɛ p b p. Using Young s inequality we can prove Höler s inequality (6) uv x u p; v p ; when the right han sie is well efine.

4 4 W. W. WONG Proof. The inequality is homogeneous in scaling of u an v. Therefore it suffices to prove for u p = v p =. Write uv x u v x u p p + v p p x p u p p; + p v p p ; = p + p = This homogeneous scaling trick is very useful in analysis of partial ifferential equations. Höler s inequality is very useful for eriving facts about the L p spaces. First we efine the localise Lebesgue spaces: a function u is sai to be in L p loc () if for every open set, u L p ( ). Consier the functions on R given by x α. These functions are not in any Lebesgue spaces, because for pα, it ecays too slowly at infinity, while for pα, it blows up too fast at the origin. The localise spaces allows one to istinguish ivergences at the bounary of, an singularities in the interior of. Also note that the local Lebesgue spaces are not norme spaces. Proposition. () L q () L p () if q p an < ; (2) L p () L q () L r () if p r q. Note that point () in particular implies that L q loc () Lp loc () for p < q an any. Proof. What we will prove are slightly stronger, quantitative versions of the above statements. For the first claim, let χ be the characteristic function of, i.e. χ (x) = if x an otherwise. Let u L q (). Observe u p p; = u p x = χ u p x an so we have (by Höler) χ /( p q ); up q/p; = p q u p q; (7) u p; p q u q; which proves the claim. For the secon statement, we write u r r; = u λr u ( λ)r x = u λr p/(λr); u ( λ)r q/( λ)r;

5 SOBOLEV SPACES 5 for λ an r = λ p + λ q. Noting that if p r q, one can always fin such a λ, we have (8) u r; u λ p; u λ q; an proving the claim. The first property above tells us that, after localising, higher L p norms control lower ones, an in particular, higher L p norms have more regularity, or that they are less singular. The secon property tells us that one can harmonically interpolate between higher an lower L p spaces to get something in between. Remark 2. The Höler inequality can be iterate to obtain that, if m (p i ) =, then f f 2 f m x m f i pi ; The localisation above can be interprete as a cut-off by the characteristic function χ. There are other ways to achieve localisation; a very useful one is to use polynomial weights. Exercise 5. Let L p s (R ) enote the Lebesgue space with s weight: that is, f L p s (R ) iff f (x) p ( + x s ) p x <. Show that L q s (R ) embes continuously into L p (R ) if an only if q > p an s > p q. (Hint: follows from juicious application of Höler inequality; follows by consiering functions of the form x t ( + x r ).) Höler s inequality can also be use to obtain the following useful inequality. Lemma 3 (Minkowski s inequality). Let u(x,y) be a measurable function efine on R R 2, where x R an y R 2. Then provie both sies of the inequality are finite, we have, for q p <, (9) u(x,y) q x R 2 R p/q y /p R i= u(x,y) p y R 2 q/p i= x Remark 4. Note that the p = case in the above inequality is trivially true. Proof. Note that by replacing u by u q, it suffices to prove the inequality for q =. The case p = q = is then just Fubini s theorem. For p >, let S(y) := u(x,y) x /q. we re-write S(y) p = R R u(z,y) zs(y) p.

6 6 W. W. WONG In this notation, S(y) p y = u(z,y) zs(y) p y R 2 (Fubini) = R 2 R R R 2 R R 2 u(z,y) S(y) p y z (Höler) = u(z,y) p y /p z S(w) (p )p w R 2 /p. Now, (p )p = p by efinition. So we have that, iviint both sies by a suitable factor of S(w) p w, that /p /p S(y) p y u(z,y) p y z R 2 precisely as claime. R.2. Frierichs mollifiers, regularisation. Now we iscuss a systematic way to approximate L p functions by smooth ones, ue to K. O. Frierichs. Definition 5. A Frierichs mollifier, or an approximation to the ientity, is a nonnegative function ψ in C c (B) (where B = B () is the ball of raius about the origin in R ) satisfying the conition that R ψ x =. R 2 A stanar example of a mollifier is the function c exp ( ) x ψ(x) = x 2 x with c chosen so that the total integral of ψ is. Now consier the expression for δ > efine by convolution () u δ (x) = δ ψ( x y )u(y) y. δ B δ (x) The right han sie of () is well-efine if u is efine, an integrable, on B δ (x). We can treat it as a sort of weighte average over the small ball. Now, suppose u L p (). By Proposition, u L loc (). So for δ >, as long as ist(x, ) > δ, u δ (x) is well-efine. Furthermore, using the property of the convolution, we see that for any with ist(, ) > δ, we have u δ C ( ). Also if u L p (R ) has compact support on R, then u δ Cc (R ). Exercise 6. Verify that for u L p (R ), u δ is also p-integrable. (Hint: by using Tonelli s theorem, it reuces to checking measurability of u(x y) on R R.) The importance of the mollifier is not just that it smooths out a function, but that it also approximates it as δ. Heuristically, as δ the expression δ ψ(x/δ) approaches the Dirac elta function, an so u δ u. To make it precise:

