Several Problems In Harmonic Analysis Associated With The Dunkl Operators

The Dunkl operators are a class of differential-reflection operators associated with root systems.The aim of this dissertation is to study several problems in harmonic analysis associated with the Dunkl operators,including the local non-tangential boundary values of the related generalized harmonic functions,boundedness of the Littlewood-Paley type functions,Hardy spaces,and the Lp-estimates of the area function and the maximal functions.The results are stated as follows.(?)A distribution estimate for the representing measures of Dunkl's intertwining operator is proved,by which some lower estimates sharper in some senses than those known for the Dunkl kernel,the associated heat kernel,and the associated Poisson kernel are obtained.(?)Local boundary behaviour of generalized harmonic functions associated with the Dunkl operators is studied,and a Lusin-type area integral operator S is introduced by means of Dunkl's generalized translation.For a generalized harmonic function u in the upper half-space and for a G-invariant subset E of the boundary,the equivalence of the following three assertions are proved:(a)u has a finite non-tangential limit at(x,0)for a.e.x?E;(b)u is non-tangentially bounded for a.e.x?E;and(c)(Su)(x)is finite for a.e.x?E.(?)Several Littlewood-Paley type square functions in the Dunkl setting are studied,and the S-function and the g?*-function associated with the Dunkl operators are introduced by means of Dunkl's generalized translation.The main results include that(1)for 1<p<?,the S-function and the related g-functions are(p,p)-bounded;(2)for 2?p<?,the g?*-function is(p,p)-bounded;and(3)for 1<p<2,the g?*-function is(p,p)-bounded for G-invariant functions in Lp.(?)The Riesz transforms associated with the Dunkl operators are introduced by an"analytic" approach and the associated Hardy space H?1(Rd)is defined in terms of these Riesz transforms.The main result is to characterize the dual of the Hardy space H?1(Rd)by the weighted Carleson type measures.(?)The area function Sb(u)associated with the Dunkl operators is introduced by means of Dunkl's generalized translation,and the Lp-estimates of the area function and the maximal functions are studied.The main results include that,for a generalized harmonic function u in the upper half-space,(1)for 1?p<?,if the non-tangential maximal function up*is in Lp,then ?Sb(u)?Lp?c?ua*?Lp;(2)for 0<p<?,if u is G-invariant and u(x,t)?0 as t??,then ?ua*??c?Sb(u)?Lp;and(3)for 0<p<? and G=Z2d,the norms of the the non-tangential maximal function and the perpendicular maximal function are equivalent.