Enji Sato

Department of Mathematical Sciences,Faculty of Science,Yamagata University, Yamagata,990-8560,Japan

Some Remarks on the Restriction Theorems for the Maximal Operators on Rd

Enji Sato?

Department of Mathematical Sciences,Faculty of Science,Yamagata University, Yamagata,990-8560,Japan

.The aim of this paper is to give a simple proof of the restriction theorem for the maximal operators on the d-dimensional Euclidean space Rd,whose theorem was proved by Carro-Rodriguez in 2012.Moreover,we shall give some remarks of the restriction theorem for the linear and the multilinear operators by Carro-Rodriguez and Rodriguez,too.

Weighted Lpspaces,Fourier multipliers,multilinear operators.

AMS Subject Classifications:42B15,42B35

1 Introduction and results

Let p be in 1≤p<∞,w(x)a nonnegative 2π periodic function in

where w(E)=REw(x)dx for E?Rdor E?Td.

Definition 1.2.For

where f is in Schwartz spaces S(Rd),and F in trigonometric polynomials P(Td)on Td,

Then,Rodoriguez[11]shows that if

for some constant C>0.

In this paper,we shall give simple proofs of Carro-Rodriguez[4,Theorem 1.3]and Rodriguez[15,Theorem 2.3]without their argument for the restriction theorems.

Our results are as follows:

2 The proofs of Propositions 1.1 and 1.2

First,we give the proof of Proposition 1.1,which is based on the idea in[13].It will be applied to the proof of Proposition 1.2,too.For the proof of Proposition 1.1,we have two lemmas.Proposition 1.1 will easily bring Theorem 1.1.So we omit the proof of Theorem 1.1.

Lemma 2.1(see[1]).Let

for a natural number J∈N.Then we have the following:

(2)

(3)There exists a constant C>0 such that

Proof.(1)This result is proved by[1,Lemma 2.2].(2)By(1),we have that

and

(3)Let G(x)be a function onTd.For t>0,we have

where j+1=(j1+1,···,jd+1).Since we have|s+2π(j+1)|≥|u+2πj|for s∈[0,2π)dand u∈[0,2π)d,we obtain that

for some constant C>0.Therefore,we have

So,we complete the proof.

and

for 1≤j≤J and F∈P(Td),where

Then it is easy to prove the following by the method of[13]and we omit the details.

Lemma 2.2.

Since

for t>0,we obtain that for a>2(2π)dΔJ,δ,

by

and the assumption of T?.After all,we have

for a>2(2π)dΔJ,δ.Hence,by(2),(3)of Lemma 2.1 and Fatou’s lemma

and by a↓0 we obtain

By applying Fatou’s lemma again,we get the desired result:

for 1≤p<∞and 0

if and only if there exists a constant C>0 independent of ε>0 such that

where φε(x)=φ(εx).

Proof.We assume‖Tφf‖Lp,q(Rd)≤C‖f‖Lp(Rd),(f∈S(Rd)).Since

we have that

where fε(x)=f(εx).Hence,by the assumption we obtain the desired result:

for all ε>0 by Proposition 1.1.Conversely,we assume that

Moreover,since we have

by limε→0X[?π,π)d(εx)=1,we have

Therefore,we obtain

So,we complete the proof.

Next,we shall give the proof of Proposition 1.2 in the same manner of the proof of Proposition 1.1.Rodriguez[15]proved Proposition 1.2,but we give an altenative proof of[15,Theorem 2.1]without using Kolmogorov’s condition.Proposition 1.2 will easily bring Theorem 1.2 whose proof is omitted.

and

for any natural number J∈N.Here,we define that

for δ1=δ2=δ/2 and δ>0,and□J,δ=max1≤j≤J‖□j,δ‖L∞(Rd)for 1≤j≤J.Then we have the following:

Lemma 2.3.

and

Also since

we obtain the following:

Here,for any ε0>0 we define I and II such that

and

Then we have

Bythe continuityof Φj(ξ,η),forany ε>0 thereexists ε0>0suchthat|Φj(m,n)?Φj(ξ,η)|< ε for|(ξ,η)?(m,n)|<ε0.Hence,we have

On the other hand,we have

Therefore,we get limsupδ→0□J,δ≤ε,and limδ→0□J,δ=0.

