ON MAXIMAL FUNCTIONS FOR MIKHLIN-HO¨RMANDER MULTIPLIERS 5 0 0 LOUKASGRAFAKOS,PETR HONZÍK,ANDREASSEEGER 2 n Abstract. Given Mikhlin-Hörmander multipliers m , i = 1,...,N, i a with uniform estimates we prove an optimal log(N+1) bound in Lp J for themaximal function sup |F−1[m f]| and related boundsfor maxi- 8 i i p mal functions generated by dilations. These improveresults in [7]. ] b A C . 1. Introduction h t a Given a symbol m satisfying m (1.1) ∂αm(ξ) C ξ −α α [ | | ≤ | | for all multiindices α, then by classical Calderón-Zygmund theory the oper- 1 ator f −1[mf] defines an Lp bounded operator. We study two types of v 7→ F 4 maximal operators associated to such symbols. 1 First we considber N multipliers m ,...,m satisfying uniformly the con- 1 N 1 ditions (1.1) and ask for bounds 1 0 (1.2) sup −1[m f] A(N) f , 5 |F i | p ≤ k kp 1≤i≤N 0 (cid:13) (cid:13) / for all f . (cid:13) b (cid:13) h ∈S Secondlyweformtwomaximalfunctionsgeneratedbydilationsofasingle t a multiplier, m : (1.3) dyadf(x)= sup −1[m(2k )f] v Mm k∈Z|F · | i X (1.4) f(x)= sup −1[m(t )f]b m r M t>0 |F · | a and ask under what additional conditions on m thbese define bounded operators on Lp. Concerning (1.3), (1.4) a counterexample in [7] shows that in general additional conditions on m are needed for the maximal inequality to hold; moreover positive results were shown using rather weak decay assumptions on m. The counterexample also shows that the optimal uniform bound in (1.2) satisfies (1.5) A(N) c log(N +1). ≥ Date: October26, 2004. p Grafakos and Seegerweresupportedin partbyNSFgrants. Honz´ık wassupportedby 201/03/0931 Grant Agency of theCzech Republic. 1 2 LOUKASGRAFAKOS,PETRHONZÍK,ANDREASSEEGER The extrapolation argument in [7] only gives the upper bound A(N) = O(log(N +1)) and the main purpose of this paper is to close this gap and to show that the upper bound is indeed O( log(N +1)). We will formulate our theorems with minimal smoothness assumptions p that will be described now. Let φ C∞(Rd) be supported in ξ :1/2 < ξ < 2 so that ∈ 0 { | | } φ(2−kξ)= 1 k∈Z X for all ξ Rd 0 . Let η C∞(Rd) so that η is even, η (x) = 1 ∈ \ { } 0 ∈ c 0 0 for x 1/2 and η is supported where x 1. For ℓ > 0 let η (x) = 0 ℓ | | ≤ | | ≤ η (2−ℓ(x)) η (2−ℓ+1x) and define 0 0 − H [m](x) = η (x) −1[φm(2k )](x). k,ℓ ℓ F · In what follows we set m := sup 2ℓα H [m] . Y(q,α) k,ℓ Lq k k k∈Z k k ℓ≥0 X Using the Hausdorff-Young inequality one gets (1.6) kmkY(r′,α) . sku∈pZkφm(2k·)kBαr,1, if 1 ≤r ≤ 2 where Br is the usual Besov space (cf. Lemma 3.3 below). Thus if m α,1 belongsto Y(2,d/2) thenm is aFourier multiplier of Lp(Rd), for 1 < p < ∞ (this follows from a slight modification of Stein’s approach in [16], ch. IV.3, see also [15] for a related endpoint