MEASURE DENSITY AND EXTENSION OF BESOV AND TRIEBEL LIZORKIN FUNCTIONS

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1 MEASURE DENSITY AND EXTENSION OF BESOV AND TRIEBEL LIZORKIN FUNCTIONS TONI HEIKKINEN, LIZAVETA IHNATSYEVA, AND HELI TUOMINEN* Absrac. We show ha a domain is an exension domain for a Haj lasz Besov or for a Haj lasz Triebel Lizorkin space if and only if i saisfies a measure densiy condiion. We use a modificaion of he Whiney exension where inegral averages are replaced by median values, which allows us o handle also he case 0 < p < 1. The necessiy of he measure densiy condiion is derived from embedding heorems; in he case of Haj lasz Besov spaces we apply an opimal Lorenz-ype Sobolev embedding heorem which we prove using a new inerpolaion resul. This inerpolaion heorem says ha Haj lasz Besov spaces are inermediae spaces beween L p and Haj lasz Sobolev spaces. Our resuls are proved in he seing of a meric measure space, bu mos of hem are new even in he Euclidean seing, for insance, we obain a characerizaion of exension domains for classical Besov spaces Bp,q, s 0 < s < 1, 0 < p <, 0 < q, defined via he L p -modulus of smoohness of a funcion. Keywords: Besov space, Triebel Lizorkin space, exension domain, measure densiy, meric measure space 1. Inroducion The resricion and exension problems for Besov spaces and Triebel Lizorkin spaces in he seing of he Euclidean space have been sudied by several auhors using differen mehods; see for example [32], [3], [25], [39], [34], [37], [7], [31], [40] and he references herein. In paricular, i is known ha if Ω R n is a Lipschiz domain or an ε, δ)- domain, hen here is a bounded exension operaor from he classical Besov space B s p,qω), defined via he L p -modulus of smoohness of a funcion, o B s p,qr n ), 0 < s, p, q < ; see [39] and [7]. The analogous exension resuls hold for Triebel Lizorkin spaces; see [37], [31] and [46]. Alhough he class of he ε, δ)-domains, defined in [23], is raher wide, i does no cover all domains which admi an exension propery for Besov spaces or for Triebel Lizorkin spaces. For example, by [35, Thm 5.1], some d-hick domains in R n, measured wih he d-dimensional Hausdorff conen, are exension domains for cerain Besov and 2010 Mahemaics Subjec Classificaion. 46E35,46B70. T.H.: Deparmen of Mahemaics, P.O. Box 11100, FI Aalo Universiy, Finland, oni.heikkinen@aalo.fi, L.I.: Deparmen of Mahemaics and Saisics, P.O. Box 35, FI Universiy of Jyväskylä, Finland, lizavea.ihnasyeva@aalo.fi H.T.: Deparmen of Mahemaics and Saisics, P.O. Box 35, FI Universiy of Jyväskylä, Finland, heli.m.uominen@jyu.fi, , * he corresponding auhor. 1

2 2 TONI HEIKKINEN, LIZAVETA IHNATSYEVA, AND HELI TUOMINEN* Triebel Lizorkin spaces. I is also known ha he race spaces of Besov and Triebel Lizorkin spaces o an n-regular se S R n can be inrinsically characerized [40, Thm 1.3, Thm 1.6], such ha S admis an exension for Besov and Triebel Lizorkin spaces defined in erms of local polynomial approximaions. See also [39] and [24] for he relaed resuls. The connecion of he n-regulariy condiion, or, in oher words, a measure densiy condiion, and he exension propery for Sobolev spaces is sudied in [40] and in [14]. By [14, Thm 5], a domain Ω R n is an exension domain for W k,p, 1 < p <, k N, if and only if Ω saisfies he measure densiy condiion and W k,p Ω) coincides wih he Calderon-Sobolev space defined via sharp maximal funcions. For fracional Sobolev spaces W s,p Ω), 0 < s < 1, 0 < p <, which are special cases of Besov and Triebel Lizorkin spaces when p = q, he measure densiy condiion characerizes exension domains by [50, Thm 1.1]. A naural quesion o ask is wheher he same saemen is rue for Besov and Triebel Lizorkin spaces wihin he full range of parameers 0 < p <, 0 < q. Moreover, if 0 < s < 1, his quesion can be sudied in a general seing of a meric measure space. The recen developmen of he heory of funcion spaces in meric measure spaces does no only provide a uniform approach for characerizing smoohness funcion spaces on opological manifolds, fracals, graphs, and Carno Carahéodory spaces, bu a he same ime i gives a new poin of view o he classical Besov spaces and Triebel Lizorkin spaces on he Euclidean space. Among several possible definiions of Besov and Triebel Lizorkin spaces in he meric seing, he definiion recenly inroduced in [28] appears o be very convenien for he sudy of exension problems. This approach is based on Haj lasz ype poinwise inequaliies; i leads o he classical Besov and Triebel Lizorkin spaces in he seing of he Euclidean space and i gives a simple way o define hese spaces on a measurable subse of R n or, more generally, on a meric measure space. Definiion 1.1. Le X, d) be a meric space equipped wih a measure µ. A measurable se S X saisfies a measure densiy condiion, if here exiss a consan c m > 0 such ha 1.1) µs Bx, r)) c m µbx, r)) for all balls Bx, r) wih x S and 0 < r 1. Noe ha in he lieraure ses saisfying condiion 1.1) are someimes called regular ses, see, for example, [41]. If he measure µ is doubling, hen he upper bound 1 for he radius r is no essenial, and we can replace i by any number 0 < R <. Roughly speaking, he measure densiy condiion means ha he se S canno be oo hin near he boundary, in paricular, by [41, Lemma 2.1], i implies ha µs \ S) = 0. In he Euclidean space, nonrivial examples of ses saisfying he measure densiy condiion are Canor-like ses such as Sierpiński carpes of posiive measure. Recall ha if A is a quasi-banach space of measurable funcions and S X, an operaor E : AS) AX) such ha Eu S = u, for all u AS), is called an exension operaor. An exension operaor E is bounded if here is a consan c E > 0 such ha Eu AX) c E u AS) for all u AS). A domain Ω X is an A-exension domain if here is a bounded exension operaor E : AΩ) AX).

3 MEASURE DENSITY AND EXTENSION OF BESOV AND T L FUNCTIONS 3 In he meric seing, a connecion beween he measure densiy condiion and he exension propery for Sobolev spaces has been sudied in [15] and in [41]. By [15, Thm 6], [41, Thm 1.3], he measure densiy condiion for a se S implies he exisence of a bounded, linear exension operaor on he Haj lasz Sobolev space M 1,p S), for all 1 p <. In a geodesic, Q-regular meric measure space, he measure densiy condiion characerizes exension domains for M 1,p, 1 p <, see [15]. Our firs main resul is an exension heorem for Haj lasz Triebel Lizorkin spaces M s p,q and for Haj lasz Besov spaces N s p,q, see Secion 2 for he definiions. Theorem 1.2. Le X be a meric measure space wih a doubling measure µ and le S X be a measurable se. If S saisfies he measure densiy condiion 1.1), hen here is a bounded exension operaor E : M s p,qs) M s p,qx), for all 0 < p <, 0 < q, 0 < s < 1. An analogous exension resul holds for Haj lasz Besov spaces N s p,q. As in he corresponding resuls for he Haj lasz Sobolev spaces M 1,p in [15] and for he fracional Sobolev spaces W s,p, 0 < s < 1, 0 < p <, in [50], he exension E is independen of he parameers s, p and q of a funcion space; only he consan c E depends on hese parameers. In general, our exension operaor is no linear. This is due o he use of a modified Whiney ype exension where inegral averages are replaced by medians; similar modificaion was previously used in [50]. Bu if p > Q/Q + s), where Q is he doubling dimension of he space, an exension operaor in Theorem 1.2 can be chosen linear by employing he classical consrucion wih inegral averages. In he Euclidean case, we have Np,qR s n ) = Bp,qR s n ) and Mp,qR s n ) = Fp,qR s n ) for all 0 < p <, 0 < q, 0 < s < 1, where Bp,qR s n ) and Fp,qR s n ) are Besov spaces and Triebel Lizorkin spaces defined via an L p -modulus of smoohness, see [10]; recall ha he Fourier analyic approach gives he same spaces when p > n/n + s) in he Besov case and when p, q > n/n + s) in he Triebel Lizorkin case. Theorem 1.2, in paricular, shows ha he race spaces of he classical Besov and Triebel Lizorkin spaces on regular ses can be characerized in erms of poinwise inequaliies. Indeed, i follows ha Bp,qR s n ) S = Np,qS) s and Fp,qR s n ) S = Mp,qS) s wih equivalen norms. The following saemen, which is a combinaion of Theorem 1.2 and Theorem 6.1, is our second main resul. Theorem 1.3. Le X be a Q-regular, geodesic meric measure space and le Ω X be a domain. The following condiions are equivalen: 1) Ω saisfies he measure densiy condiion 1.1); 2) Ω is an M s p,q-exension domain for all 0 < s < 1, 0 < p < and 0 < q ; 3) Ω is an M s p,q-exension domain for some values of parameers 0 < s < 1, 0 < p < and 0 < q. 4) Ω is an N s p,q-exension domain for all 0 < s < 1, 0 < p < and 0 < q ; 5) Ω is an N s p,q-exension domain for some values of parameers 0 < s < 1, 0 < p < and 0 < q. To our knowledge, he fac ha an exension domain for he Besov space, or for he Triebel Lizorkin space, necessarily saisfies he measure densiy condiion is also new

