Characterizations of Function Spaces via Averages on Balls

Transcription

1 Characterizations of Function Spaces via Averages on Balls p. 1/56 Characterizations of Function Spaces via Averages on Balls Dachun Yang (Joint work) School of Mathematical Sciences Beijing Normal University February 12, 2016 / Dedicated to Professor Hans Triebel on the Occasion of His 80-th Birthday

2 Characterizations of Function Spaces via Averages on Balls p. 2/56 Outline I Pointwise characterizations of Besov and Triebel-Lizorkin spaces II Ball average characterizations of second order Sobolev spaces III Ball average characterizations of second order Besov and Triebel-Lizorkin spaces IV Further remarks (Main motivation: Find new characterizations of well-known function spaces so that these new characterizations can be used as the definitions of the corresponding function spaces on metric measure spaces.)

3 Characterizations of Function Spaces via Averages on Balls p. 3/56 I. Pointwise characterizations of Besov and Triebel-Lizorkin spaces

4 Characterizations of Function Spaces via Averages on Balls p. 4/56 Sobolev Spaces Ẇm,p (R n ) & W m,p (R n ) / I Let m N := {1,2,...}, Z + := N {0} & p (1, ). f Ẇm,p (R n ) f S (R n ) (Schwartz distribution) and γ f L p (R n ) for all γ = m; moreover, f Ẇm,p (R n ) := γ f Lp (R n ). γ =m f W m,p (R n ) f L p (R n ) and γ f L p (R n ) for all γ m; moreover, f W m,p (R n ) := γ f Lp (R ). n γ m [ ] f W m,p (R n ) f Lp (R n ) + f Ẇm,p (R n )

5 Besov Spaces Ḃα p,q (Rn ) / I Let α (0, ), p, q (0, ] and ϕ S(R n ) satisfy (1.1) supp ϕ {ξ R n : 1/2 ξ 2} & Let (Triebel, 83 book) ϕ(ξ) constant > 0 if 3/5 ξ 5/3. { } S (R n ) := f S(R n ) : f(x)x α dx = 0, α Z n + R n. f Ḃα p,q(r n ) (homogeneous Besov space) f S (R n ) (dual of S (R n )) such that { 2 f Ḃα p,q (R n ) := jα ϕ j f } Lp (R n ) <, l q j Z here & hereafter, ϕ j ( ) := 2 jn ϕ(2 j ) for all j Z. Characterizations of Function Spaces via Averages on Balls p. 5/56

6 Characterizations of Function Spaces via Averages on Balls p. 6/56 Triebel-Lizorkin Spaces F α p,q (Rn ) / I f F p,q(r α n ) (homogeneous Triebel-Lizorkin space) f S (R n ) such that f F p,q(r α n ) <, where f F p,q(r α n ) := {2 jα ϕ j f} l j Z q Lp (R n ), p < and f F α,q(r n ) := sup x R n m Z 2mn B(x,2 m ) j=m 2 jα ϕ j f(y) q dy 1 q. F,2 0 (Rn ) = BMO(R n ), F p,2 0 (Rn ) = H p (R n ), p (0, ) & F p,2 m(rn ) = Ẇm,p (R n ), m Z +, p (1, ).

7 Characterizations of Function Spaces via Averages on Balls p. 7/56 Let Φ S(R n ) satisfy B α p,q(r n ) & F α p,q(r n ) / I supp Φ {ξ R n : ξ 2} & Φ(ξ) constant > 0 if ξ 5/3. The inhomogeneous Besov space Bp,q(R α n ) & Triebel-Lizorkin space Fp,q α (Rn ) are defined via replacing S (R n ) and {ϕ j } j Z in Ḃα p,q(r n ) & F p,q(r α n ), respectively, by S (R n ) and { ϕ j } j Z+, where ϕ 0 := Φ and ϕ j := ϕ j for j N. Moreover, f F α,q (R n ) := sup x R n m Z 2mn B(x,2 m ) j=max{m,0} 2 jα ϕ j f(y) q dy 1 q.

