By S. T Hu

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The useful analytic houses of Weyl transforms as bounded linear operators on $ L^{2}({\Bbb R}^{n}) $ are studied when it comes to the symbols of the transforms. The boundedness, the compactness, the spectrum and the useful calculus of the Weyl remodel are proved intimately. New effects and methods at the boundedness and compactness of the Weyl transforms when it comes to the symbols in $ L^{r}({\Bbb R}^{2n}) $ and when it comes to the Wigner transforms of Hermite services are given.

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H d . h . PI P2 III t e secon term gIve t e equatIOn 36(n - 2j)(n - 2j + l)bj + 4(3j + 8 + 2)(3n + 12(n - 2j)aj = 0, 3j + 301 + 6 + l)bi +I (4) h-Harmonic Polynomials, h-Hankel Transform, ... 41 where 0:::; j :::; [(n-1)/2]. We set 8 = -1 in (3) and (4). "( n _ 2J. b. (n-2j+1)! 4 of [371]. "( n + l)n-i _ 2J. ')' b. (n-2j+1)! ' and this gives the h-harmonic polynomial P3n+5(Z) = Z2 r3n+3C::t: (cos 9) - zr 3nH C::+ I(cos 39) = z2c~~il,a)(z3), n ~ -1, (6) of degree 3n + 5. The conjugates of the polynomials (5) and (6) are also h-harmonic.

The operators Tt. We can consider on h-harmonic polynomials the scalar product of the space ,r}(sn-I, h 2 dw). Therefore, we have the operator T;* which is adjoint to the operator T i . Since Tif)f C f)f+1 then T;* f)f C f)f-1. The aim of this section is to prove the formula Ttp(x) = (n + 2r + 21') [XiP(X) - (n + 2r + 21' - 2)-1 IxI2TiP(X)] , where P E f)~ and l' P E f)~, = 0'1 + ... + am. To prove this formula we first show that for + 2TiP(X), (2) 2)-1IxI 2Tip(X) E f)~+I. (3) ~h(XiP(X)) = Xi~hP(X) XiP(X) - (n By the product rules for ~ ~ + 2r + 21' - and \7 we have ~h(XiP(X)) = Xi~p(X) apeX) + 2~ .

Ft + (VFt, VF2 ))dx, (2) n where F I ,F2 E C 2 (n). (hh) n + (V(hh), V(fz h))] dx. (3) Chapter 1. 22 If Fl = h 1 ,F2 = fthh in (2), then we get j fth ~~ hdp, = an j [fd2hflh + (V(Jd2h), Vh)]dx. (4) n Subtracting equation (4) from equation (3) after some transformations we obtain the relation j h ~ h 2dp, = an j[hh(fl(fth) - ftflh) + h2(Vft, Vfz)]dx n which leads to formula (1). Lemma is proved. Theorem 1. If PI E f)~'P2 E f)~ and r j f:. k, then pl(X)p2(X)h(x)2dw = o. (5) S,",-1 Proof By using formula (1), the fact that the operator Dh is symmetric, and the formula 1 en j f(x)dx = j r,,-ldr j B 0 f(rx)dw(x), sn-l we obtain (degpl - degp2) j PIP2h2dw = en j(P2LhPl - PILhP2)h2dx B sn-l = en j (P2DhPl - PI DhP2)h 2dx = O.