04 - The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations.pdf

the journal of fourier analysis and applications volume 6 issue 5 2000 the banaeh envelopes of besov and triebel lizorkin spaces and applications to partial differential equations osvaldo mendez and marius mitrea communicated by c kenig abstract s n l 1 l if o p l o q 1 s ir while s n t t q s q b 1 if 0 p 1 1 q oo s er and bsp q n b i n if l p oo o q l ser with hat denoting the banach envelope of a quasi banach space we prove that s n p i 1 f q n bi n a fp q n f 1 n applications to questions regarding the global interior regularity of solutions to poisson type problems for the three dimensional lam system in lipschitz domains are presented 1 introduction the banach envelope of a quasi banach space x whose dual separates points is the minimal enlargement of x to a banach space x in particular the inclusion t 9 x x is continuous with dense range 1 1 math subject ctassijicatior v primary 46e35 42b20 46a16 secondary 35j55 47g10 73c02 keyworel and phrases banach envelope besov and triebel lizorkin spaces lipschitz domains layer potentials boundary problems acknowledgements and notes second author was supported in part by nsf grant dms 9870018 2000 biddaiiuser boston all rights reserved issn 1069 5869 504 osvaldo mendez and marius mitrea applying hat has good functorial properties such as preserving the linearity boundedness and the quality of being an isomorphism for operators between quasi banach spaces as a consequence x 1 2 the main result in the first part of this paper is a simple practical criterion for computing the banach envelope of a quasi banach space x the idea is that 1 1 and 1 2 identify the banach space uniquely up to an isomorphism when applied to the scale of besov and triebel lizorkin spaces this result yields the following formulas theorem 1 for s r o p 1 and o q l we have i i bsp q rn fp q rn b1 n e1 r n 1 3 if o p l and l q boo then s n l l q s n l l s n l l l np q if n nl n fp q l n nl n el r n 1 4 finally in the case when s e 1 p oo and 0 q 1 we have s 1 bsp q n bp n fp q rn fp 1 n 1 5 our primary motivation for establishing these results stems from problems arising at the in terface between harmonic analysis and pdes a paradigm very useful for applications is discussed below to state it for each 0 p 1 and 1 otherwise 1 t s n p l r ii t extends to an isomorphism of fp r n onto itself for each p q s e lt the proof is conceptually simple so that we can outline the main steps here details are provided in subsequent sections proof there are three basic ingredients in the proof the first one is a result from 29 cf corollary 2 for the version relevant for us here to the effect that for a linear bounded operator on some complex interpolation scale of quasi banach spaces the property of being an isomorphism is stable with respect to the scale parameter inside the scale notice that in order for this to work in the present context we need to show that fp q n s e r 0 p q 00 is a complex interpolation scale of quasi banach spaces see section 4 for the precise definition of our method of complex interpolation and for a proof of this result the banach envelopes of besov and triebel lizorkin spaces 505 the second ingredient is that properties such as being an isomorphism fredholm onto or having a finite dimensional cokernel are preserved by applying hat cf theorem 4 finally the third ingredient is the actual identification of the banach envelopes of the quasi banach spaces in the triebel lizorkin class for this see theorem 1 in order to be more specific let us assume for instance that p0 1 and that t is an isomor phism of fsp 176 rn then repeated applications of the aforementioned stability preservation results give that there exists a neighborhood g a x b x c of p0 q0 so in 0 x x 0 c x r so that t extends to an isomorphism of fp q n onto itself for each p q s 6 g invoking 1 3 and the s n l l l functorial properties of hat the conclusion is that t extends to an isomorphism of f 1 rn for each p s 6 a x c this proves the p0 1 case in i a similar pattern works in the remaining cases as well now ii is a consequence of i and duality on the scale of besov spaces we have a similar result whose proof closely parallels that of theorem 2 theorem 3 let t be a linear and bounded operator from bsp q rn into itself for each p q s in a neighborhood of a point po qo so in 0 oo x 0 x and assume that t is an isomorphism of so qo n alternatively assume that this operator is