8 Characterizations of Function Spaces via Averages on Balls p. 8/56 Sobolev Spaces A Nice Characterization / I Theorem 1.1 ([H96]). Let p (1, ) and f be a measurable function. Then f Ẇ1,p (R n ) there exists a 0 g L p (R n ) such that, for a. e. x, y R n, Moreover, f g a.e. f(x) f(y) x y [g(x)+g(y)]. This observation makes it possible to introduce the Sobolev space of order 1 on an arbitrary metric space. [H96] P. Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), = (necessity) of Theorem 1.1 is due to B. Bojarski.

9 Characterizations of Function Spaces via Averages on Balls p. 9/56 What does g look like? (1) / I Bojarski in 1991 proved that, if p (1, ) and f Ẇ1,p (R n ), then, for all x, y R n, f(x) f(y) x y [ M x y ( f)(x)+m x y ( f)(y) ]. For any R (0, ], g L 1 loc (Rn ) and x R n, let (the local Hardy-Littlewood maximal function) M R (g)(x) := sup r<r 1 r n B(x,r) where B(x,r) := {y R n : y x < r}. g(y) dy, (M the Hardy-Littlewood maximal function.)

10 Characterizations of Function Spaces via Averages on Balls p. 10/56 What does g look like? (2) / I Observe that M x y ( f) L p (R n ), since f L p (R n ) and M x y is bounded on L p (R n ). Disadvantage: M x y ( f) has no locality, which means that, if f = 0 in a ball, then M x y ( f) may not equal 0 at that ball. B. Bojarski, Remarks on some geometric properties of Sobolev mappings, Functional analysis & related topics (Sapporo, 1990), 65-76, World Sci. Publ., River Edge, NJ, 1991.

11 Characterizations of Function Spaces via Averages on Balls p. 11/56 Hajłasz-Sobolev Spaces (1) / I (X,d,µ): X nonempty set; d metric; µ regular Borel measure p (1, ), s (0,1] The homogeneous fractional Hajłasz-Sobolev space Ṁs,p (X) is defined to be the set of all measurable functions f L p loc (X) for which there exist a 0 g Lp (X) and a set E X of measure zero such that, for all x, y X \E, (1.2) f(x) f(y) [d(x,y)] s [g(x)+g(y)]. Denote by D(f) the class of all nonnegative Borel

12 Characterizations of Function Spaces via Averages on Balls p. 12/56 Hajłasz-Sobolev Spaces (2) / I functions g satisfying (1.2). Moreover, define f M s,p (X) := inf g D(f) { g L p (X)}. Let M s,p (X) := L p (X) Ṁs,p (X) and, for all f M s,p (X), let f M s,p (X) := f Lp (X) + f M s,p (X). Remarks: Ṁ1,p (X) & M 1,p (X) were introduced by Hajłasz [H96]. Ṁs,p (X) & M s,p (X) when s (0,1) were introduced by Hu [Hu03] for subsets (fractals) of R n and Yang [Y03] for metric measure spaces.

13 Characterizations of Function Spaces via Averages on Balls p. 13/56 Hajłasz-Sobolev Spaces (3) / I It was proved in [H96] that Ṁ 1,p (R n ) = Ẇ1,p (R n ) = F 1 p,2(r n ) and in [Y03] that, when s (0,1), Ṁ s,p (R n ) = F s p, (R n ) F s p,2(r n ). (There exists a gap for Triebel-Lizorkin spaces.) [Hu03] J. Hu, A note on Hajłasz-Sobolev spaces on fractals, J. Math. Anal. Appl. 280 (2003), [Y03] D. Yang, New characterizations of Hajłasz-Sobolev spaces on metric spaces, Sci. China Ser. A 46 (2003),

14 Characterizations of Function Spaces via Averages on Balls p. 14/56 Unified Description of Ṁ s,p (R n ) / I Does there exist a unified description of M s,p (R n ) for all s (0,1]? Recall that, for all s (0,1], Ẇ s,p (R n ) = F s p,2 (Rn ). For m (n+1, ), let { (1.3) A := φ M(R n ) : sup α Z n +, α 1 R n φ(x)dx = 0, sup(1+ x ) m α φ(x) 1 x R n }, where M(R n ) denotes the space of all measurable functions.

