OFF-DIAGONAL MULTILINEAR INTERPOLATION BETWEEN ADJOINT OPERATORS

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1 OFF-DIAGONAL MULTILINEAR INTERPOLATION BETWEEN ADJOINT OPERATORS LOUKAS GRAFAKOS AND RICHARD G. LYNCH 2 Abstract. We exted a theorem by Grafakos ad Tao [5] o multiliear iterpolatio betwee adjoit operators to a off-diagoal situatio. We provide a applicatio.. Itroductio ad the Mai Result Multiliear iterpolatio has proved to be a powerful ad idispesable tool i aalysis. The two mai liear iterpolatio theorems, the Marcikiewicz ad Riesz-Thori theorems, have well-established multiliear aalogs. The works [0], [6], [3], [4] provide multiliear extesios of the Marcikiewicz iterpolatio theorem. The Riesz-Thori theorem is easily adapted i the multiliear case i [2, 2, Chapter XII, 3.3)] ad [, Theorem 4.4.2]; related versios of this result have appeared i [], [8], [9]. A differet type of iterpolatio is that betwee adjoit operators. I the liear case a typical result would be as follows: if a operator ad its adjoit are of weak type, ), the the operator is L p bouded for all p, ). A multiliear versio of this result was obtaied i [5]. This theorem says that, uder a iitial coditio similar to.) below, if a m-liear operator ad all of its m adjoits are of restricted weak type,,...,, /m), the the operator is bouded from L p L to L p for all < p,..., p m < ad /m < p <. I this cotext, a m-liear operator is called of restricted weak type p,..., p m, p) if it maps L p L to L p, whe restricted to characteristic fuctios of sets of fiite measure. The jth adjoit of a m-liear operator T defied o products of simple fuctios o measure spaces X j, µ j ), j {,..., m}, ad takig values i aother measure space X 0, µ 0 )) is aother operator T j such that T χ A,..., χ Aj, χ Aj, χ Aj+,..., χ Am )dµ 0 = T j χ A,..., χ Aj, χ A0, χ Aj+, χ Am )dµ j A 0 A j for all measurable subsets A i of X i with ozero fiite measure. Whe T 0 is writte, it is uderstood to be T itself. For 0 < q <, q deotes the umber q/q ) ad =. I this ote we obtai the followig off-diagoal versio of the mai result i [5] i which the diagoal case t = ad s = /m was cosidered. Theorem.. Let t <, 0 < s, < p < t, ad t < p,..., p m < be such that /p + +/p m /p = m/t /s. Let X 0, µ 0 ), X, µ ),..., X m, µ m ) be σ-fiite measure spaces. Suppose that a m-liear operator T is defied o the space of simple fuctios o X X m ad takes values i the space of measurable fuctios o X 0. We assume that T satisfies sup A 0,A,...,A m µ 0 A 0 ) µ A ) p µ m A m ) T χ A,..., χ Am ) dµ 0 <,.) A 0 The authors ackowledge the support of the Simos Foudatio ad of the NSF ATD: 32779) 2.

2 2 LOUKAS GRAFAKOS AND RICHARD G. LYNCH 2 where the supremum is take over all measurable subsets A i of X i with ozero fiite measure. Suppose that for each j {0,,..., m}, T j is of restricted weak type t, t,..., t, s) with costat B j. The there is a costat C = Cp,..., p m, p, t, s) such that T is of restricted weak type p,..., p m, p) with orm at most where = /t + /s ). CB ) ) t 0 B t p ) B t m..2) The followig well-kow characterizatio of weak L p will be used i the proof; see for istace Propositio i [2]. Propositio.2. Let 0 < p <, A, B > 0, ad let f be a measurable fuctio o a σ-fiite measure space X, µ). i) Suppose that f L p, A. The for every measurable set E of fiite measure there exists a measurable subset E of E with µe ) µe)/2 such that f is bouded o E ad f dµ 2 p A µe) p. E ii) Suppose that a measurable fuctio f o X has the property that for ay measurable subset E of X with µe) < there is a measurable subset E of E with µe ) µe)/2 such that f is itegrable o E ad f dµ B µe) p. E The we have that f L p, B 2 2 p The Proof of Theorem. Proof. First cosider the case where µ A ) max µ 0 A 0 ) B,..., µ ) ma m ). 2.) Bm Let M be the supremum give i.). It will be eough to show that M is bouded above by the costat i.2), from which Propositio.2ii) gives the desired boudedess. Sice T is restricted weak type t,..., t, s), Propositio.2i) gives a subset A 0 of A 0 with measure µ 0 A 0) µ 0 A 0 )/2 so that T χ A,..., χ Am )dµ 0 KB 0 µ A ) t µm A m ) t µ0 A 0 ) s A 0 for some costat K = Ks). It the follows that T χ A,..., χ Am )dµ 0 T χ A,..., χ Am )dµ 0 + T χ A,..., χ Am )dµ 0 A 0 A 0 \A 0 We have that A 0 := I + II. I KB 0 µ A ) t µm A m ) t µ0 A 0 ) s

