Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators Margit Rosler Mathematisches Institut, Technische Universitat Munchen Arcisstr. 21, D-80333 Munchen, Germany e-mail:
[email protected]
Abstract Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with nite re ection groups on RN . The de nition and properties of these generalized Hermite systems extend naturally those of their classical counterparts; partial derivatives and the usual exponential kernel are here replaced by Dunkl operators and the generalized exponential kernel of the Dunkl transform. In case of the symmetric group N , our setting includes the polynomial eigenfunctions of certain Calogero-Sutherland type operators. The second part of this paper is devoted to the heat equation associated with Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem is governed by a positive one-parameter semigroup; this is assured by a maximum principle for the generalized Laplacian. The explicit solution to the Cauchy problem involves again the kernel which is, on the way, proven to be nonnegative for real arguments. K
S
K;
1991 AMS Subject Classi cation: Primary: 33C80, 35K05; Secondary: 44A15, 33C50. Running title: Hermite polynomials and the heat equation for Dunkl operators.
1 Introduction Dunkl operators are di erential-di erence operators associated with a nite re ection group, acting on some Euclidean space. They provide a useful framework for the study of multivariable analytic structures which reveal certain re ection symmetries. During the last years, these operators have gained considerable interest in various elds of mathematics and also in physical applications; they are, for example, naturally connected with certain Schrodinger operators for Calogero-Sutherlandtype quantum many body systems, see [L-V] and [B-F2], [B-F3]. For a nite re ection group G O(N; R) on RN the associated Dunkl operators are de ned as follows: For 2 RN n f0g , denote by the re ection corresponding to , i.e. in the hyperplane This paper was written while the author held a Forschungsstipendium of the DFG at the University of Virginia,
Charlottesville, VA, U.S.A.
1
orthogonal to . It is given by
xi ; (x) = x ? 2 h ; j j2
p
where h:; :i is the Euclidean scalar product on RN and jxj := hx; xi . (We use the same notations for the standard Hermitian inner product and norm on C N .) Let R be the root system associated with the re ections of G , normalized such that h ; i = 2 for all 2 R . Now choose a multiplicity function k on the root system R , that is, a G -invariant function k : R ! C , and x some positive subsystem R+ of R . The Dunkl operators Ti (i = 1; : : : ; N ) on RN associated with G and k are then given by X k( ) f (x) ? f ( x) ; f 2 C 1(RN ); T f (x) := @ f (x) + i
i
i
2R+
h ; xi
here @i denotes the i -th partial derivative. In case k = 0, the Ti reduce to the corresponding partial derivatives. In this paper, we shall assume throughout that k 0 (i.e. all values of k are non-negative), though several results of Section 3 may be extended to larger ranges of k . The most important basic properties of the Ti , proved in [D2], are as follows: Let P = C [R N ] denote the algebra of polynomial functions on RN and Pn (n 2 Z+ = f0; 1; 2 : : : g) the subspace of homogeneous polynomials of degree n . Then (1.1) The set fTi g generates a commutative algebra of di erential-di erence operators on P . (1.2) Each Ti is homogeneous of degree ?1 on P ; that is, Ti p 2 Pn?1 for p 2 Pn : Of particular importance in this paper is the generalized Laplacian associated with G and k , which P is de ned as k := Ni=1 Ti2 . It is homogeneous of degree ?2 on P and given explicitly by h i X k( ) hrf (x); i ? f (x) ? f ( x) : f (x) = f (x) + 2 k
h ; xi
2R+
h ; xi2
(Here and r denote the usual Laplacian and gradient respectively). The operators Ti were introduced and rst studied by Dunkl in a series of papers ([D1-4]) in connection with a generalization of the classical theory of spherical harmonics: Here the uniform spherical surface measure on the (N ? 1)-dimensional unit sphere is modi ed by a weight function which is invariant under the action of some nite re ection group G , namely
wk (x) =
Y
2R+
jh ; xij2k( ) ;
where k 0 is some xed multiplicity function on the root system R of G . Note that wk is homogeneous of degree 2 , with
:=
X
2R+
k( ):
In this context, in [D3] the following bilinear form on P is introduced: [p; q]k := (p(T ) q)(0) for p; q 2 P : 2