15 Characterizations of Function Spaces via Averages on Balls p. 15/56 Grand Triebel-Lizorkin Spaces onr n (1) / I For A as in (1.3), any f S (R n ) and all x R n, the grand g-function (or the grand maximal function) AĠs,a(f) is defined by AĠs,a(f)(x) := sup k Z 2 ks sup φ 2 k f(x), φ A where φ t (x) := 1 t n φ( x t ) for all t > 0 and x Rn. The grand Triebel-Lizorkin space A F s p, (Rn ), s R, p (0, ): f S (R n ) and f A F s p, (R n ) := A Ġ s,a(f) Lp (R n ) <.

16 Characterizations of Function Spaces via Averages on Balls p. 16/56 Grand Triebel-Lizorkin Spaces onr n (2) / I Theorem 1.2 ([KYZ10]). (i) For s (0,1) and p (n/(n+s), ), A F s p, (R n ) = F s p, (R n ). (ii) For s (0,1] and p (n/(n+s), ), A F s p, (R n ) = Ṁs,p (R n ). (Grand maximal function characterization) Remarks: (i) Recall that Ṁ 1,p (R n ) = Ẇ1,p (R n ) = F p,2 1 (Rn ) F p, (R 1 n )

17 Characterizations of Function Spaces via Averages on Balls p. 17/56 Some Remarks (1) / I and f F 1 p,2 (Rn ), p (0, ), if and only if f S (R n ) and f F 1 p,2(r n ) := { where ϕ is as in (1.1). k Z 2 2ks ϕ 2 k f 2 }1 2 Lp (R n ) (ii) Theorem 1.2 is also true for any RD-spaceX. <, [KYZ10] P. Koskela, D. Yang & Y. Zhou, A characterization of Hajłasz-Sobolev and Triebel-Lizorkin spaces via grand Littlewood-Paley functions, J. Funct. Anal. 258 (2010),

18 Characterizations of Function Spaces via Averages on Balls p. 18/56 Some Remarks (2) / I Remarks. (i) The choice of A in the definition of A F s p, (X) is very subtle and it depends only on the first-order derivatives of test functions, which makes it possible to generalize Theorem 1.2 to metric measure spaces. (Lipschitz regularity) (ii) We finally remark that a continuous version of the grand Littlewood-Paley g-function ( k Z sup φ A φ 2 k f 2 ) 1/2 with a different choice of A was used by Wilson [W07] to solve a conjecture of R. Fefferman and E. M. Stein on the weighted boundedness of the classical Littlewood-Paley S-function. [W07] M. Wilson, The intrinsic square function, Rev. Mat. Ibero. 23 (2007),

19 Characterizations of Function Spaces via Averages on Balls p. 19/56 RD-Spaces (1) / I A triple (X,d,µ): X is a non-empty set, d a quasi-metric (usually, for simplicity, metric), and µ a regular Borel measure. A space of homogenous type of Coifman-Weiss: if µ-doubling (µ(b(x,2r)) µ(b(x,r))). An RD-space, if µ is both doubling and reverse-doubling (µ(b(x,2r)) C 0 µ(b(x,r)) and C 0 > 1). There exist many examples of RD-spaces. Especially, all connected spaces of homogeneous type are RD-spaces.

20 Characterizations of Function Spaces via Averages on Balls p. 20/56 RD-Spaces (2) / I [HMY08] Y. Han, D. Müller & D. Yang, A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces, Abstr. Appl. Anal Art. ID , 250 pp. Cited by A. Avila (A gainer of the 2014 Fields Medal), S. Crovisier and A. Wilkinson, Diffeomorphisms with positive metric entropy, arxiv: v3. D. Yang & Y. Zhou, New properties of Besov and Triebel-Lizorkin spaces on RD-spaces, Manuscripta Math. 134 (2011),

21 Characterizations of Function Spaces via Averages on Balls p. 21/56 Grand Lusin-area Characterization (1) / I [JYY] X. Jiang, D. Yang & W. Yuan, The grand Lusin-area characterization of Hajłasz-Sobolev spaces and Triebel-Lizorkin spaces, Math. Nachr. 286 (2013), For A as in (1.3), any f S (R n ) and all x R n, the grand Lusin-area function (or the grand non-tangential maximal function) AṠs,a(f) is defined by AṠs,a (f)(x) := sup2 ks sup k Z y B(x,a2 k ) sup φ 2 k f(y). φ A

