COLLOQUIUM MATHEMATICUM

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1 COLLOQUIUM MATHEMATICUM VOL NO. TRANSFERENCE THEORY ON HARDY AND SOBOLEV SPACES BY MARIA J. CARRO AND JAVIER SORIA (BARCELONA) We show that the transference method of Cofman and Wess can be extended to Hardy and Sobolev spaces. As an applcaton we obtan the de Leeuw restrcton theorems for multplers.. Introducton. In 977, R. Cofman and G. Wess (see [CW]) proved the transference theorem n the settng of L p spaces for p. As a frst applcaton of ths result, they were able to show the classcal theorem of K. de Leeuw [D] on restrcton of multplers; namely, f m s a nce functon such that m M p (R N ), then ts restrcton (m(n)) n s n M p (Z N ), wth norm bounded by m Mp (R ), where for a general locally compact group G, we N say that m M p (G) f ts nverse Fourer transform K = m s a convoluton p operator on L (Ĝ), wth Ĝ the dual group of G. In ths case, the norm of ths convoluton operator s denoted by ether N p (K) or m Mp (G). Ths theory has been wdely extended by N. Asmar, E. Berkson and T. A. Gllespe n a collecton of papers (see [ABG] and [ABG]) where they carefully study transference for maxmal operators and transference of weak type nequaltes. On the other hand, L. Colzan (see [C]) proved, usng drect arguments, that f m s a multpler on H p (R N ) and m s a contnuous functon, then (m(n)) n s a multpler on H p (T N ), n the sense that the operator M (SP)(x) = m(n)a n e πnx n= M (wth P the trgonometrc polynomal P(x) = M n= M a ne πnx ) can be extended to a bounded operator on H p (T N ). We shall see that ths s a consequence of the fact that the transference method of Cofman and Wess can be appled to a more general class of spaces than L p, ncludng Hardy spaces and Sobolev spaces. 99 Mathematcs Subect Classfcaton: 4B30, 43A5. Ths work has been partally supported by the DGICYT: PB [47]

2 48 M. J. CARRO AND J. SORIA Ths paper s organzed as follows: In Secton, we gve the defnton of transferred space and gve several examples. Secton 3 contans the man result of ths paper for the case p and several applcatons. Secton 4 s devoted to the case 0 < p < and Secton 5 to the case of maxmal operators and maxmal spaces. Although the theory can be developed for amenable groups ([CW]), we shall restrct our attenton to locally compact abelan groups where our theory can go a lttle further and where all of our examples belong. As usual, f(u) = f(u ), (τ v f)(u) = f(uv ), and constants such as C may change from one occurrence to the next.. Transferred space. Let G be a locally compact abelan group and let L 0 (G) denote the set of all measurable functons on G. Consder a sublnear functonal S : A C, where A L 0 (G). Then, for 0 < p, we defne the space H p (S) as the completon of {f L (G) : S(τ. f) L p (G)} wth respect to the quas-norm f H p (S) = S(τ. f) L p (G). Consder now a σ-fnte measure space (M,dx) and let R be a representaton of G on L p (M) such that R s unformly bounded (see [CW]); that s, there exsts a constant A such that, for every f L p (M) and every u G, () R u f Lp (M) A f Lp (M). Defnton.. We defne the transferred space H p (S;R) of H p (S) by the representaton R as the completon of {f L (M) : S( R u f( )) L p (M)} wth respect to the quas-norm f Hp (S;R) = S( R u f( )) Lp (M). Before gong any further, we gve some nterestng examples of transferred spaces. Recall that the transferred operator T K s defned by (see [CW]) (T K f)(x) = G K(u)(R u f)(x)du. Examples.. () If S(f) = f(e), where e s the dentty element, then H p (S) = L p (G), and f R s any representaton of G actng on L p (M), then one can easly check that the transferred space s equal to L p (M). () Consder G = R, M = T, (R u f)(x) = f(x u) and S(f) = f(0) + (Hf)(0) where H s the Hlbert transform. Then H (S) = {f L (R) : Hf L (R)} = H (R),

