A Cotlar Type Maximal Function Associated With Fourier Multipliers
aa r X i v : . [ m a t h . C A ] N ov A COTLAR TYPE MAXIMAL FUNCTION ASSOCIATED WITHFOURIER MULTIPLIERS
RAJULA SRIVASTAVA
Abstract.
We prove the L p boundedness of a maximal operator associated with a dyadicfrequency decomposition of a Fourier multiplier, under a weak regularity assumption. Introduction
Consider a Mikhlin-Hörmander multiplier m on R d satisfying the assumption(1) | ∂ γ m ( ξ ) | ≤ A | ξ | −| γ | for all multi-indices γ with | γ | ≤ L for some integer L > d .Let χ ∈ C ∞ c ( R ) be supported in (1 / ,
2) such that P ∞ j = −∞ χ (2 j t ) = 1 and let φ = χ ( | . | ).For a given Schwartz function f , let Sf := F − [ m ˆ f ], and for n ∈ Z let S n be defined by(2) d S n f ( ξ ) := X j ≤ n φ (2 − j ξ ) m ( ξ ) ˆ f ( ξ ) . We are interested in bounds for the maximal function(3) S ∗ f ( x ) := sup n ∈ Z | S n f ( x ) | . The above operator was studied by Guo, Roos, Seeger and Yung in [4], in connection withproving L p bounds for a maximal operator associated with families of Hilbert transformsalong parabolas. The multiplier m in [4] was assumed to satisfy the condition(4) sup t> k φm ( t · ) k L β = B ( m ) < ∞ with β > d . Here L β is the potential space of functions g with ( I − ∆) β/ g ∈ L (we notethe analogy with condition (1) here). With the above hypothesis, the authors were able toprove a pointwise Cotlar-type inequality(5) S ∗ f ( x ) ≤ − δ ) /r ( M ( | Sf | r )( x )) /r + C d,β δ − B ( m ) M f ( x )for f ∈ L p ( R d ) and for almost every x (with r > < δ ≤ / M [ f ] denotes thestandard Hardy-Littlewood Maximal function. (5) easily implies L p and weak (1,1) boundsfor the maximal operator S ∗ .It is natural to ask if one could weaken the assumption (4) and still establish L p boundson the operator S ∗ , possibly without the intermediate step of proving a pointwise inequalityof the form (5) for all exponents r . In this paper, we answer this question in the affirmative. Date : 2019/11/12.2010
Mathematics Subject Classification.
In particular, we show that for the L p bounds on S ∗ to hold, it is enough for the multiplier m to satisfy a much weaker condition(6) sup t> Z |F − [ φm ( t · )]( x ) | (log (2 + | x | )) α dx := B ( m ) < ∞ , α > . Our main result is the following:
Theorem 1.1. S ∗ , as defined in (3) for a multiplier m satisfying (6), is of weak-type (1,1)and bounded on L p for p ∈ (1 , ∞ ) , with the respective operator norm . p B ( m ) . We remark here that it is not possible to do away with the smoothness assumptionentirely. In other words, the conditionsup t> Z |F − [ φm ( t · )]( x ) | dx < ∞ alone is not enough to guarantee L p bounds on the maximal operator S ∗ , or even on thesingular operator S . For counterexamples, we refer to [10], Section 5 and [9], Section 3.We shall denote the Littlewood-Paley pieces of m by m j . More precisely, for j ∈ Z ,we define m j ( ξ ) := φ (2 − j ξ ) m ( ξ ). Furthermore let a j ( ξ ) = m j (2 j ξ ) = η ( ξ ) m (2 j ξ ). Observethat supp ( m j ) ⊂ (2 j − , j ) and supp ( a j ) ⊂ (1 / , K j = F − [ m j ]. Then (2) can bere-written as(7) S n f ( x ) = X j ≤ n K j ∗ f ( x ) . However, in order to quantify the smoothness condition in (6), it is useful to partition K j (for each j ∈ Z ) on the space side as well (see [2]). To this effect, let η ∈ C ∞ c ( R d ) besuch that η is even, η ( x ) = 1 for | x | ≤ / η is supported where | x | ≤