22 Characterizations of Function Spaces via Averages on Balls p. 22/56 Grand Lusin-area Characterization (2) / I Theorem 1.3 ([JYY]). Let s (0,1], p (n/(n+s), ), a (0, ) and A be as in (1.3). Then f Ṁs,p (R n ) if and only if f S (R n ) and AṠs,a(f) L p (R n ). Moreover, M s,p (R n ) AṠs,a ( ) L p (R n ). Thm. 1.3 is also true for any RD-space X. Applications to real interpolation: X. Jiang, D. Yang & W. Yuan, Real interpolation for grand Besov and Triebel-Lizorkin spaces on RD-spaces, Ann. Acad. Sci. Fenn. Math. 36 (2011),

23 Characterizations of Function Spaces via Averages on Balls p. 23/56 Fractionals Hajłasz Gradient / I P. Koskela, D. Yang & Y. Zhou, Pointwise characterizations of Besov and Triebel-Lizorkin spaces and quasiconformal mappings, Adv. Math. 226 (2011), Definition 1.4. Let s (0, ) and u be a measurable function on X. A sequence of nonnegative measurable functions, g := {g k } k Z, is called a fractionals-hajłasz gradient of u if there exists E X with µ(e) = 0 such that, for all k Z and x, y X \E satisfying 2 k 1 d(x, y) < 2 k, u(x) u(y) [d(x, y)] s [g k (x)+g k (y)]. Denote by D s (u) the collection of all fractional s-hajłasz gradients of u.

24 Characterizations of Function Spaces via Averages on Balls p. 24/56 Ṁ s p,q (X) & Ṅs p,q (X) / I The homogeneous Hajłasz-Triebel-Lizorkin space M p,q(x) s is defined to be the space of all measurable functions u such that l u M := inf {g p,q(x) s j } j Z <. g D s (u) q Lp (X) The homogeneous Hajłasz-Besov space Ṅs p,q(x) is defined to be the space of all measurable functions u such that { } u N := inf p,q(x) s g D s (u) g j Lp (X) j Z <. l q

25 Characterizations of Function Spaces via Averages on Balls p. 25/56 Theorem 1.5. Let n N. Theorem 1.5 / I (i) If s (0, 1), p (n/(n+s), ) and q (n/(n+s), ], then Ṁs p,q(x) = F p,q(x). s (ii) If s (0, 1), p (n/(n+s), ) and q (0, ], then Ṅ s p,q(x) = Ḃs p,q(x). More optimal characterizations (for example, difference or sharp function) of Ṁ s p,q(x) and N s p,q(x) were established in A. Gogatishvili, P. Koskela & Y. Zhou, Characterizations of Besov and Triebel-Lizorkin spaces on metric measure spaces, Forum Math. 25 (2013),

26 Characterizations of Function Spaces via Averages on Balls p. 26/56 Theorem 1.6 / I Applying Theorem 1.5, we have the invariance under quasiconformal mappings of the following function spaces. Theorem 1.6 (i) Let n 2, s (0,1) and q (n/(n+s), ]. Then F n/s,q s (Rn ) is invariant under quasiconformal mappings of R n. (ii) Let n 2, s (0,1] and q (0, ]. Then M s n/s,q (Rn ) is invariant under quasiconformal mappings of R n. Thm. 1.6 is true for any RD-spaceX. Historical references? More recent results: H. Koch, P. Koskela, E. Saksman & T. Soto [JFA, 2014] or M. Bonk, E. Saksman & T. Soto [arxiv: ].

27 Characterizations of Function Spaces via Averages on Balls p. 27/56 More Research on Ṁs p,q (X) & Ṅs p,q (X) / I T. Heikkinen & H. Tuominen, Approximation by Hölder functions in Besov and Triebel-Lizorkin spaces, Constr. Approx. (to appear). T. Heikkinen, L. Ihnatsyeva & H. Tuominen, Measure density and extension of Besov and Triebel-Lizorkin functions, J. Fourier Anal Appl. (to appear) or arxiv: T. Heikkinen, P. Koskela & H. Tuominen, Approximation and quasicontinuity of Besov and Triebel-Lizorkin functions, arxiv:

