Besov-type spaces with variable smoothness and integrability
|
|
- Susan Charles
- 5 years ago
- Views:
Transcription
1 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
2 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
3 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
4 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
5 1. H. Triebel,Theory of function spaces. Birkhäuser, Basel, H. Triebel,Theory of function spaces. II, Birkhäuser, Basel, H. Triebel,Fractals and spectra, Basel, Birkhäuser, H. Triebel,Theory of Function Spaces. III, Birkhäuser, Basel, 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, D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 5 / 47
6 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
7 Properties D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 7 / 47
8 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
9 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
10 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
11 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), D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 9 / 47
12 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), A. El Baraka, Littlewood-Paley characterization for Campanato spaces, J. Funct. Spaces. Appl. 4 (2006), no. 2, D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 9 / 47
13 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
14 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, D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 11 / 47
15 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
16 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
17 Properties D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 14 / 47
18 Properties Quasi-Banach spaces. D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 14 / 47
19 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
20 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
21 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
22 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
23 Surpries D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 16 / 47
24 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
25 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
26 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
27 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
28 1. O. Kováčik and J. Rákosník: On spaces L p(x ) and W 1,p(x ), Czechoslovak Math. J. 41(116) (1991), 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 D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 18 / 47
29 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
30 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
31 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
32 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
33 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
34 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
35 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), , (2015)) D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 25 / 47
36 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
37 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
38 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.
39 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
40 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
41 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
42 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
43 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
44 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
45 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
46 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
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
48 J. Fu, J. Xu, Characterizations of Morrey type Besov and Triebel-Lizorkin spaces with variable exponents, J. Math. Anal. Appl., 381 (2011), 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
49 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 ( ) 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
50 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
51 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
52 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
53 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
54 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
55 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
56 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
57 1 A. Almeida and P. Hästö, Besov spaces with variable smoothness and integrability, J. Funct. Anal. 258 (2010), L. Diening, P. Hästö and S. Roudenko, Function spaces of variable smoothness and integrability, J. Funct. Anal. 256 (2009),no. 6, 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 D. Drihem, Atomic decomposition of Besov spaces with variable smoothness and integrability, J. Math. Anal. Appl. 389 (2012), no. 1, D. Drihem, Some properties of variable Besov-type spaces, Funct. Approx. Comment. Math. 52 (2015), no. 2, D. Drihem, Some characterizations of variable Besov-type spaces, Ann. Funct. Anal. 6 (2015), no.4, D. Drihem, On the duality of variable Triebel-Lizorkin spaces, arxiv: D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 46 / 47
