Besov-type spaces with variable smoothness and integrability

Besov-type spaces with variable smoothness and integrability Douadi Drihem M sila University, Department of Mathematics, Laboratory of Functional Analysis and Geometry of Spaces December 2015 M sila, Algeria D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 1 / 47

Outline Besov-type spaces with fixed exponents Variable Lebesgue spaces Spaces of variable smoothness and integrability D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 2 / 47

Besov-type spaces with fixed exponents Besov spaces. Ψ S(R n ): Ψ(x) = 1 for x 1 and Ψ(x) = 0 for x 2. We put F ϕ 0 (x) = Ψ(x), F ϕ 1 (x) = Ψ(x) Ψ(2x) and F ϕ v (x) = F ϕ 1 (2 v x) for v = 2, 3,... Then {F ϕ v } v N0 is a resolution of unity, v =0 F ϕ v (x) = 1 for all x R n. Thus we obtain the Littlewood-Paley decomposition of all f S (R n ). f = ϕ v f v =0 D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 3 / 47

Definition s R, 0 < p, q f B s p,q = ( v =0 2 vsq ϕ v f q p) 1/q <. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 4 / 47

1. H. Triebel,Theory of function spaces. Birkhäuser, Basel, 1983. 2. H. Triebel,Theory of function spaces. II, Birkhäuser, Basel, 1992. 3. H. Triebel,Fractals and spectra, Basel, Birkhäuser, 1997. 4. H. Triebel,Theory of Function Spaces. III, Birkhäuser, Basel, 2006. 5. T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications 3, Walter de Gruyter, Berlin, 1996. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 5 / 47

Besov-type spaces BMO (R n ) spaces { BMO (R n ) = } f L 1 loc (Rn 1 ) : f BMO = sup B B B f (x) m B f dx <, where, m B f = 1 B B f (y)dy. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 6 / 47

Properties D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 7 / 47

Properties and 1 f L 1 loc belongs to BMO (Rn ) if and only if R n (1 + x ) n 1 f (x) dx < sup BJ 1 B J j J F 1 ϕ j f L 2 (B J ) 2 <, J Z. Here, ϕ S(R n ), suppϕ {ξ R n : 1 2 ξ 2}, j Z ϕ(2 j ξ) = 1, ξ = 0 and ϕ j = ϕ(2 j ). D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 7 / 47

Campanato spaces Definition (Campanato spaces) Let λ 0, 1 p < +. f L p loc Lp λ (R n ) if and only if belongs to f L λ p (R n ) = 1 ( 1/p B λ/n f m B f dx) p C. B D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 8 / 47

Properties (El Baraka 2006) Let 0 λ < n + 2. f L λ 2 (Rn ) if and only if sup BJ 1 B J λ/n j J F 1 ϕ j f L 2 (B J ) 2 <, J Z. Here, ϕ S(R n ), suppϕ {ξ R n : 1 2 ξ 2}, j Z ϕ(2 j ξ) = 1, ξ = 0 and ϕ j = ϕ(2 j ). D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 9 / 47

Properties (El Baraka 2006) Let 0 λ < n + 2. f L λ 2 (Rn ) if and only if sup BJ 1 B J λ/n j J F 1 ϕ j f L 2 (B J ) 2 <, J Z. Here, ϕ S(R n ), suppϕ {ξ R n : 1 2 ξ 2}, j Z ϕ(2 j ξ) = 1, ξ = 0 and ϕ j = ϕ(2 j ). A. El Baraka, An embedding theorem for Campanato spaces, Electron. J. Diff. Eqns. 66 (2002), 1-17. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 9 / 47

Properties (El Baraka 2006) Let 0 λ < n + 2. f L λ 2 (Rn ) if and only if sup BJ 1 B J λ/n j J F 1 ϕ j f L 2 (B J ) 2 <, J Z. Here, ϕ S(R n ), suppϕ {ξ R n : 1 2 ξ 2}, j Z ϕ(2 j ξ) = 1, ξ = 0 and ϕ j = ϕ(2 j ). A. El Baraka, An embedding theorem for Campanato spaces, Electron. J. Diff. Eqns. 66 (2002), 1-17. A. El Baraka, Littlewood-Paley characterization for Campanato spaces, J. Funct. Spaces. Appl. 4 (2006), no. 2, 193-220. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 9 / 47

