Restricted weak type on maximal linear and multilinear integral maps.

Restricted weak type on maxima inear and mutiinear integra maps. Oscar Basco Abstract It is shown that mutiinear operators of the form T (f 1,..., f k )(x) = R K(x, y n 1,..., y k )f 1 (y 1 )...f k (y k )dy 1...dy k of restricted weak type (1,..., 1, q) are aways of weak type (1,..., 1, q) whenever the map x K x is a ocay integrabe L 1 (R n )-vaued function. AMS Subj. Cass: 42B35, 46B70, 42B99 Key words: restricted weak type estimated, mutiinear operators 1 Introduction and the main resut. Throughout the paper 0 < q, p 1,..., p k <,, k, n j and n = n 1 +... + n k. We write y = (y 1,..., y k ) R n = R n 1... R n k and mn (A) denotes the Lebesgue measure in R n. Given a Banach space X we write L 0 (R, X), L p (R, X) and L 1 oc (R, X) for the spaces of (strongy) measurabe functions on R with vaues in X, Bochner p-integrabe functions (0 < p < ) and ocay Bochner integrabe respectivey (we use the notation L 0 (R ), L p (R ) and L 1 oc (R ) if X = C). Let us reca that a mutiinear operator T : L p 1 (R n 1 )... L p k (R n k) L 0 (R ) is continuous if for every measurabe set E R of finite measure there exists a function C E : (0, ) R + with im λ C E (λ) = 0 such that m ({x E : T (f 1,..., f k )(x) > λ f i L p i(r n i) } C E (λ) Partiay supported by Proyecto BMF2002-04013 1

for f i L p i (R n i ), i = 1,..., k. Particuar exampes are the operators of weak type (p 1,..., p k, q), i.e. those for which there exists C > 0 such that m 1/q ({x R : T (f 1,..., f k )(x) > λ} C λ f i L p i(r n i). When the previous estimate hods ony for characteristic functions of measurabe sets, i.e. there exists C > 0 such that m 1/q ({x R : T (χ E1,..., χ Ek )(x) > λ} C λ m ni (E i ) 1/p i for measurabe sets E i in R n i, the operator is said to be of restricted weaktype (p 1,..., p k, q). Lots of exampes in Harmonic Anaysis turn out to be ony of weak type or restricted weak type (see [10]) for some tupes (p 1,..., p k, q). It is we known that interpoation techiques aow then to pass from restricted weak type in two different tupes to strong type estimates in intermediate spaces. In genera inear operators of restricted weak-type (p, q) need not be of weak-type (p, q) (see [9] for the case p > 1). It was first shown by K.H. Moon that convoution and maxima of convoution operators of restricted weak type (1, 1) are aways of weak-type (1, 1). Theorem 1.1 ([8]) Let K j L 1 (R n ) for j. Denote T j (f) = f K j and T (f) = sup j T j (f). If T is of restricted weak type (1, q) for some q > 0 then T is aso of weak type (1, q) with constant independent of the quantities K j 1. Recenty Moon s theorem has been extended to the mutiinear case by L. Grafakos and M. Mastyo. Theorem 1.2 ([5]) Let K j L 1 ((R ) k ) L ((R ) k ) for j. Define T j (f 1,..., f k )(x) = K j (x y 1,..., x y k )f 1 (y 1 )...f k (y k )dy 1...dy k (R ) k and T (f 1,..., f k )(x) = sup j T j (f 1,..., f k )(x) 2