7 SOBOLEV SPACES 7 Lemma 6. For u L p (), p <, then u δ u in L p () as δ. Proof. In the case where R, we exten u trivially by setting u = outsie. Then u L p (R ). Since u δ u p; u δ u p;r, it suffices to show the convergence for = R. ( p ) /p u δ u p = (u(x δy) u(x))ψ(y) y x ( /p (Minkowski s inequality) ψ(y) u(x δy) u(x) x) p y (Höler) sup u( y) u p y δ That the right han sie of the last inequality as δ can be checke by using the fact that the continuous functions with compact support are ense in L p for any p <. As a corollary, we have that the smooth functions are ense insie L p (). Exercise 7. Fin a counterexample to Lemma 6 in the case p =..3. Weak erivatives. First recall the formula for integrating by parts. Let u C α (), an φ C α (), then we have ( α u)φ x = ( ) α u( α φ) x. Using this formula we can efine a generalise notion of erivatives. Definition 7. Let u be locally integrable in. Then a locally integrable function v is sai to be the α th weak erivative of u, written v = D α u, if for every function φ C α (), we have () vφ x = ( ) α u( α φ) x. Note that when the weak erivative D α u exists, it is efine only up to a set of measure zero. So any point-wise statements to be mae about D α u is unerstoo to only hol almost surely. Definition 8. A locally integrable function u is sai to be k-times weakly ifferentiable if for all α k, the weak erivative D α u exists. Observe that a C k function is automatically k-times weakly ifferentiable, the classical erivatives being representatives of the equivalence class of weak erivatives. Exercise 8. (Derivatives an mollifiers commute.) Let u L loc (), an assume D α u exists. Prove that for x an δ > such that ist(x, ) > δ, we have (2) α u δ (x) = (D α u) δ (x). Exercise 9. (Approximation theorem for weak erivatives.) Let u,v L loc (), then v = D α u if an only if a sequence (u m ) of C () functions such that u m u an α u m v in L loc ().

8 8 W. W. WONG By the approximate theorem, most of classical ifferential calculus can be reprouce for weak erivatives. For example, the prouct rule D(uv) = (Du)v + u(dv) hols for every pair of weakly ifferentiable functions u,v as long as uv an (Du)v + u(dv) are both L loc. Similarly, the chain rule hols: let u L loc () be weakly ifferentiable, let y = y(x) be a C coorinate change, an let f C (R) with f boune. Then () the function v = u y is weakly ifferentiable on, with Dv = Du y ; (2) the function f u is weakly ifferentiable on, with x D(f u) = f (u)du. For more etails on the ifferential calculus, please refer to Evans or Gilbarg-Truinger..4. Sobolev spaces. We begin straight with the efinition. Definition 9. The notation W k,p () is the Sobolev space of ifferentiability k an integrability p. It consists of functions u which are k-weakly ifferentiable, such that D α u L p () for all α k. The Sobolev spaces W k,p () are Banach spaces with the norm /p (3) u p,k; = D α u p x. α k Observe that the space W,p () is just L p (). In the case that p = 2, we also introuce the notation H k () = W k,2 (). These L 2 -Sobolev spaces are Hilbert spaces uner the inner prouct (4) u,v k = D α ud α v x. α k These L 2 Sobolev spaces will be use heavily in Mihalis class; one particular reason is that Hilbert space techniques are very powerful in emonstrating existence of solutions to partial ifferential equations, another is that for hyperbolic PDEs, L 2 -base spaces are the only ones in which we can o inuctive arguments. The localise versions of these spaces are efine analogously to the case of Lebesgue spaces. There are two more variants of the Sobolev spaces that are commonly use. Definition. The spaces W k,p () (similarly H k ()) are the homogeneous Sobolev spaces. They consist of k-times weakly ifferentiable functions u such that D α u L p () for α = k. Definition. The spaces W k,p () (similarly Hk ()) are the closure of Ck () uner the Sobolev norm (3).