58 8 M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces, J. Funct. Anal. 93 (1990), no. 1, 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, 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, T. Noi, Duality of variable exponent Triebel-Lizorkin and Besov spaces, J. Funct. Spaces Appl. 2012, Art. ID , 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), D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 47 / 47
59 Thank you for your attention D. Drihem ( M sila University, Algeria) Variable Besov-type spaces 11/15 47 / 47
Jordan Journal of Mathematics and Statistics (JJMS) 9(1), 2016, pp BOUNDEDNESS OF COMMUTATORS ON HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENT
Jordan Journal of Mathematics and Statistics (JJMS 9(1, 2016, pp 17-30 BOUNDEDNESS OF COMMUTATORS ON HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENT WANG HONGBIN Abstract. In this paper, we obtain the boundedness
More informationNOTES ON THE HERZ TYPE HARDY SPACES OF VARIABLE SMOOTHNESS AND INTEGRABILITY
M athematical I nequalities & A pplications Volume 19, Number 1 2016, 145 165 doi:10.7153/mia-19-11 NOTES ON THE HERZ TYPE HARDY SPACES OF VARIABLE SMOOTHNESS AND INTEGRABILITY DOUADI DRIHEM AND FAKHREDDINE
More informationVariable Exponents Spaces and Their Applications to Fluid Dynamics
Variable Exponents Spaces and Their Applications to Fluid Dynamics Martin Rapp TU Darmstadt November 7, 213 Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, 213 1 / 14 Overview 1 Variable
More informationFunction spaces with variable exponents
Function spaces with variable exponents Henning Kempka September 22nd 2014 September 22nd 2014 Henning Kempka 1 / 50 http://www.tu-chemnitz.de/ Outline 1. Introduction & Motivation First motivation Second
More informationHardy spaces with variable exponents and generalized Campanato spaces
Hardy spaces with variable exponents and generalized Campanato spaces Yoshihiro Sawano 1 1 Tokyo Metropolitan University Faculdade de Ciencias da Universidade do Porto Special Session 49 Recent Advances
More informationHARNACK S INEQUALITY FOR GENERAL SOLUTIONS WITH NONSTANDARD GROWTH
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 37, 2012, 571 577 HARNACK S INEQUALITY FOR GENERAL SOLUTIONS WITH NONSTANDARD GROWTH Olli Toivanen University of Eastern Finland, Department of
More informationINTERPOLATION IN VARIABLE EXPONENT SPACES
INTERPOLATION IN VARIABLE EXPONENT SPACES ALEXANDRE ALMEIDA AND PETER HÄSTÖ,2 Abstract. In this paper we study both real and complex interpolation in the recently introduced scales of variable exponent
More informationarxiv: v1 [math.ap] 12 Mar 2009
LIMITING FRACTIONAL AND LORENTZ SPACES ESTIMATES OF DIFFERENTIAL FORMS JEAN VAN SCHAFTINGEN arxiv:0903.282v [math.ap] 2 Mar 2009 Abstract. We obtain estimates in Besov, Lizorkin-Triebel and Lorentz spaces
More informationarxiv: v1 [math.fa] 8 Sep 2016
arxiv:1609.02345v1 [math.fa] 8 Sep 2016 INTRINSIC CHARACTERIZATION AND THE EXTENSION OPERATOR IN VARIABLE EXPONENT FUNCTION SPACES ON SPECIAL LIPSCHITZ DOMAINS HENNING KEMPKA Abstract. We study 2-microlocal
More informationBESOV SPACES WITH VARIABLE SMOOTHNESS AND INTEGRABILITY
BESOV SPACES WITH VARIABLE SMOOTHNESS AND INTEGRABILITY ALEXANDRE ALMEIDA AND PETER HÄSTÖ,2 Abstract. In this article we introduce Besov spaces with variable smoothness and integrability indices. We prove
More informationFRANKE JAWERTH EMBEDDINGS FOR BESOV AND TRIEBEL LIZORKIN SPACES WITH VARIABLE EXPONENTS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 43, 2018, 187 209 FRANKE JAWERTH EMBEDDINGS FOR BESOV AND TRIEBEL LIZORKIN SPACES WITH VARIABLE EXPONENTS Helena F. Gonçalves, Henning Kempka and
More informationWavelets and modular inequalities in variable L p spaces
Wavelets and modular inequalities in variable L p spaces Mitsuo Izuki July 14, 2007 Abstract The aim of this paper is to characterize variable L p spaces L p( ) (R n ) using wavelets with proper smoothness
More informationAALBORG UNIVERSITY. Compactly supported curvelet type systems. Kenneth N. Rasmussen and Morten Nielsen. R November 2010
AALBORG UNIVERSITY Compactly supported curvelet type systems by Kenneth N Rasmussen and Morten Nielsen R-2010-16 November 2010 Department of Mathematical Sciences Aalborg University Fredrik Bajers Vej
More informationA Critical Parabolic Sobolev Embedding via Littlewood-Paley Decomposition
International Mathematical Forum, Vol. 7, 01, no. 35, 1705-174 A Critical Parabolic Sobolev Embedding via Littlewood-Paley Decomposition H. Ibrahim Lebanese University, Faculty of Sciences Mathematics
More informationJordan Journal of Mathematics and Statistics (JJMS) 5(4), 2012, pp
Jordan Journal of Mathematics and Statistics (JJMS) 5(4), 2012, pp223-239 BOUNDEDNESS OF MARCINKIEWICZ INTEGRALS ON HERZ SPACES WITH VARIABLE EXPONENT ZONGGUANG LIU (1) AND HONGBIN WANG (2) Abstract In
More informationA capacity approach to the Poincaré inequality and Sobolev imbeddings in variable exponent Sobolev spaces
A capacity approach to the Poincaré inequality and Sobolev imbeddings in variable exponent Sobolev spaces Petteri HARJULEHTO and Peter HÄSTÖ epartment of Mathematics P.O. Box 4 (Yliopistonkatu 5) FIN-00014
More informationVariable Lebesgue Spaces