Here p(T ) is the operator derived from p(x) by replacing xi by Ti . Property (1.1) assures that [: ; :]k is well-de ned. A useful collection of its properties can be found in [D-J-O]. We recall that [: ; :]k is symmetric and positive-de nite (in case k 0), and that [p; q]k = 0 for p 2 Pn ; q 2 Pm with n 6= m . Moreover, for all i = 1; : : : ; N; p; q 2 P and g 2 G; (1.3)
[xi p; q]k = [p; Ti q]k
and
[g(p); g(q)]k = [p; q]k ;
where g(p)(x) = p(g?1 (x)): The pairing [: ; :]k is closely related to the scalar product on L2 (RN ; wk (x)e?jxj2 =2 dx): In fact, according to [D3], (1.4)
[p; q]k = nk
Z
R
?k =2 p(x) e?k =2 q(x) wk (x)e?jxj2 =2 dx for all p; q 2 P , e N
with some normalization constant nk > 0. Given an orthonormal basis f' ; 2 ZN+ g of P with respect to [: ; :]k , an easy rescaling of (1.4) shows that the polynomials
H (x) := 2jj e?k =4 ' (x) are orthogonal with respect to wk (x)e?jxj2 dx on RN . We call them the generalized Hermite polynomials on RN associated with G; k and f' g . The rst part of this paper is devoted to the study of such Hermite polynomial systems and associated Hermite functions. They generalize their classical counterparts in a natural way: these are just obtained for k = 0 and ' (x) = ( !)?1=2 x : In the one-dimensional case, associated with the re ection group G = Z2 on R , our generalized Hermite polynomials coincide with those introduced in [Chi] and studied in [Ros]. Our setting also includes, for the symmetric group G = SN , the so-called non-symmetric generalized Hermite polynomials which were recently introduced by Baker and Forrester in [B-F2], [B-F3]. These are non-symmetric analogues of the symmetric, i.e. permutation-invariant generalized Hermite polynomials associated with the group SN , which were rst introduced by Lassalle in [L2]. Moreover, the \generalized Laguerre polynomials" of [B-F2], [B-F3], which are non-symmetric analogues of those in [L1], can be considered as a subsystem of Hermite polynomials associated with a re ection group of type BN . We refer to [B-F1] and [vD] for a thorough study of the symmetric multivariable Hermite- and Laguerre systems. After a short collection of notations and basic facts from Dunkl's theory in Section 2, the concept of generalized Hermite polynomials is introduced in Section 3, along with a discussion of the above mentioned special classes. We derive generalizations for many of the well-known properties of the classical Hermite polynomials and Hermite functions: A Rodrigues formula, a generating relation and a Mehler formula for the Hermite polynomials, analoguesof the second order di erential equations and a characterization of the generalized Hermite functions as eigenfunctions of the Dunkl transform. Parts of this section may be seen as a unifying treatment of results from [B-F2], [B-F3] and [Ros] for their particular cases. In Section 4, which makes up the second major part of this paper, we turn to the Cauchy problem for the heat operator associated with the generalized Laplacian: Given an initial distribution f 2 Cb (RN ), there has to be found a function u 2 C 2 (RN (0; T )) \ C (RN [0; T ]) satisfying 3
(1.5) Hk u := k u ? @t u = 0 on RN (0; 1); u(: ; 0) = f: For smooth and rapidly decreasing initial data f an explicit solution is easy to obtain; it involves the generalized heat kernel
2 +jyj2 )=4t x M y k ? ( j x j p p ; x; y 2 RN ; t > 0: K ?k (x; y; t) = +N=2 e ; t 2t 2t Here Mk is a positive constant and K denotes the generalized exponential kernel associated with G and k as introduced in [D3]. In the theory of Dunkl operators and the Dunkl transform, it takes over the r^ole of the usual exponential kernel ehx;yi : Some of its properties are collected in Section 2. Without knowledge whether K is nonnegative, a solution of (1.5) for arbitrary initial data seems to
be dicult. However, one can prove a maximum principle for the generalized Laplacian k , which is the key ingredient to assure that k leads to a positive one-parameter contraction semigroup on the Banach space (C0 (RN ); k:k1 ). Positivity of this semigroup enforces positivity of K and allows to determine the explicit solution of (1.5) in the general case. We nish this section with an extension of a well-known maximum principle for the classical heat operator to our situation. This in particular implies a uniqueness result for solutions of the above Cauchy problem. Acknowledgement. It is a pleasure to thank Charles F. Dunkl and Michael Voit for some valuable comments and discussions.
