Characterization of Hardy spaces for certain Dunkl operators and multidimensional Bessel operators

Transcription

1 Characterization of for certain Dunkl operators and multidimensional Bessel operators Instytut Matematyczny, Uniwersytet Wrocławski, Poland Probability & Analysis Będlewo,

2 Hardy space H 1 (R n ) H 1 (R n ) = appropriate substitute for L 1 (R n ) Several equivalent definitions by means of atoms the heat maximal operator the Poisson maximal operator Riesz transforms Remarks : other characterizations generalization to spaces of homogeneous type

3 Poisson maximal definition Poisson semigroup and kernel : e t f (x) = p t (x y) f (y) dy = e t ξ ˆf (ξ)e ix ξ dξ. R n R n Definition H 1 consists of all functions f L 1 whose Poisson maximal transform p f (x) = sup p t (x y) f (y) dy t>0 R n belongs to L 1. The nom is given by f H 1 = p f L 1.

4 Riesz transform definition Riesz transforms : R j f (x) = x j ( ) 1 2 f (x) (j =1,..., n). ξ j R j f (x) = c n ξ f (ξ)e ix ξ dξ R n Definition H 1 consists of all functions f L 1 whose Riesz transforms belong to L 1. The nom is given by f H 1 = f L 1 + n j=1 R jf L 1.

5 Remark The classical can be thought as ones associated with the Laplace operator on R n. However, one can take another semigroup e ta acting on L 1 (X, dµ) and consider HA 1 = {f L1 (X ) : sup e ta f (x) L 1 (dµ(x)} t>0 Examples A is a sub-laplacian on nilpotent Lie group (G.B. Folland and E. Stein) twisted Laplacian on C n (G.Mauceri, M. Picardello, F. Ricci) A = + 1 local (D. Goldberg) A = + V, V 0 potential J.D., J.Zienkiewicz semigroups with Davies-Gaffney estimates (S. Hofmann, G. Lu, D.Mitrea, M.Mitrea, L.X.Yan)

6 Rational Dunkl theory in dimension 1 Dunkl = deformation of Fourier Parameter : k 0 k =0 Dunkl operator Df (x) = f (x) + k x { f (x) f ( x) } Dunkl Laplacian : Lf (x) = D 2 f (x) = f (x) + 2k x f (x) k x 2 { f (x) f ( x) } Dunkl kernel normalized eigenfunctions { D x E(x, ξ) = ξ E(x, ξ) E(0, ξ) = 1 Case k =0 : D = d dx, E(x, ξ) = e xξ

7 Explicit expressions of the Dunkl kernel in dimension 1 Expression / modified Bessel functions even {}}{ E(x, ξ) = (ixξ) + j k 1 2 odd {}}{ xξ 2k+1 j k+ 1 2 (ixξ) where j ν (iz) = Γ(ν+1) + = Γ(ν+1) z ν I ν (z) 1 m=0 m! Γ(m+ν+1) ( z 2 )2m Expression / confluent hypergeometric function E(x, ξ) = Γ(k+ 1 2 ) Γ(k) Γ( 1 2 ) +1 1 du (1 u) k 1 (1+ u) k e xξu 1 = e xξ Γ(2k+1) Γ(k) Γ(k+1) dv v k 1 (1 v) k e 2xξv 0 }{{} 1F 1 (k; 2k +1; 2 xξ)

8 Properties of the Dunkl kernel in dimension 1 E(x, ξ) > 0 E(x, ξ) = E(ξ, x) E(λx, ξ) = E(x, λξ) E(x, ξ) = 1 + O( xξ ) as xξ 0 E(x, ξ) = c k e xξ (xξ) k {1 + O( 1 xξ )} as xξ + E(x, ξ) = k 2 c k e xξ xξ k 1 {1 + O( 1 xξ )} as xξ E(x, ξ) is a holomorphic function on C C

9 Dunkl transform (dimension 1) Measure : dµ(x) = w(x) {}}{ const x 2k dx Definition: Fourier-Dunkl transform F f (ξ) = f (x) E(x, iξ) dµ(x) Properties R F : L 2 (R, µ) L 2 (R, µ) isometric isomorphism F 1 g(x) = g(ξ) E(x, iξ) dµ(ξ) R F (Df )(ξ) = i ξ F f (ξ) F (xf )(ξ) = i D ξ F f (ξ)