for x R and f i L 1 (R ), i = 1,..., k. If T is of restricted weak type (1,..., 1, q) for some q > 0 then T is aso of weak type (1,...1, q) with constant independent of the quantities K j 1 and K j. Athough the proofs of the previous theorems work the same for a vaues of 0 < q <, I woud ike to point out that that ony the case q 1 is reevant. Proposition 1.3 Let T j : L 1 (R n 1 )... L 1 (R n k ) L 0 (R ) be a sequence of continuous mutiinear operators and set T (f 1,..., f k ) = sup j T j (f 1,..., f k ). If q > 1 and T is of restricted weak type (1,..., 1, q) then T is of weak type (1,..., 1, q). PROOF. It is known that weak L q (R n ) is a compete normed space for q > 1 (see [10]). Hence there exists a norm. L q, (R ) such that g L q, (R ) sup λ>0 λm 1/q ({x R : g(x) > λ}). Therefore, if f i = M i j=1 αi jχ E i j for pairwise disjoint measurabe sets E i j R n i, 1 i k, then λm 1/q C sup j C C ({x R : sup T j (f 1,..., f k )(x) > λ}) j k M i k M i j i =1 k M i j i =1 j i =1 T j (α 1 j 1 χ E 1 j1,..., α k j k χ E k jk ) L q, sup T j (αj 1 1 χ E 1 j1,..., αj k k χ E k jk ) L q, j αj 1 1 αj k k sup T j (χ E 1 j1,..., χ E k jk ) L q, j k M i C αj 1 1 αj k k m n1 (Ej 1 1 )...m nk (Ej k k ) j i =1 = C f i L 1 (R n i). 3

On the other hand, it was shown by M. Akcogu, J. Baxter, A. Beow and R.L. Jones that, in the inear case, if we repace R by Z the Moon s resut is not onger true. Theorem 1.4 ([1]) There exists a countabe set C of probabiity densities on Z such that M C f = sup g C g f for non-negative f 1 (Z) is of restricted weak type (1, 1) but not of weak type (1, 1) Making use of such a construction and the transference principe due to A. Caderón (see [2]), P.H. Hagestein and R.L. Jones have recenty shown the foowing: Theorem 1.5 ([7]) There exists a sequence of transation invariant operators T j acting on L 1 (T) such that T (f) = sup j T j (f) is of restricted weak type (1, 1) but it is not of weak type (1, 1). The operators in [7] are given by T j (f)(e iθ ) = k Z w j (k)f(e i(θ+k) ) for a sequence {w j } of probabiity measures on Z with finite support. In other words, T j (f)(e iθ ) = K j f(e iθ ) = K T j(e i(θ θ ) )f(e i(θ ) ) dθ, where K 2π j = k Z w j(k)δ k M(T). The aim of this paper is to exhibit a genera cass of the continuous mutiinear operators T j : L 1 (R n 1 ).. L 1 (R n k ) L 0 (R ) for which the restricted (1,..., 1, q)-weak type of T (f 1,..., f k ) = sup j T j (f 1,..., f k ) impies the (1,..., 1, q)-weak type of T. We sha restrict ourseves to the cass of operators T j given by T j (f 1,..., f k )(x) = K j (x, y 1,..., y k )f 1 (y 1 )...f k (y k )dy 1...dy k R n where K j : R R n 1... R n 1 C is measurabe. The reader is referred to [3] for some famiies of kernes where the restricted weak type in the inear situation aso impies better estimates. Let us start by mentioning some weak assumptions for the integra above to be we defined for amost a x R. Definition 1.6 Let T : L 1 (R n 1 ).. L 1 (R n k ) L 0 (R ) be continuous mutiinear operator. We sha say that T is an integra operator with kerne 4

K if there exists a measurabe function K : R R n 1... R n k C such that K 0 (x) = K x given by K x (y 1,..., y k ) = K(x, y 1,..., y k ) is strongy measurabe L 1 (R n )-vaued function, i.e. K 0 L 0 (R, L 1 (R n )), and T (f 1,..., f k )(x) = K(x, y 1,..., y k )f 1 (y 1 )...f k (y k )dy 1...dy k R n for amost a x R and f i L (R n i ) for i = 1,..., k. We sha write T = T K. Remark 1.1 If K : R R n 1... R n k C is measurabe and K 0 L p (R, L 1 (R n )) for some 1 p then it foows from Minkowski s inequaity that T K : L 1 (R n 1 )... L 1 (R n k ) L p (R ) is bounded and T K (f 1,..., f k ) L p (R ) K 0 L p (R,L 1 (R n )) ow we state the main resut of the paper: f i L 1 (R n i). Theorem 1.7 Let 0 < q 1 and et T j be a sequence of continuous mutiinear operators from L 1 (R n 1 )... L 1 (R n k ) L 0 (R ) with kernes K j such that K 0 j L 1 oc(r, L 1 (R n )). (1) Let T (f 1,..., f k )(x) = sup j R n K j (x, y 1,..., y k )f 1 (y 1 )...f k (y k )dy 1...dy k for f j L (R n j ), 1 j k. If T is of restricted weak type (1,..., 1, q) then T is of weak type (1,..., 1, q). Let us mention that our resut gives the foowing coroary (which seems to be new even in the inear case) when appied to a singe operator Coroary 1.8 Let 0 < q 1 and et T : L 1 (R n )... L 1 (R n ) L 0 (R n ) be mutiinear with kerne K such that K 0 C(R n, L 1 (R n )). Then T K (f 1,..., f k )(x) = K (R n ) k j (x, y 1,..., y k )f 1 (y 1 )...f k (y k )dy 1...dy k is of restricted weak type (1,..., 1, q) if and ony if it is of weak type (1,..., 1, q). Let us mention some particuar exampes where Coroary 1.8 or its maxima formuation can be appied: 5