For v Z and m Z n, let Q v,m be the dyadic cube in R n. We put Q = {Q v,m : v Z, m Z n }. Definition Let s R, τ [0, ) and 0 < p, q. The Besov-type space B s,τ p,q(r n ) is the collection of all f S (R n ) such that f B s,τ p,q = sup 1 ( ) q/p P Q P τ 2 vsq ϕ v f (x) p dx v =v + P P 1/q <, where v + P = max(0, log 2 l(p)). D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 10 / 47

Properties 1. Quasi-Banach spaces. 2. Decomposition properties (atoms, molecules,...) 3. Characterizations by Differences. 4. D. Yang and W. Yuan (2013) B s,τ p,q(r n ) = B s+n(τ 1/p), (R n ), 0 < p, s R if τ > 1/p, 0 < q or if τ = 1/p and q =. W. Sickel, D. Yang, W. Yuan, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics, vol. 2005, Springer-Verlag, Berlin, 2010. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 11 / 47

Variable Lebesgue spaces Definition Soit X un espace vectoriel sur k ( k = C ou R ). La fonctionnelle ϱ : X [0, ] est dite semi-modulaire sur X si les propriétés suivantes soient vérifiées: (1) ϱ (0) = 0. (2) ϱ (λx) = ϱ (x) pour tout x X, λ k avec λ = 1. (3) ϱ est quasi-convexe. (4) ϱ est continue à gauche. (5) ϱ (λx) = 0 pour tout λ > 0 implique x = 0. Un semi-modulaire ϱ est dit modulaire si ϱ (x) = 0 = x = 0. Une semi-modulaire ϱ est dit continue si l application λ ϱ (λx) est continue sur [0, ), pour tout x X. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 12 / 47

Variable Lebesgue spaces P 0 = {p : R n [c, ) measurable, c > 0} P = {p : R n [1, ) measurable}; p = ess-inf p (x), p + = ess-sup p (x). x R n x R n The variable exponent modular is defined by ϱ p( ) (f ) = f (x) p(x ) dx. (1) R n { } L p( ) = f : ϱ p( ) (λf ) < for some λ > 0 and ( ) f f p( ) = inf{λ > 0 : ϱ p( ) 1}. (2) λ D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 13 / 47

Properties D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 14 / 47

Properties Quasi-Banach spaces. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 14 / 47

Properties Quasi-Banach spaces. If p is canstant, then L p( ) = L p (classical Lebesgue spaces). D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 14 / 47

Properties Quasi-Banach spaces. If p is canstant, then L p( ) = L p (classical Lebesgue spaces). p + < then L p( ) is is separable. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 14 / 47

Properties Quasi-Banach spaces. If p is canstant, then L p( ) = L p (classical Lebesgue spaces). p + < then L p( ) is is separable. 1 < p p + < then L p( ) is reflexive. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 14 / 47

p P log we have Also for small balls B R n ( B 2 n ), and for large balls ( B 1). χ B p( ) χ B p ( ) B. (3) χ B p( ) B 1 p(x ), x B (4) χ B p( ) B 1 p (5) D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 15 / 47

Surpries D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 16 / 47

Surpries τ h maps L p( ) L p( ) for every h R n if and only if p is constant. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 16 / 47

Surpries τ h maps L p( ) L p( ) for every h R n if and only if p is constant. f g p( ) c f p( ) g 1, f L p( ), g L 1 if and only if p is constant. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 16 / 47

Surpries τ h maps L p( ) L p( ) for every h R n if and only if p is constant. f g p( ) c f p( ) g 1, f L p( ), g L 1 if and only if p is constant. Let p P log, ϕ L 1 and Ψ (x) := sup y x ϕ (y). We suppose that Ψ L 1. Then it was proved that ϕ ε f p( ) c Ψ 1 f p( ) for all f L p( ), where ϕ ε := 1 ε n ϕ ( ε ), ε > 0. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 16 / 47

Lemma Let p P log with 1 < p p + < and h R n. Then for all f L p( ) with supp Ff {ξ R n : ξ 2 v +1 }, v N 0, we have f ( + h) p( ) e (2+2vn h )c log (p) f p( ), where c > 0 is independent of h and v. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 17 / 47