1. For l ∈ N ,let η l ( x ) = η (2 − l x ) − η (2 − l +1 x ). For l ∈ N ∪ { } we define(8) K l,j ( x ) = η l (2 j x ) F − [ m j ]( x ) . By the assumption (6), we then have(9) k K l,j k L . B (log (2 + 2 l )) − α . Now the multiplier corresponding to K l,j is given by 2 − jd ˆ η l (2 − j · ) ∗ m j , which, unlike m j , is not compactly supported. However the rapid decay of ˆ η l still leads to the multiplier"essentially" being supported in a slightly thicker (but still compact) dyadic annulus, with theother frequency regions contributing negligible error terms. We make these ideas rigorousin Section 2. The arguments used are similar in spirit to those in [1], Section 5. Anothersource of reference is [6]. As a corollary, we prove that the singular operator(10) S l f ( x ) = X j ∈ Z K l,j ∗ f ( x )is bounded on L , with operator norm . Bl − α .In Section 3, we use Bernstein’s inequality (see [11]) to establish L p bounds for theaforementioned portion with the major contribution and with compact frequency support.We also establish a pointwise estimate on its gradient.In Section 4, using the estimates from Section 3, we prove that the Calderón-Zygmundoperator S l associated to the multiplier m is of weak type (1,1), with the operator norm . Bl − α +1 . We do so by establishing the result for the operator T l f := P j ∈ Z H l,j ∗ f , where H l,j is the portion of the kernel K l,j with the major contribution. The arguments flow COTLAR TYPE MAXIMAL FUNCTION ASSOCIATED WITH FOURIER MULTIPLIERS 3 in the same vein as those in the proof of the Mikhlin-Hörmander Multiplier Theorem (see[3],[5]). We also establish L p bounds on T l using interpolation.In Section 5, we investigate the properties of the truncated operator T ln f := P j ≤ n H l,j ∗ f (for n ∈ Z and l ≥ L p bounds for the associated maximaloperator T l ∗ f = sup | T ln f | .We wish to show that for l ≥
0, the operator T l ∗ is bounded on L p for 1 < p < ∞ withthe corresponding operator norm . p Bl − α +1 ( . B for l = 0). Since our assumption (6) on m is much weaker than that in [4], a pointwise inequality like (5) for all r > T l ∗ f ( x ) . d − δ ) /p ( M ( | T l f | p )( x )) /p + 2 ldp l − α +1 (1 + δ − /p ) B ( m )( M ( | f | p )( x )) /p . for each T l ∗ and for a large enough exponent p (here 0 < δ ≤ / p = p l ∼ l will work. In particular, p l → ∞ as l → ∞ . Using this inequality,we can conclude that T l ∗ is bounded (with norm . p Bl − α +1 ) for all p ∈ [ p l , ∞ ). This ideaof keeping track of the explicit dependence on the exponent l and using the decay in l tosum the pieces up has also been used in [2], albeit for a different maximal operator thanthe one considered here. For p ∈ (1 , p l ), however, we rely on a weak (1,1) estimate for T l ∗ (which is not hard to obtain) and then an interpolation between 1 and p l , which causes usto gain a power of l . In other words, we are only able to retain a decay of l − α +2 (hencethe assumption α > T l ∗ and obtain L p bounds for the sum P l ≥ T l ∗ , and consequently for S ∗ (we use the decay in l and the condition α > Acknowledgements.
The author would like to thank her advisor Andreas Seeger for in-troducing this problem, for his guidance and several illuminating discussions. Researchsupported in part by NSF grant 1500162.2.
The Error Terms
Let Ψ ∈ C ∞ with Ψ = 1 on { / ≤ | ξ | ≤ } and supported on { / ≤ | ξ | ≤ } . For j ∈ Z , define Ψ j ( ξ ) = Ψ(2 − j ξ ). Then(11) X j ∈ Z K l,j = X j ∈ Z K l,j ∗ F − (Ψ j ) + X j ∈ Z K l,j ∗ F − (1 − Ψ j )where K l,j is as defined in (8).In this section, we will show that the contribution from the second sum above can bemade as small as required using the rapid decay of ˆ η . To control this sum, we study thecorresponding multiplier given by X j ∈ Z − jd (1 − Ψ j )( ξ ) ˆ η l (2 − j · ) ∗ m j ( ξ ) = X j ∈ Z (1 − Ψ j ( ξ ))2 − jd Z m j ( ω ) ˆ η l (2 − j ( ξ − ω )) dω. Lemma 2.1.
