Almost everywhere convergence of inverse Dunkl transform on the real line.

1. Introduction and preliminaries

Given [alpha] [greater than or equal to] [-1/2] and a suitable function f on R, its Dunkl transform [D.sub.[alpha]] is defined by

[D.sub.[alpha]]f(y) = [[integral].sub.R] f(x)[E.sub.[alpha]]( - ixy)d[[mu].sub.[alpha]](x), y[member of]R; (1)

here

[d[mu].sub.[alpha]](x) = [1/[[2.sup.[[alpha] + 1]] [GAMMA]([alpha] + 1)]] [[|x|].sup.[2[alpha] + 1]] dx, (2)

[E.sub.[alpha]](z) = [2.sup.[alpha]] [GAMMA]([alpha] + 1) {[[[J.sub.[alpha]](iz)]/[(iz).sup.[alpha]]] + z[[[J.sub.[[alpha] + 1]](iz)]/[(iz).sup.[[alpha] + 1]]]}, (3)

where [J.sub.[alpha]] denotes the Bessel function of the first kind of order [alpha]. The inverse Dunkl transform [D.sub.[alpha]] is given by [D.sub.[[alpha]f]]([lambda]) = [D.sub.[[alpha]f]](-[lambda] (see [2] and [3]).

In this paper, we are interested in the almost everywhere convergence as R [right arrow] [infinity], of the partial sums [S.sub.R.sup.[alpha]] f(x), where

[S.sub.R.sup.[alpha]]f(x) = [[integral].sub.[|y|[less than or equal to]R] [D.sub.[alpha]]f(y)[E.sub.[alpha]](ixy)d[[mu].sub.[alpha]](y).

Recall that given [beta] [greater than or equal to] [-1/2], the Hankel transform of order [beta] of a suitable function g on (0, [infinity]) is difined by

[H.sub.[beta]g](y) = [[integral].sub.0.sup.[infinity]] g(x)[[[J.sub.[beta]](yx)]/[(yx).sup.[beta]]][x.sup.[2[beta] + 1]]dx, y > 0. (4)

Nowak and Stempak (6), found an expression of the Dunkl transform [D.sub.[alpha]] in terms of Hankel transform of orders [alpha] and [alpha] + 1.

Lemma 1.1. (See ([6]) Given [alpha] [greater than or equal to] [-1/2], we have

[D.sub.[alpha]]f(y) = [H.sub.[alpha]]([f.sub.e])([absolute value of y]) - iy[H.sub.[[alpha] + 1]] ([[f.sub.o](x)]/x)([absolute value of y]), (5)

where for a function f on R, we denote by [f.sub.e] and [f.sub.o] the restrictions to (0, [infinity]) of its even and odd parts, respectively, i.e. the functions on (0, [infinity]) defined by

[f.sub.e](x) = [1/2](f(x) + f( - x)), [f.sub.o](x) = [1/2](f(x) - f( - x)), x > 0.

Define, the partial sums [s.sub.R.sup.[beta]]g(x) by

[s.sub.R.sup.[beta]]g(x) = [[integral].sub.0.sup.R][H.sub.[beta]]g(y)[[[J.sub.[beta]](xy)]/[(xy).sup.[beta]]][y.sup.[2[beta] + 1]]dy, x > 0, (6)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

In 1988, Kanjin (4) and Prestini (7) independently proved the following.

Theorem 1.2. Let [beta] [greater than or equal to] -1/2 and 1 [less than or equal to] p < [infinity].

* If [4([beta] + 1)/2[beta] + 3 < p < 4([beta] + 1)/2[beta] + 1, then [s*.sup.[beta]] is bounded on [L.sup.p]((0, [infinity]), [x.sup.[2[beta] + 1]]).

* If p [less than or equal to] 4([beta] + 1)/2[beta] + 3 or p [greater than or equal to] 4([beta] + 1)/2[beta] + 1, then [[s*].sup.[beta]] is not bounded on [L.sup.p]((0, [infinity]), [x.sup.[2[beta] + 1]]).

Throughout this paper we use the convention that [c.sub.[alpha]] denotes a constant, depending on [alpha] and p, its value may change from line to line.