9 SOBOLEV SPACES 9 The W k,p spaces are not Banach spaces; the efining conition oes not give a norm, it merely gives a semi-norm. For example, two functions in C k that iffer by a constant are the same size uner this semi-norm. We can mo out this ambiguity by requiring that the representatives we consier has mean over, as long as <. A common reason to consier the homogeneous Sobolev spaces is that, if the omain = R, it has nice scaling properties. See 2.6. Remark 2. The W k,p an W k,p spaces are generally not equal, except when = R. Using the approximation theorem for weak erivatives an the commutation relation, we see immeiately that for u W k,p (), the mollifie α u δ D α u as δ in L p loc (). Here we erive a global approximation. Theorem 3. The subspace C () W k,p () is ense in W k,p (), for p <. Proof. Let i be an approximation of by compactly inclue subsets; that is, i i+, i i =, an j = for j. Let η j be a partition of unity suborinate to the covering { j+ \ j }. For u W k,p (), an for any ɛ >, using the result mentione before the statement of the theorem, we can choose a sequence of δ j such that the following are satisfie: δ j < ist( j+, j+3 ) (η j u) δj η j u ɛ p,k; 2 j+ Let v = (η j u) δj. By the efinition of partition of unity, an the first conition above, we have that at each x, only finitely many terms in the infinite sum is non-zero. So v C () by construction. Furthermore, using the triangle inequality u v p,k; (η j u) δj η j u ɛ p,k; an we obtain the approximation. In view of the Density Theorem 3, Definitions 9 an can be re-written to have the Sobolev spaces efine as the completion of C () an Cc () respectively in the W k,p () norm. Note that in the case = R, the spaces W k,p an W k,p coïncie, an thus Cc (R ) is ense also in W k,p ()..5. Difference quotients. The classical partial erivative is efine, for a smooth function u, as u(x + he i u(x) = lim i ) u(x). h h For consiering weak erivatives, we use a similar iea calle ifference quotients. Definition 4. Given a unit vector v an some measurable function u, the h- translate of u in the irection of v is enote as τ h v u, an is efine by τ h v u(x) = u(x + vh).

10 W. W. WONG Definition 5. For a measurable function u, the ifference quotient of u in the coorinate irection e i of length h is the measurable function enote by h i u an efine by h i u = τh e i u u. h Exercise. Derive the prouct rule for h i. The key point of this construction, unlike that of the weak erivative, is that for an arbitrary locally integrable u, h i u always exists an is locally integrable, whereas D i u may not be efine. The two notions, however, are intimately connecte. Proposition 6. Let u W,p (). Then h u L p ( ) for any with h < ist(, ). That is, h u p; Du p;. Proof. By a ensity argument, if suffices to prove the inequality for u C (), since it is clear that h u δ = ( h u) δ. Notice that the restriction to is so that the ifference quotient is well-efine. For continuously ifferentiable functions, the ifference quotient can be written as h i u(x) = h h i u(x + te i ) t. Now using the first part of Proposition, we have h i u(x) h h So integrating in L p ( ) we have h i u p p; h i u(x + te i ) t h /p h an the esire inequality follows. h τ t e i i u p p; t h The implication is also allowe to run the other way. i u(x + te i ) p t h t i u p p; Proposition 7. Let u L p () for < p <. If there exists some constant K such that for all an < h < ist(, ), we have h u p; K, then u is weakly ifferentiable an Du p; K. Proof. Let h m be an arbitrary ecreasing sequence, an m such that ist( m, ) > h. Then since h mu p;m K, we have that for each irection e i there exists v i L p () such that the weak convergence h m i u v i in L p ( ) hols for all, with v i p; K. In particular, this implies that for any φ Cc (), φ h m i u x φv i x. /p.

11 SOBOLEV SPACES For m sufficiently large, h m < ist(suppφ, ), so we can integrate by parts : φ h m i u x = φτ h m h m e i u x φu x = uτ h m h m e i φ x φu x = u h m i φ x. Since φ is classically ifferentiable, taking the limit h m gives convergence φ h m i u x u i φ x. So φv i x = u i φ for all φ C c (), so v i = D i u. 2. Imbeing Theorems an Friens In this secon part, when we write, it always enotes a boune, connecte, an open subset of R. Some results in the sequel pertains to the omain being R itself; they will be clearly marke as such. In the following we will only work with Sobolev spaces that vanish on the bounary of (for the omain being R, this conition is trivial). Quite a few of the theorems presente below will not be true in general if we consier W k,p () associate to arbitrary. This failure is usually ue to our inability to arbitrarily exten a W k,p function across in general. In the cases where has nice regularity properties (external cone conition, ifferentiability), these theorems can usually be salvage. The reaer is avise to consult the comprehensive volume of Aams for these cases. 2.. Gagliaro-Nirenberg-Sobolev inequality. The Gagliaro-Nirenberg-Sobolev inequality is an inequality for the omain being R. Theorem 8 (Gagliaro-Nirenberg-Sobolev). For u C c (R ), for any p <, there exists a constant C epening only on p an such that (5) u p p ;R C u p;r Note that by ensity, the above inequality extens to u W k,p (R ). An by the interpolation inequality in Proposition, this implies that (since p/( p) > p) for any p q p p, there is a continuous embeing W,p (R ) L q (R ). Proof. First we reuce it to the case where p =. Let v = (u 2 ) γ for γ > 2. Then v = 2γ(u 2 ) γ u u, an notice that for u C, v = when u : so v C c (R ).