Proposition 1.9 Let k 1, n 1 =... = n k = (hence n = k) and et φ L 1 (R n ) and Φ rea vaued function uniformy continuous on R R n. Define K : R R n 1... R n k C by K(x, y 1,..., y k ) = e iφ(x,y 1,...,y k ) φ(x y 1,..., x y k ). Then K 0 : R L 1 (R n ) is uniformy continuous and bounded. PROOF. Ceary K x L 1 (R n ) = φ L 1 (R n ) for a x R n. Given ɛ > 0 take δ > 0 so that e iφ(x,y 1,...,y k ) e iφ(x,y 1,..,y k ) < ɛ whenever x x + k y i y i < δ. Denoting τ xφ(y 1,..., y k ) = φ(x y 1,..., x y k ), if for x x < δ then K x K x 1 φ(x y 1,..., x y k ) φ(x y 1,..., x y k ) dy 1...dy k R + e iφ(x,y 1,...,y k ) e iφ(x,y 1,...,y k ) φ(x y 1,..., x y k ) dy 1...dy k R τ x x φ φ 1 + ɛ φ 1. ow use the fact that x τ xφ is uniformy continuous from R into L 1 (R n ) to finish the proof. In particuar one obtains Theorems 1.1 and 1.2 as particuar cases of Theorem 1.7. 2 Proof of the main theorem. Let us first estabish the approximation emmas to be used in the proof. Denote, as usua, ϕ t (u) = 1 ϕ( u) for u t n t Rn and t > 0. The proof of the foowing resut is the same as in the scaar-vaued case and it is eft to the reader (see [10]). Lemma 2.1 Let X be a Banach space and Φ L 1 (R, X). Let P t denote the Poisson kerne in R, that is P t (x) =. Then t (t 2 + x 2 ) +1 2 Φ t = P t Φ(x) = Φ(x u)p t (u)du C 0 (R, X) L 1 (R, X), R (2) sup Φ t L 1 (R,X) = Φ L 1 (R,X), (3) t>0 6

im t 0 Φ t Φ L 1 (R,X) = 0, (4) im Φ t (x) Φ(x) X = 0 for amost a x R. (5) t 0 Lemma 2.2 Let φ C 0 (R n ) L 1 (R n ), φ 0 and R n φ(y)dy = 1. Let K is a reativey compact set in L 1 (R n ). Then im sup φ t F F L 1 (R n ) = 0. (6) t 0 F K For each t > 0 the famiy {φ t F : F K} is equicontinuous, i.e. given ɛ > 0 there exists δ > 0 such that sup φ t F (y ) φ t F (y) < ɛ, y y < δ. (7) F K PROOF. It is known (see [4], Theorem 4.8.20) that a set K L 1 (R n ) is reativey compact if and ony if K is bounded, where τ y F (y ) = F (y y) and im sup τ y F F L y 0 1 (R n ) = 0, (8) F K im sup M F K y >M F (y ) dy = 0 (9) Using the standard approach one obtains the estimate φ t F (y ) F (y ) F (y y) F (y ) φ t (y)dy y <δ + F (y y) F (y ) φ t (y)dy y δ As usua, this eads to φ t F F L 1 (R n ) τ y F F L 1 (R n )φ t (y)dy + 2 F L 1 (R n ) φ t (y)dy y <δ y δ sup τ y F F L 1 (R n ) + 2 F L 1 (R n ) φ(y)dy. y <δ y δ t 7