28 Characterizations of Function Spaces via Averages on Balls p. 28/56 II. Ball average characterizations of second order Sobolev spaces

29 Characterizations of Function Spaces via Averages on Balls p. 29/56 Theorem of [AMV12] / II For any g L 1 loc (Rn ), x R n and t (0, ), let 1 B t (g)(x) := g(y) dy. B(x, t) B(x,t) ([AMV12]) Let p (1, ). Then f W 2,p (R n ) if and only if f L p (R n ) and there exists g L p (R n ) such that G(f,g)( ) := { 0 B t (f)( ) f( ) t 2 2 B t (g)( ) dt t }1 2 L p (R n ). theory of Vector-valued C-Z operators, finer estimates [AMV12] R. Alabern, J. Mateu & J. Verdera, A new characterization of Sobolev spaces on R n, Math. Ann. 354 (2012),

30 Characterizations of Function Spaces via Averages on Balls p. 30/56 Lusin-Area Funct. Charact. (1) / II ([HYY15]) (i) If p [2, ), then f W 2,p (R n ) if and only if f L p (R n ) and there exists g L p (R n ) such that { S(f,g)( ):= 0 B(,t) B t (f)(y) f(y) t 2 B t (g)(y) 2 L p (R n ). }1 dy dt 2 t n+1 (ii) If p (1,2) and n {1,2,3}, then f W 2,p (R n ) if and only if f L p (R n ) and there exists g L p (R n ) such that S(f,g) L p (R n ). [HYY15] Z. He, D. Yang & W. Yuan, Littlewood-Paley characterizations of second-order Sobolev spaces via averages on balls, Canadian Math. Bull. (to appear).

31 Characterizations of Function Spaces via Averages on Balls p. 31/56 Lusin-Area Funct. Charact. (2) / II ([DLYY]) (i) Let n [4, ) N and p ( 4+n 2n,2). Then f W 2,p (R n ) if and only if f L p (R n ) and there exists g L p (R n ) such that S(f,g) L p (R n ). 2n (ii) Let n [5, ) N and p (1, 4+n ). Then the conclusion of (i) does not hold true. In (i), if n = 4, then p (1,2). The conclusion of (i) is near sharp. Instead of one vector-valued C-Z operator, use a series of vector-valued C-Z operators [DLYY] F. Dai, J. Liu, D. Yang & W. Yuan, Littlewood-Paley characterizations of fractional Sobolev spaces via averages on balls, Submitted.

32 Characterizations of Function Spaces via Averages on Balls p. 32/56 G λ Characterization (1) / II ([HYY15]) (i) If p [2, ) and λ (1, ), then f W 2,p (R n ) f L p (R n ) and g L p (R n ) such that G λ (f,g)( ) := { 0 ( R n t t+ y B t (f)(y) f(y) t 2 }1 ) λn dy dt 2 t n+1 B t (g)(y) 2 L p (R n ). (ii) If p (1,2), λ (2/p, ) and n {1,2,3}, then f W 2,p (R n ) f L p (R n ) and g L p (R n ) such that G λ (f,g) Lp (R n ).

33 Characterizations of Function Spaces via Averages on Balls p. 33/56 G λ Characterization (2) / II ([DLYY]) (i) Let n [4, ) N, p ( 4+n 2n,2) and λ (2/p, ). Then f W 2,p (R n ) if and only if f L p (R n ) and there exists g L p (R n ) such that Gλ (f,g) Lp (R n ). 2n (ii) Let n [5, ) N, p (1, 4+n ) and λ (2/p, ). Then the conclusion of (i) does not hold true. In (i), if n = 4, then p (1,2). The conclusion of (i) is near sharp. It is still unclear on the endpoint case p = 2n 4+n.