4 4 TONI HEIKKINEN, LIZAVETA IHNATSYEVA, AND HELI TUOMINEN* in he Euclidean seing; excep for he special case p = q which was earlier proved in [50]. In order o show ha exension domains saisfy he measure densiy propery, we need a suiable Sobolev ype embedding heorem. For Haj lasz Triebel Lizorkin spaces such an embedding is easy o ge, since hey are subspaces of fracional Haj lasz Sobolev spaces. To obain an embedding heorem for Haj lasz Besov spaces, we show in Theorem 4.1 ha Haj lasz Besov spaces are inerpolaion spaces beween L p and Haj lasz Sobolev spaces, ha is, N s p,qx) = L p X), M 1,p X)) s,q, for 0 < s < 1, 0 < p < and 0 < q. We found his resul of independen ineres; in case of p > 1, q 1 i was earlier obained in [9] under an addiional assumpion ha he underlying space suppors a 1, p)-poincaré inequaliy. We close he paper wih an applicaion of Theorem 1.3 o Besov and Triebel Lizorkin spaces defined in he Euclidean space. In paricular, we obain he following resul for he classical Besov spaces B s p,q, 0 < s < 1, 0 < p < and 0 < q, defined via he L p -modulus of smoohness. Theorem 1.4. Le Ω R n be a domain. The following condiions are equivalen: 1) Ω saisfies he measure densiy condiion 1.1); 2) Ω is a B s p,q-exension domain for all 0 < s < 1, 0 < p < and 0 < q ; 3) Ω is a B s p,q-exension domain for some 0 < s < 1, 0 < p < and 0 < q ; Exension problems are closely relaed o he quesion of inrinsic characerizaion of spaces of fracional order of smoohness on subses S R n. The obained resuls show ha if S saisfies he measure densiy condiion, hen he space B s p,qs), 0 < s < 1, 0 < p <, 0 < q, can be defined via he L p -modulus of smoohness, via poinwise inequaliies or in erms of an aomic decomposiion, see [45] for he deails on he las-menioned approach; all hese definiions would lead o he same space of funcions which is he race space of he classical Besov space B s p,qr n ). We also give an analogue of Theorem 1.4 for cerain Triebel Lizorkin spaces, see Theorem 7.8. Noe ha here are several approaches o define Triebel Lizorkin spaces on domains, which, in general, give differen spaces. In Theorem 7.8, we use a definiion in he spiri of he classical definiion of F s p,qr n ) via differences; i describes, for example, he race space of F s p,qr n ) o a regular subse of he Euclidean space. Anoher version of Triebel Lizorkin ype spaces on domains was inroduced in [37] and [31]; a characerizaion of exension domains for hese spaces can be given similarly o he one for he Sobolev spaces in [14], see Theorem 7.11 and Remark The paper is organized as follows. In Secion 2, we inroduce he noaion and he sandard assumpions used in he paper and give he definiions of Haj lasz Besov spaces and Haj lasz Triebel Lizorkin spaces. In Secion 3, we presen some auxiliary lemmas needed in he proof of our exension resuls. In Secion 4, we prove inerpolaion and embedding heorems for Besov spaces. Secion 5 is devoed o he proof of Theorem 1.2. In Secion 6, we show ha Haj lasz Besov and Haj lasz Triebel Lizorkin

5 MEASURE DENSITY AND EXTENSION OF BESOV AND T L FUNCTIONS 5 exension domains saisfy measure densiy propery. In he las Secion 7, we discuss he Euclidean case. 2. Noaion and preliminaries We assume ha X = X, d, µ) is a meric measure space equipped wih a meric d and a Borel regular, doubling ouer measure µ, for which he measure of every ball is posiive and finie. These assumpions imply ha X is separable, see [19, Chaper 2.3]. The doubling propery means ha here exiss a fixed consan c D > 0, called he doubling consan, such ha 2.1) µbx, 2r)) c D µbx, r)) for every ball Bx, r) = {y X : dy, x) < r}. The doubling condiion gives an upper bound for he dimension of X since i implies ha here is a consan C = Cc D ) > 0 such ha for Q = log 2 c D, 2.2) µby, r)) µbx, R)) C r R for every 0 < r R and y Bx, R). As a special case of doubling spaces we consider Q-regular spaces. The space X is Q-regular, Q 1, if here is a consan c Q 1 such ha 2.3) c 1 Q rq µbx, r)) c Q r Q for each x X, and for all 0 < r diam X. Here diam X is he diameer of X. When we assume X o be doubling, hen Q refers o 2.2), and if X is Q-regular, hen Q comes from 2.3). A meric space X is geodesic if every wo poins x, y X can be joined by a curve whose lengh equals dx, y). By saying ha a measurable funcion u: X [, ] is locally inegrable, we mean ha i is inegrable on balls. Similarly, he class of funcions ha belong o L p B), p > 0, in all balls B, is denoed by L p loc X). The inegral average of a locally inegrable funcion u over a measurable se A wih 0 < µa) < is u A = A u dµ = 1 µa) ) Q A u dµ. The Hardy Lilewood maximal funcion of a locally inegrable funcion u is M ux) = u dµ. sup 0<r< Bx,r) If B = Bx, r) is a ball and > 0, we wrie B for he ball Bx, r). By χ E, we denoe he characerisic funcion of a se E X, and by u, he L -norm of u. The Lebesgue measure of a measurable se A R n is denoed by A. In general, C is a posiive consan whose value is no necessarily he same a each occurrence. When we wan o sress ha C depends on he oher consans or parameers a, b,..., we wrie C = Ca, b,... ). If here is a posiive consan C 1 such ha C1 1 A B C 1 A, we say ha A and B are comparable, and wrie A B.