3 TRANSFERENCE THEORY 49 f(x u) du N u /N u N π cot(πs)f(x s)ds = f(x) + (Cf)(x), N /N u and, followng the computatons n [CW], we fnd that S( R u f(x)) = f(x) + lm = f(x) + lm where Cf s the conugate functon of f. Therefore, H (S;R) = {f L (T) : Cf L (T)} = H (T). Smlarly, usng Myach s theorem (see [M]), we conclude that, for 0 < p, H p (S) = H p (R), and H p (S;R) = H p (T). (3) Consder G = R, M = T, (R u f)(x) = f(x u) and S(f) = f(0) + f (0). Then and H p (S) = {f L p (R) : f L p (R)} = W p, (R), H p (S;R) = {f L p (T) : f L p (T)} = W p, (T). That s, we get Sobolev spaces. Obvously, we can also obtan W p,k (R N ) and W p,k (T N ). (4) Consder G = Z, (R n f)(x) = f(t n x) wth T an ergodc transformaton and S((a n ) n ) = a 0 + n 0 a n/n. Then H (S) = H (Z) and H (S;R) turns out to be an ergodc Hardy space (see [CW] and [CT]) H (S;R) = { f L (M) : n n f(t n x) L (M) (5) If G = R, (R t f)(x) = w(t t x) w(x) f(t t x) wth T an ergodc transformaton on a measure space M and w a weght on M, then for S(f) = f(0) + (Hf)(0), the transferred space H (S;R) s the space of all functons F L (w) such that wf s n the ergodc Hardy space H ; ths space can be consdered as a weghted ergodc Hardy space. (6) Consder G = R N, M = T N, (R u f)(x) = f(x u) and Sf = sup t>0 ϕ t f(0), where ϕ S(R N ) andìϕ =. Then H p (S) = H p (R N ) and H p (S;R) = H p (T N ). (7) Let now G = R, M = R, the Bohr compactfcaton of R (see [HR]), and (R t f)(x) = f(x t). Then one can easly see that the transferred space of the Hardy space H (R) s the space of all functons n L (R) such that t R sgn(t) f(t)e tx s n L (R), whch s H (R). (8) Let G = R n, M = R m wth m < n and let R be the natural representaton defned by (R (x,...,x n )f)(y,...,y m ) = f(y x,...,y m x m ). }.

4 50 M. J. CARRO AND J. SORIA If T R n s the transferred operator of the Resz transform R n ( =,...,n) n R n, then T R n = 0 f = m +,...,n and T R n = R m f =,...,m. Therefore, the transferred space of H p (R n ) by ths representaton s H p (R m ) for every 0 < p. Many other examples can be gven n the settng of Trebel Lzorkn spaces, Besov spaces, etc. 3. Man results for p. Throughout ths secton we shall denote by K the convoluton operator wth kernel K, T K the transferred operator, H p (S) wll be denoted by H p (K) and the transferred space H p (S;R) by H p (T K ), whenever Sf = K f. Case of a fnte famly of kernels and p. Denote by H p ({K } =,...,n ) the completon of {f L (G) : K f L p (G), =,...,n} under the norm K f p, and smlarly for H p ({T K } =,...,n ). Theorem 3.. Let G be a locally compact abelan group and let p. Let K, {K } =,...,n and {K } =,...,m be a collecton of functons n L (G) and assume that K : H p ({K } =,...,n) H p ({K } =,...,m) has the property that there exst postve constants {A } such that m n K K f p A K f p. = Then the transferred operator s bounded, wth = T K : H p ({T K } ) H p ({T K } ) m n T K T K f p BA A T K f p, = = where A s as n () and B depends only on n and m. P r o o f. We prove ths for m =. The proof for m > s smlar. We frst recall that snce K L (G), t s known (see [CW]) that, for every v G and every f L p (M), () (R v T K f)(x) = G K(u)(R vu f)(x)du = (T K R v f)(x), a.e. x M. Also, usng the same dea, one can easly see that T K T K = T K K and therefore, we can assume wthout loss of generalty that H p (K ) = L p (G) and H p (T K ) = L p (M).

5 TRANSFERENCE THEORY 5 Now, snce K and K = K are n L (G) we can approxmate them by functons n L (G) wth compact support and hence standard arguments show that for every ε > 0 we can fnd functons K n, K,n n L (G) wth compact support such that n K n f p A K,n f p + ε f p. = Therefore, we can assume wthout loss of generalty that K and K are compactly supported functons n L (G). Let f L p (M). By (), we have T K f p = R v R v T K f p A R v T K f p. Now, as n [CW], we consder a compact set C such that the dentty element e s n C, suppk C and suppk C for every =,...,n. Also, take a neghborhood V of e such that (3) Now, by (), T K f p p µ(v C C) µ(v ) Ap µ(v ) V = Ap µ(v ) V M = Ap µ(v ) V M Ap µ(v ) M [ G Ap B µ(v ) M + R v T K f p p dv ε max(,(a f p K ) p ). GK(u)(R vu f)(x)du p dxdv GK(u)χ V C (vu )(R vu f)(x)du GK(u)χ V C (vu )(R vu f)(x)du A p χ V C (R. f)(x) p H p (K ) dx, p dxdv p ] where the last nequalty follows by applyng the hypothess to the functon h x (u) = χ V C (u)(r u f)(x). The last step s to show that, for every, (4) χ V C (R. f)(x) M p H p (K ) dx Ap µ(v ) f p H p (T K ) + εµ(v ), from whch we can easly deduce the theorem. To see ths, we observe that χ V C (R. f)(x) Hp (K ) = K h x Lp (G).