12 2 W. W. WONG Using the ensity of Cc Cc, we assume the theorem hols for v with p =, that is, v C ;R, v ;R. Plugging in the efinition we have u 2γ ;R 2γC, u 2γ 2 u u ;R 2γC, u 2γ 2 u p ;R u p;r using Höler inequality again. Then we solve When we plug it in we have 2γ = (2γ )p = 2γ = p p. u 2γ p p p ;R p C, u 2γ u p p ;R p;r. Cancelling out the reunant factors leas us to (5) as esire. It remains to prove the inequality for the case p =. Observe that for each i, so we can write u(x) x i i u(x,...,x i,y,x i+,...,x ) y u(x) x i i= i u y i We want to integrate both sies over R, an use the iterate Höler inequality (see Remark 2). To o so we first notice that x i x i x j i u y i x j = i u y i j u y j x j R i= R i j } {{ }} {{ } an sup x j x j j u y j ( ) L R = j u y j. L an that by taking the L norm of the other terms, we can pass the integral insie x i x i i u y i = i u x j y i. ; <xj <,j i So we finally arrive at the inequality u x i u x R i= R R

13 SOBOLEV SPACES 3 or u ;R i u x. i= R Now, applying the arithmetic-geometric-mean inequality, we get u ;R i u ;R = u ;R as claime. An thus we ve proven the theorem with C = p( ) ( p). i Remark 9. For imension =, if the erivative of a function is absolutely integrable, the function is absolutely continuous an boune. In other wors, we have u ;R u ;R. For higher imensions, the en-point case of p = is not true. The easiest counterexample is constructe via the Fourier transform (see Remark 2 below). The Gagliaro-Nirenberg-Sobolev theorem can be iterate. The easiest way to remember the exponents is perhaps the following observation: ( ) p (6) = p. p An so if you are willing to lose k erivatives, you have u p C p,,k α kp ;R p;r for any u C c (R ). Remark 2 (Counterexample to L enpoint Sobolev). Let us focus now in the case of L 2 -Sobolev spaces H k. We will see that in the case 2k =, we o not have the embeing H k L. Using the fact that the Fourier transform acts on L 2 by isometry (i.e. Plancherel s theorem that f 2 x = f ˆ 2 ξ), we see that the space H k is equivalently characterize as all L 2 functions u such that ( + ξ 2 ) k û(ξ) 2 ξ <. R Now let û(ξ) = α =k ( + ξ 2 ) k log(2 + ξ 2 ). û is clearly in L 2 for k > 4, an thus is the Fourier transform of some L2 function. An similarly ( + ξ 2 ) k/2 û is in L 2 if k > 2. For 2k =, ( + ξ 2 ) k [log(2 + ξ 2 )] 2 ξ C + C 2 r (logr) 2 r, R by converting to polar coorinates, an using that the function is smooth in the ball of raius. Noting that r (logr) = r (logr) 2, we have that u represents a function in H k (R ) for 2k =.

14 4 W. W. WONG On the other han, using the efinition of the inverse Fourier transform, u() = û ξ. A polar coorinate change shows that the right han sie is boune below by some constant times (r logr) r, but with the antierivative of (r logr) being log log r, the integral iverges. This is why we say that the Gagliaro-Nirenberg- Sobolev inequality iverges logarithmically as kp. As remarke earlier, Gagliaro-Nirenberg-Sobolev inequality only hols for the precise Lebesgue exponent state; one can get a range of exponents from interpolation on unboune omains. On boune omains, we can use the first part of Proposition an remove the lower boun of p q. The following in particular implies Poincaré s inequality. Corollary 2 (Sobolev inequality). For boune open omain R, we have W k,p () Lq () if q p k,p kp <. More precisely, for u W () an q as above, u q; C D α u p;. α =k Proof. This follows from Theorem 8, the ensity of Cc () in W k,p (), an Proposition. In general, any inequality which traes ifferentiability for integrability is calle a Sobolev or Sobolev-type inequality in the literature. Note that the trae is one-way. One cannot generally sacrifice integrability to gain ifferentiability, as locally integrability functions nee not have a well-efine weak erivative Morrey s inequality. Thus far we have consiere the embeing theorems for kp < n. What happens when kp > n? We know that when kp = n there is a logarithmic ivergence, but often this type of ivergences form a bounary between two ifferent regimes. (For example, the function /x iverges logarithmically uner integration, but is the bounary between locally integrable an integrable except for local efects.) Inee, for kp > n there is a ifferent class of estimates, historically attribute to Morrey. Here we will prove a slightly weakene version, originally ue to Sobolev. Theorem 22 (Morrey-Sobolev). For u C c (R ), then there exists a constant C epening on p an such that (7) sup u C suppu p u p;r if p > 2. That the support of u comes into play is natural. By rescaling the spatial variables, we can make the support larger an larger while making the function flatter an flatter, an at the same time keeping the maximum of u unchange. So without the suppu term, we can make the right han sie as small as we want, while keeping the left han sie fixe, contraicting the inequality.