Variable Lebesgue Trinity College Summer School and Workshop Harmonic Analysis and Related Topics Lisbon, June 21-25, 2010 Joint work with: Alberto Fiorenza José María Martell Carlos Pérez Special thanks
More informationA Caffarelli-Kohn-Nirenberg type inequality with variable exponent and applications to PDE s
A Caffarelli-Kohn-Nirenberg type ineuality with variable exponent and applications to PDE s Mihai Mihăilescu a,b Vicenţiu Rădulescu a,c Denisa Stancu-Dumitru a a Department of Mathematics, University of
More informationVECTOR-VALUED INEQUALITIES ON HERZ SPACES AND CHARACTERIZATIONS OF HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT. Mitsuo Izuki Hokkaido University, Japan
GLASNIK MATEMATIČKI Vol 45(65)(2010), 475 503 VECTOR-VALUED INEQUALITIES ON HERZ SPACES AND CHARACTERIZATIONS OF HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT Mitsuo Izuki Hokkaido University, Japan Abstract
More informationCOMPACT EMBEDDINGS ON A SUBSPACE OF WEIGHTED VARIABLE EXPONENT SOBOLEV SPACES
Adv Oper Theory https://doiorg/05352/aot803-335 ISSN: 2538-225X electronic https://projecteuclidorg/aot COMPACT EMBEDDINGS ON A SUBSPACE OF WEIGHTED VARIABLE EXPONENT SOBOLEV SPACES CIHAN UNAL and ISMAIL
More informationSome Applications to Lebesgue Points in Variable Exponent Lebesgue Spaces
Çankaya University Journal of Science and Engineering Volume 7 (200), No. 2, 05 3 Some Applications to Lebesgue Points in Variable Exponent Lebesgue Spaces Rabil A. Mashiyev Dicle University, Department
More informationTHE VARIABLE EXPONENT SOBOLEV CAPACITY AND QUASI-FINE PROPERTIES OF SOBOLEV FUNCTIONS IN THE CASE p = 1
THE VARIABLE EXPONENT SOBOLEV CAPACITY AND QUASI-FINE PROPERTIES OF SOBOLEV FUNCTIONS IN THE CASE p = 1 HEIKKI HAKKARAINEN AND MATTI NUORTIO Abstract. In this article we extend the known results concerning
More informationVARIABLE EXPONENT TRACE SPACES
VARIABLE EXPONENT TRACE SPACES LARS DIENING AND PETER HÄSTÖ Abstract. The trace space of W 1,p( ) ( [, )) consists of those functions on that can be extended to functions of W 1,p( ) ( [, )) (as in the
More informationSmooth pointwise multipliers of modulation spaces
An. Şt. Univ. Ovidius Constanţa Vol. 20(1), 2012, 317 328 Smooth pointwise multipliers of modulation spaces Ghassem Narimani Abstract Let 1 < p,q < and s,r R. It is proved that any function in the amalgam
More informationRiesz-Kolmogorov theorem in variable exponent Lebesgue spaces and its applications to Riemann-Liouville fractional differential equations
. ARTICLES. SCIENCE CHINA Mathematics October 2018 Vol. 61 No. 10: 1807 1824 https://doi.org/10.1007/s11425-017-9274-0 Riesz-Kolmogorov theorem in variable exponent Lebesgue spaces and its applications
More informationBOUNDEDNESS FOR FRACTIONAL HARDY-TYPE OPERATOR ON HERZ-MORREY SPACES WITH VARIABLE EXPONENT
Bull. Korean Math. Soc. 5 204, No. 2, pp. 423 435 http://dx.doi.org/0.434/bkms.204.5.2.423 BOUNDEDNESS FOR FRACTIONA HARDY-TYPE OPERATOR ON HERZ-MORREY SPACES WITH VARIABE EXPONENT Jianglong Wu Abstract.
More informationWEIGHTED VARIABLE EXPONENT AMALGAM SPACES. İsmail Aydin and A. Turan Gürkanli Sinop University and Ondokuz Mayıs University, Turkey
GLASNIK MATEMATIČKI Vol 47(67(202, 65 74 WEIGHTED VARIABLE EXPONENT AMALGAM SPACES W(L p(x,l q İsmail Aydin and A Turan Gürkanli Sinop University and Ondokuz Mayıs University, Turkey Abstract In the present
More informationBoundedness of fractional integrals on weighted Herz spaces with variable exponent
Izuki and Noi Journal of Inequalities and Applications 206) 206:99 DOI 0.86/s3660-06-42-9 R E S E A R C H Open Access Boundedness of fractional integrals on weighted Herz spaces with variable exponent
More informationPROPERTIES OF CAPACITIES IN VARIABLE EXPONENT SOBOLEV SPACES
PROPERTIES OF CAPACITIES IN VARIABLE EXPONENT SOBOLEV SPACES PETTERI HARJULEHTO, PETER HÄSTÖ, AND MIKA KOSKENOJA Abstract. In this paper we introduce two new capacities in the variable exponent setting:
More informationBILINEAR OPERATORS WITH HOMOGENEOUS SYMBOLS, SMOOTH MOLECULES, AND KATO-PONCE INEQUALITIES
BILINEAR OPERATORS WITH HOMOGENEOUS SYMBOLS, SMOOTH MOLECULES, AND KATO-PONCE INEQUALITIES JOSHUA BRUMMER AND VIRGINIA NAIBO Abstract. We present a unifying approach to establish mapping properties for
More informationTHE RADIAL LEMMA OF STRAUSS IN THE CONTEXT OF MORREY SPACES
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 39, 204, 47 442 THE RADIAL LEMMA OF STRAUSS IN THE CONTEXT OF MORREY SPACES Winfried Sickel, Dachun Yang and Wen Yuan Friedrich-Schiller-Universität
More informationFUNCTION SPACES WITH VARIABLE EXPONENTS AN INTRODUCTION. Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano. Received September 18, 2013
Scientiae Mathematicae Japonicae Online, e-204, 53 28 53 FUNCTION SPACES WITH VARIABLE EXPONENTS AN INTRODUCTION Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano Received September 8, 203 Abstract. This
More informationSharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces
Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 1/45 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces Dachun Yang (Joint work) dcyang@bnu.edu.cn
More information1. Introduction. SOBOLEV INEQUALITIES WITH VARIABLE EXPONENT ATTAINING THE VALUES 1 AND n. Petteri Harjulehto and Peter Hästö.