6 5 M. J. CARRO AND J. SORIA K h x Now, M p L p (G) dx K (u)χ V C (vu = M [ G G )(R vu f)(x)du p ] K (u)χ V C (vu = M [ V G )(R vu f)(x)du p ] [ K (u)χ V C (vu G )(R vu f)(x)du p ] + M = I + II, V C C\V where the last equalty follows snce V V C C. Let us frst estmate I: snce u C and v V, we have vu V C and therefore I K (u)(r vu f)(x)du = M [ V p ] G K (u)(r u R v f)(x)du = V [ M p ] dx dv G T K R v f V p p dv R v T K f = V p p dv A p µ(v ) T K f p p. To estmate II, we proceed as follows: [ GK (u)χ V C (vu )(R vu f)(x)du p ] II = M V C V \V K p M V C V \V G K p K (u) G [ V C V \V A p f p p K p µ(v C V \ V ). ] K (u) (R vu f)(x) p du R vu f p p dv du Therefore, χ V C (R. f)(x) M p H p (K ) dx A p µ(v ) T K f p p + Ap f p p K p µ(v C C \ V ). Now, snce, for every, µ(v C C \ V ) = µ(v C C) µ(v ) εµ(v ) (A f p K ) p,

7 TRANSFERENCE THEORY 53 we χ V C (R. f)(x) obtan M p H p (K ) dx Ap µ(v ) f p H p (T K ) + εµ(v ), as desred. Remark 3.. We observe that, as t happens n the transference theorem of [CW], the above theorem s not only a boundedness result, but the mportant thng s the norm of the transferred operator. Applcatons. We now apply the prevous results to the settng of Sobolev and Hardy spaces. A. Sobolev spaces. Let K be a functon n L (G) such that K : H p (K ) L p (G), wth norm N p (K), where K s not, n general, n L. Assume that there exsts an approxmaton of the dentty ϕ n such that ϕ n L (G) and K ϕ n s a functon n L (G). Then, f we apply the boundedness hypothess to the functon f ϕ n we get (K ϕ n ) f p N p (K) (K ϕ n ) f p, where the kernels K ϕ n and K ϕ n are functons n L (G) and hence we can transfer to deduce that T K ϕn : H p (T K ϕ n ) L p (M) s bounded wth norm less than or equal to A N p (K). The boundedness of T K from H p (T K ) nto L p (M) can be deduced, n the case of Sobolev spaces, by a lmt process, snce T K ϕ n f converges to T K f n the L p (M) norm, for every p. Theorem A.. Let p and r,s N. If m L loc s a normalzed functon such that K = m has the property that K : W p,r (R N ) W p,s (R N ) s a bounded operator wth norm N p (K), then the transferred operator T K : W p,r (T N ) W p,s (T N ) s gven by T K ( a n e πnx ) = n m(n)a ne πnx and s a bounded operator wth norm less than or equal to CN p (K), wth C only dependng on s and r. P r o o f. We prove ths n the case s = 0. The case s N s smlar. Take ϕ n (ξ) = n N ϕ(nξ) where ϕ D(R N ) s such thatìϕ = and ϕ 0. In ths case K = δ () 0 for r, and hence, K ϕ n = ϕ () n L (R N ). Therefore we get the result n the case K L (R N ). Now, for the general case we proceed as n Lemma 3.5 of [CW]. Snce m s normalzed and m L loc, we see that m n = (K ϕ n ) s also normalzed,

8 54 M. J. CARRO AND J. SORIA m n L and we can fnd a sequence (m k n) k such that m k n(ξ) m n (ξ) for every ξ R N and, f Kn k = m k n, then Kn k L, and N p (Kn) k N p (K). Also, m n (ξ) m(ξ) for every ξ R N. From ths, we deduce that T K = lm n,k T K k n and snce Kn k satsfes the rght hypothess we obtan the desred result. Smlarly, n the context of the Bohr compactfcaton R N of R N, we get the followng result: Theorem A.. Let p and r,s N. If m L loc s a normalzed functon such that for K = m the operator K : W p,r (R N ) W p,s (R N ) s bounded wth norm N p (K), then the transferred operator T K : W p,r (R N ) W p,s (R N ) s gven by T K ( a t e πtx ) = t m(t)a te πtx and s bounded wth norm less than or equal to C r,s N p (K). Theorem A.3. Let p, r,s N. If m L loc s a normalzed functon such that for K = m the operator K : W p,r (R N ) W p,s (R N ) s bounded wth norm N p (K), and K s a convoluton kernel on R M wth M < N and K(x) = m(x,0) where x = (x,x ) R M R N M, then the operator K : W p,r (R M ) W p,s (R M ) s bounded wth norm less than or equal to C r,s N p (K). P r o o f. Observe that W p,r (R M ) s the transferred space of W p,r (R N ) under the representaton of Example. (8), and argue as n Theorem A.. B. Hardy spaces (p = ). Now assume that K s a functon n L (R N ) such that (5) K : H (R N ) H (R N ) s bounded wth norm N (K). The prevous argument cannot be appled to ths case because we cannot fnd an approxmaton of the dentty ϕ n such that Hϕ s n L andìϕ =. However, we obtan the followng result (see [C]). Theorem B.. If K s such that K = m s a normalzed functon and K : H (R N ) H (R N ) s bounded wth norm N (K), then the transferred operator ( ) T K an e πnx = a n m(n)e πnx