34 Pointwise Characterization (1) / II ([DGYY15]) Let p (1, ). Then f W 2,p (R n ) f L p (R n ) and 0 g L p (R n ) and C 0 > 0 such that, for all t (0, ) and almost every x R n, f L p (R n ) and f(x) B t (f)(x) C 0 t 2 g(x) sup t (0, ) f B t (f) Lp (R n ) t 2 =: C 1 <. Not known for spaces of homogenous type. [DGYY15] F. Dai, A. Gogatishvili, D. Yang & W. Yuan, Characterizations of Sobolev spaces via averages on balls, Nonlinear Anal. 128 (2015), Characterizations of Function Spaces via Averages on Balls p. 34/56

35 Characterizations of Function Spaces via Averages on Balls p. 35/56 Pointwise Characterization (2) / II ([DGYY15]) Let p (1, ). Then f W 2,p (R n ) f L p (R n ) and 0 g L p (R n ) and c, C, C > 0 such that, for all t (0, ) and almost every x R n, Bt ( f B Ct (f)) (x) Ct 2 g(x) f L p (R n ) and 0 g L p (R n ) and c, C, C > 0 such that, for all t (0, ) and almost every x R n, B t ( f B Ct (f) ) (x) Ct 2 B ct (g)(x). The second equivalence also holds true on spaces of homogeneous type.

36 Characterizations of Function Spaces via Averages on Balls p. 36/56 Pointwise Characterization (3) / II ([DGYY15]) Let p (1, ), q [1,p), c (0, ) and K (0, ]. Then f W 2,p (R n ) f L p (R n ) and fc,q,k ( ) := sup t 2 {B t ( f B ct (f) q )( )} 1/q L p (R n ) t (0,K)

37 Characterizations of Function Spaces via Averages on Balls p. 37/56 A Key Lemma / II Let ϕ S(R n ) and C (0, ) be a constant. Then ϕ B t (ϕ) lim t 0 + t 2 = 1 2(n+2) ϕ and lim t 0 +B t with convergence in S(R n ). ( ϕ B Ct (ϕ) )( ) t 2 = C 2 2(n+2) ϕ( )

38 Characterizations of Function Spaces via Averages on Balls p. 38/56 Further Results / II For l N, t (0, ) and x R n, let B l,t (f)(x) := 2 ) ( 2l l l j=1 ( ) 2l ( 1) j B jt (f)(x). l j (Binomial coefficients; Observe that B 1,t (f) = B t (f).) All aforementioned characterizations of W 2,p (R n ) via pointwise inequalities remain true for W 2l,p (R n ), with l N and p (1, ), if we replace B t (f) by B l,t (f) therein. ([CYYZ]) Also true for Morrey-Sobolev spaces. [CYYZ] D.-C. Chang, D. Yang, W. Yuan & J. Zhang, Some recent developments of high order Sobolev-type spaces, J. Nonlinear Convex Anal. (to appear).

39 Characterizations of Function Spaces via Averages on Balls p. 39/56 Open Questions / II On spaces of homogeneous type (or even smooth domains of R n ), whether or not these Sobolev spaces coincide? (We now have several different definitions.) On spaces of homogeneous type, whether or not fractional Sobolev spaces contain the known Hajłasz-Sobolev spaces or the known Newton-Sobolev spaces? For analysis on metric measure spaces, any applications?....

40 Characterizations of Function Spaces via Averages on Balls p. 40/56 III. Ball average characterizations of second order Besov and Triebel-Lizorkin spaces

41 Characterizations of Function Spaces via Averages on Balls p. 41/56 Littlewood-Paley Characterization (1) / III ([YYZ13, DGYY15]) Let α (0, 2) and q (1, ]. (i) If p (1, ), then f Fp,q(R α n ) if and only if f L p (R n ) and { 2 kα f B 2 k(f) } l k Z + <. q Lp (R n ) (ii) If p =, then f F α,q(r n ) if and only if f C(R n ) and sup l Z x R n B 2 l k max{l,0} 2 kαq f B 2 k(f) q (x) 1 q <. C(R n ): the space of all uniformly continuous bounded functions

42 Characterizations of Function Spaces via Averages on Balls p. 42/56 Littlewood-Paley Characterization (2) / III ([YYZ13, DGYY15]) Let α (0, 2), p (1, ] and q (0, ]. Then f B α p,q(r n ) if and only if f L p (R n ) when p <, or f C(R n ) when p =, and 2 jαq f B 2 j(f) q L p (R n ) j=0 1 q <. [YYZ13] D. Yang, W. Yuan & Y. Zhou, A new characterization of Triebel-Lizorkin spaces on R n, Publ. Mat. 57 (2013), [DGYY15] F. Dai, A. Gogatishvili, D. Yang & W. Yuan, Characterizations of Besov and Triebel-Lizorkin spaces via averages on balls, J. Math. Anal. Appl. 433 (2016),