Now we proceed to the proof of Proposition 1.2 in the same way as Proposition 1.1. First we show the case 0

for F1,F2in P(Td),we have

Also by

by the assumption of T?.After all,we have

for a>2□J,δ.By(2),(3)of Lemma 2.1 and Fatou’s lemma,we have

Therefore,by a↓0 we obtain

Moreover,by applying Fatou’s lemma again,we get the desired result:

for 1≤p<∞and 0

In the same way,we can show it in the case of q=∞,and we omit the details.

if and only if there exists a constant C>0 independent of ε>0 such that

where φε(x1,···,xm)=φ(εx1,···,εxm).

we have that

where(fj)ε(x)=fj(εx),(j=1,2).Hence,by the assumption we get

Therefore,we obtain the desired result:

for all ε>0 by Theorem 1.2.Conversely,we assume that

Moreover,since we have

by limε→0X[?π,π)d(εx)=1,we have

Therefore,we obtain

3 Remarks of Theorems 1.1 and 1.2

In this section,we generalize Theorems 1.1 and 1.2 in Section 1,whose results are known in Carro-Rodriguez[4]and Rodriguez[15],but our proofs are easily rather than their proofs.

Definition 3.1.A bounded function φ defined on Rdis regulated if there exists a nonnegative function φ on Rdin S(Rd)with‖φ‖L1(Rd)=1 such that limn→∞φn?φ(m)=φ(m) for any m∈Zd,where φn(x)=ndφ(nx),(n=1,2,···).

Then,wecan showthefollowingwhoseproofisgiveninsimilar tothat of[8,Theorem 2.1].We omit the proof.

Lemma 3.1(cf.[12]).Let 1≤p<∞.Suppose k∈L1(Rd)and φj∈L∞(Rd),(j=1,2,···).Also let w be a 2π periodic weighted function.Then there exists a constant C>0 such that

where C is independent of k.

The following result is a generalization of Theorem 1.1,whose proof is different from that of Carro-Rodriguez[4].

Theorem 3.1.Let{φj}be regulated functions on Rd,and w a 2π periodic weighted function. Also let T?f(x)=supj|Tφjf(x)|.Then if there exists a constant C>0 such that

we have

Proof.Let kn,j=φn?φj.Then by Lemma 3.1,we have that for any J∈N

where C>0 is a constant independent of φ,J and f.Here,we remark kn,j∈Cb(Rd).By applying Theorem 1.1,we obtain that for any J∈N and a trigonometric polynomial F,

where C>0 is a constant independent of J and F.On the other hand,by the assumption of φjwe have

Then,by Fatou’s lemma,

for any trigonometric polynomial F,and where C>0 is a constant independent of J and F.By applying Fatou’s lemma again,we obtain that

where C>0 is a constant independent of F.Therefore,we get the desired result.

Next we give a remark about the case of bilinear Fourier multipliers in the same way of Theorem 3.1.The result is given by Rodriguez-Lopez[15],but our proof is slightly easy than that in[15].

Definition 3.2.A bounded function φ(ξ,η)defined on R2dis regulated if there exists a non-negative function φ on R2din S(R2d)with‖φ‖L1(R2d)=1 such that limn→∞φn?φ(m)= φ(m)for any m∈Z2d,where φn(x)=n2dφ(nx),(n=1,2,···).

Then,the following is proved by Rodriguez[15,Proposition 3.6].

Lemma 3.2(see[15]).Let φ∈L1(Rd)with{φj}?L∞(R2d),and p,pj,w,wj,(j=1,2)in Section 1.Also let(φ?φ)(ξ,η)=φ(ξ)φ(η).Then{(φ?φ)?φj}jsatisfies

where C>0 is a constant independent of φ and fj.

