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Appendix. Banach and Hilbert Spaces. the Lp -Spaces 395

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Appendix. Banach and Hilbert Spaces. the Lp -Spaces 395 Appendix. Banach and Hilbert Spaces. The Lp-Spaces In this appendix we shall first recall some basic concepts from calculus without proofs. After that, we shall prove some smoothing results for Lp-functions. Definition A.1. A Banach space B is a real vector space that is equipped with a norm kk that satisfies the following properties: (i) kxk >0for all x 2 B, x ¤ 0. (ii) k˛xk D j˛j kxk for all ˛ 2 R, x 2 B. (iii) kx C yk Ä kxk C kyk for all x;y 2 B (triangle inequality). (iv) B is complete with respect to kk (i.e., every Cauchy sequence has a limit in B). We recall the Banach fixed-point theorem Theorem A.1. Let .B; kk/ be a Banach space, A B a closed subset, and f W A ! B a map with f.A/ A which satisfies the inequality kf.x/ f.y/kkx yk for all x;y 2 A; for some fixed with 0 Ä <1: Then f has unique fixed point in A, i.e., a solution of f.x/ D x. For example, every Hilbert space is a Banach space. We also recall that concept: Definition A.2. A (real) Hilbert space H is a vector space over R, equipped with a scalar product .; / W H H ! R that satisfies the following properties: (i) .x; y/ D .y; x/ for all x;y 2 H. (ii) .1x1 C 2x2;y/D 1.x1;y/C 2.x2;y/for all 1;2 2 R, x1;x2;y 2 H. (iii) .x; x/ > 0 for all x ¤ 0, x 2 H. J. Jost, Partial Differential Equations, Graduate Texts in Mathematics 214, 393 DOI 10.1007/978-1-4614-4809-9, © Springer Science+Business Media New York 2013 394 Appendix. Banach and Hilbert Spaces. The Lp-Spaces (iv) H is complete with respect to the norm 1 kxk WD .x; x/ 2 : In a Hilbert space H, the following inequalities hold: – Schwarz inequality: j.x; y/j Ä kxk kyk ; (A.1) with equality precisely if x and y are linearly dependent. – Triangle inequality: kx C yk Ä kxk C kyk : (A.2) Likewise without proof, we state the Riesz representation theorem: Let L be a bounded linear functional on the Hilbert space H, i.e., L W H ! R is linear with jLxj kLk WD sup < 1: x¤0 kxk Then there exists a unique y 2 H with L.x/ D .x; y/ for all x 2 H, and kLk D kyk : The following extension is important, too: Theorem of Lax–Milgram: Let B be a bilinear form on the Hilbert space H that is bounded, jB.x; y/j Ä K kxkkyk for all x;y 2 H with K<1; and elliptic, or, as this property is also called in the present context, coercive, jB.x;x/j kxk2 for all x 2 H with >0: For every bounded linear functional T on H, there then exists a unique y 2 H with B.x; y/ D Tx for all x 2 H: Proof. We consider Lz.x/ D B.x; z/: Appendix. Banach and Hilbert Spaces. The Lp -Spaces 395 By the Riesz representation theorem, there exists Sz 2 H with .x; Sz/ D Lzx D B.x; z/: Since B is bilinear, Sz depends linearly on z. Moreover, kSzk Ä K kzk : Thus, S is a bounded linear operator. Because of kzk2 Ä B.z; z/ D .z;Sz/ Ä kzkkSzk we have kSzk kzk : So, S is injective. We shall show that S is surjective as well. If not, there exists x ¤ 0 with .x; Sz/ D 0 for all z 2 H: With z D x,weget .x; Sx/ D 0: Since we have already proved the inequality .x; Sx/ kxk2 ; we conclude that x D 0. This establishes the surjectivity of S. By what has already been shown, it follows that S 1 likewise is a bounded linear operator on H.By Riesz’s theorem, there exists v 2 H with Tx D .x; v/ D .x; Sz/ for a unique z 2 H,sinceS is bijective D B.x; z/ D B.x; S1v/: Then y D S 1v satisfies our claim. ut The Banach spaces that are important for us here are the Lp-spaces: For 1 Ä p<1, we put n Lp.˝/ WD u W ˝ ! R measurable, o R 1 p p with kukp WD kukLp .˝/ WD ˝ juj dx < 1 396 Appendix. Banach and Hilbert Spaces. The Lp-Spaces and n L1.