2. Almost everywhere convergence

Define linear operators [S.sub.R.sup.[alpha]], R > 0 and [[S*].sup.[alpha]] on the Schwartz space S (R) by

[S.sub.R.sup.[alpha]]f(x) = [[integral].sub.|y|[less than or equal to]R][D.sub.[alpha]]f(y)[E.sub.[alpha]](ixy)d[[mu].sub.[alpha]](y), (8)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

We note that, by Proposition 5 in [5], [S.sub.R.sup.[alpha]]f can be defined for f [member of] [L.sup.p](R, d[[mu].sub.[alpha]]), 1 < p < [p.sub.1], by

[S.sub.R.sup.[alpha]]f(x) = [[integral].sub.R] [[phi].sub.R]( - y)[[TAU].sub.x]f( - y)d[[mu].sub.[alpha]](y), x [member of] R, (10)

where [phis]R(x) = [c.sub.[alpha]][R.sup.[2([alpha] + 1)]]j[alpha] + 1(Rx), x [member of] R, and [T.sub.x], x [member of] R are the so-called Dunkl translation operators on R.

Lemma 2.1. Given [alpha] [greater than or equal to] -1/2, we have

[S.sub.R.sup.[alpha]](f)(x) = [s.sub.R.sup.[alpha]]([f.sub.e])([absolute value of x]) + x[s.sub.R.sup.[[alpha] + 1]]([[f.sub.o](r)]/r)([absolute value of (x)]), (11)

[S.sub.*.sup.[alpha]]f(x) [less than or equal to] [S.sub.*.sup.[alpha]]([f.sub.e])([absolute value of x]) + [absolute value of x][[s.sub.*.sup.[[alpha] + 1]] ([[f.sub.o](r)]/r)([absolute value of x]). (12)

Proof. Let x [member of] R. By (3), (8) and Lemma 1.1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We note that the second and the third integrals are equal to zero. So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus

[S.sub.*.sup.[alpha]]f(x) [less than or equal to] [S.sub.*.sup.[alpha]]([f.sub.e])(|x|) + |x|[s.sub.*.sup.[[alpha] + 1]] ([[f.sub.o](r)]/r)(|x|).

Proposition 2.2. Let [alpha] > [-1/2].

* If [p.sub.0] < p < [p.sub.1], then [S.sub.*.sup.[alpha]] is bounded on [L.sup.p] (R, d[[mu].sub.[alpha]](x)).

* If p [less than or equal to] [p.sub.0] or p [greater than or equal to] [p.sub.1], then [S.sub.*.sup.[alpha]] is not bounded on [L.sup.p] (R, d[[mu].sub.[alpha]](x)).

Proof. [S.sub.*.sup.[alpha]] cannot be bounded for p [less than or equal to] [p.sub.0] or p [greater than or equal to] [p.sub.1] (see: [4] and [7]). By Theorem 1, we have for [p.sub.0] < p < [p.sub.1],

||[[s.sub.*.sup.[alpha]]([f.sub.e])(|x|)[||.sub.Lp][(R,d[mu][alpha](x))] = 2||[[s.sub.*.sup.[alpha]]([f.sub.e])[||.sub.Lp]((0,[infinity]),[x.sup.2[[2[alpha]] + 1]]dx)]] [less than or equal to] [c.sub.[alpha]] [||[f.sub.e]||] [L.sup.p]((0,[infinity]),[x.sup.[2[alpha]+1]] dx)]] [less than or equal to][c.sub.[alpha]][||f||][L.sup.pp](R,d[mu][alpha](x))].

On the other hand, as in ((7), (8)), one gets

[absolute value of (x)][s.sub.*.sup.[[alpha] + 1]]([[f.sub.o](r)]/r)([absolute value of x])[less than or equal to][[c.sub.[alpha]]/[[absolute value of x].sup.[[alpha] + [1/2]][M + H + [~.H] + [~.C]][[f.sub.o](r)]/r[r.sup.[[alpha] + [3/2]]([absolute value of x]), (13)

where M, H, [~.H] and [~.C] denotes respectively, the maximal function, the Hilbert integral, the maximal Hilbert transform and the Carleson operator.