fredholm onto or has finite dimensional x po k cokernel then there exists a neighborhood v of po q0 so so that s n p la l qv1 i t extends to an isomorphism of b pvl i r n onto itself for each p q s e with s so respectively is fredholm of the same index onto or has finite dimensional cokernel s n p x r 1 ii t extends to an isomorphism of b 1 q r n onto itself for each p q s v with s so let us point out that other variations on this theme are possible for example some of the scale parameters may be kept fixed the operator t may depend analytically on a parameter and we can replace the euclidean space r n by the boundary of a lipschitz domain provided p0 q0 are sufficiently close to 1 and s 6 0 1 it should be remarked that there are several notable predecessors to our stability type results the general philosophy is that certain functional analytic properties of an operator t acting on a complex interpolation scale xo 0 are preserved under small changes in the scale parameter for the banach space setting see 40 6 and 47 more recently this theory has been refined to allow scales of quasi banach spaces in 29 what is new in theorems 2 and 3 is the fact that while our hy l otheses are made on t acting on xo 0 the conclusions refer to t acting on a different scale i e xo 0 in this connection it is useful to recall that while hat leaves banach spaces unchanged it fundamentally affects quasi banach spaces this is well illustrated by theorem 1 where the banach envelopes of all the quasi banach spaces in the besov and triebel lizorkin scales are identified for the first time for the applications we have in mind this is particularly useful in the second part of this paper we explain how appropriate versions of theorems 2 and 3 can be applied to pdes in nonsmooth domains below we elaborate more on this point a natural approach to the lp treatment of elliptic boundary value problems in lipschitz domains originally developed in 12 and subsequently used successfully in other important circumstances requires considering the mapping properties of certain boundary integral operators at the level of atomic hardy spaces which are quasi banach for p 0 small in section 6 we develop these ideas and establish sharp invertibility results for lain6 layer potentials in lipschitz subdomains of r 3 then based on these results we present a complete analysis of poisson type problems for the lam6 system in lipschitz subdomains of r 3 finally we indicate how these results can be applied to the problem of determining the mapping properties of the square root of the lain6 system in lipschitz subdomains of i 3 lately there has been a growing interest in these topics and similar or related programs for scalar elliptic pdes have been carried out in 25 18 1 35 2 8 in the present paper we initiate the study of such issues for systems of pdes we want to point out that our methods are rather general and work almost axiomatically whenever atomic estimates have been established these ideas have been already used in 35 for layer potentials associated with the laplace beltrami operator in a lipschitz subdomain of a riemannian manifold the organization of the paper is as follows section 2 contains functional analysis prerequisites in section 3 we recall the definition and some of the relevant properties of besov and triebel lizorkin spaces section 4 contains a discussion of the complex method of interpolation in the context of a convex quasi banach spaces here we prove that fp q r n and bsp q rn s r 0 p q oo are interpolation scales of quasi banach spaces for a natural complex interpolation method in the quasi banach range i e when p 1 or q 0 so that iix y ii 1 satisfying this last inequality is called the modulus of concavity of x the quasi normed space x ii 9 ii is said to be topologically p convex or p normable for some 0 p 0 so that ilxl x2 xnll p c llxlll p iix211 p ilxnll p 2 1 for any finite collection xl x2 x n x according to a theorem of aoki and rolewicz 30 theorem 1 3 p 7 x ii 9 ii is topologically p convex if p 1 log2k 1 where ic is the modulus of concavity of x any quasi norm on x induces a locally bounded topological vector space structure on x and conversely any such topological structure is induced by any of the equivalent quasi norms ilxlls inf r 0 x r e b x x 2 2 where b is a bounded neighborhood of the origin in x see 30 a quasi banach space is a locally bounded topological vector space x which is complete with respect to the quasi norms 2 2 if we prefer to stress the