˝/ WD u W ˝ ! R measurable, o with kuk1 WD kukL1.˝/ WD sup juj < 1 . Here sup jujWDinffk 2 R Wfx 2 ˝ Wju.x/j >kg is a null setg is the essential supremum of juj. Occasionally, we shall also need the space ˚ « p R p 0 0 Lloc.˝/ WD u W ˝ ! measurable with u 2 L .˝ / for all ˝ ˝ ; 1 Ä p 1. In those constructions, one always identifies functions that differ on a null set. (This is necessary in order to guarantee (i) from Definition A.1.) We recall the following facts: p Lemma A.1. The space L .˝/ is complete with respect to kkp Inductively and hence is a Banach space, for 1 Ä p 1. L2.˝/ is a Hilbert space, with scalar product Z .u; v/L2.˝/ WD u.x/v.x/dx: ˝ Any sequence that converges with respect to kkp contains a subsequence that converges pointwise almost everywhere. For 1 Ä p<1, C 0.˝/ is dense in Lp.˝/; i.e., for u 2 Lp.˝/ and ">0, there exists w 2 C 0.˝/ with ku wkp <": (A.3) H¨older’s inequality holds: If u 2 Lp.˝/,v2 Lq .˝/, 1=p C 1=q D 1,then Z uv Ä kukLp .˝/ kvkLq.˝/ : (A.4) ˝ Inequality (A.4) follows from Young’s inequality ap bq 1 1 ab Ä C ; if a; b 0; p; q > 1; C D 1: (A.5) p q p q Appendix. Banach and Hilbert Spaces. The Lp -Spaces 397 To demonstrate this, we put A WD kukp ;BWD kvkq ; ju.x/j jv.x/j and without loss of generality A; B ¤ 0. With a WD A , b WD B ,(A.5)then implies Z ju.x/v.x/j 1 Ap 1 Bq Ä C D 1; AB p Ap q Bq i.e., (A.4). p1 pm Inductively, (A.4) yields that if u1 2 L ;:::;um 2 L , Xm 1 D 1; p iD1 i then Z u1 um Ä ku1kLp1 kumkLpm : (A.6) ˝ By Lemma A.1,for1 Ä p<1, C 0.˝/ is dense in Lp.˝/ with respect to p 1 p the L -norm. We now wish to show that even C0 .˝/ is dense in L .˝/.For that purpose, we shall use so-called mollifiers, i.e., nonnegative functions % from 1 C0 .B.0; 1// with Z % dx D 1: Here, ˚ « B.0; 1/ WD x 2 Rd W jxj Ä 1 : The typical example is 8 Á < c exp 1 for jxj <1; jxj21 %.x/ WD : 0 for jxj 1; R where c is chosen such that % dx D 1.Foru 2 Lp.˝/, h>0,wedefinethe mollification of u as Z Á 1 x y u .x/ WD % u.y/ y; h d d (A.7) h Rd h 398 Appendix. Banach and Hilbert Spaces. The Lp-Spaces wherewehaveputu.y/ D 0 for y 2 Rd n ˝. (We shall always use that convention in the sequel.) The important property of the mollification is 1 Rd uh 2 C0 for a bounded ˝. 0 Lemma A.2. For u 2 C .˝/,ash ! 0,uh converges uniformly to u on any ˝0 ˝. Proof. Z Á 1 x y u .x/ D % u.y/ y h d d h jxyjh h Z (A.8) x y D %.z/u.x hz/dz with z D : jzj1 h Thus, if ˝0 ˝ and 2h < dist.˝0;@˝/, employing Z u.x/ D %.z/u.x/dz jzj1 R (this follows from jzj1 %.z/dz D 1), we obtain Z sup ju uhj Ä sup %.z/ ju.x/ u.x hz/j dz; ˝0 x2˝0 jzj1 Ä sup sup ju.x/ u.x hz/j : x2˝0 jzj1 Since u is uniformly continuous on the compact set fx W dist.x; ˝0/ Ä hg,it follows that sup ju uhj ! 0 for h ! 0: ut ˝0 Lemma A.3. Let u 2 Lp.˝/, 1 Ä p<1.Forh ! 0, we then have ku uhkLp.˝/ ! 0: Moreover, uh converges to u pointwise almost everywhere (again putting u D 0 outside of ˝). Proof. We use Holder’s¨ inequality, writing in (A.8) 1 1 %.z/u.x hz/ D %.z/ q %.z/ p u.x hz/ Appendix. Banach and Hilbert Spaces. The Lp -Spaces 399 with 1=p C 1=q D 1, to obtain ÂZ Ã p Z q p p juh.x/j Ä %.z/dz %.z/ ju.x hz/j dz jzj1 jzj1 Z D %.z/ ju.x hz/jp dz: jzj1 We choose a bounded ˝0 with ˝ ˝0. If 2h < dist.˝; @˝0/, it follows that Z Z Z p p juh.x/j dx Ä %.z/ ju.x hz/j dz dx ˝ ˝ jzj1 Z Â Z Ã D %.z/ ju.x hz/jp dx dz jzj1 ˝ Z Ä ju.y/jp dy (A.9) ˝0 (with the substitution y D x hz). For ">0, we now choose w 2 C 0.˝0/ with ku wkLp.˝0/ <" (cf. Lemma A.1). By Lemma A.2, for sufficiently small h, kw whkLp .˝0/ <": Applying (A.9)tou w, we now obtain Z Z p p juh.x/ wh.x/j dx Ä ju.y/ w.y/j dy ˝ ˝0 and hence ku uhkLp .˝/ Ä ku wkLp .˝/ C kw whkLp.˝/ C kuh whkLp .˝/ Ä 2" C ku wkLp .˝0/ Ä 3": Thus uh converges to u with respect to kkp.
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