Let K = M + H + [~.H] + [~.C] and w [member of] [A.sub.p] (R), p > 1. It is well known that

||Kf||[.sub.Lp][(R,w(x)dx)][less than or equal to][c.sub.p]||f||[.sub.Lp][(R,w(x)dx)]. (14)

Hence

[||[absolute value of x][[s *].sup.[[alpha] + 1]]([[f.sub.o](r)]/r)([absolute value of x])]||.sub.[L.sup.p](R,d[[mu].sub.[alpha]](x))][less than or equal to][c.sub.[alpha]]||[[absolute value of x].sup.-[alpha]-[1/2]]]]K[[[f.sub.o](r)]/r[r.sup.[[alpha] + [3/2]]]]([absolute value of x])]||.sub.[L.sup.p](R,d[[mu].sub.[alpha]](x))][less than or equal to][c.sub.[alpha]]||K[[f.sub.o](r)]/r[r.sup.[[alpha] + 3/2]]]([absolute value of x])]||.sub.[L.sup.p](R,w(x)dx)],

with w(x) = [[absolute value of x].sup.[2[alpha] + 1 - p([alpha] + 1/2)]]]

Since [p.sub.0] < p < [p.sub.1] if and only if -1 < 2[alpha] + 1-p([alpha] + 1/2) < p - 1, then w [member of] [A.sub.p] (R), and by (13)

[||[absolute value of x][[s *].sup.[[alpha] + 1]]([[f.sub.o](r)]/r)([absolute value of x])]||.sub.[L.sup.p](R,d[[mu].sub.[alpha]](x))][less than or equal to][c.sub.[alpha]][||[[[f.sub.0]([absolute value of x])]/[[absolute value of x]]][absolute value of x][.sup.[[alpha] + 3/2]||].sub.[L.sup.p](R,w(x)dx)] [less than or equal to][c.sub.[alpha]][||[f.sub.o](x)||.sub.[L.sup.p](R,d[[mu].sub.[alpha]](x))] [less than or equal to][c.sub.[alpha]][||[f.sub.o](x)||.sub.[L.sup.p](R,d[[mu].sub.[alpha]](x))].

We conclude by Lemma 2.1.

Using Proposition 2.2, and since almost everywhere convergence holds for functions on S(R), which is a dense subset of [L.sup.p] (R, [d[mu].sub.[alpha]]), (see [3]), we obtain

Corollary 2.3. For every f [member of] [L.sup.p] (R, [d[mu].sub.[alpha]]), if [p.sub.0] < p < [p.sub.1], then

[S.sub.R.sup.[alpha]]f(x)[right arrow] or f(x) a.e. as R[right arrow] or [infinity].

3. Endpoint estimates

We recall that the Lorentz space [L.sup.p,q] (X, [mu]), is the set of all measurable functions f on X satisfying

[||f||.sub.p,q] = [(q/p[[integral].sub.0.sup.[infinity]][([t.sup.[1/p]]f * (t)).sup.q][dt]/t).sup.[1/q]]<[infinity]

when, 1 [less than or equal to] p < [infinity], 1 [less than or equal to] q < [infinity], and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when, 1 [less than or equal to] p [less than or equal to] [infinity] and q = [infinity]. Here f * denotes the nonincreasing rearrangement of f, i.e.

f * (t) = inf {s>0/[d.sub.f](s)[less than or equal to]t}, [d.sub.f](s) = [mu]{x[member of]X/|f(x)|>s}.

In 1991, Romera and Soria (8) (see also Colzani and all (1)) proved the following

Theorem 3.1. Let [alpha] > -1/2, then [[s *].sup.[alpha]] is bounded from the Lorentz space [L.sup.pi,1]((0, [infinity]), [x.sup.[2[alpha] + 1]] dx) into [L.sup.pi,[infinity]]((0, [infinity]), [x.sup.[2[alpha] + 1]]dx), i = 0,1.

Using this result, we will see that Proposition 2.2 can be srengthened. More precisely we obtain

Proposition 3.2. Let [alpha] > -1/2, then [[S *].sup.[alpha]] is bounded from the Lorentz space [L.sup.pi,1] (R, [d[mu].sub.[alpha]]) into [L.sup.pi,[infinity]] (R, [d[mu].sub.[alpha]]), i=0,1.

