FOURIER TRANSFORMS OF LIPSCHITZ FUNCTIONS ON THE HYPERBOLIC PLANE H =

The purpose of the present work is to study the order of magnitude of the Fourier transforms f(A) for larse A of complex-valued functions ,f(z) sating certain Lipschitz conditions in


INTRODUCTION
The relation between smoothness conditions imposed on functions f(z) and the behavior of its Fourier transforms f near infinity is well known in the literature.
In fact, the Fourier transforms of Lipschitz functions defined on various domains have been extensively studied over the last decades.The purpose of the present research is to trace the behavior of the Fourier transforms of complex-valued functions satisfying Lipschim conditions in the hyperbolic plane H9-.This will pave the ground for handling the transforms of Lipschitz functions defined on other domains such as SL(2.R) and SL(2.C) in particular.

DEFINITIONS AND NOTATIONS
Our main reference on the Fourier analysis on H 2 is the book by Helgason ([2], p. 29 infra); reference [7] will be consulted (especially Chapter 10) as well.In the beginning we would like to make clear an idea which will be frequently encoumered in the sequel.Let D be the open disc Izl < 1 in the plane R2.Then a horocycle in D is a circle tangential to the boundary B OD at a certain point b.This horocycle is denoted by .I f z is a point on , the distance d(0, z) from the origin 0 of D to z is denoted by (z, b).Let d(0, z) r.Then r d(0, z) log 1 -iz----This gives z[ tanh r.We indicate here that tanh r is of the order of r near r 0 and that it approaches 1 as r goes to infinity.These two limits will occur frequently in due course.In this section we introduce the basic definitions necessary for the proof of the main theorems.Other definitions will be given when they are needed.Thus we start with DEFINITION 2.1.Let f(z) be a complex-valued function defined on the unit disc D when endowed with the Pdemannian structure.Then f(z) is said to belong to the Lipschitz class Lip(c, 2) if II.f(z + h) .f(z)llO(Ihl) (2) as h 0, 0 < a < 1.Where I[-[[2 is the usual L norm.The generalization ofthis definition for higher differences Ahlf(z of order k in step h takes the form ll/khl](z)ll2 O(lh[() 0 < a _< k + 1.

MAIN THEOREMS
Our main result may be viewed as the non-Euclidean analogue ofthe following theorem.
as X oo are equivalent.The main theorem ofthis section is stated as THEOREM 3.2.Let f(z) be a complex-valued function on the unit disc D. Then the conditions aslh --,0, O< a < land as ov are equivalent.
PROOF.By definition of f (A, b) it is easily seen that the transform of (f ) r and hence r --0 with [hi where Ihl tanhr.So, the factor e {h,b) e tends to one as r goes to zero and therefore can be suppressed without harm.Also, since E R + in the right side of the Parseval's identity ( [7], p. 376), (10) could be simplified so that the transform of If(z / h) .f(z)l is given as 12 sin ]'1.Thus 2sin--f tanh dAdb O(tanhr) 2a O(r2a).
Observe that tanh(-) tends to be a bounded constant for large A. This yields (as in the proof of It is clear that the integral db I when the normalized Haar measure is in action.The last quantity is bounded as , -oo if 1-a/-/ < 0, so that </3 <_ 2, giving the condition a > 0 for/9 1. This shows that the Fourier transform f(,, b) of f(z) converges absolutely for any c, greater than zero.
This reflects the strength of the conclusion in Theorem 3.2 in contrast to those of similar theorems proved for Lipschitz functions in R, R and T for example.This is mainly due to the presence of the weight A in the inversion formula as well as in one side of Parseval's identity.Had it not been for this extra factor in the present situation, the conclusion of Theorem 3.2 would have been exactly the same as that of Theorem 3.1 as far as the order of magnitude and the absolute convergence of f are concerned.We shall encounter the same situation when dealing with the spherical Furier transforms of Lipschitz functions in H 2 which is the subject matter ofthe next section.