Publ. Mat. 52 (2008), 347 363 SOBOLEV INEQUALITIES WITH VARIABLE EXPONENT ATTAINING THE VALUES AND n Petteri Harjulehto and Peter Hästö Dedicated to Professor Yoshihiro Mizuta on the occasion of his sixtieth
More informationDecompositions of variable Lebesgue norms by ODE techniques
Decompositions of variable Lebesgue norms by ODE techniques Septièmes journées Besançon-Neuchâtel d Analyse Fonctionnelle Jarno Talponen University of Eastern Finland talponen@iki.fi Besançon, June 217
More information2-MICROLOCAL BESOV AND TRIEBEL-LIZORKIN SPACES OF VARIABLE INTEGRABILITY
2-MICROLOCAL BESOV AND TRIEBEL-LIZORKIN SPACES OF VARIABLE INTEGRABILITY HENNING KEMPKA Abstract. We introduce 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability and give a characterization
More informationContinuity of weakly monotone Sobolev functions of variable exponent
Continuity of weakly monotone Sobolev functions of variable exponent Toshihide Futamura and Yoshihiro Mizuta Abstract Our aim in this paper is to deal with continuity properties for weakly monotone Sobolev
More informationAPPROXIMATE IDENTITIES AND YOUNG TYPE INEQUALITIES IN VARIABLE LEBESGUE ORLICZ SPACES L p( ) (log L) q( )
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 35, 200, 405 420 APPROXIMATE IDENTITIES AND YOUNG TYPE INEQUALITIES IN VARIABLE LEBESGUE ORLICZ SPACES L p( ) (log L) q( ) Fumi-Yuki Maeda, Yoshihiro
More informationTHÈSE. Présentée pour l obtention du diplôme de Doctorat troisième cycle THÈME
RÉPUBLIQUE ALGÉRIENNE DÉMOCRATIQUE ET POPULAIRE MINISTÈRE DE L ENEEIGNEMENT SUPÉRIEURE ET DE LA RECHERCHE SCIENTIFIQUE UNIVERSITÉ MOHAMED BOUDIAF- M'SILA FACULTÉ DE MATHÉMATIQUES ET DE L'INFORMATIQUE DÉPARTEMENT
More informationMath. Res. Lett. 16 (2009), no. 2, c International Press 2009 LOCAL-TO-GLOBAL RESULTS IN VARIABLE EXPONENT SPACES
Math. Res. Lett. 6 (2009), no. 2, 263 278 c International Press 2009 LOCAL-TO-GLOBAL RESULTS IN VARIABLE EXPONENT SPACES Peter A. Hästö Abstract. In this article a new method for moving from local to global
More informationResearch Article Local Characterizations of Besov and Triebel-Lizorkin Spaces with Variable Exponent
Function Spaces, Article ID 417341, 8 pages http://dxdoiorg/101155/2014/417341 Research Article Local Characterizations of Besov and Triebel-Lizorkin Spaces with Variable Exponent Baohua Dong and Jingshi
More informationThe variable exponent BV-Sobolev capacity
The variable exponent BV-Sobolev capacity Heikki Hakkarainen Matti Nuortio 4th April 2011 Abstract In this article we study basic properties of the mixed BV-Sobolev capacity with variable exponent p. We
More informationarxiv: v3 [math.ca] 9 Apr 2015
ON THE PRODUCT OF FUNCTIONS IN BMO AND H 1 OVER SPACES OF HOMOGENEOUS TYPE ariv:1407.0280v3 [math.ca] 9 Apr 2015 LUONG DANG KY Abstract. Let be an RD-space, which means that is a space of homogeneous type
More informationGeneralized pointwise Hölder spaces
Generalized pointwise Hölder spaces D. Kreit & S. Nicolay Nord-Pas de Calais/Belgium congress of Mathematics October 28 31 2013 The idea A function f L loc (Rd ) belongs to Λ s (x 0 ) if there exists a
More informationThe p(x)-laplacian and applications
The p(x)-laplacian and applications Peter A. Hästö Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland October 3, 2005 Abstract The present article is based
More informationLORENTZ SPACE ESTIMATES FOR VECTOR FIELDS WITH DIVERGENCE AND CURL IN HARDY SPACES
- TAMKANG JOURNAL OF MATHEMATICS Volume 47, Number 2, 249-260, June 2016 doi:10.5556/j.tkjm.47.2016.1932 This paper is available online at http://journals.math.tku.edu.tw/index.php/tkjm/pages/view/onlinefirst
More informationThe maximal operator in generalized Orlicz spaces