1. O. Kováčik and J. Rákosník: On spaces L p(x ) and W 1,p(x ), Czechoslovak Math. J. 41(116) (1991), 592 618. 2. L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Berlin 2011. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 18 / 47

Mixed Lebesgue-sequence space Let p, q P 0. l q( ) (L p( ) ) is defined on sequences of L p( ) -functions by the modular ( ) ϱ l q( ) (L p( ) ) ((f f v v ) v ) = inf{λ v > 0 : ϱ p( ) 1}. (6) v (f v ) v l q( ) (L p( ) ) = inf{µ > 0 : ϱ l q( ) (L p( ) ) If p, q are constants, then (f v ) v l q( ) (L p( ) ) = (f v ) v lq (L p ) = ( v =0 λ 1/q( ) v ( ) 1 µ (f v ) v f v q p) 1/q. 1}. (7) D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 19 / 47

Let p, q P 0. Then l q( ) (L p( ) ) is a semimodular. Additionally it is a modular if p + <, and it is continuous if p +, q + <. Theorem (Almeida, Hästö, 2010) Let p, q P. If either 1 p + 1 q constant, then l q( ) (L p( ) ) is a norm. 1 or q is a Theorem (Kempka, Vybíral, 2013) Let p, q P. If 1 q p, then l q( ) (L p( ) ) is a norm. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 20 / 47

log-hölder continuity condition p(x) p(y) abbreviated p C log loc. log-hölder decay condition p(x) p c log (p) log(e + 1/ x y ), x, y Rn, (8) c log log(e + x ), x Rn. (9) Abbreviated p C log, if it is locally log-hölder continuous and satisfies the log-hölder decay condition. P log = {p P : 1p } is globally log-hölder continuous. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 21 / 47

Spaces of variables smoothness and integrability Besov spaces of variables smoothness and integrability Definition (Almeida, Hästö 2010) For α : R n R and p, q P 0, the Besov space B α( ) is defined by p( ),q( ) ( ) } B {f α( ) p( ),q( ) = S (R n ) : 2 v α( ) ϕ v f <. l q( ) (L p( ) ) v D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 22 / 47

Lemma ( ) (f v ) v l q( ) (L p( ) p( ) ) = sup f v {P Q, P 1} P 1/p( ) χ P ( ) (f v ) v l τ( ),q( ) (L p( ) ) = sup f v χ P Q χ P P τ( ) v v P l q( ) (L p( ) ) v v + P l q( ) (L p( ) ) Let p P log with 1 < p p + < and q, τ P log 0 with 0 < q q + <. (i) For m > 2n + c log (1/τ), there exists c > 0 such that (η v,m f v ) v l τ( ),q( ) (L p( ) ) c (f v ) v l τ( ),q( ) (L p( ) ). (ii) For m > 2n + c log (1/p), there exists c > 0 such that (η v,m f v ) l v q( ) (L p( ) p( ) ) c (f v ) v l q( ) (L p( ) p( ) )... D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 23 / 47

Besov-type spaces of variables smoothness and integrability Ψ, ϕ S(R n ): and suppf Ψ B(0, 2) and F Ψ(ξ) c if ξ 5 3 suppf ϕ B(0, 2)\B(0, 1/2) and F ϕ(ξ) c if 3 5 ξ 5 3, c > 0. Definition α : R n R, p, q, τ P 0 and ϕ v = 2 vn ϕ(2 v ). Besov-type space B α( ),τ( ) p( ),q( ). f B α( ),τ( ) p( ),q( ) ( ) = 2 v α( ) ϕ v f <. l τ( ),q( ) (L p( ) ) v D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 24 / 47

Definition α : R n R, p, q P 0 and ϕ v = 2 vn ϕ(2 v ). Besov-type space B α( ),p( ) p( ),q( ). f B α( ),p( ) p( ),q( ) ( = sup P Q 2 v α( ) ϕ v f P 1/p( ) χ P ) v v + P l q( ) (L p( ) ) <. Independently, D. Yang, C. Zhuo and W. Yuan, studied the function spaces B α( ),τ( ) (Besov-Type Spaces with Variable Smoothness and p( ),q( ) Integrability, J. Funct. Anal. 269(6), 1840 1898, (2015)) D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 25 / 47