Let l ≥ . For any N ∈ N , multi-index γ and ξ = 0 , we have the estimate (12) X j ∈ Z − jd | (1 − Ψ j ) ∂ γξ ( ˆ η l (2 − j · ) ∗ m j )( ξ ) | . η,N,γ,d B l ( d + | γ |− N ) | ξ | −| γ | . COTLAR TYPE MAXIMAL FUNCTION ASSOCIATED WITH FOURIER MULTIPLIERS 4
Proof.
Fix ξ = 0. Let k ∈ Z be such that 2 k − ≤ | ξ | < k +1 .As Ψ k = 1 on { k − ≤ | ξ | ≤ k +2 } , we can split the sum under consideration into two parts X j ∈ Z − jd | (1 − Ψ j ) ∂ γξ ( ˆ η l (2 − j · ) ∗ m j )( ξ ) | ≤ X j>k +2 − jd | ∂ γξ ( ˆ η l (2 − j · ) ∗ m j )( ξ ) | + X j We have X j>k +2 − jd | ∂ γξ ( ˆ η l (2 − j · ) ∗ m j )( ξ ) | ≤ X j>k +2 − jd Z j − ≤| ω |≤ j +1 | m j ( ω ) || ∂ γξ ˆ η l (2 − j ( ξ − ω )) | dω. Now we observe that for 2 j − ≤ | ω | ≤ j +1 , we have | ω − ξ | ≥ | ω | / − j | ω | / ∼ η yields X j>k +2 − jd | ∂ γξ ( ˆ η l (2 − j · ) ∗ m j )( ξ ) | . η,N,γ X j>k +2 − jd Z j − ≤| ω |≤ j +1 k m k L ∞ l ( d + | γ | ) − j | γ | (cid:16) l − j | ω | / (cid:17) − N dω . d,η,N,γ X j>k +2 − jd k m k L ∞ l ( d + | γ |− N ) − j | γ | Z j − ≤| ω |≤ j +1 dω . d,η,N,γ X j>k +2 d k m k L ∞ l ( d + | γ |− N ) − j | γ | . d,η,N,γ B l ( d + | γ |− N ) − ( k +2) | γ | . d,η,N,γ B l ( d + | γ |− N ) | ξ | −| γ | . Second Term: In this case for 2 j − ≤ | ω | ≤ j +1 we have that | ω − ξ | ≥ | ξ | / − j | ξ | / ≥ 1. Hence X j Theorem 2.2. Let l ≥ . For any N ∈ N , p ∈ (1 , ∞ ) and Schwartz function f , we have k X j ∈ Z K l,j ∗ F − (1 − Ψ j ) ∗ f k L p . d max ( p, ( p − − ) C η,N ,p, Ψ B − log k f k L p . Furthermore, we also have k X j ∈ Z K l,j ∗ F − (1 − Ψ j ) ∗ f ∈ L , ∞ k L , ∞ . d C η,N ,p, Ψ B − log k f k L . COTLAR TYPE MAXIMAL FUNCTION ASSOCIATED WITH FOURIER MULTIPLIERS 5 Proof. We need to prove that | X β ≤ γ c β,γ X j ∈ Z ∂ β (1 − Ψ j )( ξ ) ∂ γ − β (2 − jd ˆ η l (2 − j · ) ∗ m j )( ξ ) | . d C η,N,p, Ψ B − log | ξ | −| γ | where γ is a multi-index with | γ | ≤ d/ ξ = 0. We observe that for all values of j except for j , j where 2 j +2 ≤ | ξ | < · j or when 2 j / ≤ | ξ | < j − , we can use lemma2.1 directly (as then | (1 − Ψ j )( ξ ) | is a constant). Further, for the two remaining cases, weobserve that | ξ | ∼ j k ( k = 1 , | ∂ β (1 − Ψ j k ) | by C Ψ | ξ | −| β | and apply the previous lemma to the term | ∂ γ − β (2 − jd ˆ η l (2 − j · ) ∗ m j )( ξ ) | to bound it aboveby C η,N B l ( d + | γ |−| β |− N −| γ | ) | ξ | − ( | γ |−| β | ) . The result then follows by summing up. (cid:3) As a consequence, we obtain the L boundedness of S l (as defined in (10)), with poly-nomial decay in l . Theorem 2.3. For f ∈ L and l > , we have k S l ( f ) k L . Bl − α k f k L . We also have k S ( f ) k L . B k f k L . Proof. It is enough to prove the above for a Schwartz function f . Now S l ( f ) = X j ∈ Z K l,j ∗ f = X