6 6 TONI HEIKKINEN, LIZAVETA IHNATSYEVA, AND HELI TUOMINEN* 2.1. Haj lasz Besov and Haj lasz Triebel Lizorkin spaces. Besov and Triebel Lizorkin spaces are cerain generalizaions of Sobolev spaces o he case of fracional order of smoohness. There are several ways o define hese spaces in he Euclidean seing and spaces of Besov ype and of Triebel Lizorkin ype in he seing of a meric space equipped wih a doubling measure. For various definiions in a meric measure seing, see [9], [10], [16], [28], [33], [38], [49] and he references herein. In his paper, we mainly use he approach based on poinwise inequaliies, inroduced in [28]. An advanage of he poinwise definiion is ha i provides a simple way o inrinsically define funcion spaces on subses. Definiion 2.1. Le S X be a measurable se and le 0 < s <. A sequence of nonnegaive measurable funcions g k ) k Z is a fracional s-gradien of a measurable funcion u: S [, ] in S, if here exiss a se E S wih µe) = 0 such ha ux) uy) dx, y) s g k x) + g k y) ) for all k Z and all x, y S \ E saisfying 2 k 1 dx, y) < 2 k. The collecion of all fracional s-gradiens of u is denoed by D s u). For 0 < p, q and a sequence f = f k ) k Z of measurable funcions, we define fk ) L k Z = fk ) p S, l q ) k Z l q L p S) and where fk ) l k Z = ) f q L p S)) k L p S) k Z l q, fk ) k Z l q = { k Z f k q) 1/q, when 0 < q <, sup k Z f k, when q =. Definiion 2.2. Le S X be a measurable se. Le 0 < s < and le 0 < p, q. The homogeneous Haj lasz Triebel Lizorkin space Ṁ s p,qs) consiss of measurable funcions u: S [, ], for which he semi)norm u Ṁ s p,q S) = inf g D s u) g L p S, l q ) is finie. The inhomogeneous) Haj lasz Triebel Lizorkin space M s p,qs) is Ṁ s p,qs) L p S) equipped wih he norm u M s p,q S) = u L p S) + u Ṁ s p,q S). Similarly, he homogeneous Haj lasz Besov space Ṅ p,qs) s consiss of measurable funcions u: S [, ], for which u Ṅ s p,q S) = inf g k ) D s u) g k) l q L p S)) is finie, and he Haj lasz Besov space Np,qS) s is Ṅp,qS) s L p S) equipped wih he norm u N s p,q S) = u L p S) + u Ṅ s p,q S).

7 MEASURE DENSITY AND EXTENSION OF BESOV AND T L FUNCTIONS 7 When 0 < p < 1, he semi)norms defined above are acually quasi-semi)norms, bu for simpliciy we call hem, as well as oher quasi-seminorms in his paper, jus norms. Remark 2.3. Observe ha for inhomogeneous Haj lasz Triebel Lizorkin and Haj lasz Besov spaces he norms defined above are equivalen o u L p S) + inf g k) k N L p g D s S, l q ) and u L p S) + inf g k) k N l q u) g D s S, L p ) u) respecively, ha is, i is enough o ake ino accoun only he coordinaes of g wih posiive indices. Indeed, if x, y S \ E and 2 k 1 dx, y) < 2 k wih k 0, hen ux) uy) ux) + uy) 2 k+1)s dx, y) s ux) + uy) ). Hence, if g k ) k Z D s u), hen g k ) k Z, where g k = g k for k > 0 and g k = 2k+1)s u for k 0, belongs o D s u). Calculaing he norm, for example, for Haj lasz Triebel Lizorkin spaces, we obain ha g L p S, l q ) C ) g k) k N L p S, l q ) + g k) k 0 L p S, l q ) 0 ) 1/q, = C g k ) k N L p S, l q ) + C u L p S) 2 k+1)sq k= where he consans C depend on p and q only. This implies ha inf g g D s L p S, l q ) C inf g ) k) k N L p u) g D s S, l q ) + u L p S). u) If X suppors a 1, p)-poincaré inequaliy wih p 1, ), hen for all q 0, ), he spaces Mp,qX) 1 and Np,qX) 1 are rivial, ha is, hey conain only consan funcions, see [10, Thm 4.1]. Recall ha X suppors a 1, p)-poincaré inequaliy, if here exis consans c P > 0 and τ 1 such ha ) 1/p u u B dµ c P r g p dµ B for all locally inegrable funcions u, for all upper gradiens g of u and balls B X wih radius r > 0. For he definiion of an upper gradien, see, for example, [18], [13], [19]. The definiions of funcion spaces formulaed above are, in paricular, moivaed by Haj lasz s approach o he definiion of Sobolev spaces M 1,p X) on a meric measure space; see [12] and [13]. The fracional spaces M s,p X) were inroduced in [47], and were sudied, for example, in [22] and [20]. Definiion 2.4. Le S X be a measurable se. Le s 0 and le 0 < p <. A nonnegaive measurable funcion g is an s-gradien of a measurable funcion u in S if here exiss a se E S wih µe) = 0 such ha for all x, y S \ E, 2.4) ux) uy) dx, y) s gx) + gy)). The collecion of all s-gradiens of u is denoed by D s u) and he 1-gradiens shorly by Du). The homogeneous Haj lasz space Ṁ s,p S) consiss of measurable funcions u for which u Ṁ s,p S) = inf g g D s L p S) u) τb

8 8 TONI HEIKKINEN, LIZAVETA IHNATSYEVA, AND HELI TUOMINEN* is finie. The Haj lasz space M s,p S) is Ṁ s,p S) L p S) equipped wih he norm u M s,p S) = u L p S) + u Ṁ s,p S). Recall ha, by [12], M 1,p R n ) = W 1,p R n ) when p > 1, whereas for n/n+1) < p 1, M 1,p R n ) coincides wih he Hardy Sobolev space H 1,p R n ) by [26, Thm 1]. Noice also ha M 0,p X) = L p X) and ha M s,p X) coincides wih he Haj lasz Triebel Lizorkin space M s p, X), see [28, Prop. 2.1] for a simple proof of his fac On differen definiions of Besov and Triebel Lizorkin spaces. In he Euclidean seing he mos common ways o define Besov and Triebel Lizorkin spaces, via he L p -modulus of smoohness differences) and by he Fourier analyic approach, lead o he same spaces of funcions wih comparable norms when p > n/n + s) in he Besov case and when p, q > n/n + s) in he Triebel Lizorkin case. See, for example, [43, Chaper 2.5] and [17]. The space Mp,qR s n ) given by he meric definiion coincides wih he Triebel Lizorkin space F s p,qr n ), defined via he Fourier analyic approach, when 0 < s < 1, n/n+s) < p < and n/n + s) < q, and Mp, R 1 n ) = M 1,p R n ) = F 1 p,2r n ), when n/n + 1) < p <. Similarly, Np,qR s n ) coincides wih Besov space B s p,qr n ) for 0 < s < 1, n/n + s) < p < and 0 < q, see [28, Thm 1.2 and Remark 3.3]. For he definiions of F s p,qr n ) and B s p,qr n ), we refer o [43], [44], [28, Secion 3] Modulus of smoohness and Besov spaces. In addiion o he definiion based on poinwise inequaliies, we will someimes use a generalizaion o he meric seing of he classical definiion of he Besov spaces via he L p -modulus of smoohness; his general version was inroduced in [9]. Recall ha he L p -modulus of smoohness of a funcion u L p R n ) is 2.5) ωu, ) p = sup h u, ) L p R n ), h where > 0 and h u, x) = ux + h) ux). For 0 < s < 1 and 0 < p, q <, he Besov space Bp,qR s n ) consiss of funcions u L p R n ) for which 1 ) ) 1/q u B s p,q R n ) = u L p R n ) + s q d ωu, ) p is finie wih he usual modificaions when p = or q = ). For 1 s <, he definiion involves higher order differences of a funcion; see for example [7], [44]. Noe ha he inegral over he inerval 0, 1) can be replaced by he inegral over 0, ), since ωu, ) p C u L p R n ). Following [9] and [10], we define a modulus of smoohness which does no rely on he group srucure of he underlying space and which, for a funcion u L p R n ), is comparable wih ωu, ) p. Definiion 2.5. Le > 0, 0 < s < and 0 < p, q <. Le 1/p. 2.6) E p u, ) = ux) uy) dµy)dµx)) p X Bx,) 0