The maximal operator in generalized Orlicz spaces Peter Hästö June 9, 2015 Department of Mathematical Sciences Generalized Orlicz spaces Lebesgue -> Orlicz -> generalized Orlicz f p dx to ϕ( f ) dx to
More informationSome functional inequalities in variable exponent spaces with a more generalization of uniform continuity condition
Int. J. Nonlinear Anal. Appl. 7 26) No. 2, 29-38 ISSN: 28-6822 electronic) http://dx.doi.org/.2275/ijnaa.26.439 Some functional inequalities in variable exponent spaces with a more generalization of uniform
More informationPERIODIC SOLUTIONS FOR A KIND OF LIÉNARD-TYPE p(t)-laplacian EQUATION. R. Ayazoglu (Mashiyev), I. Ekincioglu, G. Alisoy
Acta Universitatis Apulensis ISSN: 1582-5329 http://www.uab.ro/auajournal/ No. 47/216 pp. 61-72 doi: 1.17114/j.aua.216.47.5 PERIODIC SOLUTIONS FOR A KIND OF LIÉNARD-TYPE pt)-laplacian EQUATION R. Ayazoglu
More informationOn a compactness criteria for multidimensional Hardy type operator in p-convex Banach function spaces
Caspian Journal of Applied Mathematics, Economics and Ecology V. 1, No 1, 2013, July ISSN 1560-4055 On a compactness criteria for multidimensional Hardy type operator in p-convex Banach function spaces
More informationON CERTAIN COMMUTATOR ESTIMATES FOR VECTOR FIELDS
ON CERTAIN COMMUTATOR ESTIMATES FOR VECTOR FIELDS JAROD HART AND VIRGINIA NAIBO Abstract. A unifying approach for proving certain commutator estimates involving smooth, not-necessarily divergence-free
More informationOne-sided operators in grand variable exponent Lebesgue spaces
One-sided operators in grand variable exponent Lebesgue spaces ALEXANDER MESKHI A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, Georgia Porto, June 10, 2015 One-sided operators
More information1. Introduction The α-modulation spaces M s,α
BANACH FRAMES FOR MULTIVARIATE α-modulation SPACES LASSE BORUP AND MORTEN NIELSEN Abstract. The α-modulation spaces M p,q (R d ), α [,1], form a family of spaces that include the Besov and modulation spaces
More informationPERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 207 222 PERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS Fumi-Yuki Maeda and Takayori Ono Hiroshima Institute of Technology, Miyake,
More informationarxiv: v1 [math.cv] 3 Sep 2017
arxiv:1709.00724v1 [math.v] 3 Sep 2017 Variable Exponent Fock Spaces Gerardo A. hacón and Gerardo R. hacón Abstract. In this article we introduce Variable exponent Fock spaces and study some of their basic
More informationarxiv: v1 [math.ap] 28 Mar 2014
GROUNDSTATES OF NONLINEAR CHOQUARD EQUATIONS: HARDY-LITTLEWOOD-SOBOLEV CRITICAL EXPONENT VITALY MOROZ AND JEAN VAN SCHAFTINGEN arxiv:1403.7414v1 [math.ap] 28 Mar 2014 Abstract. We consider nonlinear Choquard
More informationBrøndsted-Rockafellar property of subdifferentials of prox-bounded functions. Marc Lassonde Université des Antilles et de la Guyane
Conference ADGO 2013 October 16, 2013 Brøndsted-Rockafellar property of subdifferentials of prox-bounded functions Marc Lassonde Université des Antilles et de la Guyane Playa Blanca, Tongoy, Chile SUBDIFFERENTIAL
More informationSharp estimates for a class of hyperbolic pseudo-differential equations
Results in Math., 41 (2002), 361-368. Sharp estimates for a class of hyperbolic pseudo-differential equations Michael Ruzhansky Abstract In this paper we consider the Cauchy problem for a class of hyperbolic
More informationThe 2D Magnetohydrodynamic Equations with Partial Dissipation. Oklahoma State University
The 2D Magnetohydrodynamic Equations with Partial Dissipation Jiahong Wu Oklahoma State University IPAM Workshop Mathematical Analysis of Turbulence IPAM, UCLA, September 29-October 3, 2014 1 / 112 Outline
More informationarxiv: v2 [math.ap] 30 Jan 2015