Boundedness of the ϕ-transform Let Φ and ϕ satisfy: suppf Φ B(0, 2) and F Φ(ξ) c if ξ 5 3 (10) and 3 suppf ϕ B(0, 2)\B(0, 1/2) and F ϕ(ξ) c if 5 ξ 5 3, c > 0. (11) Let Ψ S(R n ) satisfying (10) and ψ S(R n ) satisfying (11) such that for all ξ R n F Φ(ξ)F Ψ(ξ) + j=1 F ϕ(2 j ξ)f ψ(2 j ξ) = 1, ξ R n. (12) D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 26 / 47

The ϕ-transform S ϕ is defined by setting where Ψ m (x) = Ψ(x m) and (S ϕ ) 0,m = f, Ψ m (S ϕ ) v,m = f, ϕ v,m where ϕ v,m (x) = 2 vn/2 ϕ(2 v x m) and v N. The inverse ϕ-transform T ψ is defined by T ψ λ = λ 0,m Ψ m + m Z n v =1 where λ = {λ v,m C : v N 0, m Z n }. m Z n λ v,m ψ v,m, D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 27 / 47

Q v,m is the dyadic cube in R n. χ v,m is the characteristic function of Q v,m. Definition Let p, q, τ P 0 and let α : R n R. Then for all complex valued sequences λ = {λ v,m C : v } N 0, m Z n } we define b {λ α( ),p( ) p( ),q( ) = : λ b α( ),p( ) < where p( ),q( ) 2 v (α( )+n/2) λ v,m χ v,m λ b α( ),p( ) = sup m Z n p( ),q( ) P Q P 1/p( ) χ P } and b {λ α( ),τ( ) p( ),q( ) = : λ α( ),τ( ) b < where p( ),q( ) 2 v (α( )+n/2) λ v,m χ v,m λ α( ),τ( ) b = sup m Z n p( ),q( ) P Q χ P τ( ) χ P v v + P l q( ) (L p( ) ) v v + P l q( ) (L p( ) ) D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 28 / 47.

We use a α( ),τ( ) to denote either bα( ),τ( ) p( ),q( ) p( ),q( ) or b α( ),p( ) p( ),q( ). Lemma Let α C log loc log and p, q, τ P0 and Ψ, ψ S(R n ) satisfy, respectively, (10) and (11). Then for all λ a α( ),τ( ) p( ),q( ) converges in S (R n ). T ψ λ = λ 0,m Ψ m + m Z n v =1 m Z n λ v,m ψ v,m, D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 29 / 47

We use A α( ),τ( ) α( ),τ( ) to denote either B p( ),q( ) p( ),q( ) or B α( ),p( ) p( ),q( ). Theorem Let α C log log loc and p, q, τ P0, 0 < q + <. Suppose that Φ, Ψ S(R n ) satisfying (10)and ϕ, ψ S(R n ) satisfy (11) such that (12) holds. The operators and S ϕ : A α( ),τ( ) p( ),q( ) aα( ),τ( ) p( ),q( ) T ψ : a α( ),τ( ) p( ),q( ) Aα( ),τ( ) p( ),q( ) are bounded. Furthermore, T ψ S ϕ is the identity on A α( ),τ( ) p( ),q( ). Corollary The definition of the spaces A α( ),τ( ) p( ),q( ) and ϕ. is independent of the choices of Φ D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 30 / 47

Lemma Let α C log log loc and p, q, τ P0 with τ (0, p ] and 0 < q + <. (i) f S (R n ) belongs to B α( ),τ( ) if and only if, p( ),q( ) f # B α( ),τ( ) p( ),q( ) = sup {P Q, P 1} ( 2 v α( ) ϕ v f χ P τ( ) (ii)f S (R n )belongs to B α( ),p( ) p( ),q( ) ( f α( ),p( ) = 2 # B v α( ) ϕ v f p( ),q( ) χ P if and only if, ) v ) v v P l q( ) (L p( ) ) <. l q( ) (L p( ) p( ) ) <, D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 31 / 47

Theorem Let α C log loc and p, q, τ P log 0 with τ (0, p ] and 0 < q + <. If (1/τ 1/p) > 0 or then and (1/τ 1/p) 0 and q, B α( ),τ( ) p( ),q( ) = B, α( )+n(1/τ( ) 1/p( )) B α( ),p( ) p( ), = B α( ), D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 32 / 47