9 MEASURE DENSITY AND EXTENSION OF BESOV AND T L FUNCTIONS 9 The homogeneous Besov space B p,qx) s consiss of funcions u L p loc X) for which u B s p,qx) = 0 s E p u, ) ) q d is finie wih he usual modificaion when q = ). B p,qx) s L p X) wih he norm u B s p,q X) = u L p X) + u B s p,q X). ) 1/q The Besov space B s p,qx) is By he comparabiliy of ωu, ) p and E p u, ), he space Bp,qR s n ) coincides wih he classical space Bp,qR s n ) when 0 < s < 1. By [10, Thm 1.2], Ṅp,qX) s = B p,qx) s for all 0 < s < and 0 < p, q, and 2.7) u Ṅ s p,q X) u B s p,qx). As above, he inegral over he inerval 0, ) in he norm u B s p,q X) can be replaced by he inegral over 0, 1). I also follows by he resuls in [10] ha, for 0 < s < 1, 0 < p, q, he Haj lasz Triebel Lizorkin space M s p,qr n ) coincides wih he classical Triebel Lizorkin space F s p,qr n ) defined using differences. This space consiss of funcions u L p R n ), for which he norm u F s p,q R n ) = u L p R n ) + g L p R n ), where 1 gx) = and 0 < r < min{p, q}, is finie. 0 s ux + h) ux) r dh B0,) 3. Lemmas ) ) 1/r q d This secion conains lemmas needed in he proofs of he exension resuls. Below we will frequenly use he following simple inequaliy, which holds whenever a i 0 for all i Z and 0 < p 1, ) p 3.1) a i a p i. i Z i Z ) 1/q The firs lemma is used o esimae he norms of fracional gradiens. Lemma 3.1. Le 1 < a <, 0 < b < and c k 0, k Z. There is a consan C = Ca, b) such ha ) b a j k c j C c b j. k Z Proof. If b 1, hen he Hölder inequaliy for series implies ha ) b a j k c j C a j k c b j. j Z j Z j Z j Z

10 10 TONI HEIKKINEN, LIZAVETA IHNATSYEVA, AND HELI TUOMINEN* If 0 < b < 1, hen, by 3.1), ) b a j k c j a b j k c b j. j Z j Z Thus, denoing b = min{b, 1}, we obain ) b a j k c j C a b j k c b j C j Z j Z j Z k Z which proves he claim. k Z c b j a b j k C c b j, k Z j Z Nex we recall he Poincaré ype inequaliies which are valid for funcions and fracional gradiens, give a definiion of median values and lis some of heir properies and obain cerain norm esimaes for Lipschiz funcions Poincaré ype inequaliies. The definiion of he fracional s-gradien implies he validiy of some Sobolev Poincaré ype inequaliies. A similar reasoning as in he proof of [28, Lemma 2.1] in R n gives our firs inequaliy. Lemma 3.2. Le X be a Q-regular, geodesic meric measure space. Le 0 < s <. Le u be a locally inegrable funcion and le g j ) D s u). Then, for every x X and k Z, k 3.2) inf u c dµ C2 ks g j dµ. c R Bx,2 k ) j=k 3 Bx,2 k+2 ) Noe ha we will apply Lemma 3.2 for funcions in N s p,qx) wih sp > Q and wih sp = Q, and for hese values of parameers funcions in N s p,qx) are locally inegrable. Lemma 3.3 [10], Lemma 2.1). Le 0 < s < and 0 < < Q/s. Then for every ε and ε wih 0 < ε < ε < s, here exiss a consan C > 0 such ha for all measurable funcions u wih g j ) D s u), x X and k Z, 3.3) inf c R Bx,2 k ) where ε) = Q/Q ε). ) 1/ uy) c ε) ε) dµy) C2 kε j k 2 2 js ε ) Bx,2 k+1 ) g j dµ) 1/, If u is locally inegrable, g j ) D s u) and 0 < ε < ε < s <, hen 3.3) wih = Q/Q + ε) and he Hölder inequaliy imply ha for p Q/Q + ε), ) u u Bx,2 ) dµ C2 k kε 2 js ε ) 1/p. 3.4) g p j dµ Bx,2 k ) j k 2 Bx,2 k+1 ) While working wih he Haj lasz Triebel-Lizorkin spaces M s p,qx) we ofen use an embedding of hese spaces ino he space M s,p X) and employ he following Sobolev- Poincaré inequaliy for s-gradiens.

11 MEASURE DENSITY AND EXTENSION OF BESOV AND T L FUNCTIONS 11 Lemma 3.4 [10], Lemma 2.2). Le 0 < s < and 0 < < Q/s. There exiss a consan C > 0 such ha for all measurable funcions u wih g D s u), x X and r > 0, 3.5) inf c R Bx,r) where s) = Q/Q s). ) 1/ uy) c s) s) dµy) Cr s 1/, g dµ) Bx,2r) For s = 1 inequaliy 3.5) is given by [13, Thm 8.7], as well as for s 0, 1), since in his case d s is a disance in X Median values. Using inegral averages of a funcion is a sandard echnique in consrucing an exension operaor for a locally inegrable funcion. Since we are dealing wih L p -inegrable funcions, possibly wih 0 < p < 1, i is convenien o replace in he argumen inegral averages by median values, as for example in [50]. This allows o handle in he same way spaces of funcions wih inegrabiliy parameer 0 < p < ; a cerain disadvanage of his uniform reamen is ha he resuling exension operaor is, in general, non-linear. Definiion 3.5. The median value of a measurable funcion u on a se A X is { 3.6) m u A) = max µ {x A : ux) < a} ) µa) }. a R 2 The following properies of medians jusify heir role of counerpars for he inegral averages in he conex. Lemma 3.6 [50], Lemma 2.2; [10], 2.4)). Le 0 < η 1 and u L η loc X). Then 1/η. 3.7) m u B) c 2 u c dµ) η for all balls B and all c R. Moreover, 3.8) ux) = lim r 0 m u Bx, r)) a every Lebesgue poin x X. Remark 3.7. Propery 3.8) follows from 3.7) by he Lebesgue differeniaion heorem. The proof of [50, Lemma 2.2] shows ha inequaliy 3.7) holds for all measurable ses E wih posiive and finie measure. In paricular, 3.7) holds for every se B S, where S saisfies he measure densiy condiion 1.1) and B is a ball cenered a S. This, ogeher wih he measure densiy condiion and he Lebesgue differeniaion heorem, implies ha, ux) = lim r 0 m u Bx, r) S), for almos all x S. By combining 3.7) and Lemma 3.3, we obain he following resul, which is frequenly used in he proof of Theorem 1.2. B

12 12 TONI HEIKKINEN, LIZAVETA IHNATSYEVA, AND HELI TUOMINEN* Lemma 3.8. Le 0 < < and 0 < ε < s < 1. Le k Z, x X and le B be a ball such ha B Bx, 2 k ) and µb) µbx, 2 k )). Then here exiss a consan C > 0 such ha for all measurable funcions u wih g j ) D s u), 3.9) m u B) m u Bx, 2 k )) C2 kε 2 js ε ) 1/. gj dµ) j k 2 Bx,2 k+1 ) Proof. Le 0 < ε < ε and le η = ε) = Q/Q ε) if Q/Q + ε), and η = 1 oherwise. Using 3.3) and he Hölder inequaliy, we obain ) 1/η 3.10) inf uy) c η dµy) C2 kε 2 js ε ) 1/. gj dµ) c R Bx,2 k ) Le c R. By 3.7), we have j k 2 Bx,2 k+1 ) m u B) m u Bx, 2 k )) m u B) c + c m u Bx, 2 k )) ) 1/η 2 u c η dµ + 2 u c η dµ B C Bx,2 k ) u c η dµ) 1/η. Bx,2 k ) The claim follows by aking he infimum over c R and applying 3.10). ) 1/η Remark 3.9. If a se S saisfies he measure densiy condiion 1.1), hen he induced space S, d, µ S ) saisfies he doubling condiion 2.1) locally, ha is, for radii 0 < r 1, and we can replace small balls B, which are cenered in S, wih B S in inequaliy 3.9) Leibniz ype rules and norm esimaes for Lipschiz funcions. We finish his secion by proving a Leibniz ype rule for fracional s-gradiens and some norm esimaes for Lipschiz funcions. These norm esimaes are used laer o show ha he exension propery for Besov spaces, or for Triebel Lizorkin spaces, implies he measure densiy condiion 1.1). Lemma Le 0 < s < 1, 0 < p < and 0 < q, and le S X be a measurable se. Le u: X R be a measurable funcion wih g k ) D s u) and le ϕ be a bounded L-Lipschiz funcion suppored in S. Then sequences h k ) k Z and ρ k ) k Z, where ρ k = g k ϕ + 2 ks 1) L u ) χ supp ϕ and h k = g k + 2 sk+2 u ) ϕ χ supp ϕ are fracional s-gradiens of uϕ. Moreover, if u M s p,qs), hen uϕ M s p,qx) and uϕ M s p,q X) C u M s p,q S). Proof. For he firs claim, le x, y X, and le k Z such ha 2 k 1 dx, y) < 2 k. By he riangle inequaliy, we have ux)ϕx) uy)ϕy) ux) ϕx) ϕy) + ϕy) ux) uy).