9 TRANSFERENCE THEORY 55 can be extended to a bounded operator from H (T N ) nto H (T N ) wth norm less than or equal to N (K). P r o o f. Frst assume that K L and N = (a smlar proof works for N > ). Let P be a trgonometrc polynomal of degree such that P(0) = 0. Let φ H (R) be such that φ(n) = for every 0 < n. Then both K φ and Hφ are functons n L (R) and therefore T K φ P H (T) N (K)( T φ P + T Hφ P ). Snce T K φ = T K T φ, T φ P = P and T Hφ = T H T φ, we obtan the desred result. Fnally, every convoluton kernel on H (R) s also a convoluton kernel on L (R) and therefore m L (R). Moreover, m N (K). Hence, f a(x) =, then (T K )a(x) = m(0) and thus T K a H (T) = m(0) m N (K). To consder the general case K L, we need the followng techncal lemma. Lemma. If K s a convoluton operator on H (R N ) wth norm N (K), then there exsts a sequence (K n ) n of compactly supported functons n L (R N ) such that m n (ξ) = K n (ξ) m(ξ) for every ξ R N and N (K n ) N (K). P r o o f. We prove ths for N =. The general case s smlar. Frst, we know that m s a contnuous functon on R\{0}. Let ϕ S(R) wth compact support and ϕ(ξ) = for every ξ [,]. Defne ϕ k (x) = ϕ(x/k) and ϕ k (x) = ϕ(kx). Set m k (x) = m(x)ϕ k (x)( ϕ k (x)). Then m k (x) m(x) as k for every x 0, and m k s a multpler on H (R) wth norm less than or equal to CN (K), wth C only dependng on ϕ. Choose Ψ S(R) wth compact support such that Ψ(0) = and set Ψ n (ξ) = Ψ(ξ/n). Let φ n (ξ) = e πξ Ψ n (ξ), and consder ( x m n,k (x) = s φ x ) n m k (s) ds ( ) x dt t φ n (t)m k s t t. Then ( ( ) ) x dt m n,k (x) m k (x) = t φ n (t) m k m k (x) t t wth δ to be chosen. = δ + +δ δ s = +, +δ

10 56 M. J. CARRO AND J. SORIA Now, usng the decay of φ n we obtan δ ( ( ) ) x dt t φ n (t) m k m k (x) t t C m δ n M and n tends to nfnty. Smlarly forì +δ t ( δ, + δ), m k (x/t) m k (x) ε and hence +δ ( ( ) x t φ n (t) m k m k ) dt (x) t t Cε δ Now, snce K n,k (x) = m n,k (x) = φ n (sx)m k(s)ds, t M dt, and the above expresson converges to zero whenever M s large enough. For the second term we use the contnuty of m k to deduce that gven ε there exsts δ such that for every φ n (t) dt = Cε. φ n has compact support and m k (s) = 0 n a neghborhood of zero and for s large enough, we nfer that K n,k has compact support and obvously s n L (R). Fnally, K n,k f Rm n,k (ξ) = R f(ξ)e πxξ dξ dx t φ n (t)m k (ξ/t) dt ] f(ξ)e πxξ dξ dx = R R = R R R [ R t φ n (t) m k (ξ/t) R [ R f(ξ)e ] πxξ dξ dt dx φ n Rm k (ξ/t) (t) R f(ξ)e πxξ dξdxdt φ n Rm k (y)t (t) R f(ty)e πxty dy dxdt and hence, K n,k satsfes the lemma. N (K k ) f H (R) CN (K) f H (R), The proof of Theorem B. now follows by standard approxmaton arguments. Smlarly, we get Theorem B.. If m s a normalzed functon such that for K = m the operator K : H (R N ) H (R N )

11 TRANSFERENCE THEORY 57 s bounded wth norm N (K), then the operator T K : H (R N ) H (R N ) defned by T K ( a t e πtx ) = t m(t)a te πtx s bounded wth norm less than or equal to N (K). Theorem B.3. If m s a normalzed functon such that for K = m the operator K : H (R N ) H (R N ) s bounded wth norm N (K), and K s a convoluton kernel on R M wth M < N and K(x) = m(x,0) where x = (x,x ) R M R N M, then the operator K : H (R M ) H (R M ) s bounded wth norm less than or equal to N (K). If we want to use the technques of Theorem B. to cover the case of Example.(4), that s, to transfer the boundedness of a convoluton operator from H (R) to an ergodc Hardy space H (M), we observe that, n general, t s not the case that, for every f n a dense set of H (M), there exsts ϕ H (R) such that T ϕ f = f wth T ϕ the transference operator of the convoluton operator ϕ. Therefore, we can only show that, f N (K) s the norm of the convoluton operator K n H (R), then, for every ϕ H (R), T K ϕ f H (M) N (K)( T ϕ f L (M) + T ϕ T H f L (M)), where T H s the transference operator of the Hlbert transform. From ths, we can deduce that f m = K has compact support away from zero, then T K f H (M) N (K) ϕ ( f L (M) + T H f L (M)), snce, n ths case, there exsts ϕ H (R) such that K ϕ = K. 4. Case of a fnte famly of kernels and 0 < p <. Now consder a σ-fnte measure space (M,dx) and let R be a representaton of G on L p (M) and on L (M) such that R s unformly bounded; that s, there exst constants A and B such that, for every f L p (M) and every u G, and, for every f L (M), R u f Lp (M) A f Lp (M), R u f L (M) B f L (M). Under ths last condton, the transferred operator T K s well defned n a dense subset of the transferred space. We observe that n ths case the boundedness of T K s not trval even n the case of K L wth compact support snce the Mnkowsk ntegral nequalty does not hold.