10 Generalizations Product case R n x = (x 1,..., x n ) σ j (x 1,..., x j,..., x n ) = (x 1,..., x j,..., x n ) reflections k = (k 1,..., k n ), we consider k j 0 dµ(x) = const x 1 2k 1 x 2 2k2... x n 2kn dx 1 dx 2...dx n D j f (x) = x j f (x) + k [ j x j f (x) f (σj x) ] (j =1,..., n) E(x, y) = E k1 (x 1, y 1 ) E kn (x n, y n ) Lf (x) = n j=1 D2 j Lf (x) = n j=1 f (x) Dunkl Laplacian ( ) x 2 j f (x) + 2k j x j xj f (x) k j (f (x) f (σ xj 2 j x)) Dunkl-Fourier transform F f (ξ) = R E(x, iξ)f (x)dµ(x) n F f L 2 (µ) = c k f L 2 (µ) isometry on L 2 (µ) f (x) = c k R F f (ξ)e(x, iξ) dµ(ξ) inversion formula n

11 Dunkl heat semigroup e tl f (x) = F 1 (e t ξ 2 F f (ξ))(x) = R n h t (x, y)f (y) dµ(y), where h t (x, y) > 0 is C but is has no Gaussian bounds: h t (x, y) µ(b(x, t )) 1 if t >0, x R n and y = ( x 1, x 2,..., x n ). t t + x y 2 Dunkl Poisson semigroup P t = e t L e t L f (x) = F 1 (e t ξ F f (ξ))(x) = R n p t (x, y)f (y) dµ(y), p t (x, y) = 1 π 0 e u h t 2 /4u(x, y) du u. Riesz transforms R j f (x) = i ξ j R n ξ F f (ξ)e(x, iξ) dµ(ξ) = D j( L) 1/2 f (x).

12 Definition. Hardy space for Dunkl operator f H 1 L if and only if h f (x) = sup t>0 h t f (x) L 1 (µ). Then f H 1 L = h f L 1 (µ). Equivalent characterizations. Theorem (J.D.). Poisson semigroup characterization f H 1 L if and only if p f (x) = sup t>0 p t f (x) L 1 (µ). Then f H 1 L p f L 1 (µ). Theorem (J.D). Riesz transform characterization f H 1 L if and only if R jf L 1 (µ). Then f H 1 L f L 1 (µ) + n j=1 R jf L 1 (µ).

13 Atomic characterization Atoms A function a is an atom if there is a ball (Euclidean) such that supp a B a µ(b) 1 a(x) dµ(x) = 0. Theorem (J-Ph.Anker, N.Ben Salem, N.Hamda, J.D.) f H 1 L f (x) = j λ ja j (x), j λ j <, where a j (x) are atoms. Moreover, f H 1 L inf λ j, where the infimum is taken over all representations.

14 Proof of the Riesz transform characterization The implication f HL 1 = R jf L 1 (µ) is easier. The implication R j f L 1 (µ) = f HL 1 is harder. We try to follow the classical proof of Fefferman-Stein, which for the classical Riesz transforms R j = xj ( ) 1/2. For each σ G -the group of reflections set f σ (x) = f (σ(x)), F σ = (P t (f σ )(x), P t R 1 (f σ )(x),..., P t R n (f σ )(x)) = (u σ 0, u σ 1,..., u σ n ) and built the very long vector F = {F σ } σ G. Since R j (f σ ) = (R j f ) σ L 1 (µ), we get F(x) = F(σx). For each σ the system (u σ 0 (t, x), uσ 1 (t, x),..., uσ n (t, x)) satisfies the Cauchy-Riemmann type equations: D j u 0 (t, x) = t u j (t, x), j = 1,..., n; D j u l (t, x) = D l u j (t, x), j, l = 1,..., n; n t u 0 (t, x) + D j u j (t, x) = 0. j=1

15 Using the C-R equations after some calculations we obtain that ( 2 t + L)( F q ) 0. Maximum principle for 2 t + L subharmonic functions. Weak (M. Rösler) : 2 t + L subharmonic function attains its maximum on the boundary of a bounded set. Strong: If 2 t + L subharmonic G invariant function attains maximum in the interior of a bounded connected set, then the function is constant in the set. Having: subharmonicity of F q ; the maximum principle (weak or strong); + some properties of the Poisson kernel p t (x, y); we can follow the ideas of the proof of C.Fefferman and E.Stein for classical to obtain the Riesz transforms characterization for the Hardy space for the Dunk operator.

16 Thank you!