13 MEASURE DENSITY AND EXTENSION OF BESOV AND T L FUNCTIONS 13 We consider four cases depending on wheher x or y belongs o supp ϕ or no. x, y supp ϕ, hen ux)ϕx) uy)ϕy) ux) Ldx, y) + ϕ dx, y) s g k x) + g k y)) dx, y) s 2 ks 1) L ux) + ϕ g k x) + g k y)) ) or alernaively, dx, y) s ρ k x) + ρ k y)), ux)ϕx) uy)ϕy) 2 ϕ ux) + ϕ L dx, y) s g k x) + g k y)) dx, y) s ϕ 2 2 sk+1) ux) + g k x) + g k y) ) dx, y) s h k x) + h k y)). Hence, in his case, ρ k ) k Z and h k ) k Z saisfy he required inequaliy. The remaining wo cases are considered in he same, even simpler, way. This shows ha ρ k ) k Z and h k ) k Z are fracional s-gradiens of uϕ. To prove he second claim, suppose ha u Mp,qS) s and g k ) D s u). By he firs par of he proof, he sequence g k ) k Z, { g k h k, if k < k L, = ρ k, if k k L, where k L is an ineger such ha 2 kl 1 < L 2 k L, is a fracional s-gradien of uϕ. Concerning he norm, if 0 < q <, we have ) [ 1/q k L 1 ) 1/q g k q C ϕ g k + 2 sk+2 u ) q and hence k Z [ C k= ) ] 1/q + g k ϕ + 2 ks 1) L u ) q k=k L ϕ k Z + L u ) 1/q k L 1 g k q + ϕ u k=k L 2 kqs 1) ) 1/q ], k= 2 sk+2)q ) 1/q g L p X, l q ) C ϕ g L p S, l q ) + 4 ϕ u L p S)2 sk L + L u L p S)2 k Ls 1) ) C ϕ g L p S, l q ) + 4 ϕ + 1)2 s L s u L p S)). Since g k ) k Z is an arbirary fracional s-gradien of u, he claim follows. The case q = follows using similar argumens. Remark An analogue of Lemma 3.10 holds also for funcions from Haj lasz Besov spaces N s p,qs). To prove his, i remains o show ha g l q X,L p ) <, wih he If

14 14 TONI HEIKKINEN, LIZAVETA IHNATSYEVA, AND HELI TUOMINEN* corresponding bound for he norm, whenever g D s u) is such ha g l q S,L p ) <. Indeed, when 0 < q <, we have ) 1/q g l q X,L p ) = g k q L p X) k Z kl 1 = ϕ k=k L k= gk + 2 sk+2 u ) 1/q q L p S) + gk ϕ + 2 ks 1) L u q L p S) ) 1/q C ϕ k Z g k q L p S)) 1/q + ϕ u L p S) + L u L p S) ) 1/q 2 kqs 1) k=k L C ϕ g l q S,L p ) + L s ϕ + 1) u L p S)), which implies he claim. The case q = follows similarly. kl 1 k= 2 sk+2)q ) 1/q By selecing u 1 and g k 0 for all k Z in he proof of) Lemma 3.11, we obain norm esimaes for Lipschiz funcions suppored in bounded ses. Corollary Le 0 < s < 1, 0 < p < and 0 < q. Le Ω X be a measurable se and le ϕ: Ω R be an L-Lipschiz funcion suppored in a bounded se F Ω. Then ϕ M s p,qω) and 3.11) ϕ M s p,q Ω) C1 + ϕ )1 + L s )µf ) 1/p, where he consan C > 0 depends only on s and q. The claim holds also wih M s p,qω) replaced by N s p,qω). 4. Inerpolaion and embedding heorems for Besov spaces In his secion, we prove new inerpolaion and embedding heorems for Besov spaces. We recall some essenial definiions and properies of he real mehod of inerpolaion; see, for example, he classical references [1], [2] for he deails. Le A 0 and A 1 be quasi-semi)normed spaces coninuously embedded ino a opological vecor space A. For every f A 0 + A 1 and > 0, he K-funcional is Kf, ; A 0, A 1 ) = inf { f 0 A0 + f 1 A1 : f = f 0 + f 1 }. Le 0 < s < 1 and 0 < q. The inerpolaion space A 0, A 1 ) s,q consiss of funcions f A 0 + A 1, for which f A0,A 1 ) s,q = 0 s Kf, ; A 0, A 1 ) ) ) q d 1/q, if q < sup >0 s Kf, ; A 0, A 1 ), if q =,

15 MEASURE DENSITY AND EXTENSION OF BESOV AND T L FUNCTIONS 15 is finie. The following heorem is he main resul of his secion. We will apply i laer only in he case q =, bu since his inerpolaion resul is of independen ineres, we prove i in full generaliy. The case 1 p <, 1 q was earlier obained in [9, Cor. 4.3] using a version of he Korevaar Schoen definiion for he Sobolev spaces in he meric seing. Theorem 4.1. Le X be a meric space wih a doubling measure µ. Le 0 < p <, 0 < q and 0 < s < 1. Then 4.1) Ṅ s p,qx) = L p X), Ṁ 1,p X)) s,q and 4.2) N s p,qx) = L p X), M 1,p X) ) s,q wih equivalen norms. Proof. To prove 4.1), we firs show ha here exiss a consan C > 0 such ha for all > 0, 1/ p 4.3) C 1 E p f, ) Kf, ; L p X), Ṁ 1,p X)) C 2 k p Epf, p 2 )) k, where p = min{p, 1} and E p f, ) is as in 2.6). We begin wih he firs inequaliy in 4.3). Le f = g + h, where g L p X) and h Ṁ 1,p X), and le > 0. Then k=0 E p f, ) CE p g, ) + E p h, )), where, by he Fubini heorem, Epg, p ) 2 p gx) p dµx) + 2 p 4.4) = 2 p X X C g p L p X). gx) p dµx) + 2 p X X Bx,) gy) p gy) p dµy)dµx) By,) 1 dµx) dµy) µbx, )) The las esimae follows using he doubling propery of µ and he fac ha By, ) Bx, 2) for each x By, ). By he definiion of he 1-gradien and by a similar argumen as in 4.4), for every ρ Dh) L p X), we have, Eph, p ) = hx) hy) p dµy)dµx) X X Bx,) Bx,) dx, y)) p ρx) + ρy)) p dµy)dµx) C p ρx) p dµx) + X C p ρ p L p X). X Bx,) ) ρy) p dµy)dµx)

16 16 TONI HEIKKINEN, LIZAVETA IHNATSYEVA, AND HELI TUOMINEN* By aking he infimum over all represenaions of f in L p X) + Ṁ 1,p X), we have ha E p f, ) CKf, ; L p X), Ṁ 1,p X)). To prove he second inequaliy in 4.3), le f L p loc X) and le > 0. By a sandard covering argumen, here is a covering of X by balls B i = Bx i, /6), i N, such ha χ i 2B i N wih he overlap consan N > 0 depending only on he doubling consan of µ. Le {ϕ i } i N be a collecion of C 1 -Lipschiz funcions ϕ i : X [0, 1] such ha supp ϕ i 2B i and i ϕ ix) = 1 for all x X, his is a so called pariion of uniy subordinae o he covering {B i } i N, see also he beginning of Secion 5). Le h: X R be a funcion defined using median values 3.6) of f, hx) = i N m f B i )ϕ i x), for all x X, and le g = f h. Le x X and le I x = {i : x 2B i }. By he properies of he pariion of uniy, gx) = fx) mf B i ) ) ϕ i x) = fx) m f B i ))ϕ i x). i N i Ix Since he number of elemens in I x is bounded by he overlap consan N independen of x and, and, for every i I x, B i Bx, ) 8B i, using 3.7) and he doubling propery of µ we obain 4.5) gx) 2 ) 1/p 1/p, fx) fz) p dµz) C fx) fz) dµz)) p i I B i x which implies ha 4.6) g L p X) CE p f, ). Bx,) Nex we esimae h in he Ṁ 1,p -norm. Le > 0 and le x, y X. We consider wo cases. Case 1: If dx, y), hen B i Bx, 2) 20B i, for every i I x I y. Since i N ϕ ix) ϕ i y)) = 0 for all x, y X and ϕ i is suppored in 2B i for each i, we have hx) hy) = i Nm f B i ) fx))ϕ i x) ϕ i y)) = i I x Iy m f B i ) fx))ϕ i x) ϕ i y)), which ogeher wih he C 1 -Lipschiz coninuiy of he funcions ϕ i, 3.7) and he doubling propery of µ implies ha dx, y) ) 1/p hx) hy) C fx) fz) p dµz) i I B x I i y dx, y) 1/p. C fx) fz) dµz)) p Bx,2)