Given ɛ > 0, using (8) there exists δ > 0 so that sup sup τ y F F L 1 (R n ) < ɛ/2. y <δ F F For such a δ one has sup φ t F F L 1 (R n ) ɛ/2 + 2 sup F L 1 (R n ) F F F F y δ t φ(y)dy. Taking imit as t 0 one gets (6). To obtain (7) use that F φ t F is continuous from L 1 (R n ) to C 0 (R n ). Hence {φ t F : F K} is reativey compact in C 0 (R n ). Proof of Theorem 1.7. Assume T is of restricted weak type (1,..., 1, q). Let, λ > 0 and et f i 0 be a non-negative simpe functions on R n i for 1 i k and denote f(y) = f 1 (y 1 )...f k (y k ). Let us show that there exists C > 0 (independent of ) m 1/q ({ x : sup T Kj (f 1,..., f k )(x) > λ}) C λ Let t > 0 and 1 j and et us use the notation f i L 1 (R n i) (10) K j, (x, y) = K j (x, y)χ { x } (x), K t,j, (x, y 1,..., y k ) = P t (x u)k j, (u, y 1,..., y k )du. R Consider the Banach space X = L 1 (R n, ) = {(g j ) j=1 : sup g j (y) dy < }, and Φ : R X given by R n Φ (x) = (K 0 j (x)χ { x } ) j=1. (11) From the assumption (1) one has K 0 j, L1 (R, L 1 (R n ))). Hence Φ L 1 (R, X ) and ( K 0 t,j, ) j=1 = P t Φ. 8

Taking into account (5) in Lemma 2.1 one obtains that there exists A R with m (A) = 0 and if x / A then im t 0 R n sup P t K j, (x, y 1,..., y k ) K j, (x, y 1,..., y k ) dy 1...dy k = 0. Therefore, if x / A then im sup t 0 ow, for any η > 0, T Kt,j, (f 1,..., f k )(x) = sup T Kj, (f 1,..., f k )(x) m ({ x : sup T Kj (f 1,..., f k )(x) > η}) = m ({x / A : sup T Kj, (f 1,..., f k )(x) > η}) im inf ({ x : sup T K1/, (f 1,..., f k )(x) > η}). M Let M be fixed. Using (2) in Lemma 2.1 one has that K 0 1/, : { x } L 1 (R n ) is continuous for a 1 j. Hence F i,j, = {( K 1/, ) x : x } is reativey compact in L 1 (R n ) for each 1 j. Seect, for instance, 1 φ(y) = in Lemma 2.2 and define H t (1+ y 2 ) (n+1)/2 (x, y) = φ t ( K 1/, ) x for 1 j. Let us denote T H t (f 1,..., f k )(x) = H(x, t y 1,..., y k )f 1 (y 1 )...f k (y k )dy 1...dy k. R n If 1 j and x then T K1/, (f 1,..., f k )(x) T H t (f 1,..., f k )(x) K 1/, (x, y) H(x, t y) f(y) dy R n f L (R n ) φ t ( K 1/, ) x ( K 1/, ) x L 1 (R n ) For a given ɛ > 0, from (6), there exists t = t(m) > 0 such that sup ( K 1/, ) x φ t ( K 1/, ) x L 1 (R n ) <, x 9 ɛ. f L (R n )

