Jiaxi Jiu and Zhongkai Li

Department of Mathematics,Shanghai Normal University,Shanghai 200234,China

Abstract. We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator D in the upper half-plane = R×(0,∞),where

Key Words: Dunkl operator, Dunkl transform, harmonic function, non-tangential limit, area integral.

1 Introduction and main results

For given λ>0,the rank-one Dunkl operator on the line R is defined by

A C2function u in the upper half-plane= R×(0,∞) is said to be λ-harmonic if∆λu=0,where

The operator ∆λis called the λ-Laplacian,and can be written explicitly by

Some aspects of harmonic analysis in the upper half-planeassociated to the Dunkl operator D were studied in[25]and their analogues in the unit disk D, associated with Dunkl-Gegenbauer expansions,were developed in[26]. These are generalizations of the seminal work of Muckenhoupt and Stein[31]on the Bessel operator and the Gegenbauer expansions. In this paper we study the local existence of boundary values of λ-harmonic functions in the upper half-plane

It is well known that,if u is a harmonic function in the unit disk D and E is a subset of positive measure of the boundary ∂D,then the existence of non-tangential limit at almost every eiθ∈E of u can be characterized by non-tangential boundedness of u at almost every eiθ∈E, and also by finiteness of Lusin’s area integral of u at almost every eiθ∈E.The former,as a local version of Fatou’s theorem,was owed to Privalov[38],and the latter was proved by Marcinkiewicz and Zygmund[30]and Spencer[42]. One of the basic tools in these works is the conformal mapping,which introduces technical difficulties in extending them to more variables and other settings.Calder´on[5,6]made a breakthrough and generalized Privalov’s theorem and Marcinkiewicz and Zygmund’s theorem to Euclidean half-spaces of several variables by the real-variable method. A generalization of the theorem of Spencer [42] to several variables was obtained in Stein [43]. Since then,criteria on existence of non-tangential boundary limits of harmonic functions in many different contexts, in terms of non-tangential boundedness or one-side non-tangential boundedness or finiteness of area integrals have been intensively studied;see,for example,[1-4,7,14-22,24,32-37,39]and[46].

As usual,we denote by Γα(x)the positive cone of aperture α>0 with vertex(x,0)∈=R,and(x)the truncated one with height h>0,that is,

where τx, acting on the first argument, is the associated (generalized) translation in the Dunkl setting (see Section 2). We note that Su was first defined in [27] (see [28] also),where it is used to characterize the Hardy spaces associated to the Dunkl operator.

Our purpose is to characterize the local existence of non-tangential boundary limits of λ-harmonic functions inwhich generalizes the theorems of Privalov [38],Marcinkiewicz and Zygmund[30],Spencer[42],Calder´on[5,6]and Stein[43]. The main results are stated in the following theorem.

Theorem 1.1. Assume that u is a λ-harmonic function in,and E is a measurable subset of positive measure of ∂= R and is symmetric about y-axis. Then the following assertions are equivalent:

(i) u has a finite non-tangential limit at(x,0)for almost every x ∈E;

(ii) u is non-tangentially bounded at(x,0)for almost every x ∈E;

(iii) the area integral(Sα,hu)(x)is finite for almost every x ∈E with some α,h>0.

Several remarks are given in order.

(a) For a λ-harmonic function u in R2+,from[29,(2.2)]we have

(d) The more general setting of the Dunkl theory is on the study of multivariable analytic structures associated with finite reflection groups, of which the basic tools are the Dunkl transform and the Dunkl operators invariant under a given group(cf. [8-13,41,45]). During the last decades, it has gained considerable interest in various fields of mathematics and also in physical applications; for example, the Dunkl operators for the symmetric group Sdon Rdare naturally connected with the analysis of quantum many body systems of Calogero-Moser-Sutherland type,which describe algebraically integrable systems in one dimension(cf.[23]).

The paper is organized as follows.Section 2 contains some basic facts on the rank-one case of the Dunkl theory which will be relevant for the sequel. The proof of that(ii)⇒(i)of Theorem 1.1 is given in Section 3,and the equivalence of parts(ii)and(iii)in Theorem 1.1 is proved in Section 4.