17 MEASURE DENSITY AND EXTENSION OF BESOV AND T L FUNCTIONS 17 Case 2: Le dx, y) >. Since hx) hy) fx) fy) + gx) + gy), i suffices o esimae he erms on he righ side. The assumpion dx, y) > and 4.5) imply ha dx, y) 1/p, gx) C fz) fx) dµz)) p Bx,) and a corresponding upper bound holds for gy). Using 3.7) and he doubling propery of µ and wriing R = dx, y), we obain Hence where fx) fy) fx) m f Bx, R)) + fy) m f Bx, R)) ) 1/p 2 fz) fx) p dµz) Bx,R) + C By,2R) fz) fy) p dµz)) 1/p. fx) fy) Cdx, y)f x) + f y)), f x) = sup r 1 1/p fz) fx) dµz)) p. r Bx,r) Collecing he esimaes, we obain, in boh cases, ha hx) hy) Cdx, y)f x) + f y)), which shows ha f Dh). Hence i suffices o esimae f L p X). Using he definiion of f and he doubling propery of µ, we have f 1 ) 1/p x) sup fz) fx) p dµz) k=0 2 k 1 <r 2 k r Bx,r) C 2 k 1/p fz) fx) dµz)) p. k=0 Bx,2 k ) If 0 < p 1, we use inequaliy 3.1) and obain f p L p X) C 2 kp fz) fx) p dµz) dµx) p 4.7) k=0 X Bx,2 k ) = C 2 kp E pf, p 2 k ). p k=0 When p > 1, we have, by he Minkowski inequaliy, ha 4.8) f L p X) C 2 k E p f, 2 k ). k=0

18 18 TONI HEIKKINEN, LIZAVETA IHNATSYEVA, AND HELI TUOMINEN* Thus, he desired inequaliy ) 1/ p Kf, ; L p X), Ṁ 1,p X)) C 2 k p Epf, p 2 k ) follows using 4.6)-4.8) and he definiion of he K-funcional. Inerpolaion resul 4.1) for Ṅ p,qx): s The equivalence of Besov norms 2.7), he definiion of he norm f B and he firs inequaliy in 4.3) imply ha p,qx) s k=0 f Ṅ s p,q X) C f L p X),Ṁ 1,p X)) s,q. To obain he reverse esimae, we use he second inequaliy of 4.3). If q <, hen ) 1/q f L p X),Ṁ 1,p X)) s,q = s Kf, ; L p X), Ṁ 1,p X)) ) q d 0 ) q/ p ) 1/q C s p 2 k p Epf, p 2 k d ) =: A. 0 k=0 If q p, hen using 3.1) and change of variables, we obain A 2 kq k=0 = 2 kq k=0 0 0 s E p f, 2 k ) ) 1/q = 2 kqs 1) k=0 ) q d ) 1/q ) q 2 k τ) s dτ E p f, τ) τ 0 ) 1/q ) q τ s dτ E p f, τ) τ ) 1/q. If q p, hen, using Minkowski s inequaliy for inegrals, see [42, Appendix A1, p.271.], and changing variables, we have A k=0 0 = k=0 0 s p 2 k p E p pf, 2 k ) ) 1/ p = 2 k ps 1) k=0 ) q/ p d ) p/q ) 1/ p ) q/ p 2 k τ) s p 2 k p Epf, p dτ τ) τ 0 ) p/q ) 1/ p ) ) q 1/q τ s dτ E p f, τ). τ

19 MEASURE DENSITY AND EXTENSION OF BESOV AND T L FUNCTIONS 19 In he case q =, we have, using 4.3), f L p X),Ṁ 1,p X)) s, = sup s Kf, ; L p X), Ṁ 1,p X)) >0 s C sup >0 = C sup >0 k=0 k=0 k=0 ) 1/ p 2 k p Epf, p 2 k ) ) ) p 1/ p 2 2 k ps 1) k ) s E p f, 2 k ) ) 1/ p C 2 k ps 1) sup s E p f, ). >0 Since 0 < s < 1 and he norms in Besov spaces given by differen definiions are comparable, see 2.7), we have f L p X),Ṁ 1,p X)) s,q C f Ṅ s p,q X). Inerpolaion resul 4.2) for N s p,qx): We sar by showing ha 4.9) Kf,, L p X), M 1,p X)) Kf, ; L p X), Ṁ 1,p X)) + min{1, } f L p X) for each f L p X) + M 1,p X) and every > 0. Le f be such a funcion and le > 0. The definiion of he K-funcional and he spaces Ṁ 1,p X) and M 1,p X) imply ha Kf, ; L p X), Ṁ 1,p X)) Kf,, L p X), M 1,p X)). For every g L p X) and h M 1,p X) wih f = g + h, we have min{1, } f L p X) C ) min{1, } g L p X) + min{1, } h L p X) which implies ha C g L p X) + h M 1,p X)), min{1, } f L p X) CKf,, L p X), M 1,p X)). This implies one direcion of inequaliy 4.9). For he oher direcion, assume firs ha > 1. Then he claim follows from he fac ha Kf,, L p X), M 1,p X)) f L p X). If 0 < < 1, le g L p X) and h Ṁ 1,p X) be such ha f = g + h. Then h L p X) wih h L p X) C f L p X) + g L p X)) and Kf,, L p X), M 1,p X)) g L p X) + h M 1,p X) = g L p X) + h L p X) + h Ṁ 1,p X) C g L p X) + h Ṁ 1,p X) + f L p X)), from which he claim follows by aking he infimum over such decomposiions f = g+h.

20 20 TONI HEIKKINEN, LIZAVETA IHNATSYEVA, AND HELI TUOMINEN* Since 4.9) implies ha 0 s min{1, } ) q d f L p X),M 1,p X)) s,q and, hence, 4.2) follows from 4.1). < and sup s min{1, } <, >0 f L p X),Ṁ 1,p X)) s,q + f L p X), Remark 4.2. Since each linear operaor which is bounded in L p and in M 1,p, is bounded in he inerpolaion space, he exension resuls for Besov spaces wih p 1 follow from Theorem 4.1 and he exension resuls in [15]. Theorem 4.1 and he reieraion heorem [21, Thm 3.1] imply he following inerpolaion heorem for he Haj lasz Besov spaces. In he Euclidean seing, his resul was proved in [6] using differen mehods. For relaed inerpolaion resuls in he meric seing, see [48], [16] and [9]. Theorem 4.3. Le X be a meric space wih a doubling measure µ. Le 0 < p <, 0 < q, q 0, q 1, 0 < s 0, s 1, λ < 1, and s = 1 λ)s 0 + λs 1. Then Ṅ s 0 p,q 0 X), Ṅ s 1 p,q 1 X) ) = Ṅ s λ,q p,qx) and wih equivalen norms. N s 0 p,q 0 X), N s 1 p,q 1 X) ) λ,q = N s p,qx) 4.1. An Embedding heorem for he Haj lasz Besov spaces. Our inerpolaion heorem implies a Sobolev ype embedding resul for he Haj lasz Besov spaces. The embedding is ino he Lorenz spaces. Recall ha he Lorenz space L p,q X), 0 < p <, 0 < q, consiss of measurable funcions u: X [, ], for which he quasi)norm when q <, and u L p,q X) = 0 q µ {x X : ux) } ) q/p d u L p, X) = sup µ {x X : ux) > } ) 1/p, >0 ) 1/q, when q =, is finie. Using he Cavalieri principle, i is easy o see ha L p,p X) = L p X) wih equivalen norms). Moreover, L p, X) equals weak L p X)-space and L p,q X) L p,r X) when r > q. In he Euclidean seing, embedding Bp,qR s n ) L p s),q R n ) wih p s) = np/n sp) was obained in [17, Thm 1.15] using an aomic decomposiion of Bp,qR s n ). In he meric case, he embedding of Besov spaces Bp,qX), s when p > 1 and q 1, o Lorenz spaces was proved in [9] under he assumpion ha X suppors a 1, p)- Poincaré inequaliy. The idea of our proof comes from [9, Thm 5.1]. For he readers convenience, we give he proof wih all deails.