2 Preliminaries The purpose of this section is to establish our basic notations and collect some further facts on Dunkl operators and the Dunkl transform which will be of importance later on. General references here are [D3], [D4] and [dJ]. First of all we note the following product rule, which is con rmed by a short calculation: For each f 2 C 1 (RN ) and each g 2 C 1 (RN ) which is invariant under the natural action of G ,
Ti (fg) = (Ti f )g + f (Tig) for i = 1; : : : ; N: We use the common multi-index notation; in particular, for = (1 ; : : : ; N ) 2 ZN+ and x = (x1 ; : : : ; xN ) 2 RN we set x := x11 : : : xNN , ! := 1 ! : : : N ! and j j := 1 + : : : + N : If P f : RN ! C is analytic with f (x) = a x ; the operator f (T ) is de ned by X X f (T ) := a T = a T11 : : : TNN :
(2.1)
We restrict its action to if f is a polynomial of degree k and to P otherwise. The following formula will be used frequently: 2.1. Lemma. Let p 2 Pn . Then for c 2 C and a 2 C n f0g;
C k (RN )
?
?
eck p (ax) = an ea?2 c k p(x) for all x 2 RN :
In particular, for p 2 Pn we have
(2.2)
p ? p e?k =2 p ( 2x) = 2 n e?k =4 p (x):
?
4
Proof. For m 2 Z+ with 2m n; the polynomial m k p is homogeneous of degree n ? 2m: Hence ?
eck p (ax) =
bX n=2c m c
bX n=2c m c
m=0
m=0
m m! (k p)(ax) =
?
n?2m (m p)(x) = an ea?2 c k p (x): a k m!
A major tool in this paper is the generalized exponential kernel K (x; y) on RN RN , which generalizes the usual exponential function ehx;yi . It was rst introduced in [D3] by means of a certain intertwining operator. By a result of [O1] (see also [dJ]), the function x 7! K (x; y) may be characterized as the unique analytic solution of the system Ti f = yi f (i = 1; : : : ; N ) on RN with f (0) = 1: Moreover, K is symmetric in its arguments and has a holomorphic extension to P1 C N C N . Its power series can be written as K = n=0 Kn , where Kn (x; y ) = Kn (y; x) and Kn is a homogeneous polynomial of degree n in each of its variables. Note that K0 = 1 and K (z; 0) = 1 for all z 2 C N . For the re ection group G = Z2 on R; the multiplicity function k is characterized by a single parameter 0, and the kernel K is given explicitly by
K (z; w) = j?1=2 (izw) + 2zw+ 1 j+1=2 (izw); z; w 2 C ;
where for ?1=2, j denotes the normalized spherical Bessel function
j (z) = 2 ?( + 1) J z( z)
= ?( + 1)
1 X
(?1)n (z=2)2n : n=0 n! ?(n + + 1)
For details and related material we refer to [D4], [R], [R-V] and [Ros]. We list some further general properties of K and the Kn (all under the assumption k 0) from [D3], [D4] and [dJ]: For all z; w 2 C N and 2 C , (2.3) K (z; w) = K (z; w); (2.4) jKn (z; w)j n1! jz jn jwjn and jK (z; w)j ejzjjwj : For all x; y 2 RN and j = 1; : : : ; N;
(2.5) (2.6)
p
jK (ix; y)j jGj ;
Tjx Kn (x; y) = yj Kn?1 (x; y) and Tjx K (x; y) = yj K (x; y); here the superscript x denotes that the operators act with respect to the x -variable. In [dJ], exponential bounds for the usual partial derivatives of K are given. They imply in particular that for each 2 ZN+ there exists a constant d > 0; such that (2.7) j@x K (x; z )j d jz jj j ejxj jRe zj for all x 2 RN ; z 2 C N . 5