bound). Theorem 1.1. Suppose that 1 r < 2 and suppose that the multipliers m , i ≤ i= 1,...,N satisfy the condition (1.7) sup mi Y(r′,d/r) B < . k k ≤ ∞ i Then for r < p < ∞ sup −1[m f] C B log(N +1) f . F i p ≤ p,r k kp i=1,...,N (cid:13) (cid:12) (cid:12)(cid:13) p By (1.6) w(cid:13)e immed(cid:12)iately getb(cid:12)(cid:13) Corollary 1.2. Suppose that 1 < r < 2, and (1.8) sup sup φmi(t ) Br A. 1≤i≤N t>0 k · k d/r,1 ≤ Then for r < p < ∞ sup −1[m f] C A log(N +1) f . F i p ≤ p,r k kp i=1,...,N (cid:13) (cid:12) (cid:12)(cid:13) p (cid:13) (cid:12) b(cid:12)(cid:13) ON MAXIMAL FUNCTIONS FOR MIKHLIN-HO¨RMANDER MULTIPLIERS 3 Remark. If one uses Y( ,d + ε) in (1.7) or B1 in (1.8) one can use ∞ d+ε,1 Calderón-Zygmund theory (see [8], [7]) to prove the H1 L1 boundedness − and the weak type (1,1) inequality, both with constant O( log(N +1)). dyadp Our second result is concerned with the operators , generated m m M M by dilations. Theorem 1.3. Suppose 1 < p< , q = min p,2 . ∞ { } (i) Suppose that (1.9) kφm(2k·)kLqα ≤ ω(k), k ∈ Z, holds for α > d/q and suppose that the nonincreasing rearrangement ω∗ satisfies ∞ ω∗(l) (1.10) ω∗(0)+ < . l√logl ∞ l=2 X Then dyad is bounded on Lp(Rd). m M (ii) Suppose that (1.10) holds and (1.9) holds for α > d/p+1/p′ if 1 < p 2 or for α > d/2+1/p if p > 2. Then is bounded on Lp(Rd). m ≤ M If (1.9), (1.10) are satisfied with q = 1, α > d then is of weak type m M (1,1), and maps H1 to L1. m M Thisimproves theearlier resultin[7]wheretheconclusion isobtainedun- der the assumption ∞ ω∗(l)/l < , however somewhat weaker smooth- l=2 ∞ ness assumptions were made in [7]. P In 2 we shall discuss model cases for Rademacher expansions. In 3 we § § shall give the outline of the proof of Theorem 1.1 which is based on the exp(L2) estimate by Chang-Wilson-Wolff [5], for functions with bounded Littlewood-Paley square-function. The proof of a critical pointwise inequality is given in 4. The proof of Theorem 1.3 is sketched in 5. Some open § § problems are mentioned in 6. § Acknowledgement: The second named author would like to thank Luboˇs Pick for a helpful conversation concerning convolution inequalities in rearrangement invariant function spaces. 2. Dyadic model cases for Rademacher expansions Before we discuss the proof of Theorem 1.1 we give a simple result on expansions for Rademacher functions r on [0,1] which motivated the proof. j Proposition 2.1. Let ai ℓ2. and let ∈ F (s) = air (s), s [0,1]. i j j ∈ j X Then sup F . sup ai log(N +1). | i| L2[0,1] k kℓ2 iλ 2exp λ2 { ∈ | i | } ≤ − 4kaik2ℓ2 Set u = (4log(N(cid:0)+1))1/2sup ai (cid:1). Then (cid:0) (cid:1) N 1≤i≤N ℓ2 k k N ∞ sup F 2 u2 +2 λmeas s : F (s) > λ dλ | i| 2 ≤ N { | i | } i=1,...,N i=1ZuN (cid:13) (cid:13) X (cid:0) (cid:1) (cid:13) (cid:13)N ∞ ≤ u2N +4 λe−λ2/(4kaik2ℓ2)dα ≤ u2N +4 sup kaik2ℓ2Ne−u2N/4 i=1ZuN i=1,...,N X which is bounded by (1+4log(N +1))sup ai 2 . The claim follows. (cid:3) ik kℓ2 There is a multiplier interpretation to this inequality. One can work with a single function f = a r and a family of bounded sequences (or j j multipliers) bi and one forms F (s) = bia r (s). The norm then grows { } P i j j j j as a square root of the logarithm of the number of multipliers; i.e. we have P Corollary 2.2. sup bia r . sup bi log(N +1) a r . i=1,...,N j j j L2([0,1]) i k k∞ j j L2([0,1]) (cid:13) (cid:12)Xj (cid:12)(cid:13) p (cid:13)Xj (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:12) (cid:12) (cid:13)We shall now consi(cid:13)der a dyadic model case for the(cid:13)maximal(cid:13)operators generated by dilations. Proposition 2.3. Consider a sequence b = bi i∈Z which satisfies { } A b∗(l) . ≤ (log(l+2))1/2 Then for any sequence a = a ∞ we have { n}n=1 ∞ sup b a r CA a . j−k j j 2 k∈Z 2 ≤ k k (cid:13) (cid:12)Xj=0 (cid:12)(cid:13) (cid:13) (cid:13) (cid:12) (cid:12) Proof. We may assum(cid:13)e that both a and(cid:13)b are real valued sequences. Let ∞ H (s)= b a r (s). k j−k j j j=1 X Then by orthogonality of the Rademacher functions ∞ H 2 = [b a ]2. k kk2 j−k j j=1 X WeshallusearesultofCalderón[4]whichstatesthatifsomelinearoperator is boundedon L1(µ) and on L∞(µ) on a space with σ-finite measure µ, then it is bounded on all rearrangement invariant function spaces on that space. ON MAXIMAL FUNCTIONS FOR MIKHLIN-HO¨RMANDER MULTIPLIERS 5 In our case the intermediate space is the Orlicz space expℓ, which coincides with the space of all sequences γ = γj j∈Z that satisfy the condition { } C (2.2) γ∗(l) , l 0, ≤ log(l+2) ≥ and the best constant in 2.2 is equivalent to the norm in exp(ℓ). We apply Calderón’s result to the operator T defined by ∞ [Tγ] = γ a2 k j−k j j=1 X and get suplog(l+2)(Tγ)∗(l) C a2 suplog(l+2)γ∗(l). ≤ { n} ℓ1 l≥0 l≥0 (cid:13) (cid:13) Let c = H ([T(b2)] )1/2 where(cid:13)b2 st(cid:13)ands for the sequence b2 ; then k k kk2 ≡ k { j} by our bound for Tγ and the assumption on b it follows that (2.3) c∗(l) C A a log(2+l) −1/2. 1 ℓ2 ≤ k k We can proceed with the proof as (cid:0)in Proposit(cid:1)ion 2.1, using again (2.1), i.e. meas( s [0,1] : Hk(s) > α ) 2e−α2/4c2k. { ∈ | | } ≤ Then we obtain for u> 0 ∞ sup|Hk| 2 ≤ u2+4 αe−α2/4c2k k k Zu (cid:13) (cid:13) X (cid:13) (cid:13) ≤ u2+8 c2ke−u2/(4c2k) k X = u2+8 (c∗(l))2e−u2/4(c∗(l))2. l≥0 X We set the cutoff level to be u= 10C A a and obtain 1 2 k k sup H 2 u2+C2A2 (2+l)−5/2 . A2 a 2 k | k|k2 ≤ 1 k k2 k l≥0 X which is what we wanted to prove. (cid:3) Remark: Since the Lp norm of a r is equivalent to the ℓ2 norm of a j j j { } one can also prove