21 MEASURE DENSITY AND EXTENSION OF BESOV AND T L FUNCTIONS 21 Theorem 4.4. Le X be a Q-regular meric space, Q 1. Le 0 < s < 1, 0 < p < Q/s and 0 < q. There is a consan C > 0 such ha 4.10) inf c R u c L p s),q X) C u Ṅ s p,q X), where p s) = Qp/Q sp). Proof. By Lemma 3.4, for every ball Bx, r) X and for every u Ṁ 1,p X), we have ) 1/p 1/p, uy) c p dµy) Cr g dµ) p inf c R inf c R Bx,r) Bx,r) Bx,2r) where p = Qp/Q p), whenever g L p X) is a 1-gradien of u. Since he measure µ is Q-regular, we obain ) 1/p 1/p, uy) c p dµy) C 0 g dµ) p Bx,2r) where he consan C 0 > 0 is independen of r. I follows ha, for every k 1, here is c k R such ha u c k L p Bx,k)) 2C 0 u Ṁ 1,p X). Since u c k L p Bx,1)) u c k L p Bx,k) 2C 0 u Ṁ 1,p X), and X is Q-regular, we have ha c k c 1/p Q u ck L p Bx,1)) + u L p Bx,1))) ) u ck L p Bx,1)) + u c 1 L p Bx,1)) + c 1 c 1/p Q c 1/p Q C u Ṁ 1,p X) + C, where he consan C > 0 does no depend on k. As a bounded sequence in R, c k ) has a subsequence c kj ) ha converges o some c R. Now, for a fixed m, and for each k j m, we have u c L p Bx,m)) C ) u c kj L p Bx,m)) + c kj c L p Bx,m)) C ) u c kj L p Bx,k j )) + c kj c L p Bx,m)) C u Ṁ 1,p X) + µbx, m))1/p c kj c ). By leing firs j and hen m, we conclude ha u c L p X) C u Ṁ 1,p X). Since 1 s + s = 1, an inerpolaion heorem from [21, Thm 4.3] ogeher wih he p p p s) fac ha L r,r X) = L r X) for each r, saes ha L p s),q X) = L p X), L p X)) s,q. Thus, using Theorem 4.1, we obain inf c R u c L p s),q X) C inf c R u c L p X),L p X)) s,q C u L p X),Ṁ 1,p X)) s,q C u Ṅ s p,q X).

22 22 TONI HEIKKINEN, LIZAVETA IHNATSYEVA, AND HELI TUOMINEN* 5. The proof of Theorem 1.2 In order o prove Theorem 1.2, we use a modificaion of he Whiney exension mehod, which has been sandard in he sudy of exension problems saring from work [23]. We sar by recalling basic properies of he Whiney covering and he corresponding pariion of uniy; see, for example, [15]. We also refer o [5, Thm III.1.3] and [29, Lemma 2.9] for he proofs of hese properies. Le U X be an open se and, for each x U, le rx) = disx, X \ U)/10. There exiss a counable family B = {B i } i I of balls B i = Bx i, r i ), where r i = rx i ), such ha B is a covering of U and he balls 1/5B i are disjoin. The nex lemma easily follows from he definiion of he Whiney covering B and he doubling propery of he measure µ. Lemma 5.1. Le B be a Whiney covering of an open se U. There is M N such ha for all i N, 1) 5B i U, 2) if x 5B i, hen 5r i < disx, X \ U) < 15r i, 3) here is x i X \ U such ha dx i, x i ) < 15r i, 4) i I χ 5B i x) M for all x U. Le {ϕ i } i I be a Lipschiz pariion of uniy subordinaed o he covering B wih he following properies: i) supp ϕ i 2B i, ii) ϕ i x) M 1 for all x B i, where M is he bounded overlap consan of B, iii) here is a consan K > 0 such ha each ϕ i is Kr 1 i -Lipschiz, iv) i I ϕ ix) = χ U x). Noe ha if 5B i 5B j, hen 1/3r i r j 3r i and dx i, x j) 80r i, where he poins x i, x j are as in Lemma 5.1 3). As i was already menioned, we consruc an exension operaor using median values of a funcion. By his echnique, we can prove he resul for all 0 < p <, bu our exension operaor appears o be non-linear. If p > Q/Q + s), a linear exension can be obained by replacing medians wih inegral averages in he definiion of he local exension 5.2). This is easy o show employing 3.4) in he proof; we leave he deails o he reader The proof of Theorem 1.2. Le S X be a se saisfying measure he densiy condiion 1.1). We may assume ha S is closed, because µs \ S) = 0 by [41, Lemma 2.1]. Assume firs ha u M s p,qs) and ha g k ) k Z D s u) wih g k ) L p S, l q ) < 2 inf h k) L p h k ) D s S, l q ). u) Alhough he funcions g k are defined on S only, we idenify hem wih funcions defined on X by assuming ha each g k = 0 on X \ S.

23 MEASURE DENSITY AND EXTENSION OF BESOV AND T L FUNCTIONS 23 Le B = {B i } i I, B i = Bx i, r i ), be a Whiney covering of X \ S and le {ϕ i } i I be he associaed Lipschiz pariion of uniy. Define B 1 = {B i } i J as he collecion of all balls from B wih radius less han 1, and noe ha he measure densiy condiion holds for balls in B 1. For each i J, le x i be he closes poin of x i in S as in Lemma 5.1, le B i = Bx i, r i ), and for each x 2B i, i J, le B x = Bx, 25rx)) = Bx, 5 disx, S)). 2 Since 4r i 5rx) 6r i, we have Bi B x 47Bi and, by he measure densiy condiion and he doubling propery of µ, 5.1) µb x ) CµB i S). A local exension o a neighborhood of S: We will firs consruc an exension of u wih norm esimaes o he se For each x V \ S, le V = {x X : disx, S) < 8}. I x = {i I : x 2B i }. By Lemma 5.1, he number of elemens in I x is bounded by M. Moreover, if i I \ J, hen r i 1 and hence dis2b i, S) 8r i 8. Thus 2B i V = and i I x. Accordingly, I x J and herefore ϕ i x) = ϕ i x) = ϕ i x) = 1 for x V \ S. i I x i I i J Define he local exension Ẽu of u by { ux), if x S, 5.2) Ẽux) = i J ϕ ix)m u Bi S), if x X \ S. We begin by showing ha 5.3) Ẽu L p V ) C u L p S). If x X \ S, applying 3.7) wih 0 < η < p and using he fac ha Bi B x and 5.1), we obain Ẽux) ϕ i x) m u Bi S) C u η dµ i I x i I B x i S ) 1/η C u η dµ C M u η x) ) 1/η. B x Now norm esimae 5.3) follows from he definiion of Ẽu and he boundedness of he Hardy Lilewood maximal operaor in L p/η. A fracional s-gradien for he local exension: ) 1/η