Let us nally recall a useful reproducing kernel property of K from [D4] (it is rescaled with respect to the original one, thus tting better in our context of generalized Hermite polynomials): De ne the probability measure k on RN by
dk (x) := ck e?jxj2 wk (x)dx;
with ck =
Z
R
?jxj2 wk (x)dx e N
?1
:
P Moreover, for z 2 C N set l(z ) := Ni=1 zi2 : Then for all z; w 2 C N ;
(2.8)
Z
RN
K (2z; x)K (2w; x) dk (x) = el(z)+l(w) K (2z; w):
The generalized exponential function K gives rise to an integral transform, called the Dunkl transform on RN ; which was introduced in [D4] and has been thoroughly studied in [dJ] for a large range of parameters k . The Dunkl transform associated with G and k 0 is de ned by
Dk : L1(RN ; wk (x)dx) ! C (RN ); Dk f () :=
Z
RN
f (x) K (?i; x) wk (x)dx ( 2 RN ):
In [dJ], many of the important properties of Fourier transforms on locally compact abelian groups are proved to hold true for Dk : In particular, Dk f 2 C0 (RN ) for f 2 L1 (RN ; wk (x)dx); and there holds an L1 -inversion theorem, which we recall for later reference: If f 2 L1 (RN ; wk (x)dx) with Dk f 2 L1(RN ; wk (x)dx); then f = 4? ?N=2 c2k Ek Dk f a:e:; where Ek f (x) = Dk f (?x): (Note that Dk (e?jxj2=2 )(0) = 2 +N=2 c?k 1 , which gives the connection of our constant ck with that of de Jeu.) Moreover, the Schwartz space S (RN ) of rapidly decreasing functions on RN is invariant under Dk , and Dk can be extended to a Plancherel transform on L2 (RN ; wk (x)dx). For details see [dJ].
3 Generalized Hermite polynomials and Hermite functions Let f' ; 2 ZN+ g be an orthonormal basis of P with respect to the scalar product [: ; :]k such that L ' 2 Pjj and the coecients of the ' are real. As P = n0 Pn and Pn ? Pm for n 6= m , the ' with j j = n can for example be constructed by Gram-Schmidt orthogonalization within Pn from an arbitrary ordered real-coecient basis of Pn . If k = 0, the Dunkl operator Ti reduces to the usual partial derivative @i , and the canonical choice of the basis f' g is just ' (x) := ( !)?1=2 x : As in the classical case, we have the following connection of the basis f' g with the generalized exponential function K and its homogeneous parts Kn :
3.1. Lemma. (i) Kn (z; w) = (ii) K (x; y) =
X
2ZN+
X
j j=n
' (z) ' (w) for all z; w 2 C N :
' (x) ' (y) for all x; y 2 RN ,
where the convergence is absolute and locally uniform on RN RN : Proof. (i) It suces to consider the case z; w 2 RN : So x some w 2 RN . As a function of z , the polynomial Kn (z; w) is homogeneous of degree n . Hence we have
Kn (z; w) =
X
j j=n
c; w ' (z) with c;w = [Kn (: ; w); ' ]k : 6
Repeated application of formula (2.6) for Kn gives
c;w = ' (T z )Kn (z; w) = ' (w) K0 (z; w) = ' (w): Thus part (i) is proved. For (ii), rst note that by (2.4) we have jKn (x; x)j n1! jxj2n and hence, as the ' (x) are real, j' (x)j p1n! jxjn for all x 2 RN and all with j j = n: It follows that for P each M > 0 the sum ZN+ j' (x)' (y)j is majorized on f(x; y) : jxj; jyj M g by the convergent ? P series n0 n+Nn ?1 M 2n =n! : This yields the assertion. For homogeneous polynomials p; q 2 Pn , relation (1.4) can be rescaled (by use of formula (2.2)): (3.1)