2 Some facts on the rank-one case of the Dunkl theory

where Eλis the Dunkl kernel(cf.[11,40])

and jα(z)is the normalized Bessel function

Since j−1/2(z) = cos z, j1/2(z) = z−1sin z,it follows that E0(iz) = eizand F0agrees with the usual Fourier transform F. In what follows,we assume that λ>0.

The Dunkl transform shares many of the important properties with the usual Fourier transform, part of which are listed as follows. These conclusions extend those on the Hankel transform and are special cases on the general Dunkl transform studied in[8,12].

and

if x,t ∈R{0},and 0 otherwise.

For two appropriate functions f and g on R,their λ-convolution f ∗λg is defined by

The properties of τtand ∗λare listed as follows(cf.[25,40]).

3 The proof of the main theorem

In this section,we shall prove that(ii)⇒(i)in Theorem 1.1,which is restated as follows.

it follows that,for some c>0,

Then from(3.1)we have

and since

Obviously|ϕk|≤1 on R for all k ≥1,and thus,one can find a function ϕ with|ϕ|≤1 on R and a subsequence{ϕkj}so that{ϕkj}converges weakly∗to ϕ. In particular,for their λ-Poisson integrals,we have

If we write

then

Further we claim that,for each k,

where H is the function given in Lemma 3.3.In fact,we shall prove the following stronger version of(3.6)

If there were some (x0,y0) ∈Ω1so that |ψk(x0,y0)|+|ψk(−x0,y0)| > 2H(x0,y0), we consider the set

where

The choice of plus or minus sign depends on

which contradicts the nonnegativity of H by Lemma 3.3. The claim(3.7),and so(3.6),are proved.

Finally,taking k= kjin(3.6)and letting j →∞yields|ψ(x,y)|≤2H(x,y)for(x,y)∈Ω1, and by Lemma 3.3, for almost every x ∈E, ψ(t,y) tends to zero as (t,y) ∈Γα(x)approaching to (x,0). Further by Proposition 2.4, (Pϕ)(x,y) has a finite non-tangential limit for almost every x ∈E, and hence, by(3.5), u(x,y) = ψ(x,y)+(Pϕ)(x,y) has the desired assertion in the lemma.

Now we turn to the proof of Theorem 3.1:

Proof. We assume that the set E is bounded, without loss of generality. By Lemma 3.1,for each k ∈N,there exists a compact set Ek⊂E,such that|EEk|0< 1/k,and for any α>0,there is a constant cα,k>0,so that|u(x,y)|≤cα,k,(x,y)∈x0∈EkΓ2α(x0). If we put E0=Ek,then|EE0|0= 0. Since E is symmetric about y-axis,we may choose each Ekpreserving this property; and otherwise, Ek∩σEkcould be used instead of Ek. Thus applying Lemma 3.4 to u/cα,k,it follows that,for almost every x ∈Ek, u(t,y)/cα,khas a finite limit as(t,y)∈Γα(x)approaching to(x,0),and hence,u has a finite non-tangential limit at(x,0)for almost every x ∈E. The proof is completed.

4 The proof of the main theorem(continue)

We recall that,for a C2function u in,the Lusin-type area integral Su= Sα,hu for some α,h>0 is defined by

In this section,we shall prove the equivalence of parts(ii)and(iii)in Theorem 1.1,which is reformulated in the following two theorems.

Theorem 4.1. Assume that u is a λ-harmonic function in,and E is a measurable subset of positive measure of ∂=R and is symmetric about y-axis. If u is non-tangentially bounded at(x,0)for every x ∈E,then for arbitrary α,h>0,the area integral(Sα,hu)(x)is finite for almost every x ∈E.

Theorem 4.2. Assume that u is a λ-harmonic function,and E is a measurable subset of positive measure of ∂= R and is symmetric about y-axis. If for every x ∈E, there exist some α,h > 0,so that the area integral(Sα,hu)(x)is finite,then u is non-tangentially bounded at(x,0)for almost every x ∈E.