12 58 M. J. CARRO AND J. SORIA Ths secton s organzed as follows: frst we prove the transference theorem f one of the followng condtons holds: (a) G s compact. (b) G s dscrete. (c) M s of fnte measure. Then, f G and M are ether R N, Z m or T k, we can transfer as n the followng dagram: Z m R N T k Z m and hence t remans to transfer from R N to Z m, or more generally from R N to any measure space M. The next step wll be to show that under some condtons on the representaton we can transfer from R N to any measure space M ether va the factorzaton R N T k M and/or usng the dlaton structure of the group R N. Theorem 4.. Let G be ether a compact or a dscrete abelan group, or let M be of fnte measure, and let 0 < p <. Let K, {K } =,...,n and {K } =,...,m be a collecton of functons n L (G) wth compact support and assume that K : H p ({K } =,...,n ) H p ({K } =,...,m ) has the property that there exst postve constants {A } such that m n K K f p A K f p. = Then the transferred operator s bounded, wth = T K : H p ({T K } ) H p ({T K } ) m n T K T K f p DA A T K f p, = = where A s as n () and D depends only on n and m. P r o o f. As n Theorem 3., we prove ths for m =. (a) Assume frst that G s compact. Then we proceed as n Theorem 3. but, n ths case, we can take V = G and then the term II s zero.

13 TRANSFERENCE THEORY 59 (b) If G s dscrete, and we argue as n Theorem 3., t remans to show that II/µ(V ) can be made small enough. Now, snce p <, [ II K (u)χ V C (vu = M V C V \V G )(R vu f)(x)du p ] [ ] K (u) M p (R vu f)(x) p dudx dv V C C\V G K (u) G p R vu f p p dv du V C C\V A p f p p K p µ(v C C \ V ), µ(c)/(/p) and hence we can choose V n such a way that II/µ(V ) s arbtrarly small. (c) If M s of fnte measure, and we assume that R acts on L (M), then II (m(m)µ(v C C \ V )) /(/p) [ M[ ] ] p K (u)χ V C (vu )(R vu f)(x) du V C C\V G ( m(m)µ(v C C \ V ) ) /(/p) µ(v C V \ V ) p K p R vu f p m(m)µ(v C C \ V ) K p Bp f p, and ths expresson converges to zero on choosng V approprately. Transference from R N to M. Let us now consder the case of transference from R N to a general measure space M. Let R be a representaton from R N nto L p (M). Assume that one of the followng two condtons hold: () For every f n a dense subset of H p ({T K } ), there exsts M > 0 such that R M f = f. Then, f we defne (Rθ Mf)(x) = (R Mθf)(x) for θ [ /,/] N = T N, we fnd that R M s a unformly bounded representaton of T N n L p (M). If (R M f)(x) = f(s M x), then M may also depend on f and x. () For every f n a dense subset of H p ({T K } ), there exst C > 0 and M 0 > 0 such that, for every M M 0, M ( M N (R u f)(x) du ( M,M) N ) p dx C. In the frst case, consder a kernel K L (R N ) and set K M (x) = M N K(x/M). Let K M (θ) = m Z N K M (θ + m)

14 = 60 M. J. CARRO AND J. SORIA be the perodc extenson. Then for the transferred operator we have (T K RM f)(x) K M M (θ)(r θ M f)(x)dθ T N M N K(Mθ + Mm)(R Mθ f)(x)dθ T N m = K(u + Mm)(R u f)(x)du [ M/,M/] N m = K(u)(R u f)(x)du [ M/,M/] N + K(u + Mm)(R u f)(x)du [ M/,M/] N m 0 = I M + II M. Now, snce the representaton R acts on L (M) we see that, by the Mnkowsk ntegral nequalty, II M A f K(u + Mm) du [ M/,M/] N m 0 = A f K(u) du, u M/ and therefore II M converges to zero as M tends to nfnty. Therefore, there exsts a subsequence M k such that II Mk converges to zero almost everywhere. Snce I M converges to the transferred operator TK R, we get (T R Kf)(x) = lm k (T RM k K Mk f)(x). From ths, we can deduce the followng result, Theorem 4.. Let G = R N and let M be a σ-fnte measure space. Let 0 < p <. Let K, {K } =,...,n and {K } =,...,m be a collecton of functons n L (G) wth compact support and assume that K : H p ({K } =,...,n ) H p ({K } =,...,m ) has the property that there exst postve constants {A } such that m n K K f p A K f p. = If the representaton R satsfes condton () then the transferred operator = T K : H p ({T K } ) H p ({T K } )