SPHERICAL TRANSFORbIS IN H 2
In this section we prove an analogue of Theorem 3.2 for the spherical Fourier transforms of Lipschitz functions.These are related to spherical functions which (by their very nature) are radial eigenfunctions ofthe Laplacian on the unit disc (see [2], pp. 38, 39for details).They are defined by x(z) f el-'x+l><z'b>db.
(13) If f(z) is a radial complex-valued function in D, then its spherical-Fourier transform is given by JD (14) with this in hand, we state the following THEOREM 4.1.Let f(z) be radial in D. Then the conclusion of Theorem 3.2 holds with the integral as hl " 0 and the proof is complete.REMARK 3.3.We indicate here that if the k-th difference Ahlf(z is employed in Definition 2.2 of the Lipschitz condition, then this would result in the appearance ofthe factor [rAI 2k in place of [rAI in the estimates given by ( 10), ( 11), but this will not affect the final conclusion of Theorem 3.2.This is due to the way in which r and are tied up in their variation.REMARK 3.4.Applying HOlder's inequality to (12) for/ff < 2 we get as A tends to infinity, this proves the first part ofthe theorem.
On the other hand, given (12), then by following the reverse argument as in the proof of Theorem 3.1 one can arrive easily at the estimate PROOF.The proof will carried briefly; we mainly point out the necessary modifications which will suit the present situation.One essential aspect here is the presence of the Harish-Chandra c-function both in the inversion formula as well as in the Parseval's identity.Thus one has c(a)=r (15) The transform of f(z + h) is fo.f(z + h)_x(z)dz fo f(w) [f e(-iX+l){u'-h'b>db] dw e(-iA+l)(h.b)T()" Tng imo aoum e or of e (h,b) e r tends to ro d tt IC()l -= o() for l$e A, e prf of eorem 3.2 could be appfi (mo word for word) to @eld the r reslt of e theory.
We retook at s pot tt the deee of sim bn the 1 o r is not t mdsg ew of the f t e te over the d B of e t di does not play y sifit role pnen to the ord of mde d in the ablute nvergen of bo e FoYer d the sph trsfo of ons in H2.Ts imeg cod y suppress out dge to the cose of the proof.Thus as f as e probl of e order of mde d e solute nvg e nm e or of f(A, b) d f(A) is ost the e 4.2.We would fike to t out t e prous ysis appfies ost exay to the sphec sfo on the Lobaschows space sociat th e complex goup SL(2.C) (s [7], pp. 40002).r trident he 11 be ve brief.
In ts ce the sph trsfo is de to be I () As h 0 sinh(x h) O(sinhx) and the transform of f(t + h) f(t) is equal to (e -'vh 1)]'Iv as usual and this leads to the final estimate L ITl,2d, o(,-,) or equivalently as , goes to infinity.The other part of the proof is very clear by now.Moreover the range of for which f [f[d, is bounded is better than that found in the case the spherical transforms for functions on H2.In this case HOlder's inequality for/ _< 2 leads to which in turn gives 2 <_<2 for the boundedness of the last quantity as v oo.
. CONCLUDING REMARKS It would be convenient to end with a few comments-of a rather heuristic nature-which might cast some fight on the treatment of the problem on other domains.Our first target in this respect is the group SL(2.R) denoted here by G for brevity.Thus for a function f(g)g E G a small translation suitable for the Lipschitz conditions in this case is given by f(gh),h G, Ih O. Thus f belongs to Lip(a, 2) on G if Ilf(gh) f(g)ll2 O(Ihl) 0 < a < 1, ]hi --, 0. With this in hand, one can apply Parseval's identity (see [4], p. 346, [6], Vol. 2, p. 53) in order to obtain the required estimates for the Fourier (spherical) transforms f corresponding to the principal and discrete series representations of G.The occurrence of the two integrals and the one summation on the right side of the identity causes no problem because on the one hand coth A tends to one for large A, thus the two integrals are treated in the same manner.On the other hand the three parts are majorized by the same quantity O(Ihl); this enables us to deal with each part on its own.This would be easier; besides it will not affect the final conclusions.
Secondly, we hint that the previous analysis is amenable to treatment of the problem on other classical groups and their allied symmetric and homogeneous spaces (see [1], Chapter X for examples of those groups).It goes without saying that there would be some modifications in the main course of the proof when handling concrete situations such as SL(2,C) for example.Thirdly, one could explore the validity of the present line of thoughts on semi-simple Lie groups in general.A clue to investigating this problem lies in an estimate for the Harish-Chandra c-function near infinity in that case (see [2], p. 450 and [3], p. 183).These points will be taken up in a forthcoming paper.
B for which this integral exists.The Parseval's identity in this case is written as Theorem 3.1) Z-Z, .l,aeaeo(-o) o(-o), s idemity reads in this case If(z)ldz I]'1 IC(A) dA.