43 Characterizations of Function Spaces via Averages on Balls p. 43/56 Littlewood-Paley Characterization (3) / III Lusin-area type function: For any f L 1 loc (Rn ) and x R n, A r (f)(x) := { 2 kαq [B 2 k ( f B 2 k(f) r )(x)] q r }1 q. k=1 ([CLYY15]) Let α (0,2), p (1, ), q (1, ] and r [1,q). Then f F α p,q(r n ) if and only if f L p (R n ) and A r (f) L p (R n ). [CLYY15] D.-C. Chang, J. Liu, D. Yang & W. Yuan, Littlewood-Paley characterizations of Hajłasz-Sobolev and Triebel-Lizorkin spaces via averages on balls, Submitted.

44 Characterizations of Function Spaces via Averages on Balls p. 44/56 Littlewood-Paley Characterization (4) / III ([CLYY15]) Let α (0,2) and p (1, ), q (1, ]. (i) If f L p (R n ) and A q (f) L p (R n ), then f F α p,q(r n ). (ii) If p [q, ) and α (0,2), or p (1,q) and α (n(1/p 1/q),1), then f F α p,q (Rn ) implies that f L p (R n ) and A q (f) L p (R n ). In case when q = 2 and α < 1, i. e., F α p,2 (Rn ) = W α,p (R n ), then (ii) is not true when α < n(1/p 1/2). In case when p (1,q) and α [1,2), (ii) is unknown. Part of aforementioned results are also true for Bp,q α,τ (R n ) & Fp,q α,τ (R n ), in progress.

45 Characterizations of Function Spaces via Averages on Balls p. 45/56 Pointwise Characterization (1) / III ([YY15]) Let α (0, ) and f L 1 loc (Rn ). A sequence g := {g j } j 0 of non-negative measurable functions is called an α-order Hajłasz type gradient sequence of f if, for each j, there exists a set E j R n with measure zero such that f(x) B 2 jf(x) 2 jα g j (x), x R n \E j. Each g j satisfying the above is called an α-order Hajłasz type gradient of f at level j. [YY15] D. Yang & W. Yuan, Pointwise characterizations of Besov and Triebel-Lizorkin spaces in terms of averages on balls, Trans. Amer. Math. Soc. (to appear).

46 Characterizations of Function Spaces via Averages on Balls p. 46/56 Pointwise Characterization (2) / III ([YY15]) Let α (0,2) and p, q (1, ]. Then f Fp,q(R α n ) if and only if f L p (R n ) when p (0, ) or f C(R n ) when p =, and there exists an α-order Hajłasz type gradient sequence g := {g k } k=0 of f such that { } 1/q 2 kαq g k q <, p < k=0 Lp (R n ) and sup l Z x R n B 2 l k max{l,0} 2 kαq g k q (x) 1/q <, p =.

47 Characterizations of Function Spaces via Averages on Balls p. 47/56 Pointwise Characterization (3) / III ([YY15]) Let α (0,2), p (1, ] and q (0, ]. Then f Bp,q(R α n ) if and only if f L p (R n ) when p (0, ), or f C(R n ) when p =, and there exists an α-order Hajłasz type gradient sequence g := {g k } k=0 of f such that { k=0 2 kαq g k Lp (R n )} 1/q <. Let l N. All aforementioned pointwise characterizations of B α p,q(r n ) and F α p,q(r n ) remain true when α (0,2l) if we replace {f B 2 j(f)} j by {f B l,2 j(f)} j. Applications?