15 TRANSFERENCE THEORY 6 s bounded, wth m T K T K f p CA = n A T K f p, = where A s as n () and C depends only on n and m. P r o o f. As always, take m = and H p ({K } =,...,m) = L p. Usng the dlaton structure of R N we see that, for every M > 0, n K M f p A (K ) M f p. = Snce T N s a measure space of fnte measure, we can apply Theorem 4. to deduce that we can transfer the boundedness of K M to L p (T N ) va the natural representaton (S u f)(θ) = f(θ u). Hence, for every M, TK S M s a bounded operator wth TK S M f p CA A T (K ) M f p. But, snce K and K have compact support, for M large enough we have K M (x) = K M (x) for every x T N and smlarly for the kernels K. Now, snce T S KM = K M, we see that f we take M large enough such that ths condton holds and also that R M f = f, we get T K f p p = K(y)(R y f)( )dy p K M (y)(r p= yf)( )dy M p p R N T N = KM (y)(r y M f)( )dy p A p (K ) M(y)(R y M f)( )dy p p p T N T N = A p K (y)(r yf)( )dy p p. R N Theorem 4.3. Under the hypothess of Theorem 4., f the representaton R satsfes condton (), then the transferred operator s bounded, wth T K : H p ({T K } ) H p ({T K } ) m n T K T K f p CA A T K f p, = = where A s as n () and C depends only on n and m. P r o o f. We follow the same steps as for Theorem 3.. Let f L p (M). Take M large enough such that the supports of the functons K M and (K ) M = (K ) M are contaned n ( ε,ε) N for ε > 0 and

16 6 M. J. CARRO AND J. SORIA () holds. Then, f V = (,) N, we get T K f p p = K(y)(R y f)( )dy p = p R N Ap K M (y)(rv y M µ(v f)(x)dy p dxdv ) V M ( ε,ε) N = Ap µ(v ) V M Ap [ ( ε,ε) N K M (y)(r M yf)( )dy K M (y)χ ( ε,+ε) N(v y)(r M v yf)(x)dy p dxdv ( ε,ε) N µ(v ) M (,) N [ [ = M R N [ M (,) N + M V ε [ = I + II. Ap C µ(v ) M K M (y)χ ( ε,+ε) N(v y)(rv y M f)(x)dy p ] ( ε,ε) N A p χ ( ε,+ε) N(R Ṃ f)(x) p H p ((K ) M ) dx. Now, f we take h x (y) = χ ( ε,+ε) N(y)(Ry Mf)(x) and V ε = ( ε, + ε) N \ (,) N, we get (K ) M h x M p L p (R N ) dx ] (K ) M (y)χ ( ε,+ε) N(v y)(r M v yf)(x)dy p dv ( ε,ε) N (K ) M (y)(rv y M f)(x)dy p ] ( ε,ε) N (K ) M (y) (R M v yf)(x) dy p ] ( ε,ε) N To estmate I we proceed as n Theorem 3., and for the second term, II V ε p M (K V ε ] p ) M (y) (Rv y M f)(x) dy ( ε,ε) ( N ) p Cε N K p (Ry M M f)(x) dy dx (,) ( N ) p = Cε N K p M M N (R y f)(x) dy dx C(f) K p ( M,M) N Lettng ε tend to zero, we are done. p p dx εn.

17 TRANSFERENCE THEORY 63 For the examples, t wll be very convenent to get rd of the hypothess of K beng wth compact support. Because of the lack of the Mnkowsk ntegral nequalty, we cannot argue as n the case p. However, we are gong to show that whenever M s of fnte measure we do not need that condton on K. Then we shall prove that under certan condtons on the representaton we can restrct ourselves to ths case. Assume then that M s of fnte measure. Then, f K n s a sequence of functons n L (G) such that K n has compact support and K n converges to K n the L norm, we have Now, T K f p p T K Kn f p p + T Kn f p p D T K Kn f p + T K n f p p Dε f + T Kn f p p. T Kn f p p D µ(v ) V M K n (u)χ V C (vu G )(R vu f)(x)du p dxdv D G(K n K)(u)χ µ(v ) V C (vu [ M V )(R vu f)(x)du p K(u)χ V C (vu G )(R vu f)(x)du p ] + M V D ( M µ(v ) µ(v ) p K n K p (R u f)(x)du dx V C + D µ(v ) A p M G K (u)χ V C (vu G )(R vu f)(x)du p. Followng the deas n Theorem 3. and usng the fact that, by densty, we can consder f L (M), we get the result by lettng ε tend to zero. Defnton 4.4. We say that R acts locally on L p (M) f the followng condton holds: Gven a compact set C, and gven ε > 0, there exsts V such that µ(v C ) ( + ε)µ(v ) and, for every fnte famly {K } of kernels n L, there exsts a postve constant B such that, gven any measurable set M n M of fnte measure and gven any u G, there exsts a measurable set M u such that R u f Lp (M) B f Lp (M u ) for every f n a dense subset of H p (T K ) and, for every neghborhood V of the dentty there exsts another measurable set M V such that M v M V for every v V and M V p µ(v C C \ V ) µ(v ) ε. In ths case, we can reduce ourselves to the case of M of fnte measure ) p