Therefore, for 1 j and x, T K1/, (f 1,..., f k )(x) T H t (f 1,..., f k )(x) < ɛ. (12) On the other hand, from (7) there exists δ > 0 such that sup H(x, t y) H(x, t y ) <, x ɛ, y y < nδ. (13) f L 1 (R n ) ow consider, for 1 i k, R n i = s I s (i) where I s (i) are disjoint cubes with ength side δ (in particuar, m ni (I s (i) ) = δ n i and diam(i s (i) ) < n i δ) and write f i = M i s for some α s (i) > 0. Denote α (i) = f i. Since α (i) s s=1 α(i) χ I (i) s α (i) and the Lebesgue measure is non-atomic we can then find J s (i) I s (i) such that α (i) m ni (J s (i) ) = α s (i) m ni (I s (i) ) = α s (i) δ n i. Hence, denoting E (i) = M i s=1j s (i) and E = E (1)... E (k), one gets f i 1 = α (i) m ni (E (i) ) and f 1 = α (1)...α (k) m n (E). Let us write I (j1,...,j k ) = I (1) j 1... I (k) j k and J (j1,...,j k ) = J (1) j 1... J (k) j k for 1 j M and 1 k. One has = T H t (f 1,..., f k )(x) T H t (α (1) χ E (1),..., α (k) χ E (k))(x) = M 1 j 1 =1... M k j k =1 ( α (1) j 1...α (k) j k T H t (χ I (1) j 1,..., χ I (k) j k )(x) ) α (1)...α (k) T H t (χ (1) J,..., χ (k) j J )(x) 1 j k M 1 M k ( =... α (1) j 1...α (k) j k H(x, t y)dy j 1 =1 j k =1 α (1)...α (k) J (j1,...,j k ) I (j1,...,j k ) ) H(x, t y)dy ow, denoting α (j1,...,j k ) = α (1) j 1...α (k) j k and α = α (1)...α (k) one has that 10

α (j1,...,j k )δ n = α (j1,...,j k )m n (I (j1,...,j k )) = αm n (J (j1,...,j k )). Therefore = = T H t (f 1,..., f k )(x) T H t (α (1) χ E (1),..., α (k) χ E (k))(x) = M 1 M k ( 1... α (j1,...,j k )m n (I (j1,...,j k )) H m j 1 =1 j k =1 n (I (x, t y)dy (j1,...,j k )) I (j1,...,j k ) 1 ) H m n (J (x, t y)dy (j1,...,j k )) M 1 j 1 =1... M k j k =1 J (j1,...,j k ) α (j1,...,j k )m n (I (j1,...,j k )) ( 1 m n (I (j1,...,j k ))m n (J (j1,...,j k )) I (j1,...,j k ) J (j1,...,j k ) (H t (x, y)dy H t (x, y )dydy ) ow observe that y I (j1,...,j k ) and y J (j1,...,j k ) then y y < nδ. Hence (13) shows that, for 1 j and x, = T H t (f 1,..., f k )(x) αt H t (χ E (1),..., χ E (k))(x) M 1 j 1 =1... M k j k =1 α (j1,...,j k )m n (I (j1,...,j k )) ( 1 m n (I (j1,...,j k ))m n (J (j1,...,j k )) ɛ ( M i ) α (i) j f i m(i (i) j i ) 1 ɛ f 1 j i =1 f i 1 = ɛ. I (j1,...,j k ) J (j1,...,j k ) Therefore, using (12) and the previous estimate one gets m ({ x : sup T K1/, (f 1,..., f k )(x) > λ + 3ɛ} m ({ x : sup T H t (f 1,..., f k )(x) > λ + 2ɛ}) H t (x, y) H t (x, y ) dydy ) m ({ x : sup T H t (α (1) χ E (1),..., α (k) χ E (k))(x) > λ + ɛ} m ({ x : sup T K1/, (χ E (1),..., χ E (k))(x) > λ α }. 11

From the restricted weak type assumption we concude that m ({ x : sup T K1/, (χ E (1),..., χ E (k))(x) > λ α } m ({ x : sup T Kj (χ E (1),..., χ E (k))(x) > λ 2α } + m ({ x : P 1/M Φ (x) Φ (x) X > λ 2α } m ({ x : sup T Kj (χ E (1),..., χ E (k))(x) > λ 2α } + 2α λ P 1/M Φ Φ L 1 (R,X ) C αq λ q mq n(e) + 2α λ P 1/M Φ Φ L 1 (R,X ) = C f q L 1 (R n ) λ q + 2α λ P 1/M Φ Φ L 1 (R,X ). Taking im inf M and combining a the previous estimates one gets m 1/q ({ x : sup T Kj (f 1,..., f k )(x) > λ + 3ɛ} C f L 1 (R n ). λ q Finay, since ɛ > 0 is arbitrary one gets (10). Using that { x : sup T Kj (f 1,..., f k )(x) > λ} is an increasing sequence, one concudes k m 1/q ({x R : T (f 1,..., f k )(x) > λ} C f i L 1 (R n i) λ (14) for a simpe functions f i 0, 1 i k. Let us now extend (14) for integrabe functions f i. Let f i 0 be an arbitrary integrabe function in L 1 (R n i ) with f i 1 = 1 for i = 1,.., k. For each, j denote C j, (λ) = sup m ({ x : T Kj (g 1,..., g k )(x) > λ}. g i L 1 (R n i ) =1 Given ɛ > 0 there exists λ 0 > 0 such that C j, (η) < 1 j. ɛ k for η > λ 0 and 12