15 SOBOLEV SPACES 5 Proof. Perform the rescaling where u (x) = u(λx), then u (x) = λ( u)(λx). An u p = λ p u p. On the other han suppu = λ suppu, so the inequality to prove is invariant uner rescaling of the spatial variables. Therefore it suffices to prove the inequality for functions u such that suppu =. Furthermore, by rescaling u λu, we see that the inequality is also invariant, an so we can assume u p =. It then suffices to show that there exists some constant C epening on p an such that sup u C uner these assumptions. Now observe that p > 2, so p = p p < = 2. Fix γ >, we first make use of the Gagliaro-Nirenberg-Sobolev inequality u γ ;R γ u γ u ;R. Applying Höler inequality to the right han sie, we get u γ γ ;R γ u γ (γ )p ;R u p;r, where, by our assumption, the last factor in the right han sie is. Now, since (γ )p γp, an that u is supporte on a set of measure, we can use the first part of Proposition an arrive at u γ ;R γ/γ u γ γp ;R Now take η = /p > an set γ = η k. For k >, we have u η k ;R ηk/ηk η u k. η k ;R For the base case k = we use the Gagliaro-Nirenberg-Sobolev inequality again, together with Höler s inequality to get An so iterating, we have u ;R u ;R suppu /p u p;r =. u η k ;R = u η k ;suppu η jη j = C(p,). Lastly we apply the fact that (8) lim u p p x for any boune omain, we have that as claime. /p sup u = sup u C suppu = sup u Exercise. Prove (8). In Theorem 22, we see that the fact u has compact support is essentially use. This means that only knowing the erivative of a function is p-integrable oes not guarantee the function itself is boune. A simple example is obtaine by consiering u(x) = ( + x ) /3 in one imension. It has one erivative that is L p integrable for all p 2, but it is not a boune function.

16 6 W. W. WONG However, all is not lost! If we also know that the function itself is in L p, we can break the scaling an get control on bouneness. In the proof we will also see an illustration of the useful technique in harmonic/functional analysis calle optimisation. Corollary 23. For u C (R ) an p >, there exists a constant C epening on p an such that (9) sup u C u p p u p;r p;r. Proof. Let t > be arbitrary, an enote by t the set { u > t}. Let v be the function such that v = outsie t an v = u t on t. Clearly v W,p ( t). Using that u is smooth, v = ±u t on each of the connecte components of t. Therefore v p;t u p;r. Observe that Applying Theorem 22 we get sup u t + sup v. sup v C t p v p;t. We can estimate the size of t using the L p -weak-l p estimate (also known as Chebyshev s inequality or Markov s inequality) which states that Thus we arrive at t t p u p p;r. sup v Ct p p u u p;r p;r which implies ( (2) sup u t + Ct p p ) u u p;r p;r. We now optimise by choosing t as a function of u p;r an u p;r. In particular we can choose t such that the term insie the parenthesis in (2) is a constant: we set p C u u p;r p;r = t p. This implies that as claime. sup u 2C p u p p u p;r Corollary 24. () For p >, the inclusion W,p () C( ) is continuous. (2) Similarly the inclusion W k,p () Cl ( ) for l < k /p. (3) If u W k,p (R ), then for any with compact closure we have u C l () if l < k /p. Exercise 2. Prove Corollary 24. (Hint: for x,y, consier the function v(y) = ψ(y)(u(y) u(x)) where ψ(y) C c () an ψ(y) = on. By letting an suppv one can recover continuity from Theorem 22.) p;r