[p; q]k = 2n
Z
RN
e?k =4 p(x) e?k =4 q(x) dk (x):
This suggests to de ne a generalized multivariable Hermite polynomial system on RN as follows:
3.2. De nition. The generalized Hermite polynomials fH ; 2 ZN+ g associated with the basis f' g on RN are given by bjX j=2c
(?1)n n ' (x): k n n=0 4 n! Moreover, we de ne the generalized Hermite functions on RN by
(3.2)
(3.3)
H (x) := 2jj e?k =4 ' (x) = 2jj
h (x) := e?jxj2 =2 H (x); 2 ZN+ :
Note that H is a polynomial of degree j j satisfying H (?x) = (?1)j j H (x) for all x 2 R N . A standard argument shows that P is dense in L2 (R N ; dk ). Thus by virtue of (3.1), the f2?jj=2 H ; 2 ZN+ g form an orthonormal basis of L2(RN ; dk ): Let us give two immediate examples: 3.3. Examples. (1) In the classical case k = 0 and ' (x) := ( !)?1=2 x ; we obtain N N j j Y Y 1 2 2 =4 i ? @ i e (xi ) = p Hb (x ); H (x) = p ! i=1 ! i=1 i i
where the Hbn ; n 2 Z+ denote the classical Hermite polynomials on
R
de ned by
dn ?e?x2 : e?x2 Hb n(x) = (?1)n dx n
(2) For the re ection group G = Z2 on R and multiplicity parameter 0, the polynomial basis f'n g on R with respect to [: ; :] is determined uniquely (up to sign-changes) by suitable normalization of the monomials fxn ; n 2 Z+g . One obtains Hn (x) = dn Hn (x); where dn 2 R n f0g are constants and the Hn ; n 2 Z+ are the generalized Hermite polynomials on R
7
as introduced e.g. in [Chi] and studied in [Ros] (in some di erent normalization). They are orthogonal with respect to jxj2 e?jxj2 and can be written as 8 0, and the corresponding weight function is given by Q wS (x) = i 0. A short calculation, using Prop. 3.10 or Lemma 4.11 of [dJ], shows that (Dk Fk )(; t) = e?tjj2 :
(4.3)
By use of the reproducing formula (2.8) one therefore obtains from (4.2) the representation k e?(jxj2 +jyj2 )=4t K px ; py : L?k y Fk (x; t) = M t +N=2 2t 2t
(4.4)
4.4. De nition. The generalized heat kernel ?k is given by
k e?(jxj2 +jyj2 )=4t K px ; py ; x; y 2 RN ; t > 0: ?k (x; y; t) := M t +N=2 2t 2t
4.5. Lemma. The heat kernel ?k has the following properties on RN RN (0; 1) : 2
Z
(1) ?k (x; y; t) = +ckN=2 e?tjj2 K (ix; ) K (?iy; ) wk ()d : N 4 R N (2) For xed y 2 R , the function u(x; t) := ?k (x; y; t) solves the generalized heat equation Hk u = 0 on RN (0; 1) . (3)
Z
RN
?k (x; y; t) wk (x)dx = 1:
k e?(jxj?jyj)2 =4t : (4) j?k (x; y; t)j M + N= t 2
17
Proof. (1) is clear from the above derivation. For (2), remember that xk K (ix; ) = ?j j2 K (ix; ): Hence the assertion follows at once from representation (1) by taking the di erentiations under the integral. This is justi ed by the decay properties of the integrand and its derivatives in question (use estimation (2.7) for the partial derivatives of K (ix; ) with respect to x .) To obtain (3), we employ formula (4.4) as well as (4.3) and write Z
R
? ?k (x; y; t) wk (x)dx = Dk L?k y Fk (0; t) = K (?iy; 0)(Dk Fk )(0; t) = 1: N
Finally, (4) is a a consequence of the estimate (2.4) for K . Remarks. 1. For integer-valued multiplicity functions, Berest and Molchanov [B-M] constructed the heat kernel for the G -invariant part of Hk (in a conjugated version) by shift-operator techniques.