3 OFF-DIAGONAL MULTILINEAR INTERPOLATION BETWEEN ADJOINT OPERATORS 3 i view of 2.) ad µ 0 A 0 ) µ A ) p µ m A m ) i view of.). Cosequetly, KB 0 B II Mµ A ) p µ m A m ) = µ 0 A 0 ) µ A ) p µ m A m ) M KB 0 B ) t p B m ad sice M < by assumptio.), we have t ) ) t p ) t p B m ) 2 µ 0A 0 ) ) M 2 ) t + M 2 K M B 2 0 B B t The precedig iequality is a implicatio of the fact that m t ) = p p m s ) = p s + p ) m. ) ) t ) = t ). There are m more cases left i each of which µ j A j )/Bj is iterchaged with µ 0 A 0 )/B0 i 2.) for some j {,..., m}. Fixig such a j we recall the assumptio that the jth adjoit T j of T is also of restricted weak type t,..., t, s). Settig p 0 =, we otice that.) ca be writte as sup A 0,A,...,A m µ j A j ) j ) µ 0 A 0 ) p 0 µ i A i ) p i i j T j χ A,..., χ A0,..., χ Am )dµ j A j <, i which the m+)-tuple p,..., p j, p 0, p j+, j) replaces p,..., p m, p), ad the idetity p 0 + i j p i ) = m replaces j t s p + + p m = m. The argumet i this case p t s follows by a idetical repetitio of the argumet i the precedig case, uder this chage of otatio ad cocludes the proof i all cases. Remark. Oe may woder whether hypothesis.) weakes the statemet of the mai theorem. As i [5], it is a essetial elemet of the proof, but i most applicatios it does ot preset ay sigificat restrictio. I fact, i most cases, oe may work with trucated versios of a operator T for which.) holds with costats depedig o the trucatio. The boudedess is obtaied for trucated operators with bouds idepedet of the trucatio ad a limitig argumet implies the same coclusio for the origial operator T. Remark 2. It is also worth otig that if.) holds for every poit p,..., p m ) with < p < t, t < p,..., p m < ad /p + + /p m /p = m/t /s, the oe obtais restricted weak type estimates at every poit i the ope covex hull H of these poits combied with the poit,...,, ). The by the multiliear Marcikiewicz iterpolatio theorem see t t s for istace [4]), it follows that T satisfies strog type bouds i H.

4 4 LOUKAS GRAFAKOS AND RICHARD G. LYNCH 2 3. A Applicatio Let 0 < α <. Cosider the biliear fractioal itegral I α f, g)x) = fx + y)gx y) y α dy, 3.) R defied for positive fuctios f, g o R. It was show i [3] ad [7] that I α maps the product L p R ) L p 2 R ) to L p R ) wheever p + p 2 = + α ad p p, p 2, ) lies i the ope covex p hull of the poits,, ),,, ),,, ),,, ), ad,, ). The proof α α α α 2 α is achieved i two steps: a) restricted weak type estimates are prove at the aforemetioed five poits first; b) the multiliear iterpolatio is used to obtai boudedess i the ope covex hull H of these five poits. I this ote we provide a simpler proof of the boudedess of I α i H by reducig it to a restricted weak type estimate at oly the poit,, ) for I 2 α α ad its two adjoits. We will use Theorem. with t = ad s = for which 2 α p + p 2 = 2. To satisfy p t s coditio.) we itroduce the followig trucated versio of I α : Iα ɛ,n,m f, g)x) = χ x M fx + y)gx y) y α dy. For < p < it is easy to see that I ɛ,n,m α χ A, χ A2 ) L p C ɛ,n,m mi, A ) p mi, A2 ) p mi, A, A 2 ) C ɛ,n,m A p A 2 p 2, ad from this.) follows for Iα ɛ,n,m via Hölder s iequality. Here = p ad 0 < p p + p 2 <. Aalogous estimates hold for the two adjoits of Iɛ,N,M p α. For istace ) I ɛ,n,m h, α g)x) = hx y)gx 2y) y α χ x y M dy, which is bouded by χ x M+N hx y)gx 2y) y α dy, whe g, h 0, ad thus a similar estimate holds for it. The Theorem. ad Remark 2 yield boudedess for Iα ɛ,n,m i H with bouds as i.2) i.e., idepedet of ɛ, N, M. Lettig ɛ 0 ad N, M we obtai the same coclusio for I α, via the Lebesgue mootoe theorem. Refereces [] J. Bergh ad J. Löfström, Iterpolatio Spaces, A Itroductio, Grudlehre der Mathematische Wisseschafte, No. 223, Spriger-Verlag, Berli New York, 976. [2] L. Grafakos, Moder Fourier Aalysis, Third editio, Graduate Texts i Math., o 250, Spriger, New York, 204. [3] L. Grafakos ad N. Kalto, Some remarks o multiliear maps ad iterpolatio, Math. A ), [4] L. Grafakos, L. Liu, S. Lu, ad F. Zhao, The multiliear Marcikiewicz iterpolatio theorem revisited: The behavior of the costat J. Fuct. Aal ), [5] L. Grafakos ad T. Tao, Multiliear iterpolatio betwee adjoit operators, J. Fuct. Aal ),

5 OFF-DIAGONAL MULTILINEAR INTERPOLATION BETWEEN ADJOINT OPERATORS 5 [6] S. Jaso, O iterpolatio of multiliear operators, Fuctio Spaces ad Applicatios Lud, 986), pp , Lect. Notes i Math. 302, Spriger, Berli New York, 988. [7] C. Keig ad E. M. Stei, Multiliear estimates ad fractioal itegratio, Math. Res. Lett ), 5. [8] R. Sharpley, Iterpolatio of pairs ad couterexamples employig idices, J. Approx. Theory 3 975), [9] R. Sharpley, Multiliear weak type iterpolatio of m-tuples with applicatios, Studia Math ), [0] R. Strichartz, A multiliear versio of the Marcikiewicz iterpolatio theorem, Proc. Amer. Math. Soc ), [] M. Zafra, A multiliear iterpolatio theorem, Studia Math ), [2] A. Zygmud, Trigoometric Series, Vol. II, 2d ed., Cambridge Uiversity Press, New York, 959. Departmet of Mathematics, Uiversity of Missouri, Columbia MO 652, USA address: grafakosl@missouri.edu, rglz82@mail.missouri.edu ad rilych37@gmail.com