17 SOBOLEV SPACES Prouct estimates. By putting together the Höler inequality an the Sobolev inequalities, we get Theorem 25 (Sobolev prouct estimates). Let u W k,p (R ) (or W k,p ()) an u W k 2,p 2 (R ) (W k 2,p 2 ()). Then u u 2 W k,p (R ) (W k,p ()), whenever k min(k,k 2 ) an (2) p k > p k + p 2 k 2. In other wors, there exists a constant C = C(k,k 2,k,p,p 2,p,) such that for any u,u 2 C c (R ), u u 2 p,k;r C u p,k ;R u 2 p2,k 2 ;R. Proof. (Hereon C stans for a constant that may change line by line, but only epens on the parameters specifie above.) By the prouct rule for ifferentiation, α (u u 2 ) = β u γ u 2. So using the triangle inequality For a pair (β,γ), we note that u u 2 p,k;r β+γ=α β+γ k β u γ u 2 p;r. β u γ u 2 p;r β u r;r γ u 2 q;r where p = q + r by Höler. Using Sobolev inequality, when k > β an q > p whenever as claime. β u r;r C u p,k ;R k β. So β u γ u 2 p;r C u p,k ;R u 2 p2,k 2 ;R p > k β + k 2 γ k p p 2 p + k 2 p 2 + k A irect consequence is Corollary 26. The spaces W k,p for kp > are algebras, i.e. if u,u 2 are in W k,p, so is the prouct u u Rellich-Konrachov compactness. The continuous inclusion of one Banach space B into another B 2 is sai to be compact if the image of the unit ball in B is precompact in B 2. That is, for every sequence {f k } B with f k B, there exists a subsequence which is Cauchy in B 2. Compactness is particularly useful for variational problems: to minimize a functional, one is le to stuy a minimizing sequence. A suitable compactness result allows one to state that the minimizing sequence actually converges, though often in a less regular space (which is not as ba as it souns, since quite often, especially in elliptic situations, we can gain back the regularity once we have a weak solution).

18 8 W. W. WONG For a function space efine over subsets of R, there are generally three ways for compactness to fail, which are all relate to the symmetries of R : () ivergence in scale; (2) ivergence in physical support; an (3) ivergence in frequency support. The ivergence in physical support is easiest to illustrate. Consier u a function with support containe within the ball of raius. Then for any translation invariant norm on R (any non-weighte Sobolev or Lebesgue norm will be one such), the sequence of functions u m (x) = u(x + 4me ) is boune, but has no Cauchy subsequence. This is because the supports of u m,u n for m n are isjoint, so u m u n = 2 u. On compact/boune sets, a sequence cannot run off to infinity. But with a proper rescaling we can still have infinitely many functions with isjoint support fitting in the set. A classic example is given when = (,) R. In L (), consier the sequence of functions u m (x) = 2 m χ (2 m,2 m+ )(x) with u m ;(,) =. This sequence has again isjoint supports an thus contains no Cauchy subsequence. Another example is given by the failure of the Sobolev embeing W.p p () L p () to be compact. Lemma 27. The Sobolev embeing W,p p () L p () is not compact. Proof. Since is open, it contains some metric ball. Without loss of generality, we assume it contains the unit ball B about the origin. Take a function u in C c (B) such that u = on the ball of raius /2. The functions in the sequence u m (x) = 2 m( p ) u(2 m x) can be shown by explicit computation to all have the same norms u m p = u p p ;B p ;B, u m p;b = u p;b. So this is a boune sequence in W,p (). On the other han, we can compute u m u n p p ;B = u m n u p p ;B ( ) (2 2 m n ( p ) m n B ) p p ( ) 2 p 2 p B p p an so the sequence amits no Cauchy subsequence. Remark 28. Ha we examine instea the imbeing W,p () Lq () where q < p p, we woul ve foun that the above scaling construction leas to u m in L q. The ivergence in frequency support is similar to the ivergence in physical support, except for a conjugation via the Fourier transform. Here I ll merely given