2. In contrast to the classical case, it is not yet clear at this stage that the kernel ?k is generally nonnegative. In fact, it is still an open conjecture that the function K (iy; :) is positive-de nite on RN for each y 2 RN (c.f. the remarks in [dJ] and [D3]). This would imply a Bochnertype integral representation of K (iy; :) and positivity of K on RN RN as an immediate consequence. In the one-dimensional case this conjecture is true, and the Bochner-type integral representation is explicitly known (see [Ros] or [R]). By one-parameter semigroup techniques, it will however soon turn out that K is at least positive on RN RN .
4.6. De nition. For f 2 Cb(RN ) and t 0 set (4.5)
8Z >
RN
?k (x; y; t)f (y) wk (y)dy if t > 0;
:f (x)
if t = 0
Part (4) of Lemma 4.5 assures that for each t 0; H (t)f is well-de ned and continuous on RN . It provides a natural candidate for the solution to our Cauchy problem. Indeed, when restricting to initial data from the Schwartz space S (RN ), we easily obtain the following:
4.7. Theorem. Suppose that f 2 S (RN ): Then u(x; t) := H (t)f (x); (x; t) 2 RN [0; 1); solves
the Cauchy-problem (4.1) for each T > 0 . Morover, it has the following properties:
(i) H (t)f 2 S (RN ) for each t > 0 . (ii) H (t + s)f = H (t)H (s)f for all s; t 0 . (iii) kH (t)f ? f k1;RN ! 0 with t ! 0 . Proof. Using formula (1) of Lemma 4.5 and Fubini's theorem, we can write 2 Z Z u(x; t) = H (t)f (x) = +ckN=2 N N K (ix; )K (?iy; ) e?tjj2 f (y) wk ()wk (y) ddy 4 R R 2 Z ?tjj2 c = +kN=2 N e (4.6) Dk f ()K (ix; ) wk ()d 4 R
18
for all t > 0. (Remember that S (RN ) is invariant under the Dunkl transform). This makes clear that (i) is satis ed. As before, it is seen that di erentiations may be taken under the integral in (4.6), and that Hk u = 0 on RN (0; 1). Moreover, in view of the inversion theorem for the Dunkl transform, (4.6) holds for t = 0 as well. Using (2.5), we thus obtain the estimation 2 Z ? c kH (t)f ? f k1;RN jGj 4 +kN=2 N jDk f ()j 1 ? e?tjj2 wk ()d; R and this integral tends to 0 with t ! 0: This yields (iii). In particular, it follows that u is continuous ? on RN [0; 1). To prove (ii), note that Dk H (t)f ( ) = e?tjj2 Dk f ( ) : Therefore p
?
?
?
Dk H (t + s)f () = e?tjj2 Dk H (s)f f () = Dk H (t)H (s)f (): The statement now follows from the injectivity of the Dunkl transform on S (RN ). We are now going to show that indeed, the linear operators H (t) on S (RN ) extend to a positive contraction semigroup on the Banach space C0 (RN ); equipped with its uniform norm k:k1 . To this end, we consider the generalized Laplacian k as a densely de ned linear operator on C0 (RN ) with domain S (RN ). 4.8. Theorem. (1) The operator k on C0 (RN ) is closable, and its closure k generates a positive, strongly continuous contraction semigroup fT (t); t 0g on C0 (RN ) . (2) The action of T (t) on S (RN ) is given by T (t)f = H (t)f . Proof. (1) We apply a variant of the Lumer-Phillips theorem, which characterizes generators of positive one-parameter contraction semigroups (see e.g. [A], Cor. 1.3). It requires two properties:
(i) The operator k satis es the following \dispersivity condition": Suppose that f 2 S (RN ) is real-valued with maxff (x) : x 2 RN g = f (x0 ): Then k f (x0 ) 0: (ii) The range of I ? k is dense in C0 (RN ) for some > 0: Property (i) is an immediate consequence of Lemma 4.1. Condition (ii) is also satis ed, because I ? k maps S (RN ) onto itself for each > 0; this follows from the fact that the Dunkl transform ? is a homeomorphism of S (RN ) and Dk (I ? k )f ( ) = ( + j j2 )Dk f ( ). The assertion now follows by the above-mentioned theorem. (2) It is known from semigroup theory that for every f 2 S (RN ); the function t 7! T (t)f is the unique solution of the abstract Cauchy problem (4.7)