Using Marcinkiewicz's interpolation theorem in terms of Lorentz space we retrieve Proposition 2.2 as a corollary.

Proof. By Lemma 2.1, we have

[[mu].sub.[alpha]]{x[member of]R/[[S *].sup.[alpha]]f(x)>[lambda]}[less than or equal to][[mu].sub.[alpha]]{x[member of]R/[[s *].sup.[alpha]][f.sub.e]([absolute value of x])>[[lambda]/2]} + [[mu].sub.[alpha]]{x[member of]R/[absolute value of x][[s *].sup.[alpha] + 1]]([[f.sub.o](r)]/r)([absolute value of x])>[[lambda]/2]}. = I + II.

By Theorem 2.4, we get:

[[mu].sub.[alpha]] {x [member of] [R/[s.sub.*.sup.[alpha]]] [f.sub.e]([absolute value of x]) > [[lambda]/2]} = 2[[mu].sub.[alpha]] {x [member of] (0,[infinity])/[s.sub.*.sup.[alpha]][f.sub.e](x) > [[lambda]/2]} [less than or equal to] [[c.sub.[alpha]]/[[lambda].sup.pi]][||[f.sub.e]||.sub.pi,1] [less than or equal to] [[c.sub.[alpha]]/[[lambda].sup.pi]] [||f||.sub.pi,1].

To estimate II, we follow closely [8] and we sketch a proof for completeness. We decompose the set

{x [member of] R/[absolute value of x][[s.sub.x.sup.[alpha]] + 1]]([[f.sub.o](r)]/r)([absolute value of x])>[[lambda]/2]} = [[union].[k[member of]Z]] {x[member of]R/[absolute value of x][member of][I.sub.k],[absolute value of x][s.sub.*.sup.[alpha]] + 1]]([[f.sub.o](r)]/r)([absolute value of x]) > [[lambda]/2]},

where [I.sub.k] = [[2.sup.k], [2.sup.[k + 1]][.

Put g(r) := [[[f.sub.o](r)]/r] = [g.sub.k.sup.1](r) + [g.sub.k.sup.2](r), with [g.sub.k.sup.1] = [gXI.sub.k.sup.*], [g.sub.k.sup.2] = gX[[([I.sub.k.sup.*])].sup.c] where [I.sub.k] = [[2.sup.k],[2.sup.[k + 1]] [.

By (12), we have

[absolute value of x][s.sub.*.sup.[[alpha] + 1]]([g.sub.k.sup.1](r))([absolute value of x]) [less than or equal to] [[c.sub.[alpha]]/[[absolute value of x].sup.[[alpha] + 1/2]]]] K([g.sub.k.sup.1](r)[r.sup.[[alpha] + 3/2]])([absolute value of x]).

By ([8], p: 1021), we have for 1 < p < [infinity],

[summation over (k[member of]Z)] [[mu].sub.[alpha]] {x [member of] [R/[absolute value of x]] [member of] [I.sub.k],[1/[[absolute value of x].sup.[[alpha] + 1/2]]]]K([g.sub.k.sup.1](r)[r.sup.[[alpha] + 3/2]])([absolute value of x]) > [[lambda]/2]} [less than or equal to] [[c.sub.[alpha]]/[[lambda].sup.p]] [||[f.sub.o]||.sub.[[L.sup.p](R,d[[mu].sub.[alpha]](x))].sup.p] [less than or equal to] [[c.sub.[alpha]]/[[lambda].sup.p]][||f||.sub.[[L.sup.p](R,d[[mu].sub.[alpha]](x))].sup.p][less than or equal to][[c.sub.[alpha]]/[[lambda].sup.p]][||f||.sub.[p,1].sup.p].