Theorem Let α C log loc or then and p, q, τ P log 0 with 0 < q + <. If f B α( ),τ( ) p( ),q( ) (1/τ 1/p) + < 0 (1/τ 1/p) + 0 and q, ( = sup P Q 2 v α( ) ϕ v f χ P τ( ) is an equivalent quasi-norm in B α( ),τ( ) p( ),q( ). χ P ) v 0 l q( ) (L p( ) ), D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 33 / 47

Embeddings Theorem Let α C log loc and p, q 0, q 1, τ P log 0. (i) If q 0 q 1, then A α( ),τ( ) p( ),q 0 Aα( ),τ( ) ( ) p( ),q 1 ( ). (ii) If (α 0 α 1 ) > 0, then Theorem A α 0( ),τ( ) p( ),q 0 ( ) Aα 1( ),τ( ) p( ),q 1 ( ). Let α 0, α 1 C log loc and p 0, p 1, q, τ P log 0 with 0 < q + <. If α 0 > α 1 n ( ) and α 0 (x) p 0 (x ) = α 1(x) p 1 (x ) with p0 p 1 < 1, then n A α 0( ),τ( ) p 0 ( ),q( ) Aα 1( ),τ( ) p 1 ( ),q( ). D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 34 / 47

Theorem Let α C log log loc and p, q, τ P0 with 0 < q q + <. If (p 2 p 1 ) + 0, then B α( )+ n p 2 ( ),q( ) τ( ) + n p 2 ( ) n p 1 ( ) B α( ),τ( ) p 1 ( ),q( ). Theorem Let α C log loc and p, q, τ P log 0 with 0 < q q + <. Then B α( ),τ( ) p( ),q( ) B α( )+ n τ( ) n p( ),. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 35 / 47

Theorem Let α C log loc and p, q, τ P log 0 with 0 < q + <. Then S(R n ) A α( ),τ( ) p( ),q( ) S (R n ). D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 36 / 47

Let 0 < u p. The Morrey space M p u is defined to be the set of all u-locally Lebesgue-integrable functions f on R n such that f M p u ( 1/u = sup B p 1 u 1 f (x) dx) u <, B B where the supremum is taken over all balls B in R n. Definition Let {F ϕ v } v N0 be a resolution of unity, α : R n R, 0 < u p and 0 < q. The Besov-Morrey space Np,q,u α( ) is the collection of all f S (R n ) such that f N α( ) p,q,u = ( v =0 2 v α( ) ϕ v f ) 1/q q <. M p u D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 37 / 47

J. Fu, J. Xu, Characterizations of Morrey type Besov and Triebel-Lizorkin spaces with variable exponents, J. Math. Anal. Appl., 381 (2011), 280-298. Proposition Let α C log loc, 0 < q < and 0 < u < p <. (i) For 0 < q < we have the continuous embeddings N α( ) p,q,u B α( ),( 1 u 1 p ) 1 p,q. (ii) We have N α( ) p,,u = B α( ),( 1 u 1 p ) 1 p,. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 38 / 47

Definition Let 0 < p p + < and 0 < q q + <. The Triebel-Lizorkin space F α( ) p( ),q( ) is the collection of all f S (R n ) such that f α( ) F := p( ),q( ) ( ) 2 α( ) ϕ v f v 0 <. (13) L p( ) (l q( ) ) M. Izuki and T. Noi (2012-2014) have obtained the duality of F α( ) p( ),q( ), for 1 < p p + < and 1 < q q + <. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 39 / 47

Theorem Let α C log loc and q P log with 1 < q q + <. Then ( ) F α( ) 1,q( ) = B α( ),q ( ) q ( ),q ( ). In particular, if g B α( ),q ( ) q ( ),q ( ), then the map, given by l g (f ) = f, g, defined initially for f S(R n ) extends to a continuous linear functional on F α( ) 1,q( ) with g B α( ),q ( ) l g ( q ( ),q F α( ) ) and every l ( ) 1,q( ) l = l g for some g B α( ),q ( ) q ( ),q ( ). ( F α( ) 1,q( )) satisfies D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 40 / 47

Atomic decomposition Definition Let K N 0, L + 1 N 0 and let γ > 1. A function a C K (R n ) is called [K, L]-atom centered at Q v,m, v N 0 and m Z n, if supp a γq v,m (14) D β a(x) 2 v ( β +1/2), for 0 β K, x R n (15) and if R n x β a(x)dx = 0, for 0 β L and v 1. (16) D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 41 / 47