Clearly Theorem 4.1 is a stronger version of that(ii)⇒(iii)in Theorem 1.1. To prove Theorems 4.1 and 4.2,we need several lemmas.

Lemma 4.1. Let E be a measurable and bounded set of ∂=R,symmetric about y-axis. Then for any є>0,there exists a compact set Eє,symmetric about y-axis,satisfying

(i) Eє⊂E,|E −Eє|λ< є;

(ii)for η ∈(0,1),there exists some δ>0,such that for x ∈Eєand 0

Proof. For f ∈Lλ,loc(R),we define the weighted maximal function Mλf by

Since Mλis of weak-type(1,1),by a standard process one can prove that

for a.e. x ∈R.

Now taking f = χE,it follows that

By Egorov’s theorem, for given є > 0 there exists Eє⊂ E, |E −Eє|λ< є, so that|(x −r,x+r)∩E|λ/|(x −r,x+r)|λtends to 1 uniformly for x ∈Eєas r →0+. Clearly one may take Eєto be a closed subset of E.Since E is symmetric about y-axis,the uniform convergence above is true also for x ∈σEє,and so is for x ∈Eє∪σEє. Thus we may take Eєto be symmetric about y-axis,as desired.

Lemma 4.2. Let u(x,y)be a λ-harmonic function in,E a compact subset of ∂=R being symmetric about y-axis,and for α,h>0,Ω= ΩE(α,h). Then

whenever the right hand side above is finite.

Substituting this into(4.4)and in consideration of the symmetry of Ω about y-axis,(4.1)is proved.

Lemma 4.3. Let u(x,y) be a λ-harmonic function in, E a measurable and bounded set of∂R2+= R being symmetric about y-axis, and let β,κ > 0 be given. Then for any є > 0, there exists a compact set Eє,symmetric about y-axis,satisfying

(i) Eє⊂E,|E −Eє|λ< є;

(ii)for fixed α ∈(0,β)and h ∈(0,κ),there exists some c= c(є,α,β,h,κ)>0,such that

whenever the right hand side above is finite.

Proof. For given є > 0, by Lemma 4.1 there exist a compact subset Eєof E, symmetric about y-axis,and some δ>0,satisfying|E −Eє|λ< є and

for x ∈Eєand 0

which concludes(4.9);if(x −t)2+2|xt|≥β2y2,one has

which again concludes(4.9).

Now we show that,for x satisfying(4.8),

If|t|≤2βy,then

thus(4.10)is verified. If|t|>2βy,we have

and then,collecting these estimates verifies(4.10)again.

Applying(4.9)and(4.10)to(4.7)we obtain,for(t,y)∈ΩEє(α,h),

yields

Finally,inserting this into(4.6)we get

The proof of the lemma is completed.

Lemma 4.4. Let E be a compact subset of ∂R2+= R, symmetric about y-axis, and for α,h >0, Ω = ΩE(α,h). Then there exists a family of y-symmetric regions {Ωє}є∈(0,h/3), with the following properties:

(i)Ωє⊂Ω,and Ωє1⊂Ωє2if є2< є1;

(ii)Ωє→Ω as є →0+(i.e.,∪Ωє= Ω);

Further,for Ω= ΩE(α,h),there exists a family of y-symmetric regions{Ωє}є>0satisfying(i)-(iv)in Lemma 4.4,and hence,we only need to prove that,there exists some constant c>0 independent of є ∈(0,h/3),so that completed once we prove that, for a compact and y-symmetric set E ⊂∂R2+= R, u is non-tangentially bounded at(x,0)for almost every x ∈E under the conditions

for some α,h>0,and

By Proposition 2.4, v(x,y)is non-tangentially bounded at(x,0)for almost every x ∈R,and hence,the same is true for u(x,y)and for almost every x ∈E. The proof of Theorem 4.2 is completed.

Acknowledgements

The work was supported by the National Natural Science Foundation of China(No.11371258).