FOURIER MULTIPLIERS ON WEIGHTED Lp SPACES SEBASTIAN KROL´ Abstract. The paper provides a complement to the classical results on Fourier multipliers on Lp spaces. In particular, we prove that if q ∈ (1, 2) and a func- tion m : R → C is of bounded q-variation uniformly on the dyadic intervals p in R, i.e. m ∈ Vq (D), then m is a Fourier multiplier on L (R,wdx) for every p ≥ q and every weight w satisfying Muckenhoupt’s Ap/q-condition. We also obtain a higher dimensional counterpart of this result as well as of a result by E. Berkson and T.A. Gillespie including the case of the Vq(D) spaces with q > 2. New weighted estimates for modified Littlewood-Paley functions are also provided. 1. Introduction and Statement of Results For an interval [a,b] in R and a number q ∈ [1, ∞) denote by Vq ([a,b]) the space of all functions m :[a,b] → C of bounded q-variation over [a,b], i.e., kmkVq([a,b]) := sup |m(x)| + kmkVarq ([a,b]) < ∞, x∈[a,b] n−1 q 1/q where kmkVarq ([a,b]) := sup{( i=0 |m(ti+1) − m(ti)| ) } and the supremum is taken over all finite sequences a =: t0 <t1 < ... < tn := b (n ∈ N). We write D for the dyadic decomposition of RP, i.e., D := ±(2k, 2k+1]: k ∈ Z , and set R C Vq(D) := m : → : sup km|I kVq (I) < ∞ (q ∈ [1, ∞)). I∈D Moreover, let Ap(R) (p ∈ [1, ∞)) be the class of weights on R which satisfy the R Muckenhoupt Ap condition. Denote by [w]Ap the Ap-constant of w ∈ Ap( ). If w ∈ A∞(R) := ∪p≥1Ap(R) we write Mp(R, w) for the class of all multipliers on Lp(R, w) (p> 1), i.e., ∞ p Mp(R, w) := {m ∈ L (R): Tm extends to a bounded operator on L (R, w)} . Here Tm stands for the Fourier multiplier with the symbol m, i.e., (Tmf) = mf (f ∈ arXiv:1403.4477v2 [math.CA] 13 May 2014 R R S( )). Note that Mp( , w) becomes a Banach space under the norm kmkMp(R,w) := R kTmkL(Lp(R,w)) (m ∈ Mp( , w)). b b The main result of the paper is the following complement to results due to D. Kurtz [18], R. Coifman, J.-L. Rubio de Francia, S. Semmes [8], and E. Berkson, T. Gillespie [4]. Theorem A. (i) Let q ∈ (1, 2]. Then, Vq(D) ⊂ Mp(R, w) for every p ≥ q and every Muckenhoupt weight w ∈ Ap/q (R). R 1 1 −1 (ii) Let q > 2. Then, Vq(D) ⊂ Mp( , w) for every 2 ≤ p < ( 2 − q ) and every 1 1 −1 Muckenhoupt weight w ∈ Ap/2 with sw > (1 − p( 2 − q )) . 2010 Mathematics Subject Classification. 42B25 (42B15). Key words and phrases. weighted Fourier multipliers, weighted inequalities, Littlewood-Paley square functions, Muckenhoupt weights. The author was supported by the Alexander von Humboldt Foundation. 1 2 SEBASTIAN KROL´ Here, for every w ∈ A∞(R), we set sw := sup{s ≥ 1 : w ∈ RHs(R)} and we write w ∈ RHs(R) if 1/s −1 1 b 1 b sup w(x)sdx w(x)dx < ∞. b − a b − a a<b Za ! Za ! Recall that, by the reverse H¨older inequality, sw ∈ (1, ∞] for every Muckenhoupt weight w ∈ A∞(R). For the convenience of the reader we repeat the relevant material from the liter- ature, which we also use in the sequel. Recall first that in [18] D. Kurtz proved the following weighted variant of the classical Marcinkiewicz multiplier theorem. Theorem 1 ([18, Theorem 2]). V1(D) ⊂ Mp(R, w) for every p ∈ (1, ∞) and every Muckenhoupt weight w ∈ Ap(R). As in the unweighted case, Theorem 1 is equivalent to a weighted variant of the Littlewood-Paley decomposition theorem, which asserts that for the square D D function S corresponding to the dyadic decomposition D of R, kS fkp,w h kfkp,w p (f ∈ L (R, w)) for every p ∈ (1, ∞) and w ∈ Ap(R); see [18, Theorem 1], and also [18, Theorem 3.3]. Here and subsequently, if I is a family of disjoint intervals in R, we write SI for the Littlewood-Paley square function corresponding to I, i.e., I 2 1/2 2 R S f := I∈I |SI f| (f ∈ L ( )). Recall also that in [26] J.-L. Rubio de Francia proved the following extension of the classicalP Littlewood-Paley decomposition theorem. Theorem 2 ([26, Theorem 6.1]). Let 2 <p< ∞ and w ∈ Ap/2(R). Then for an arbitrary family I of disjoint intervals in R the square function SI is bounded on Lp(R, wdx). Applying Rubio de Francia’s inequalities, i.e. Theorem 2, R. Coifman, J.