8
0; : u(0) = f
within the class of all (strongly) continuously di erentiable functions u on [0; 1) with values in the Banach space (C0 (RN ); k:k1 ): By property (i) of Theorem 4.6 we have H (t)f 2 C0 (RN ) for f 2 S (RN ). Moreover, from representation (4.6) of H (t)f it is readily seen that t 7! H (t)f is continuously di erentiable on [0; 1) and solves (4.7). This nishes the proof. 19
4.9. Corollary. The heat kernel ?k is strictly positive on generalized exponential kernel K satis es
RN
RN (0; 1): In particular, the
K (x; y) > 0 for all x; y 2 RN : Proof. For any initial distribution f 2 S (RN ) with f 0 the last theorem implies that Z
RN
?k (x; y; t)f (y) wk (y)dy = T (t)f (x) 0 for all (x; t) 2 RN [0; 1):
As y 7! ?k (x; y; t) is continuous on RN for each xed x 2 RN and t > 0, it follows that ?k (x; y; t) 0 for all x; y 2 RN and t > 0: Hence K is nonnegative as well. Now recall again the reproducing identity (2.8), which says that
e(jxj2 +jyj2 )K (2x; y) = ck
Z
R
?jzj2 dz K ( x; 2 z ) K ( y; 2 z ) w ( z ) e k N
for all x; y 2 RN : The integrand on the right side is continuous, non-negative and not identically zero (because K (x; 0)K (y; 0) = 1). Therefore the integral itself must be strictly positive.
4.10. Corollary. The semigroup fT (t)g on C0 (RN ) is given explicitly by T (t)f = H (t)f; f 2 C0 (RN ): Proof. This is clear from part (2) of Theorem 4.8 and the previous corollary, which implies that the operators H (t) are continuous { even contractive { on C0 (RN ): Remark. The generalized Laplacian also leads to a contraction semigroup on the Hilbert space H := L2(RN ; wk (x)dx); this generalizes the results of [Ros] for the one-dimensional case. In fact, let M denote the multiplication operator on H de ned by Mf (x) = ?jxj2 f (x) and with domain D(M ) = ff 2 H : jxj2 f (x) 2 Hg . M is self-adjoint and generates the strongly continuous contraction semigroup M (t)f (x) = e?tjxj2 f (x) (t 0) on H . For f 2 S (RN ), we have the identity Dk (k f ) = M (Dk f ): As S (RN ) is dense in D(M ); this shows that k has a self-adjoint extension e k on H , namely e k = Dk?1 M Dk , where here Dk denotes the Plancherel-extension of the Dunkl transform to H . The domain of e k is the Sobolev-type space D(e k ) = ff 2 H : j j2 Dk f ( ) 2 Hg: Being unitarily equivalent with M; the operator e k also generates a strongly continuous contraction semigroup fTe(t)g on H which is unitarily equivalent with fM (t)g; it is given by
Te(t)f (x)
=
Z
R
?tjj2 Dk f ( ) K (ix; ) wk ( )d: e N
The knowledge that ?k is nonnegative allows also to solve the Cauchy problem (4.1) in its general setting:
4.11. Theorem. Let f 2 Cb(RN ) . Then u(x; t) := H (t)f (x) is bounded on solves the Cauchy problem (4.1) for each T > 0:
20
RN
[0; 1) and
Proof. In order to see that u is twice continuously di erentiable on RN (0; 1) with Hk u = 0, we only have to make sure that the necessary di erentiations of u may be taken under the integral in (4.5). One has to use again the estimations (2.7) for the partial derivatives of K ; these provide sucient decay properties of the derivatives of ?k , allowing the necessary di erentiations of u under the integral by use of the dominated convergence theorem. Boundedness of u is clear from the positivity and normalization (Lemma 4.5(3)) of ?k ; in fact, ju(x; t)j kf k1;RN on RN [0; 1): Finally, we have to show that H (t)f (x) ! f ( ) with x ! and t ! 0. We start by the usual method: For xed > 0, choose > 0 such that jf (y) ? f ( )j < for jy ? j < 2 and let M := kf k1;RN : Keeping in mind the positivity and normalization of ?k , we obtain for jx ? j < the estimation Z
jH (t)f (x)?f ()j
?