Lp analogues of the two propositions, for 0 < p < . P ∞ 3. Proof of Theorem 1.1 To prove (1.2) we may assume that f is compactly supported in Rd 0 \{ } and thus we may assume that the multipliers m are compactly supported i on a finite union of dyadic annuli. Inbview of the scale invariance of the assumptions we may assume without loss of generality that (3.1) m (ξ) = 0, ξ 2N,i = 1,...,N. i | | ≤ 6 LOUKASGRAFAKOS,PETRHONZÍK,ANDREASSEEGER In the case of Fourier multipliers the inequality (2.1) will be replaced by a “good-λ inequality” involving square-functions for martingales as proved by Chang, Wilson and Wolff [5]. To fix notation let, for any k 0, Q k ≥ denote the family of dyadic cubes of sidelength 2−k; each Q is of the form d [n 2−k,(n +1)2−k). Denote by E the conditional expectation, i=1 i i k 1 Q E f(x)= χ (x) f(y)dy k Q Q QX∈Qk | | ZQ and by D the martingale differences, k D f(x)= E f(x) E f(x). k k+1 k − The square function for the dyadic martingale is defined by 1/2 S(f)= D f(x)2 ; k | | (cid:16)Xk≥0 (cid:17) one has the inequality S(f) C f for 1 < p < (see [3], [2] for the p p p k k ≤ k k ∞ general martingale case, and for our special case cf. also Lemma 3.1 below). The result from [5] says that there is a constant c > 0 so that for all d λ> 0, 0 < ε< 1, one has (3.2) meas x:sup E g(x) E g(x) > 2λ, S(g) < ǫλ ) k 0 | − | k≥0 (cid:0)(cid:8) c (cid:9)(cid:1) Cexp( d)meas x: sup E g(x) > ελ ; ≤ −ǫ2 | k | k≥0 (cid:0)(cid:8) (cid:9)(cid:1) see [5] (Corollary 3.1 and a remark on page 236). To use (3.2) we need a pointwise inequality for square functions applied to convolution operators. Choose a radial Schwartz function ψ which equals 1 on the support of φ (defined in the introduction) and is compactly supported in Rd 0 , and \{ } define the Littlewood-Paley operator L by k (3.3) L f(ξ) = ψ(2−kξ)m(ξ)f(ξ) k Let M be the Hardy-Littlewood maximal operator and define the operator d b M by r M = (M(f r))1/r. r | | Denote byM =M M M thethree-fold iteration of themaximal operator. ◦ ◦ Now define 1/2 (3.4) G (f) = M[L f r] 2/r . r k | | (cid:16)Xk∈Z(cid:0) (cid:1) (cid:17) From the Fefferman-Stein inequality for vector-valued maximal functions [9], (3.5) G (f) C f , 1 < r < 2, r < p < . r p p,r p k k ≤ k k ∞ ON MAXIMAL FUNCTIONS FOR MIKHLIN-HO¨RMANDER MULTIPLIERS 7 Lemma 3.1. Let Tf = −1[mf] and let 1 4λ )dλ . | i | p { | i | } (cid:13)1≤i≤N (cid:13) (cid:16) Z0 i (cid:17) Now(cid:13) by Lemma(cid:13)3.1 one gets the pointwise bound (3.8) S(T f) A BG (f). i r r ≤ We note that x : sup T f(x) > 4λ E E E i λ,1 λ,2 λ,3 { | | } ⊂ ∪ ∪ 1≤i≤N where with c 1/2 d (3.9) ε := N 10log(N +1) (cid:16) (cid:17) we have set ε λ E = x: sup T f(x) E T f(x) >2λ,G (f)(x) N , λ,1 i 0 i r { | − | ≤ A B} 1≤i≤N r ε λ N E = x: G (f)(x) > , λ,2 r { A B} r E = x: sup E T f(x) > 2λ . λ,3 0 i { | | } 1≤i≤N By (3.8), N (3.10) E x : T f(x) > 2λ,S(T f) ε λ , λ,1 i i N ⊂ { | | ≤ } i=1 [ and thus using the good-λ inequality (3.2) we obtain N meas(E ) meas x : T f(x) E T f(x) > 2λ,S(T f) ε λ λ,1 i 0 i i N ≤ { | − | ≤ } i=1 X (cid:0) (cid:1) N Cexp( cd )meas( x :sup E (T f) > λ ). ≤ −ε2 { | k i | } N k i=1 X 8 LOUKASGRAFAKOS,PETRHONZÍK,ANDREASSEEGER Hence ∞ 1/p p λp−1meas(E )dλ λ,1 (cid:16) Z0 (cid:17) N 1/p . exp( cd ) sup E (T f) p −ε2 | k i | p N k (cid:16)Xi=1 (cid:13) (cid:13) (cid:17) N (cid:13) (cid:13) 1/p . exp( cd ) T f p −ε2 i p N (cid:16)Xi=1 (cid:13) (cid:13) (cid:17) (3.11) . B N exp( cd )(cid:13)1/p (cid:13)f . B f −ε2 k kp k kp N (cid:0) (cid:1) uniformly in N (by our choice of ε in (3.9)). N Next, by a change of variable, ∞ 1/p A B p λp−1meas(E )dλ = r G (f) λ,2 ε r p (cid:16) Z0 (cid:17) N (cid:13) (cid:13) (3.12) . B lo(cid:13)g(N +(cid:13)1) f p k k Finally, from Lemma 3.2 and the Feffermanp-Stein inequality N meas(E ) meas x: E T f(x) > 2λ λ,3 0 i ≤ { | | } i=1 X (cid:0) (cid:1) and thus ∞ 1/p p λp−1meas(E )dλ = 2 sup E (T f) λ,3 | 0 i | p (cid:16) Z0 (cid:17) (cid:13)i=1,...,N (cid:13) N 1/p (cid:13) (cid:13) (3.13) 2 E (T f) p . BN1/p2−N/r f . B f . ≤ 0 i p k kp k kp (cid:16)Xi=1(cid:13) (cid:13) (cid:17) (cid:13) (cid:13) The asserted inequality follows from (3.11), (3.12), and (3.13). (cid:3) For completeness we mention the well known relation of the Y(r′,α) conditions with Besov and Sobolev norms. Lemma 3.3. Let 1 r 2 and α > d/r. Then ≤ ≤ m Y(r′,d/r) . sup φm(2k ) Br k k k · k d/r,1 k . sup φm(2k ) . sup φm(2k ) k · kLrα k · kL2α k k Proof. By the Hausdorff-Young inequality and the definition of the Besov space we have ∞ ∞ 2ℓd/r Hk,ℓ r′ . 2ℓd/r [φm(2k )] ηℓ r . φm(2k ) Br . k k k · ∗ k k · k d/r,1 ℓ=0 ℓ=0 X X b ON MAXIMAL FUNCTIONS FOR MIKHLIN-HO¨RMANDER MULTIPLIERS 9 By elementary imbedding properties g Br . g Lr if γ > d/r. Finally k k d/r,1 k k γ φm(2k ) . C′ φm(2k ) , if 1 < r 2. In this last inequality we k · kLrγ rk · kL2γ ≤ used that for χ ∈ Cc∞ we have kχgkLrγ0 . kgkLrγ1 for r0 ≤ r1, γ ≥ 0; this is trivial for integers γ from Hölder’s inequality and follows for all γ 0 by ≥ interpolation. (cid:3) 4. Proofs of Lemma 3.1 and Lemma 3.2 ChoosearadialSchwartzfunctionβ withthepropertythatβ issupported in x : x 1/4 so that β(ξ) = 0 in ξ :1/4 ξ 4 and β(0) = 0. Now { | |≤ } 6 { ≤ | |≤ } choose a function ψ C∞ so that ψ(ξ)(β(ξ))2 = 1 for all ξ bsupp φ, here ∈ c ∈ φ is as in the formulation of the theorem. Define operators T , B , L by k k k e e T f(ξ) = φ(2−kξ)m(ξ)f(ξ) k e B f(ξ) = β(2−kξ)f(ξ) dk b L f(ξ) = ψ(2−kξ)f(ξ). dk b Then T = T = B2L T L and we write k k k edk k k ke b (4.1) P DkTfP= e(DkBk+n)(Bk+nLk+n)Tk+nLk+nf. n∈Z X Sublemma 4.1. e (4.2) B L f(x) . Mf(x). k k | | Proof. Immediate. (cid:3) e Sublemma 4.2. For s 0, ≥ (4.3) E B f(x) + E B f(x) . 