The following result is a generalization of Theorem 1.2,whose proof is different from that inRodriguez[4],and is slightly simple,because we useourTheorem1.2 forthe proof without applying[11,Theorem3.1].

we have that

where C>0 is a constant independent of φ,J and fj.Here,we remark(φn?φn)?Φj∈Cb(Rd).By applying Theorem 1.2,we obtain that for any J∈N and a trigonometric polynomial F1,F2,

whereC>0is aconstantindependentof φ,J and Fj.Ontheotherhand,bytheassumption of Φjwe have

Then,by Fatou’s lemma,

for any trigonometric polynomial F1,F2,where C>0 is a constant independent of φ,J and F1,F2.By applying Fatou’s lemma again,we obtain that

where C>0 is a constant independent of F1,F2.Therefore,we get the desired result.

Acknowledgments

The author was supported partly by Grant-in-Aid for Scientific Research(C).

[1]K.F.Anderson,P.Mohanty,Restriction andextension ofFourier multipliers between weighted Lpspaces on Rnand Tn,Proc.Amer.Math.Soc.,137(2009),1689–1697.

[2]N.Asmar,E.Berkson and J.Bourgain,Restriction from Rnto Znof weak type(1,1)multipliers,Studia Math.,108(1994),291–299.

[3]E.Berkson and T.A.Gillespie,On restrictions of multipliers in weighted settings,Indiana Univ.Math.J.,52(2003),927–961.

[4]M.J.Carro and S.Rodriguez-Lopez,On restriction of maximal multipliers in weighted settings,Trans.Amer.Math.Soc.,364(2012),2241–2266.

[5]K.de Leeuw,On LpMultipliers,Ann.Math.,81(1965),364–379.

[6]D.Fan,Multipliers on certain function spaces,Rend.Circ.Mat.Plermo,43(1994),449–463

[7]D.Fan and Z.J.Wang,Transference of maximal multipliers on Hardy spaces,Proc.Amer. Math.Soc.,123(1995),3169–3174.

[8]D.Fan and Z.J.Wang,Transference of multipliers on certain Hardy-type spaces,Integral Equations Operator Theory,27(1997),184–194.

[9]D.Fan and S.Sato,Transference on certain multilinear operators,J.Aust.Math.Soc.,70 (2001),37–55.

[10]J.Garcia-CuervaandJ.L.Rubio de Francia,Weighted Norm Inequalities andRelated Topics, North-Holland Mathematics Studies,Vol.116,North-Holland Publishing Co.,Amsterdam, 1985.

[11]S.Igari,Functions of Lp-multipliers,Tohoku Math.J.,21(1969),304–320.

[12]M.Kaneko and E.Sato,Notes on transferenceof continuity from maximal Fourier multiplier operators on Rnto those on Tn,Interdisciplinary Information Sciences,4(1998),97–107.

[13]Y.Kanjin,A.Kanno and E.Sato,A note on the restriction of Fourier multipliers from weighted Lpspaces to Lorentz spaces,Bull.Yamagata Univ.Nat Sci.,17(2013),17–26.

[14]C.E.Kenig and P.A.Tomas,Maximal operators defined by Fourier multipliers,Studia Math.,68(1980),79–83.

[15]S.Rodrigues-Lopez,Restriction results for multilinear multipliers on weighted settings, Proc.Roy.Soc.Edinburgh Sect.A,145(2015),391–405.

[16]E.M.Stein and G.Weiss,Introduction to Fourier Analysis on Euclidean Spaces,Princeton University Press,Princeton,NJ,1971.

[17]K.Wozniakowski,A new proof of the restriction theorem for weak type(1,1)multipliers on Rn,Illinois J.Math.,40(1996),479–483.

?Corresponding author.Email address:esato@sci.kj.yamagata-u.ac.jp(E.Sato)

http://www.global-sci.org/ata/123○c2015 Global-Science Press

Received 13 November 2014;Accepted(in revised version)24 March 2015