19 SOBOLEV SPACES 9 an example. By Proposition, we have a continuous inclusion L q () L p () is p < q. Lemma 29. The inclusion L q () L p () is not compact. Proof. Using that is open, without loss of generality, we assume that the cube [,]. Consier the functions sin(2πmx u m (x) = ) x [,] x [,] where are boune on an hence boune in any L q. On the other han, the interpolation inequality gives us While u m u n 2 2; = u m u n 2 2; u m u n ; u m u n ; 2 u m u n ;. [,] u 2 m + un 2 2u m u n x = sin(2πmy) 2 + sin(2πny) 2 y =. So the sequence has no Cauchy subsequence in L (), an hence no Cauchy subsequence in L p () for any p. So much for negative results. Here we ll give one positive result ue to Rellich in the p = 2 case an Konrachov in general. Theorem 3 (Rellich-Konrachov lemma). On a boune open set, the nonenpoint Sobolev imbeings where q < p p is compact. W,p () Lq () Proof. It suffices to prove it for q =. Then since the inclusion W,p p () L p () is continuous, we get compactness for all q < p by Höler interpolation (Proposition ). Let A be the unit ball in W,p (); fix δ >, ψ a mollifier, an let A δ Cc (R ) be its corresponing regularisation {u δ u A} (exten u outsie by ). First we show that A δ is uniformly close to A in L (). u u δ ; = sup z δ ψ(z)(u(x) u(x δz)) z x R u( z) u ;R using the same argument as in the proof of Lemma 6. But now we observe that using the ifference quotient techniques of Proposition 6, we have that sup z δ I thank Denis Serre for this example. u( z) u ;R suph h u ;R δ Du ;. h<δ

20 2 W. W. WONG The last term on the right han sie is boune by δ /p from Proposition an that A is the unit ball in W,p (). Therefore it suffices to show that A δ is precompact for every δ >. To o so we note that u δ (x) δ ψ( x y δ ) u(y) y δ supψ u ; an u δ (x) δ B δ (x) B δ (x) ( ψ)( x y δ )u(y) y δ sup ψ u ;. Therefore A δ is a boune, equicontinuous subset of C( ), an thus precompact in C( ) by Arzelà-Ascoli Theorem. Therefore A δ is precompact in L (). Exercise 3. Show that the imbeing of W,p () Cl () for l < p of Corollary 24 is compact. (Hint: it suffices to show that the image of the unit ball is equicontinuous; this follows from the fact that in Theorem 22 there is a term.) 2.5. Trace theorems. In the stuy of partial ifferential equations, we often nee to consier the bounary value problem, where the restriction of a function u in our Sobolev space to is require to take certain prescribe values. In the case where u is classical, this restriction operation is not a problem. But in the case where u is a measurable function, as the bounary of an open set has measure, one can always freely moify u on that bounary. In this section we try to make sense of this trace of a measurable function onto a positive coimension submanifol. The main trace theorem is a combination of the Gagliaro-Nirenberg-Sobolev Theorem 8 an the Morrey-Sobolev Theorem 22. Theorem 3. For u C c (R ), consier its restriction u (M) onto a co-imension n hyperplane M for n. Then there exists some constant C = C(p,q,k,,n) such that (22) u (M) q;m C u p,k;r whenever (23) k > ( p ) + n q q. Proof. Write R = R n x R m y, where the subscripts enote the variables we will use. In other wors, we write u C c (R ) = u(x,y), where x R n an y R m. Consier the following C c (R n x) function v(x) = u(x, ) q;r m y. By the Gagliaro-Nirenberg-Sobolev inequality we have that v(x) C u(x, ) m k ( p q ) p;r m y mp m kp. k m k ( p yu(x, ) q ) p;r m y C u(x, ) p,k;r m y for p q Similarly we use Corollary 23 of the Morrey-Sobolev inequality on x, an have v(x) C v p,l;r n x

21 SOBOLEV SPACES 2 if lp > n. Chaining the two estimates together an ifferentiating uner the integral sign, we have that sup u q;r m y C u p,j;r R n x whenever j > ( p q ) + n q. By a ensity argument this implies the continuous imbeing W k,p (R ) L q (M), an similarly W k,p () Lq (M ), for almost every hyperplane M. One can similarly generalize this to traces where M can be any compact smooth submanifol of R ; it suffices to use a partition of unity argument, an the fact that M can be covere by finitely many balls an in each ball there exist a iffeomorphism with boune first k erivatives bringing M to a hyperplane. We leave such natural generalisations to the reaer. Notice that if we set the Lebesgue integrability the same on both sies (p = q), we see that for each rop in imension one traes a little more than /p egree of ifferentiability. This little more can be remove in some cases. We give an example below whose proof is base on the Fourier isometry of L 2. Theorem 32. Let u C (R ), an u (M) again, its trace onto a co-imension n hyperplane. Then if j >, we have u (M) 2,j;M C u 2,j+ n 2 ;R. Proof. By rotation an translation, we can assume that M is the plane x,...,x n =. Write û for the full Fourier transform of u on R, an ũ for the Fourier transform of u (M) on M. The Fourier inversion formula gives ũ(ξ n+,...,ξ ) = û(ξ,...,ξ ) ξ. ξ,...,ξ n We use the short han η = (ξ,...,ξ n ) an ζ = (ξ n+,...,ξ ). So u (M) 2 2,j;M = ( + ζ 2 ) j ũ(ζ) 2 ζ = ( + ζ 2 ) j 2 û(η,ζ) η ζ ( )( = ( + ζ 2 ) j ( + η + ζ 2 ) j+ n 2 û 2 η ( + η + ζ 2 ) j n 2 η ) Consier the last term in the right han sie. Writing t 2 = + ζ 2 an r 2 = η 2 /t 2, we have ( + η + ζ 2 ) j n 2 η = ( + η 2 + ζ 2 ) j n 2 η = t 2j ( + r 2 ) j n 2 ω n r n r = ct 2j ζ