RN
Z
?k (x; y; t)jf (y) ? f ( )jwk (y)dy +
jy?xj
Z
jy?xj>
?k (x; y; t)jf (y) ? f ( )jwk (y)dy
?k (x; y; t)wk (y)dy:
It thus remains to show that for each > 0, lim(x;t)!(;0)
Z
jy?xj>
For abbreviation put
I (x; t) :=
?k (x; y; t)wk (y)dy = 0:
Z
jy?xj
?k (x; y; t)wk (y)dy:
As I (x; t) 1, it suces to prove that lim inf (x;t)!(;0) I (x; t) 1: For this, choose some positive constant 0 < and h 2 S (RN ) with 0 h 1; h( ) = 1 and such that h(y) = 0 for all y with jy ? j > ? 0 : Then for each x with jx ? j < 0 the support of h is contained in fy 2 RN : jy ? xj g ; therefore Z
RN
h(y)?k (x; y; t)wk (y)dy I (x; t)
for all (x; t) with jx ? j < 0 : But according to Theorem 4.7 we have lim(x;t)!(;0) This nishes the proof.
Z
RN
h(y)?k (x; y; t) wk (y)dy = h() = 1:
It is still open whether our solution of the Cauchy problem (4.1) is unique within an appropriate class of functions. As in the classical case, this follows from an maximum principle for the generalized heat operator on RN (0; 1). The rst step is the following weak maximum principle for Hk on bounded domains. It is proved by a similar method as used in Theorem 4.2. By virtue of Lemma 4.1, this proof is literally the same as the standard proof in the classical case (see e.g. [J]) and therefore omitted here. 21
4.12. Proposition. Suppose that RN is open, bounded and G -invariant. For T > 0 set
T := (0; 1) and @ T := f(x; t) 2 @ T : t = 0 or x 2 @ g : Assume further that u 2 C 2 ( T ) \ C ( T ) satis es Hk u 0 in T . Then
max T (u) = max @ T (u) : Under a suitable growth condition on the solution, this maximum principle may be extended to the case where = RN . The proof is adapted from the one in [dB] for the classical case.
4.13. Theorem. (Weak maximum principle for Hk on RN .) Let ST := RN (0; T ) and suppose that u 2 C 2 (ST ) \ C (S T ) satis es 8 r: Then
sup S T (u) supRN (f ):
Proof. Let us rst assume that 8T < 1: For xed > 0 set 2 1 j x j v(x; t) := u(x; t) ? (2T ? t) +N=2 exp 4(2T ? t) ; (x; t) 2 RN [0; 2T ):
By Lemma 4.3, v satis es Hk v = Hk u 0 in ST . Now x some constant > r and consider the bounded cylinder T = (0; T ) with = fx 2 RN : jxj < g: Setting M := supRN (f ), we have v(x; 0) < u(x; 0) M for x 2 : Moreover, for jxj = and t 2 (0; T ] v(x; t) Ce2 ? (2T ) 1 +N=2 e 2 =8T :
As < (8T )?1 , we see that v(x; t) M on @ T , provided that is large enough. Then by Prop. 4.12 we also have v(x; t) M on T : As > r was arbitrary, it follows that v(x; t) M on S T . As > 0 was arbitrary as well, this implies that u(x; t) M on S T . If 8T 1, we may subdivide ST into nitely many adjacent open strips of width less than 1=8 and apply the above conclusion repeatedly.
4.14. Corollary. The solution of the Cauchy problem (4.1) according to Theorem 4.11 is unique within the class of functions u 2 C 2 (ST ) \ C (S T ) which satisfy the following exponential growth condition: There exist positive constants C; ; r such that
ju(x; t)j C ejxj2 for all (x; t) 2 ST with jxj > r: 22
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