f (i) On the other hand, for each 1 i k, we can find a simpe (i) 0 such that f f i and f i f (i) L 1 (R n i) < ɛ λ 0. Denote and, for 2 i, B (i) B (1) j () = {x R : T Kj (f 1 f (1), f 2,..., f k ) > ɛ}, j () = {x R : T Kj (f (1) (i 1),...f, f i f (i), f i+1,..., f k ) > ɛ}. Set B (i) () = j=1b (i) j () and B() = k B (i) (). ote that m (B()) Since j=1 j=1 k m (B (i) j ()) k i 1 C j, (ɛ/ f (j) L 1 (R nj ) f i f (i) L 1 (R n i)) < ɛ. j=1 T Kj (f 1,..., f k ) = T Kj (f 1 f (1), f 2,..., f k ) k 1 + T Kj (f (1) (i 1),..., f, f i f (i), f i+1,..., f k ) i=2 + T Kj (f (1) then, for each x / B(), one has Therefore + T Kj (f (1) (k),..., f ), sup T Kj (f 1,..., f k )(x) sup (k 1),..., f, f k f (k) ) T (f (1) T Kj (f (1) (k),..., f )(x) + kɛ (k),..., f )(x) + kɛ. 13

m ({x R : sup T Kj (f 1,..., f k )(x) > λ + kɛ}) m ({x / B() : sup T Kj (f 1,..., f k )(x) > λ} + m (B()) m ({x / B() : T (f (1) (k),..., f )(x) > λ} + ɛ ( C f (i) λ L 1 (R n i)) q + ɛ Cq λ (1 + ɛ ) qk + ɛ. q λ 0 Finay using the fact sup T Kj (f 1,..., f k ) sup +1 T Kj (f 1,..., f k ) and mutiinearity we concude that m 1/q ({x R : T (f 1,..., f k )(x) > λ}) C λ f i L 1 (R n i) for non negative integrabe functions f i. The case of compex-vaued functions in now immediate using the mutiinearity of the operators. References [1] M. Akcogu, J Baxter, A. Beow, R.L. Jones On restricted weak type (1,1): the discrete case Israe J. Math. 124 (2001), 285-297. [2] A. P. Caderón Ergodic theory and transation invariant operators Proc. at. Acad. Sci. U.S.A. 59 (1968), 349-353. [3] M. J. Carro From restricted weak type to strong type estimates. J. London Math. Soc. (2) 70 (2004), 750-762. [4]. Dunford, J.T. Schwartz Linear operators, Part I, John Wiey and sons. ew York, 1958. [5] L. Grafakos, M. Mastyo, Restricted weak type versus weak type, Proc. Amer. Math. Soc. 133(4) (2005), 185-190. 14

[6] L. Grafakos, R. Torres, Mutiinear Carderón-Zygmund theory, Adv. in Math. 165 (2002), 124-164. [7] P.A. Hagestein, R.L. JonesOn restricted weak type (1,1):The continuous case, Proc. Amer. Math. Soc. 133(1) (2005), 148-152. [8] H.K. Moon, On restricted weak type (1,1), Proc. Amer. Math. Soc. 42 (1974), 148-152. [9] E.M. Stein, G. Weiss An extension of a theorem of Marcinkiewicz and some of its appications, J. Math. Mech. 8 (1959), 263-284. [10] E.M. Stein, G. Weiss Introduction to Fourier Anaysis on Eucidean Spaces, Princeton Univ. Press, Princeton, J, 1971. [11] A. Zygmund, Trigonometric series, Cambrigde Univ. Press, ew York, 1959. Departamento de Anáisis Matemático Universidad de Vaencia 46100 Burjassot Vaencia Spain obasco@uv.es 15