22 22 W. W. WONG where c = c(n,j) is finite whenever j >. Plugging this back in we have irectly u (M) 2 2,j;M c ( + η + ζ 2 ) j+ n 2 û 2 η ζ = c u 2,j+ n 2 ;R. Examining the proof, we see that the failure of the embeing in the case j = is, again, given by a logarithmic ivergence. Thus we have the continuous embeing H k (R ) H j (R n ) when k j n 2 an j except the case where both equalities are satisfie Numerology. In this last section we re-visit the scaling property an use it to obtain a quick heuristic for checking whether a propose inequality is reasonable. Consier the function u : R R as a map between two vector spaces. The scaling symmetry of a vector space inuces two types of scaling on u. The first is the magnification of the source vector space u(x) u(λx) an the secon is the amplification of the target vector space u(x) κu(x). In the following, the phrase v is a scaling of u by factor (λ,κ) will mean the replacement v(x) = κu(λx). Each homogeneous Sobolev space W k,p (R ) has, associate to it, a natural scaling law L = L(λ,κ), which we efine as Definition 33. The natural scaling law L = L(λ,κ) associate to function satisfying W k,p (R ) is the whenever v is a (λ,κ)-scaling of u. D k v p = L D k u p For a fixe k,p,, the set {(λ,κ) R 2 + L(λ,κ) = } is calle the set of invariant scalings of W k,p (R ). This notion of the scaling law an the invariant scalings can be extene to other homogeneous function spaces. We can compute explicitly what L is for Sobolev spaces. First D k v(x) = κλ k D k u(λx). So taking the p th power an integrating, D k v p x = κ p λ kp So the scaling law is L (k,p,) (λ,κ) = κλ k p. D k u p λ y. Now, how o we use this scaling law? Suppose we are aske to check whether one Sobolev space embes continuously into another, which boils own to checking whether an inequality of the form D k 2 u p2 ;R 2 C D k u p ;R

23 SOBOLEV SPACES 23 is plausible with some constant epening only on C = C(k,k 2,p,p 2,, 2 ). What we o is consier the (λ,κ) scaling v of u: if this inequality hols for u, then by efinition L (k 2,p 2, 2 ) Dk 2 v p2 ; 2 C L (k,p, ) Dk v p ;. Now consier the invariant scaling set for k 2,p 2, 2. Solving for κ we get κ = λ 2 p2 k 2. Then the restriction of L (k,p, ) to this set becomes L = λ 2 p2 k 2 +k p = λ c. So if c, by choosing a scaling corresponing to either a really large or really small λ, we can make the right han sie arbitrarily small, while fixing the left han sie to be constant size, an thus obtaining a contraiction. Hence a necessary conition for a embeing of one homogeneous Sobolev space into another is the equality of their scaling laws. Let us test this on the trace theorem of the previous section. Then =, 2 = n, k = k, p = p, k 2 =, p 2 = q. Then a homogeneous embeing will require n + k q p = or k = ( p q ) + n q precisely the logarithmically ivergent en-point case which we isallow. (Notice that insofar as scaling laws are concerne, log(x) x.) For another example we can look at the interpolate Sobolev inequality. Suppose we want u q;r C u t p;r k u t p;r the scaling law of the left han sie is invariant when κ = λ /q. The right han sie has scaling by (L (,p,) ) t (L (k,p,) ) t = κλ ( t) p λ t(k p ) so scaling invariance requires q + tk p = which tells us that the interpolation exponent must be t = k ( p q ) (which was use in the proof of Theorem 3), an gives us precisely the range of amissible Sobolev inequalities (as long as q < ). This last example here also emonstrates how one is to treat embeing inequalities with non-homogeneous Sobolev spaces: instea of consiering u p,k;r, we shoul consier the interpolation u t p;r k u t p;r which will have a homogeneous scaling law, while also be roughly equivalent to the inhomogeneous Sobolev norm. Department of Pure Mathematics an Mathematical Statistics, University of Cambrige, Cambrige, UK aress: ww278@pmms.cam.ac.uk