LOCAL WELL-POSEDNESS FOR THE HALL-MHD EQUATIONS WITH FRACTIONAL MAGNETIC DIFFUSION DONGHO CHAE 1, RENHUI WAN 2 AND JIAHONG WU 3 arxiv:144.486v2 [math.ap] 3 Jan 215 Abstract. The Hall-magnetohydrodynamics
More informationATOMIC DECOMPOSITIONS AND OPERATORS ON HARDY SPACES
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Volumen 50, Número 2, 2009, Páginas 15 22 ATOMIC DECOMPOSITIONS AND OPERATORS ON HARDY SPACES STEFANO MEDA, PETER SJÖGREN AND MARIA VALLARINO Abstract. This paper
More informationHARNACK S INEQUALITY AND THE STRONG p( )-LAPLACIAN
HARNACK S INEQUALITY AND THE STRONG p( )-LAPLACIAN TOMASZ ADAMOWICZ AND PETER HÄSTÖ Abstract. We study solutions of the strong p( )-Laplace equation. We show that, in contrast to p( )-Laplace solutions,
More informationHardy-Littlewood maximal operator in weighted Lorentz spaces
Hardy-Littlewood maximal operator in weighted Lorentz spaces Elona Agora IAM-CONICET Based on joint works with: J. Antezana, M. J. Carro and J. Soria Function Spaces, Differential Operators and Nonlinear
More informationHarnack Inequality and Continuity of Solutions for Quasilinear Elliptic Equations in Sobolev Spaces with Variable Exponent
Nonl. Analysis and Differential Equations, Vol. 2, 2014, no. 2, 69-81 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/nade.2014.31225 Harnack Inequality and Continuity of Solutions for Quasilinear
More informationTHE L 2 -HODGE THEORY AND REPRESENTATION ON R n
THE L 2 -HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 -Hodge theory on whole R n based on the minimization principle of the calculus of variations and some
More informationOn pointwise estimates for maximal and singular integral operators by A.K. LERNER (Odessa)
On pointwise estimates for maximal and singular integral operators by A.K. LERNER (Odessa) Abstract. We prove two pointwise estimates relating some classical maximal and singular integral operators. In
More informationNONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian
More informationON THE EXISTENCE OF THREE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEM. Paweł Goncerz
Opuscula Mathematica Vol. 32 No. 3 2012 http://dx.doi.org/10.7494/opmath.2012.32.3.473 ON THE EXISTENCE OF THREE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEM Paweł Goncerz Abstract. We consider a quasilinear
More informationLocal Well-Posedness for the Hall-MHD Equations with Fractional Magnetic Diffusion
J. Math. Fluid Mech. 17 (15), 67 638 c 15 Springer Basel 14-698/15/467-1 DOI 1.17/s1-15--9 Journal of Mathematical Fluid Mechanics Local Well-Posedness for the Hall-MHD Equations with Fractional Magnetic
More informationOn the local existence for an active scalar equation in critical regularity setting
On the local existence for an active scalar equation in critical regularity setting Walter Rusin Department of Mathematics, Oklahoma State University, Stillwater, OK 7478 Fei Wang Department of Mathematics,
More informationGRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS
LE MATEMATICHE Vol. LI (1996) Fasc. II, pp. 335347 GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS CARLO SBORDONE Dedicated to Professor Francesco Guglielmino on his 7th birthday W
More informationWeak Solutions to Nonlinear Parabolic Problems with Variable Exponent
International Journal of Mathematical Analysis Vol. 1, 216, no. 12, 553-564 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ijma.216.6223 Weak Solutions to Nonlinear Parabolic Problems with Variable
More informationBesov regularity for operator equations on patchwise smooth manifolds
on patchwise smooth manifolds Markus Weimar Philipps-University Marburg Joint work with Stephan Dahlke (PU Marburg) Mecklenburger Workshop Approximationsmethoden und schnelle Algorithmen Hasenwinkel, March
More informationHardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus.
Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus. Xuan Thinh Duong (Macquarie University, Australia) Joint work with Ji Li, Zhongshan
More informationarxiv: v1 [math.fa] 15 Aug 2018
VARIABLE EXPONENT TRIEBEL-LIZORKIN-MORREY SPACES arxiv:808.05304v [math.fa] 5 Aug 208 ANTÓNIO CAETANO AND HENNING KEMPKA Abstract. We introduce variable exponent versions of Morreyfied Triebel- Lizorkin
More informationCONDITIONALITY CONSTANTS OF QUASI-GREEDY BASES IN SUPER-REFLEXIVE BANACH SPACES
CONDITIONALITY CONSTANTS OF QUASI-GREEDY BASES IN SUPER-REFLEXIVE BANACH SPACES G. GARRIGÓS, E. HERNÁNDEZ, AND M. RAJA Abstract. We show that in a super-reflexive Banach space, the conditionality constants
More informationEXISTENCE AND REGULARITY RESULTS FOR SOME NONLINEAR PARABOLIC EQUATIONS
EXISTECE AD REGULARITY RESULTS FOR SOME OLIEAR PARABOLIC EUATIOS Lucio BOCCARDO 1 Andrea DALL AGLIO 2 Thierry GALLOUËT3 Luigi ORSIA 1 Abstract We prove summability results for the solutions of nonlinear
More informationWELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL FLUID EQUATIONS WITH DEGENERACY ON THE BOUNDARY
Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 13, pp. 1 15. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu WELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL
More informationEXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 98, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE
More informationYoshihiro Mizuta, Takao Ohno, Tetsu Shimomura and Naoki Shioji
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 35, 2010, 115 130 COMPACT EMBEDDINGS FOR SOBOLEV SPACES OF VARIABLE EXPONENTS AND EXISTENCE OF SOLUTIONS FOR NONLINEAR ELLIPTIC PROBLEMS INVOLVING
More informationVANISHING VISCOSITY IN THE PLANE FOR NONDECAYING VELOCITY AND VORTICITY
VANISHING VISCOSITY IN THE PLANE FOR NONDECAYING VELOCITY AND VORTICITY ELAINE COZZI Abstract. Assuming that initial velocity and initial vorticity are bounded in the plane, we show that on a sufficiently
More informationREGULARITY CRITERIA FOR WEAK SOLUTIONS TO 3D INCOMPRESSIBLE MHD EQUATIONS WITH HALL TERM
Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 10, pp. 1 12. ISSN: 1072-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EGULAITY CITEIA FO WEAK SOLUTIONS TO D INCOMPESSIBLE
More informationMinimization problems on the Hardy-Sobolev inequality
manuscript No. (will be inserted by the editor) Minimization problems on the Hardy-Sobolev inequality Masato Hashizume Received: date / Accepted: date Abstract We study minimization problems on Hardy-Sobolev
More informationMicro-local analysis in Fourier Lebesgue and modulation spaces.
Micro-local analysis in Fourier Lebesgue and modulation spaces. Stevan Pilipović University of Novi Sad Nagoya, September 30, 2009 (Novi Sad) Nagoya, September 30, 2009 1 / 52 Introduction We introduce
More informationErratum to Multipliers and Morrey spaces.