24 24 TONI HEIKKINEN, LIZAVETA IHNATSYEVA, AND HELI TUOMINEN* Le 0 < δ < 1 s, 0 < ε < s and 0 < < min{p, q}. We define he sequence g k ) k Z, a candidae for he fracional s-gradien of Ẽu, as follows 5.4) g k x) = k 1 j= 2 j k)δ M g jx) ) 1/ + j=k 6 2 k j)s ε ) M g jx) ) 1/. We will spli he res of he proof of he heorem ino several seps. Lemma 5.2. There is a consan C > 0 such ha C g k ) k Z, where funcions g k are given by formula 5.4), is a fracional s-gradien of Ẽu. Proof. Le k Z and le x, y V be such ha 2 k 1 dx, y) < 2 k. We consider he following four cases: Case 1: Since, clearly, almos everywhere on S he inequaliy g k g k holds, for almos every x, y S, Ẽux) Ẽuy) = ux) uy) dx, y)s g k x) + g k y)). Case 2: x V \ S, y S. Then rx) = disx, S)/10 < 2 k /10 and here is x B x S such ha dx, x ) < 15rx). Le m Z be such ha 2 m 1 50rx) < 2 m and le Then B x B x and 2B x 9B x. Now B x = Bx, 2 m ). 5.5) Ẽux) Ẽuy) Ẽux) m ub x S) + uy) m u B x S), and we begin wih he firs erm of 5.5). Using 3.9), see also Remark 3.9, and he fac ha Bi B x B x wih comparable measures, we have Ẽux) m ub x S) = ϕ i x)m u Bi S) m u B x S)) i I x where Since Bx,2 m+1 ) C2 mε j m 2 2 js ε ) Bx,2 m+1 ) g j dµ) 1/, ) 1/ ) 1/ gj dµ C gj dµ C M g jx) ) 1/. 9B x 2 m 100rx) = 10 disx, S) 10dx, y) < 10 2 k, we have ha m k 4, and hence Ẽux) m ub x S) C2 ks j k 6 Cdx, y) s 2 k j)s ε ) M g jx) ) 1/ 2 k j)s ε ) M gjx) ) 1/. j k 6

25 MEASURE DENSITY AND EXTENSION OF BESOV AND T L FUNCTIONS 25 Nex we esimae he second erm in 5.5): uy) m u B x S) uy) m u By, 2 k ) S) + m u By, 2 k ) S) m u 2 l B x S) + m u 2 l B x S) m u B x S) = a) + b) + c), where l is an ineger which will be chosen laer. To esimae a), le y be such ha 3.8) holds almos every poin is such a poin). Using 3.9) and esimaing he geomeric series in he hird row by is firs erm 2 kε, we obain a) m u By, 2 i k ) S) m u By, 2 i+1) k ) S) C C i=0 2 i k)ε i=0 j=k 2 C2 kε C2 ks Cdx, y) s j=i+k) 2 2 js ε ) 2 js ε ) M g jy) ) 1/ j=k 2 j=k 2 i=0 2 js ε ) M g jy) ) 1/ 2 k j)s ε ) M g jy) ) 1/ j=k 2 By,2 i k+1 ) 2 i k)ε 2 k j)s ε ) M g jy) ) 1/. ) 1/ gj dµ Le now l be he smalles ineger such ha By, 2 k ) 2 l B x. By he minimaliy of l, he radius of he ball 2 l 1 B x is a mos 2 k+2. This implies ha l m k + 3 and ogeher wih he esimae dx, x ) 3 2 k 1, ha 2 l+1 B x Bx, 18 2 k ). Moreover, by he fac ha 2 k 1 dx, y) < 2 k and he selecion of l, we have 2 k 1 dx, y) dx, x ) + dy, x ) < 2 l m+1. Hence he radius 2 l m of he ball 2 l B x is comparable o 2 k, 5.6) 2 k 2 2 l m 2 k+3. For erm b), we use 3.9) and he fac By, 2 k ) 2 l B x 16By, 2 k ) o obain b) = m u By, 2 k ) S) m u 2 l B x S) C2 m l)ε 2 js ε ) 1/. gj dµ) j m l 2 2 l+1 B x

26 26 TONI HEIKKINEN, LIZAVETA IHNATSYEVA, AND HELI TUOMINEN* Since 5.6) ogeher wih he preceding discussion implies ha we have 2 l+1 B x 18Bx, 2 k ) C2 l+1 B x, b) C2 kε C2 ks j k 5 j k 5 Cdx, y) s 2 js ε ) 18Bx,2 k ) 2 k j)s ε ) M g jx) ) 1/ j k 5 ) 1/ gj dµ 2 k j)s ε ) M g jx) ) 1/. For he hird erm c), using similar esimaes as above, we obain, c) = m u 2 l B x S) m u B x S) l 1 m u 2 i B x S) m u 2 i+1 B x S) i=0 l 1 C 2 m i 1)ε i=0 j m i 3 l 1 C2 m l)s 2 i l+1)ε i=0 2 js ε ) j m i 3 2 i+2 B x ) 1/ gj dµ 2 m l j)s ε ) M g jx) ) 1/, where he las inequaliy follows from he inclusion B x B x 5B x and he doubling propery of he measure µ. Changing he order of summaion and using 5.6), we have c) C2 m l)s C2 ks j k 5 Cdx, y) s j m l 2 2 m l j)s ε ) M gjx) ) 1/ l 1 2 ε i l+1). 2 k j)s ε ) M g jx) ) 1/ j k 5 2 k j)s ε ) M g jx) ) 1/. Case 3: x, y V \ S, dx, y) min{disx, S), disy, S)}. We begin wih he inequaliy Ẽux) Ẽuy) Ẽux) m ub x S) + Ẽuy) m ub y S) + m u B x S) m u B y S) = 1) + 2) + 3), where y B y S and B y are chosen similarly as poin x and ball B x for x in he beginning of case 2. The radii of balls B x and B y are denoed by 2 mx and 2 my. We may assume ha disx, S) disy, S). Then ry) = 1 1 disy, S) dx, y) + disx, S)) 1 dx, y), i=0

27 MEASURE DENSITY AND EXTENSION OF BESOV AND T L FUNCTIONS 27 and hence disy, S) 2dx, y), dx, x ) < 2 k+1 and dy, y ) < 3 2 k. Hence esimaes for 1) and 2) follow similarly as for he firs erm of 5.5). For he las erm 3), le K 0 be he smalles ineger such ha he radius of he ball 2 K B y is a leas 2 k, ha is, 5.7) 2 K 1 my < 2 k 2 K my. Then m u B x S) m u B y S) m u B y S) m u 2 K B y S) + m u 2 K B y S) m u B x S) = α) + β). We begin wih α). If K = 0, hen α) = 0. If K > 0, hen he radius of 2 K 1 B y is a mos 2 k. This ogeher wih he fac ha dy, y ) < 3 2 k implies ha Hence, using 3.9), we have α) K 1 i=0 2 K+1 B y By, 7 2 k ) 5 2 K+1 B y. m u 2 i B y S) m u 2 i+1 B y S) K 1 C 2 my i 1)ε C i=0 K 1 i=0 j m y i 3 2 my i 1)ε j m y i 3 2 js ε ) 2 i+2 B y 2 js ε ) M g jy) ) 1/. ) 1/ gj dµ As in he case 2 c), we change he order of summaion, use esimae 5.7) and obain α) Cdx, y) s 2 k j)s ε ) M gjy) ) 1/. j k 2 For β), le L 0 be he smalles ineger such ha 2 K B y 2 L B x. By he selecion of L and K, we have L m x k + 4 and hence 2 L B x 22 2 K B y. We may assume ha here exiss a poin z 2 L B x \ 2 K B y, oherwise we have 2 K B y = 2 L B x and he argumen for esimaing β) jus becomes shorer. By 5.7), 2 k 2 K my dz, y ) dz, x ) + dx, y ) 2 L mx + 2 L mx = 2 L mx+1. Combining wo previous esimaes, we obain 5.8) 2 k 1 2 L mx < 2 k+4, and hence 2 L B x 18Bx, 2 k ) and Bx, 2 k ) 6 2 L B x. Now β) = m u 2 K B y S) m u B x S) m u 2 K B y S) m u 2 L B x S) + m u 2 L B x S) m u B x S),