Revista de la Uni6n Matematica ArgeBtina Volumen 38, 1993.

ABOUT THE LP-BOUNDEDNESS OF SOME INTEGRAL OPERATORS

1 2 T. Godoy - M. Urciuolo

192

ABSTRACT. In this note we prove the LP-boundedness, l<p<oo, and the weak type 1-1

of integral operators with kernels of the form n(x,y) j x-y j.a j x+y j'D+a, with n

in L oo(1R2D) and O<a<n.

1. INTRODUCTION.

In [R-S], authors show the boundedness in L2(1R) of the operator

Tf(x) = J jx_yj·a jx+yja.1 f(y) dy (1.1)

for 0<0;<1.

The fundamental tool for the proof of this result is the following

generalization of Schur's boundedness criterion.

LEMMA I. Let IQ() be a measurable function on R2n. Assume that there exist a

function g E LI (~) with g >0 a.e.such that loe

Tg(x) = J K(x,y) g(y) dy S c g(x) a.e.

and

T* g(x) = J K(y,x) g(y) dy S c g(x) a.e.

Then the operator T defined by the kernel K is bounded on L2(~).

1 Partially supported by CONICOR and SECYT (U.N.C.)

2 Partially supported by CONICOR and SECYT (U.N.C.)

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For the proof of lemma 1 see [B-H-S]. As this is an n-dimensional result, we can

expect that operators of the form

Tf(x) = J O(x,y)lx-yl-a Ix+yl-D+a f(y) dy (1.2)

with 0 in Loo(~2n) and O<a<l1, be bounded on L2(lRn). Indeed, we obtain the

boundedness of (1.2) on LP(lRn), 1 <p<oo.

2.THE MAIN RESULT.

THEOREM I. Let 0 be afunction of Loo(R2n). Then for O<a<n, the operator given by

(1.2) is bounded on LP(~), 1 <p<oo.

PROOF. I Tf(x) I :s; 1101100 J Ix-yl-a Ix+YI-n+a If(y) I dy. We will show that the

operator TI with kernel I x-y I-a I x+y r n+a is bounded on L2(1R0) and of weak type

1-1. The Marcinkiewicz interpolation theorem will then provide the boundedness

of TI on LP(iR") for 1 <p<2. As it is self adjoint, it is also bounded on LP(iR")

for 2<p<00, and the theorem follows. See ([S]) for details.

To study TI, we set k(x,y) = Ix-yl-a I x+yrn+a . This kernel satisfies the

hypothesis of lemma 1. We take g(x) = I x I-~. for some O<~n. It is enough to

obtain, for some constant c>O, J k(x,y) g(yj dy :s; c g(x), for almost every x.

~ x*Q J I x-y I-a I x+y I-o+a I y I-~ dy = J + J A + J A + J

. Al 2 3 ~ with Al = { y:ly-x l:S;l x ll2 }, A2 = { y:ly+xl:S;lxll2 },

A3 = { y:lyl:S;l x ll2} and A4 = (Alu A2u A/.

Now, if y belongs to AI' Iy+xl = I y+2x-x I ~ 2Ixl-ly-xl ~ Ixl and

Iyl ~ Ixl-Ix-YI ~ Ix ll2· Then

J A Ix-yl-a Ix+yl-O+a Iyl-~ dy:S; 2~lxl-o+a-~ J Jr-a+n-I dr dcr =

I 1: O<r<lxll2

=2~+a-n(n-arll1: II x I-~ = C I g(x).

Similarly, J I x-y I-a I x+y I-o+a I y I-~ dy :s; c2 g(x). A2

For yeA3, Ix-yl ~ Ixl-Iyl ~ Ixll2, and also, Ix+yl ~ Ix1/2. So

194

J -~+n-I dr r = c3 g(x).

0< r< I x 112

To estimate the fourth integral, we define B = { y: I y I <31 x I } and we observe

that for YEBc, Ix-YI ~ Iyl-lxl ~ 21y1/3; also Ix+yl ~ 21y1/3. So

I Ix-yl-(l Ix+yl-n+(l IYI-~dy = J + J ~ A An BAn BC

4 4 4

2n+~lxl-n-~ J. dy + (3/2)n J IYI-n-~dy = c' g(x). B BC 4

Then le~ma 1 implies the boundedness of T on L2(lRri). It remains to check the 1 1 2 3

weak type 1-1 of T1·We set EA = { x: I T/(x) I > A}. Then EA !;;; EA u EA u EA, where

= Ix I (l-n L J I x-y I -(l I f(y) I dy ~ j E N 2 - j -I I x I < I x-y I <2 - j I x I

I x I-n L 20+1)(X J I f(y) I dy ~ 2(lMf(x)

j E N I x _ y I <2 -j I x I

L ~«l-n)

JEN

where Mf denotes the Hardy Littlewood maximal function of f. As it is of weak

type 1-1, we get IE~I ~ alllfllllA. , for some positive constant aC

In a similar way we obtain, for XE~, ').)3 ~ 2n-(lMf( -x) L 2-j(l, and so

JEN IE~I ~ a2 IIfll llA., for another positive

If XE~, ').)3 ~ 2nlxl-nllflll and so xEB(O,2(3I1fIl 11A.)11n), then I~I ~ a3 IIfU/A,

for some a3>O.

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So T is of weak type 1-1 and the theorem follows. • 1

REFERENCES.

[B-H-S] Brown, A., Halmos, P.R., and Shields, A.L. : Cesaro

operators. Acta Szeged 26, 125-137, 1965.

[R-S] Ricci,F., Sjogren,P. : Two parameter maximal functions in

the Heisemberg group, Math. Z. vol 199, 4, 565-575,1988.

[S] Stein, E. Singular integrals and differentiability

properties of functions, Princeton Univ. Press Princeton,

1970.

Facultad de Matematica, AstrOllomia y Fisica

Universidad Nacional de Cordoba

Argentina

Recibido en noviembre de 1992.