On the other hand as in ([8], p: 1021), we have

[absolute value of x][s.sub.*.sup.[[alpha] + 1]]([g.sub.k.sup.2](r))([absolute value of x])[less than or equal to][[c.sub.[alpha]]/[|x[|.sup.[[alpha] + 1/2]]]][[integral].sub.0.sup.[infinity]][[[s.sup.[[alpha] + 3/2]]|[f.sub.o](s)|]/[s([absolute value of x] + s)]]ds[less than or equal to][[c.sub.[alpha]]/[|x[|.sup.[[alpha] + 3/2]]]][[integral].sub.0.sup.[infinity]]|[f.sub.o](s)|[s.sup.[[alpha] + 1/2]]ds[less than or equal to][[c.sub.[alpha]]/[|x[|.sup.[[alpha] + 3/2]]]][[integral].sub.R]|[f.sub.o](s)|[1/[|s|.sup.[[alpha] + 1/2]]]d[[mu].sub.[alpha]](s).

Remark that we have considered [f.sub.0] as a function defined on R. As the same we get,

[absolute value of x][s.sub.*.sup.[[alpha] + 1]] ([g.sub.k.sup.2](r))([absolute value of x]) [less than or equal to] [[c.sub.[alpha]]/[[absolute value of x].sup.[[alpha] + 1/2]]] [[integral].sub.0.sup.[infinity]]|[f.sub.o](s)|[s.sup.[[alpha] - 1/2]]ds [less than or equal to] [[c.sub.[alpha]]/[[absolute value of x].sup.[[alpha] + 1/2]]] [[integral].sub.R]|[f.sub.o](s)|[1/[|s|.sup.[[alpha] + 3/2]]]d[[mu].sub.[alpha]](s).

Using the following facts

[1/[[absolute value of x].sup.[[alpha] + [1/2]]]][member of][L.sup.p1,[infinity]](R,d[[mu].sub.[alpha]](x)), [1/[[absolute value of x].sup.[[alpha] + [3/2]]]][member of][L.sup.p0,[infinity]](R,d[[mu].sub.[alpha]](x)),

and Holder's inequality for the Lorentz spaces, we arrive to

[[mu].sub.[alpha]]{x[member of]R/[absolute value of x][[s*].sup.[[alpha] + 1]]([g.sub.k.sup.2](r))([absolute value of x])>[[lambda]/2]}[less than or equal to][[c.sub.[alpha]]/[[lambda].sup.pi]][[||f||.sub.o].sub.[pi,1].sup.[pi]] [less than or equal to] [[c.sub.[alpha]]/[[lambda].sup.pi]][||f||.sub.[pi,1].sup.[pi]],

which completes the proof.

Acknowledgment

We are grateful to Professor K. Stempak for sending us the preprint [6].

References

(1) L. Colzani, A. Crespi, G. Travaglini, and M. Vignati, Equiconvergence theorems for Fourier-Bessel expansions with applications to the harmonic analysis of radial functions in euclidean and noneuclidean spaces, Trans. Amer. Math. Soc., 338, N.1(1993), 43-55.

(2) C. F. Dunkl, Hankel transforms associated to finite reflexion groups, Contemp.Math., 138(1992), 123-138.

(3) M. F. F. de Jeu, The Dunkl transform, Inv. Math., 113(1993), 147-162.

(4) Y. Kanjin, Convergence and divergence almost everywhere of spherical means for radial functions, Proc. Amer. Math. Soc., 103, N.4(1988), 1063-1069.

(5) L. Kamoun, Besov-type spaces for the Dunkl operator on the real line, J. Comp. Appl. Math., 199(2007), 56-67.

(6) A. Nowak and K. Stempak, Relating transplantation and multipliers for Dunkl and Hankel transforms, to appear in Math. Nachr.

(7) E. Prestini, Almost everywhere convergence of the spherical partial sums for radial functions, Mh. Math., 105(1988), 207-216.

(8) E. Romera and F. Soria, Endpoint estimates for the maximal operator associated to spherical partial sums on radial functions, Proc. Amer. Math. Soc., 111, N.4(1991), 1015-1022.

J. El Kamely [dagger] and Ch. Yacoub [double dagger]

Department of Mathematics, Faculty of Sciences of Monastir, 5019 Monastir, TUNISIA

Received November 3, 2007, Accepted February 19, 2008.

* 2000 Mathematics Subject Classification. 42B15, 42C10, 42B10, 31B10, 31B20.

[dagger] E-mail: [e-mail]

[double dagger] E-mail: [e-mail]