18 64 M. J. CARRO AND J. SORIA and therefore we do not need the hypothess of K beng of compact support. To see ths we ust have to start computng T K f Lp (M) for any M of fnte measure. Then T K f p L p (M) Ap R v T K f p µ(v L ) V p (M v ) dv A p R v T K f p µ(v L ) V p (M V ) dv, and the rest of the proof follows as usual. One can easly check that f R s the representaton of Example.(8), then R acts locally on L p (R m ), and therefore, we can transfer from R N to R m (m < N) wth 0 < p <. C. Hardy spaces (p < ). Let K be such that m = K s a normalzed functon wth m(0) = 0. Assume that K f p C Hf p, for some p <. As n Theorem B., let P be a trgonometrc polynomal of degree such that P(0) = 0. Let φ H (R N ) be such that φ(n) = for every 0 < n. Take φ n convergng to φ n the L norm and such that Hφ n has compact support for every n. Then K φ n and Hφ n are functons n L (R N ) and the latter has compact support. Therefore T K φn P Hp (T N ) C T Hφn P p. But, snce m s normalzed, we have T K φn = T K T φn. Takng the lmt as n and usng the fact that T φ P = P, we get the followng result: Proposton C.. Let K be such that m = K s a normalzed functon wth m(0) = 0. If K f p C Hf p, then m(n)an e πnx C sgn(n)an e Lp(T) πnx. Lp(T) 5. Maxmal operators and maxmal spaces. In ths secton, we consder the case where the operator S s determned by an nfnte collecton of kernels K n L (G) wth compact support; namely Sf = sup K f(0). In ths case, we wrte H p (S) = H p ({K } ). Hence, f we have two collectons of functons satsfyng the above condtons, {K } and {K }, and K L (G) has the property that the convoluton operator K : H p ({K } ) H p ({K } )

19 TRANSFERENCE THEORY 65 s bounded wth norm less than or equal to N p (K), then the maxmal operator sup K : H p ({K } ) L p (G) s bounded wth norm less than or equal to N p (K). Therefore, we can reduce ourselves to the case of a maxmal operator actng on a maxmal space. Obvously ths maxmal space can be L p (G) and then our case wll nclude the maxmal transference of [ABG]. For that reason, throughout ths secton we consder only representatons such that () R u s separaton-preservng for every u G, () there exsts B such that R u f Hp ({T K } ) B f Hp ({T K } ) for every u G and every f L p (M), and () f 0 < p <, then the representaton R also acts nto L (M). Theorem 5.. Let G be a compact abelan group and let 0 < p <. Let K and K be two collectons of functons n L (G) and let N p (K) be the norm of the convoluton operator sup K : H p ({K } ) L p (G). If R s a representaton from G nto L p (M) satsfyng () (), then the transferred operator sup T K : H p ({T K } ) L p (M) s bounded, wth norm less than or equal to ABN p (K), where A s as n () and B as n (). P r o o f. Snce R s separaton-preservng, we get (see [ABG]) and therefore sup R v (sup (T K f)(x) ) sup (T K R v f)(x), T K f p p A p G sup T K R v f p p dv = A p G M sup K G (u)(r vu f)(x)du p dxdv A p M [ G sup G K (u)(r vu f)(x)du p dv ]dx (AN p (K)) p M (R. f)(x) p H p ({K } ) dx,

20 66 M. J. CARRO AND J. SORIA where the last nequalty follows by applyng the hypothess to the functon h x (u) = (R u f)(x). The last step s to show that (R. f)(x) M p H p ({K } ) dx Bp f p H p ({T K } ). Now, (R. f)(x) M p H p ({K } dx sup K ) = M [ G G (u)(r vu f)(x)du p ] sup K G (u)(r u R v f)(x)du p ] dx dv G [ M = G R v f p H p ({T K } ) dv Bp f p H p ({T K } ), where the last nequalty follows by (). If the group G s not compact, the proof s not so clear. Moreover, the natural extenson of Theorem 3. does not work n general snce condton (4) fals. However, we can formulate a qute general result that wll be useful for our purpose. Theorem 5.. Let G be a locally compact abelan group and let 0<p<. Let K and K be two collectons of compactly supported functons n L (G) and let N p (K) be the norm of the convoluton operator sup K : Hp ({K } ) L p (G). Let R be a representaton from G nto L p (M) satsfyng () (). Let f H p ({T K } ) satsfy the followng condton: there exsts B > 0 so that, for every compact E G large enough, there exsts ϕ E such that ϕ E (u) = for every u E, and (6) M ϕ E (R. f)(x) p H p ({K } ) dx Bp µ(e) f p H p ({T K } ). Then wth B as n (6) and A as n (). sup T K f p BA f H p ({T K } ), P r o o f. Frst, by Fatou s lemma, t s enough to estmate the norm sup =,...,N T K f p. Consder C such that suppk C for every =,...,N. Then we can adapt the proof of Theorem 3. qute easly to get