choice of a particular quasi norm we shall simply write x ii 9 ii when s e and either 0 p 1 or 0 q 0 x r co b 2 3 defines a norm on x moreover a different choice of b generates an equivalent norm define 7 ii 9 iic the banach envelope of x i1 ii to be the completion of x in the norm i1 iic see 30 when no reference to the norm is necessary we simply write x in place of x ii 9 iic note that x x continuously and densely 2 4 also if x is banach then x x consider next two quasi banach spaces xi ii 9 ilxi i 1 2 and denote by xl x2 the collection of all linear continuous maps from x1 into x2 clearly for each bounded neighborhood b of the origin in x1 iitus sup itxllx2 t s x1 x2 2 5 xeb defines a quasi norm on e x1 x2 as expected other choices of b yield equivalent quasi norms in the end s x2 is a quasi banach space when furnished with 2 5 moreover when x2 is actually banach then so is x1 x2 a simple but useful observation iscontained below proposition 1 let xi ii 9 xi i 1 2 be quasi banach spaces then any t e xi x2 extends to an operator 7 e x1 x2 and the mapping s t i e s xi y2 2 6 is continuous and injective in particular for any quasi banach space x and any banach space y proof the verification of the first part is straightforward of also 18 this also gives the left to right inclusion in 2 7 the opposite one is a consequence of the fact that t x 6 s y for each t 6 x y theorem 4 let xi ii 9 ilxi i 1 2 be quasi banach spaces then the following are true i ift 2 x1 x2 is an isomorphism then so is 7 e 2 xi x2 ii ift 2 x1 x2 is onto then so is 7 2 x1 x2 iii if t e x1 x2 is compact then so is 1 s x2 iv ift e e xi x2 is fredholm then so is 7 e xi 2 furthermore index t x1 x2 index i xi 2 9 v 2 8 ift e x1 x2 hasafinitedimensionalcokernel thenthesameholdsfor t e z xi x2 508 osvaldo mendez and marius mitrea proof for i apply proposition 1 to t and t 1 in order to obtain 1 t 1 consider next ii let y 6 x2 with ilyll z 1 then there exists 9 6 co bxz such that ily yll 2 1 2 thus jyj with yj e bx2 0 3 j 1 j 1 by hypothesis there exist x k x1 x2 t oo and xj xi so that xjllxl tc and txj yj for each j if we set x y ljxj x1 then ilxll x and tx y in particular ily txll 1 2 this suffices to conclude that 7 6 1 3 1 x2 is actually onto cf e g 29 lemma 2 4 going further iii follows more or less directly from the fact that t commutes with the operation of taking the convex hull and definitions in turn iii readily implies that fredholmness is preserved under hat hence to finish the proof of iv we are left with proving 2 8 to this end let e be a finite dimensional subset of x2 so that e t x1 x2 and denote by f the kernel of t 6 xi x2 then introduce the linear operator t x1 f e x2 t x y tx y 2 9 which clearly is an isomorphism thus by i it remains an isomorphism after applying the hat to the spaces in 2 9 since hat commutes with and leaves e f invariant cf theorem 5 below it follows that t xi f 9 e 3 2 is an isomorphism 2 10 this easily gives 2 8 finally with the aid of ii v is proved in a similar fashion to iv we leave the details to the interested reader in order to continue we need to explain one more thing specifically if xi ii 9 iix i 1 2 are quasi banach spaces so that x1 fq x2 is dense in xi ii 9 ilxi i 1 2 we say that xl ii 9 iix x2 ii 9 iix2 if any a x1 ii 9 iix has the property that aixinx2 extends to some element k x2 i1 iixy with 9 ii xt ll llxt x2 ii iix2 0independent of a also we say that x1 i1 iix x2 i1 iix2 if xl ii iix0 x2 i1 iix2 and x2 i1 iix2 xl i1 ilx corollary 1 for any quasi banach space x there holds x x proof it follows directly from proposition 1 we now turn to the main result of this section which can be thought of as a converse to corollary 1 in the class of banach spaces satisfying 2 4 see also 30 for a different criterion theorem 5 let x i1 ii x be a quasi banach space and y ii 9 ii y be a banach space then x ii x y i1 u r if and only if the following two conditions are fulfilled i the inclusion map t x ii iix y ii 9 ii y is well defined continuous and with dense image ii x i1 iix y i1 iiy proof the necessity is contained in corollary 1 as for sufficiency since the completion of x with respect to ii 9 iir and ii 9 i1 of 2 3 is y and x respectively we only need to prove that ii llc ii lly on x 2 12 the banach envelopes of besov and triebel lizorkin spaces 509 to see this we use the hahn banach theorem in order to find for each fixed x 9 x a functional a ax a 9 2 ii i1 x ii iix y ii lit 2 13 with iiaii u iir 1 and ia x l ilxllo thus iiailcr