2−s/q′M f(x) k+1 k+s k k+s q | | | | and (4.4) D B f(x) . 2−sMf(x). k k−s | | Proof. We give the proof although the estimates are rather standard (for similar calculations in other contexts see for example [6], [12], [10], [13]). For (4.3) first note this inequality is trivial if s is small and assume, say, s 10. For Q Q , s > 0 let b (Q) be the set of all x Q for which the k s ≥ ∈ ∈ ℓ∞ distance to the boundary of Q is 2−k−s+1. ≤ Fix a cube Q Q . If Q′ is a dyadic subcube of sidelength 2−k−s+1 0 k+1 ∈ subcube which is not contained in bs(Q) then Bk+s[fχQ′] is supported in Q0 and usingthe cancellation of −1[β] we see that Ek+1Bk+s[χQ′g] = 0 for F all g. Let (Q ) be the union over all dyadic cubes of sidelength 2−k−s+1 s 0 V whose closures intersect the boundary of Q . Then 0 E B [χ g] = E B [gχ ] k+1 k+s Q0 k+1 k+s Vs(Q0) forallg. Inviewofthesupportpropertiesofβ wenotethatB [gχ ]is k+s Vs(Q0) also supported in (Q ). Observe that this set has measure O(2−kd2−s). s−1 0 V b 10 LOUKASGRAFAKOS,PETRHONZÍK,ANDREASSEEGER It follows that for x Q 0 ∈ E B f(x) 2d Q −1 B [χ f](y)dy | k+1 k+s | ≤ | 0| | k+s Vs(Q0) | ZVs−1(Q0) . Q −1 f(y)qdy 1/q2−(kd+s)/q′ 0 | | | | ZQ0 . 2−s/q′ (cid:0)M(f q) 1/q (cid:1) | | By the same argument one obtai(cid:0)ns this bo(cid:1)und also for E B f and thus k k+s | | (4.3) follows. The inequality (4.4) D B f is a simple consequence of the smoothness k k−s of the convolution kernel of B and the cancellation properties of the k−s operator D = E E . (cid:3) k k+1 k − Sublemma 4.3. Let 1< r < . We have ∞ (4.5) Tkf(x) C m Y(r′,d/r)Mrf(x). | | ≤ k k Proof. We may decompose T using the kernels H and obtain k k,l ∞ T f(x) = 2kdH (2ky)f(x y)dy k k,ℓ | | − (cid:12)Xℓ=0Z (cid:12) (cid:12)(cid:12)∞ 2kd H (2ky)r′dy 1/r′(cid:12)(cid:12)2kd f(x y)rdy 1/r k,ℓ ≤ Xℓ=0(cid:16) Z | | (cid:17) (cid:16) Z|y|≤2−k+ℓ| − | (cid:17) ∞ 2ℓd/r Hk,ℓ r′ M(f r)(x) 1/r. (cid:3) ≤ k k | | ℓ=0 X (cid:0) (cid:1) Proof of Lemma 3.1. To estimate the terms in (4.1) we use Sublemma 4.1 to bound B L , Sublemma 4.2 to bound D B and Sublemma 4.3 k+n k+n k k+n to bound T . This yields that k+n e |DkBk2+nLk+nTk+nLk+nf(x)| . kmkY(r′,d/r) 2−n/q′M M M (L f)(x) if n 0 e q ◦ ◦ r k+n ≥ ×(2nM M Mr(Lk+nf)(x) if n < 0, ◦ ◦ and straightforward estimates imply the asserted bound. (cid:3) Proof of Lemma 3.2. We split E Tf = E B2L T , and by the 0 k≥N−2 0 k k k sublemmas we get P e |E0Bk2LkTkf(x)|. 2−k/rkmkY(r′,d/r)Mr ◦M ◦Mr(f)(x) which implies the assertion. (cid:3) e

On maximal functions for Mikhlin-Hoermander multipliers

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