Theorem Let α C log log loc and p, q, τ P0 with 0 < q q + <. Let 0 < p p + and let K, L + 1 N 0 such that respectively K ([α + + n/p ] + 1) + and K ([α + + n/τ ] + 1) +, (17) 1 L max( 1, [n( min(1, p ) 1) α ]). (18) Then f S (R n ) belongs to B α( ),τ( ) p( ),q( ), respectively to B α( ),p( ), if and p( ),q( ) only if it can be represented as f = v =0 where ϱ v,m are [K, L]-atoms and m Z n λ v,m ϱ v,m, converging in S (R n ), (19) D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 42 / 47

λ = {λ v,m C : v N 0, m Z n } b α( ),τ( ) p( ),q( ), respectively λ b α( ),p( ) p( ),q( )., where the infimum, respectively inf λ b α( ),τ( ) p( ),q( ) Furthermore, inf λ α( ),τ( ) b p( ),q( ) is taken over admissible representations (19), is an equivalent quasi-norm in B α( ),τ( ) p( ),q( ), respectively B α( ),p( ) p( ),q( ). D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 43 / 47

Characterization by ball means of differences Let f be an arbitrary function on R n and x, h R n. and h f (x) = f (x + h) f (x), M h +1 f (x) = h ( M h f )(x), M N. M h f (x) = M j=0 ( 1) j Cj M f (x + (M j)h). We put (ball means of differences) f (x) = t n M h f (x) dh d M t h t t > 0, M N Let L p( ) be the collection of functions f Lp( ) τ( ) loc (Rn ) such that f p( ) L = sup τ( ) {P Q, P 1} f χ P <, p, τ P 0, χ P τ( ) p( ) D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 44 / 47

We define f B α( ),τ( ) p( ),q( ) = f p( ) L + sup τ( ) P Q ( 2 k α( ) χ P τ( ) d M 2 k f χ P ) k v + P l q( ) (L p( ) ). D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 45 / 47

We set δ p,τ = n min ( 0, ( 1 p 1 ) ). τ Theorem Let α C log log loc, M N, τ, q P0 and p P log, with p > 1. Assume Then B α( ),τ( ) p( ),q( ) 0 < α α + < M + δ p,τ. is equivalent quasi-norm on B α( ),τ( ) p( ),q( ). D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 46 / 47

1 A. Almeida and P. Hästö, Besov spaces with variable smoothness and integrability, J. Funct. Anal. 258 (2010), 1628 1655. 2 L. Diening, P. Hästö and S. Roudenko, Function spaces of variable smoothness and integrability, J. Funct. Anal. 256 (2009),no. 6, 1731 1768. 3 L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Berlin 2011. 4 D. Drihem, Atomic decomposition of Besov spaces with variable smoothness and integrability, J. Math. Anal. Appl. 389 (2012), no. 1, 15 31. 5 D. Drihem, Some properties of variable Besov-type spaces, Funct. Approx. Comment. Math. 52 (2015), no. 2, 193 221. 6 D. Drihem, Some characterizations of variable Besov-type spaces, Ann. Funct. Anal. 6 (2015), no.4, 255 288. 7 D. Drihem, On the duality of variable Triebel-Lizorkin spaces, arxiv:1506.01642. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 46 / 47

8 M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces, J. Funct. Anal. 93 (1990), no. 1, 34 170. 9 H. Kempka and J. Vybíral, A note on the spaces of variable integrability and summability of Almeida and Hästö, Proc. Amer. Math. Soc. 141 (2013), no. 9, 3207 3212. 10 H. Kempka and J. Vybíral, Spaces of variable smoothness and integrability: Characterizations by local means and ball means of differences, J. Fourier Anal. Appl. 18, (2012), no. 4, 852 891. 11 T. Noi, Duality of variable exponent Triebel-Lizorkin and Besov spaces, J. Funct. Spaces Appl. 2012, Art. ID 361807, 19 pp. 12 M. Izuki and T. Noi, Duality of Besov, Triebel-Lizorkin and Herz spaces with variable exponents, Rend. Circ. Mat. Palermo. 63 (2014), 221 245. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 47 / 47

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