Erratum to Multipliers Morrey spaces. Pierre Gilles Lemarié Rieusset Abstract We correct the complex interpolation results for Morrey spaces which is false for the first interpolation functor of Calderón,
More informationParaproducts and the bilinear Calderón-Zygmund theory
Paraproducts and the bilinear Calderón-Zygmund theory Diego Maldonado Department of Mathematics Kansas State University Manhattan, KS 66506 12th New Mexico Analysis Seminar April 23-25, 2009 Outline of
More informationON THE REGULARITY OF WEAK SOLUTIONS OF THE 3D NAVIER-STOKES EQUATIONS IN B 1
ON THE REGULARITY OF WEAK SOLUTIONS OF THE 3D NAVIER-STOKES EQUATIONS IN B 1, A. CHESKIDOV AND R. SHVYDKOY ABSTRACT. We show that if a Leray-Hopf solution u to the 3D Navier- Stokes equation belongs to
More informationDIV-CURL TYPE THEOREMS ON LIPSCHITZ DOMAINS Zengjian Lou. 1. Introduction
Bull. Austral. Math. Soc. Vol. 72 (2005) [31 38] 42b30, 42b35 DIV-CURL TYPE THEOREMS ON LIPSCHITZ DOMAINS Zengjian Lou For Lipschitz domains of R n we prove div-curl type theorems, which are extensions
More informationRemarks on the blow-up criterion of the 3D Euler equations
Remarks on the blow-up criterion of the 3D Euler equations Dongho Chae Department of Mathematics Sungkyunkwan University Suwon 44-746, Korea e-mail : chae@skku.edu Abstract In this note we prove that the
More informationCRITICAL POINT METHODS IN DEGENERATE ANISOTROPIC PROBLEMS WITH VARIABLE EXPONENT. We are interested in discussing the problem:
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LV, Number 4, December 2010 CRITICAL POINT METHODS IN DEGENERATE ANISOTROPIC PROBLEMS WITH VARIABLE EXPONENT MARIA-MAGDALENA BOUREANU Abstract. We work on
More informationEXISTENCE AND UNIQUENESS OF p(x)-harmonic FUNCTIONS FOR BOUNDED AND UNBOUNDED p(x)
UNIVERSITY OF JYVÄSKYLÄ DEPARTMENT OF MATHEMATICS AND STATISTICS REPORT 30 UNIVERSITÄT JYVÄSKYLÄ INSTITUT FÜR MATHEMATIK UND STATISTIK BERICHT 30 EXISTENCE AND UNIQUENESS OF p(x)-harmonic FUNCTIONS FOR
More informationarxiv: v2 [math.ap] 6 Sep 2007
ON THE REGULARITY OF WEAK SOLUTIONS OF THE 3D NAVIER-STOKES EQUATIONS IN B 1, arxiv:0708.3067v2 [math.ap] 6 Sep 2007 A. CHESKIDOV AND R. SHVYDKOY ABSTRACT. We show that if a Leray-Hopf solution u to the
More informationON A MAXIMAL OPERATOR IN REARRANGEMENT INVARIANT BANACH FUNCTION SPACES ON METRIC SPACES
Vasile Alecsandri University of Bacău Faculty of Sciences Scientific Studies and Research Series Mathematics and Informatics Vol. 27207), No., 49-60 ON A MAXIMAL OPRATOR IN RARRANGMNT INVARIANT BANACH
More informationON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 167 176 ON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES Piotr Haj lasz and Jani Onninen Warsaw University, Institute of Mathematics
More informationA NOTE ON WAVELET EXPANSIONS FOR DYADIC BMO FUNCTIONS IN SPACES OF HOMOGENEOUS TYPE
EVISTA DE LA UNIÓN MATEMÁTICA AGENTINA Vol. 59, No., 208, Pages 57 72 Published online: July 3, 207 A NOTE ON WAVELET EXPANSIONS FO DYADIC BMO FUNCTIONS IN SPACES OF HOMOGENEOUS TYPE AQUEL CESCIMBENI AND
More informationResearch Article Function Spaces with a Random Variable Exponent
Abstract and Applied Analysis Volume 211, Article I 17968, 12 pages doi:1.1155/211/17968 Research Article Function Spaces with a Random Variable Exponent Boping Tian, Yongqiang Fu, and Bochi Xu epartment
More informationSOBOLEV S INEQUALITY FOR RIESZ POTENTIALS OF FUNCTIONS IN NON-DOUBLING MORREY SPACES
Mizuta, Y., Shimomura, T. and Sobukawa, T. Osaka J. Math. 46 (2009), 255 27 SOOLEV S INEQUALITY FOR RIESZ POTENTIALS OF FUNCTIONS IN NON-DOULING MORREY SPACES YOSHIHIRO MIZUTA, TETSU SHIMOMURA and TAKUYA
More informationc 2014 Society for Industrial and Applied Mathematics
SIAM J. MATH. ANAL. Vol. 46, No. 1, pp. 588 62 c 214 Society for Industrial and Applied Mathematics THE 2D INCOMPRESSIBLE MAGNETOHYDRODYNAMICS EQUATIONS WITH ONLY MAGNETIC DIFFUSION CHONGSHENG CAO, JIAHONG
More informationLOWER BOUNDS FOR HAAR PROJECTIONS: DETERMINISTIC EXAMPLES. 1. Introduction
LOWER BOUNDS FOR HAAR PROJECTIONS: DETERMINISTIC EXAMPLES ANDREAS SEEGER TINO ULLRICH Abstract. In a previous paper by the authors the existence of Haar projections with growing norms in Sobolev-Triebel-Lizorkin
More informationEXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem
Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
More information