21 TRANSFERENCE THEORY 67 sup =,...,N Ap µ(v ) V T K f p sup =,...,N = Ap sup µ(v =,...,N ) V M = Ap sup µ(v =,...,N ) V M Ap µ(v ) M [ G sup T K R v f p H p ({T K } ) dv G K (u)(r vu f)(x)du p dxdv K G (u)ϕ V C (vu )(R vu f)(x)du p dxdv K G (u)ϕ V C (vu )(R vu f)(x)du p ] (AN p(k)) p µ(v ) M ϕ V C (R. f)(x) p H p ({K } ) dx, where the last nequalty follows by applyng the hypothess to the functon h x (u) = ϕ V C (u)(r u f)(x). The last step s to show that ϕ V C (R. f)(x) M p H p ({K }) dx Bp A p µ(v )( + ε) f p H p ({T K } ), but ths follows by (6) and the choce of V such that µ(v C )/µ(v ) + ε. As n Sectons and 4, f p or f M s of fnte measure (or t can be reduced to ths case) and p <, we do not need the hypothess on the support of K but we do need t for the support of K (see also [ABG]). D. Maxmal spaces and maxmal operators. We start wth the result of [C] we mentoned n the ntroducton. Theorem D.. Let 0 < p <. If K s such that K = m s a normalzed functon and the operator K : H p (R N ) H p (R N ) s bounded wth norm N p (K), then the operator T K ( an e πx) = a n m(n)e πx wth m = K can be extended to a bounded operator from H p (T N ) nto H p (T N ) wth norm less than or equal to N p (K). P r o o f. As for the case p =, frst assume that K L.

22 68 M. J. CARRO AND J. SORIA Snce K s a convoluton kernel n L, we see by nterpolaton that K s also a convoluton kernel n H. Now, snce we are transferrng to a measure space of fnte measure, we do not need any condton on the support of K but, f we want to apply Theorem 4., we need to have that restrcton on the kernels that defne the space H p (R N ). Snce these kernels do not have compact supports, we are forced to use Theorem 5.. Take a to be an atom n H p (T N ). Then ether a = or a s a (p,q)-atom. For the frst case, we proceed as n Theorem B., snce T K a Hp (T N ) = m(0) m N p (K), and for a general atom we observe that f T N = (,) N, then the functon χ ( M,M) Na s an atom n H p (R N ) and χ ( M,M) Na Hp (R N ) CM N a Hp (T N ) and therefore condton (6) holds. Hence T K a Hp (T N ) N p (K) a Hp (T N ). For the general case of a normalzed multpler we argue as n Theorem B.. Take a trgonometrc polynomal P n H p (T N ) wth P(0) = 0 and degree and let φ be as n B.. Then K φ s n L and snce T φ P = P and P satsfes condton (6), we can apply Theorem 5. to obtan the result. Smlarly, we can obtan the analogue to Theorem B.3 for 0 < p < observng that f a H p (R) s a (p, )-atom and for every compact E R we defne ϕ E (u,u ) = whenever (u,u ) E and ϕ E (u,u ) = 0 f (u,u ) Ẽ, where Ẽ = E + (0,n) s such that E Ẽ =, then ϕ E (u,u )a(x u ) s a (p, )-atom of H p (R ) satsfyng condton (6). A fnal applcaton we want to menton s the followng: Assume that we have two equvalent norms n a fxed space H p. Then we can try to transfer the dentty operator to obtan two equvalent norms n the transferred space. We llustrate ths stuaton wth the followng example. Let ϕ L wth compact support and consder the atomc and maxmal versons of the space H p (R N ) whch of course are equvalent. Then, f H p (R N ) denotes the atomc space, the operator sup ϕ t : H p (R N ) L p (R N ) t s bounded. Now, snce the atomc verson of H p (T N ) satsfes condton (6) we can transfer ths maxmal operator to obtan sup t ϕ t F Lp (T N ) C F Hp (T N ), whch s a well-known result.

23 TRANSFERENCE THEORY 69 REFERENCES [ABG] N.Asmar,E.BerksonandT.A.Gllespe,Transferenceofstrongtype maxmal nequaltes by separaton-preservng representatons, Amer. J. Math. 3(99), [ABG],,, Transference of weak type maxmal nequaltes by dstrbutonally bounded representatons, Quart. J. Math. Oxford 43(99), [CT] R.CaballeroandA.dela Torre,AnatomctheoryforergodcH p spaces, Studa Math. 8(985), [CW] R.CofmanandG.Wess,TransferenceMethodsnAnalyss,CBMSRegonalConf.Ser.nMath.3,Amer.Math.Soc.,977. [CW],,MaxmalfunctonsandH p spacesdefnedbyergodctransformatons, Proc. Nat. Acad. Sc. U.S.A. 70(973), [C] L.Colzan,FourertransformofdstrbutonsnHardyspaces,Boll.Un.Mat. Ital.A(6)(98), [D] K.de Leeuw,OnL pmultplers,ann.ofmath.8(965), [HR] E.HewttandK.A.Ross,AbstractHarmoncAnalyss,Vol.I,Sprnger,963. [M] A.Myach,OnsomeFourermultplersforH p (R n ),J.Fac.Sc.Unv.Tokyo 7(980), Departament de Matemàtca Aplcada Anàls Unverstat de Barcelona 0807 Barcelona, Span E-mal: carro@cerber.mat.ub.es sora@cerber.mat.ub.es Receved 9 Aprl 996; revsed7august996and7november996