lly c so that ilxllc 5 iiail v ll llr ilxllr 0 independent of x 9 x conversely there exists a a a 9 y ii 9 lit x ii 9 iix ii 9 iic so that iia ll v lt llr 1 and ia x i ilxllr hence ilxllr iia ll ll llc llxllc 0 independent of x 9 x this proves 2 12 and finishes the proof of the theorem to exemplify these ideas let us give a short proof of the well known fact that p e 1 for 0 p c 0 uniformly for 151 2 supp t 9 5 9 n 89 i l c 0 uniformly for 3 151 3 i z ijr 9 2i5 12 1 if 5 0 here f denotes the fourier transform of f set oi x 2incp 2ix i 9 z for s 9 1t and 0 p q oo the inhomogeneous besov spaces are defined as follows bsp q n f 9 st l n i fllbp q rn idp fi lp 3o l i 1 510 osvaldo mendez and marius mitrea whereas for s r 0 p oo and 0 q oo the inhomogeneous triebel lizorkin spaces are defined as fp q n f e s r n ilfller i1 flilp cx e 2iskoi fj q lp 1 but only quasi banach when either p 1 of q 1 note that the diagonals of the two scales coincide i e b p r n f p r n for 0 p c and s jr there are also homogeneous versions of these spaces denoted by q r n and fr rn which are obtained by dropping the requirement that f ii lp oo and extending the sum in i to the range i z following 20 we also introduce a discrete version of the triebel lizorkin scale of spaces by defining j q for s r 0 p oo and 0 q oo as the collection of all sequences s sq q indexed by dyadic cubes q c r n such that q l q q dyadic 3 3 where xq is the characteristic function of q and i qi is the euclidean volume of q there is also an appropriate version of this definition when p oo specifically f 1 l q ilsll q sup e ial 1 2 s n p dyadic j isalxa x qdx 3 4 qcp observe that j sq q iqi a nsq q is an isomorphism between j p q and p a q for each ot r also as is well known each f q is a quasi banach lattice the inhomogeneous sequence space f q is defined as in 3 3 except that this time one insists on indexing the sequence and performing the sum only over dyadic cubes q c r n with qi 1 these enjoy similar properties as their homogeneous counterparts next as in 22 section 5 the sequence spaces q associated with the besov scale are introduced for s e r and 0 p q oo as the collection of all sequences s so q indexed by dyadic cubes q c ir n satisfying isii q e e l 2 s n q lel is lxq l l q v z iqi 2 nu e e qi i 2 s n i pisqi p q p l q v z iqi 2 nv 3 5 again the inhomogeneous spaces bsp q are defined similarly except that the sum only involves cubes q satisfying i qi 1 they are all quasi banach spaces for the indicated ranges of indices according to 22 theorem 7 20 the continuous versions of the spaces introduced above are actually isomorphic to their respective discrete versions via a wavelet transform nonetheless the occasional advantage of working with the sequence spaces is that they are quasi banach lattices in the sequel we shall make repeated use sometimes tacitly of this observation in the next several theorems we collect some basic properties of besov and triebel lizorkin spaces cf e g 42 39 and the references therein the banach envelopes of besov and triebel lizorkin spaces 511 theorem 6 embeddings for o po pl oo so sl e r o qo ql o0 with so n sl n the po pl inclusion bso qo l sl ql po r p r 1 is continuous with dense range moreover the same holds for the inclusion fpso qo 17st ql r n p r provided 0 po pl oo 0 qo ql oo and so sl po theorem 7 duality for s lr o p l and o q one has s n l 1 q n nsp q n no and and n pi ffi q rn foo rn b rn also for l p oo o q oo and s r s q n np q n p e q rn f s q r n 3 6 3 7 3 8 3 9 3 10 3 11 since 3 11 with 0 q fp q 3 12 is well defined linear bounded and has a continuous right inverse for any s e r 0 p c 0 q oo see 20 theorem 12 2 given the aforementioned result from 46 theorem 5 gives that fp q n fp l n if l p foo 0 q l se 3 13 thus the hypotheses of proposition 2 are verified and hence a fp q rn f l rn forl p or 0 q l s6r 3 14 in particular we get that fp q r n and f 1 rn have the same dual for the ranges of indices specified in 3 14 now the fact that fp i rn f s s 176 n is known see 20 remark 5 14 p 80 p and 39 p 20 to state the next result let 79 denote the collection of all polynomials in r n theorem 8 lifting property for f e s n 79 ands 1 0 p q oo the two assertions below are equivalent 512 osvaldo mendez and marius mitrea i f fp q n ii okf e fp l q rn foreach 1 k n with the obvious equivalence of norms also for each tempered distribution f s r following three statements are equivalent iii f fp q rn iv f lp r n a fp q rn v f e fp l q r n and o