48 IV. Further remarks Characterizations of Function Spaces via Averages on Balls p. 48/56

49 Characterizations of Function Spaces via Averages on Balls p. 49/56 Newton Spaces (1) / IV N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana 16 (2000), Defined via upper gradients (J. Heinonen & P. Koskela [Acta Math., 1998]; P. Koskela & P. MacManus [Studia Math., 1998]). Instead of straight lines by curves γ: f(γ(a)) f(γ(b)) γ gds. Advantage: Strong locality (If a function is constant on a measurable set, then we can take upper gradient to be zero almost everywhere on that set; however, we cannot take the Hajłasz gradient to be zero almost everywhere on that set.) N. Shanmugalingam, D. Yang & W. Yuan, Newton-Besov spaces and Newton-Triebel-Lizorkin spaces, Positivity 19 (2015),

50 Characterizations of Function Spaces via Averages on Balls p. 50/56 Newton Spaces (2) / IV Newtonian-type Orlicz-Sobolev spaces on metric measure spaces H. Tuominen, Orlicz-Sobolev spaces on metric measure spaces, Dissertation, University of Jyväskylä, Jyväskylä, Ann. Acad. Sci. Fenn. Math. Diss. No. 135 (2004), 86 pp. Hajłasz-type Orlicz-Sobolev spaces and Newtonian-type Orlicz-Sobolev spaces T. Ohno & T. Shimomura, Musielak-Orlicz-Sobolev spaces on metric measure spaces, Czechoslovak Math. J. 65 (140) (2015),

51 Characterizations of Function Spaces via Averages on Balls p. 51/56 Sphere Average Charact. / IV P. Hajłasz & Z. Liu, A Marcinkiewicz integral type characterization of the Sobolev space, Publ. Mat. or arxiv: v2. Let p (1, ). Then f W 1,p (R n ) if and only if f L p (R n ) and 0 f( ) 1 S(,t) L p (R n ), S(,t) 2 dt f(y)dσ(y) t 3 where S(x,t) denotes the sphere centered at x with the radius t. (Not so useful for metric measure spaces) 1/2

52 Characterizations of Function Spaces via Averages on Balls p. 52/56 Weighted Sobolev Spaces / IV A new and simplified proof of the characterization of W 1,p (R n ) with p (1, ) ([Theorem 1, AMV12]) S. Sato, Littlewood-Paley operators and Sobolev spaces, Illinois J. Math. 58 (2014), Generalize [Theorem 1, AMV12] to the weighted case: W α,p w (R n ), α (0,2), p (1, ) and w A p (R n ) S. Sato, Littlewood-Paley equivalence and homogeneous Fourier multipliers, arxiv:

53 Characterizations of Function Spaces via Averages on Balls p. 53/56 Morrey-Sobolev Spaces / IV Morrey-Sobolev Spaces on Metric Measure Spaces Let 0 < p q. Recall that the Morrey space M q p(x) is defined to be the space of all measurable functions f on X such that f M q p (X) [ := sup[µ(b)] 1/q 1/p B X B f(x) p dµ(x)] 1/p <, where the supremum is taken over all balls B in X. Replace L p (X) by M q p(x) Y. Lu, D. Yang & W. Yuan, Morrey-Sobolev spaces on metric measure spaces, Potential Anal. 41 (2014),

54 Characterizations of Function Spaces via Averages on Balls p. 54/56 Haroske and Triebel / IV Triebel [T10] introduced the higher version of Hajłasz-Sobolev spaces on R n via higher differences, and some very interesting applications are given in [T11] and [HT11]: [HT11] D. D. Haroske & H. Triebel, Embeddings of function spaces: a criterion in terms of differences, Complex Var. Elliptic Equ. 56 (2011), [T10] H. Triebel, Sobolev-Besov spaces of measurable functions, Studia Math. 201 (2010), [T11] H. Triebel, Limits of Besov norms, Arch. Math. 96 (2011),

55 Characterizations of Function Spaces via Averages on Balls p. 55/56 Sobolev Spaces Associated with Operators / IV L. Yan & D. Yang, New Sobolev spaces via generalized Poincaré inequalities on metric measure spaces, Math. Z. 255 (2007), S. Hofmann, S. Mayboroda & A. McIntosh, Second order elliptic operators with complex bounded measurable coefficients in L p, Sobolev and Hardy spaces, Ann. Sci. École Norm. Sup. (4) 44 (2011), F. Bernicot, T. Coulhon & F. Dorothee, Sobolev algebras through heat kernel estimates, arxiv:

56 Future of Function Spaces in China? / IV (Son of Liguang and Wen) (Whose newest book?) Thank you for your attention. Characterizations of Function Spaces via Averages on Balls p. 56/56