-L. Ru- bio de Francia, and S. Semmes [8] proved the following extension and improvement of the classical Marcinkiewicz multiplier theorem. (See Section 2 for the definition of R2(D).) Theorem 3 ([8, Th´eor`eme 1 and Lemme 5]). Let 2 ≤ q < ∞. Then, Vq(D) ⊂ R 1 1 1 Mp( ) for every p ∈ (1, ∞) such that | p − 2 | < q . Furthermore, R2(D) ⊂ M2(R, w) for every w ∈ A1(R). Subsequently, a weighted variant of Theorem 3 was given by E. Berkson and T. Gillespie in [4]. According to our notation their result can be formulated as follows. Theorem 4 ([4, Theorem 1.2]). Suppose that 2 ≤ p< ∞ and w ∈ Ap/2(R). Then, there is a real number , depending only on and , such that 1 1 1 s> 2 p [w]Ap/2 s > 2 − p and Vq(D) ⊂ Mp(R, w) for all 1 ≤ q<s. Note that the part (i) of Theorem A fills a gap which occurs in Theorem 1 and the weighted part of Theorem 3. The part (ii) identifies the constant s in Berkson- 1 1 −1 ′ sw Gillespie’s result, i.e., Theorem 4, as ( − ′ ) , where s := , and in general, 2 swp w sw−1 this constant is best possible. Except for some details, the proofs given below reproduce well-known argu- ments from the Littlewood-Paley theory; in particular, ideas which have been FOURIER MULTIPLIERS ON WEIGHTED Lp SPACES 3 presented in [18], [8], [26], and [29]. A new point of our approach is the follow- I ing result on weighted estimates for modified Littlewood-Paley functions Sq (·) := ′ ′ q 1/q ( I∈I |SI (·)| ) (q ∈ (1, 2]), which may be of independent interest. TheoremP B. (i) Let q ∈ (1, 2), p > q, and w ∈ Ap/q (R). Then, there exists a constant C > 0 such that for any family I of disjoint intervals in R I p R kSq fkp,w ≤ Ckfkp,w (f ∈ L ( , wdx)). R Moreover, for every q ∈ (1, 2), p > q and V ⊂ Ap/q ( ) with supw∈V [w]Ap/q < ∞ I R sup kSq fkp,w : w ∈V, I a family of disjoint intervals in , kfkp,w =1 < ∞. (ii) For any family I of disjoint intervals in R and every Muckenhoupt weight R I 2 R 2 R w ∈ A1( ), the operator S2 maps L ( , wdx) into weak-L ( , wdx), and I sup kS fk 2,∞ : w ∈V, I a family of disjoint intervals in R, kfk 2 =1 < ∞ 2 Lw Lw n R o for every V ⊂ A1( ) with supw∈V [w]A1 < ∞. Moreover, if q ∈ (1, 2), then for any well-distributed family I of disjoint intervals R R I q R in and every Muckenhoupt weight w ∈ A1( ), the operator Sq maps L ( , wdx) into weak-Lq(R, wdx). Recall that a family I of disjoint intervals in R is well-distributed if there exists λ> 1 such that supx∈R I∈I χλI (x) < ∞, where λI denotes the interval with the same center as I and length λ times that of I. P 2 I Note that the validity of the A1-weighted L -estimates for square function S = I R S2 corresponding to an arbitrary family I of disjoint intervals in , i.e., I 2 R R kS2 fk2,w ≤ Cwkfk2,w (f ∈ L ( , wdx), w ∈ A1( )), is conjectured by J.-L. Rubio de Francia in [26, Section 6, p.10]; see also [12, Section 8.2, p. 187]. Theorem B(ii), in particular, provides the validity of the weak variant of Rubio de Francia’s conjecture. Notice that in contrast to the square function I I operators S2 , in general, operators Sq (q ∈ [1, 2)) are not bounded on (unweighted) Lq(R); see [9]. Moreover, in [24] T.S. Quek proved that if I is a well-distributed ′ R I q R q,q R family of disjoint intervals in , then the operator Sq maps L ( ) into L ( ) for ′ every q ∈ (1, 2). Note that this result is in a sense sharp, i.e., Lq,q (R) cannot be replaced by Lq,s(R) for any s < q′; see [24, Remark 3.2]. Therefore, Theorem B provides also a weighted variant of this line of researches. Cf. also relevant results given by S.V. Kisliakov in [17]. Furthermore, as a consequence of our approach we also get a higher dimensional analogue of Theorem A, see Theorem C in Section 4, which extends earlier results by Q. Xu [29]; see also M. Lacey [19, Chapter 4]. Since the formulation of Theorem C is more involved and its proof is essentially the iteration of one-dimensional arguments we refer the reader to Section 4 for more information. The part (ii) of Theorem A is a quantitative improvement of [4, Theorem 1.2] due to E. Berkson and T.