j ∈ Z K l,j ∗ F − (Ψ j ) ∗ f + X j ∈ Z K l,j ∗ F − (1 − Ψ j ) ∗ f. As we have already established the L boundedness of the second term in Theorem 2.2 (withas good a decay in l as required), we only need to prove the theorem for the first term. Tothis effect, let f = X k ∈ Z ∆ k f be a Littlewood-Paley decomposition of f . Then k X j ∈ Z K l,j ∗ F − (Ψ j ) ∗ f k L . k X j ∈ Z X k ∈ Z K l,j ∗ F − (Ψ j ) ∗ ∆ k f k L . k X j ∈ Z K l,j ∗ F − (Ψ j ) ∗ (∆ j − + ∆ j + ∆ j +1 ) f k L where we have used the fact that the frequency support of K l,j ∗ F − (Ψ j ) is contained in { j / ≤ | ξ | ≤ . j } . From (9), we also have that k K l,j ∗ F − (Ψ j ) k L ≤ k K l,j k L kF − (Ψ j ) k L . B (log (2 + 2 l )) − α k Ψ k L ∞ . Hence k X j ∈ Z K l,j ∗ F − (Ψ j ) ∗ f k L . B (log (2 + 2 l )) − α k X j ∈ Z (∆ j − + ∆ j + ∆ j +1 ) f k L . B (log (2 + 2 l )) − α k f k L which proves the theorem. (cid:3) COTLAR TYPE MAXIMAL FUNCTION ASSOCIATED WITH FOURIER MULTIPLIERS 6 Estimates for the Majorly Contributing Portion of the Kernel We now turn our attention to the first term in (11), which is the one with the maincontribution. The main advantage we have now is that the jth term in X j ∈ Z K l,j ∗ F − (Ψ j )has frequency supported in the annulus { j / ≤ | ξ | ≤ · j } . Hence, we can use Bernstein’sinequality to get bounds on the L p norm (of each term, with 1 < p < ∞ ) and L ∞ norm (ofthe derivative). Proposition 3.1. Let q ∈ (1 , ∞ ) , and let q ′ denote the Hölder conjugate exponent of q .Then for all l ∈ N ∪ { } and j ∈ Z , we have(1) k K l,j ∗ F − (Ψ j ) k L q ′ . B jdq (log (2 + 2 l )) − α .(2) For x ∈ R d , |∇ ( K l,j ∗ F − (Ψ j ))( x ) | . j | K l,j ∗ F − (Ψ j )( x ) | .Proof. For the first part, we have k K l,j ∗ F − (Ψ j ) k L q ′ ≤ k K l,j k L kF − (Ψ j ) k L q ′ . B (log (2 + 2 l )) − α k Ψ j k L q . Ψ B (log (2 + 2 l )) − α jd/q where in the second step we have used (9) and Hausdorff-Young’s inequality.For the second part, we recall that the Fourier transform of ∇ ( K l,j ∗F − )(Ψ j ) is suppor-ted on a dyadic annulus of radius 2 j . The assertion then follows from Bernstein’s inequality(see [11], Proposition 5.3). (cid:3) Weak (1,1) Boundedness of S l Let l be the characteristic function of the set { x : 2 l − ≤ | x | ≤ l +1 } for l > { x : | x | ≤ } for l = 0. We will denote l (2 j x ) by l,j ( x ). Then (11) can berewritten as(13) X j ∈ Z K l,j = X j ∈ Z K l,j l,j = X j ∈ Z K l,j ∗ F − (Ψ j ) l,j + X j ∈ Z K l,j ∗ F − (1 − Ψ j ) l,j . The advantage of (13) over (11) is that it preserves information about the compact supportof the kernel K l,j , a property we will exploit quite often in the forthcoming proofs. Now for f ∈ L ( R d ), we can estimate k X j ∈ Z K l,j ∗ f k L , ∞ ≤ k X j ∈ Z H l,j ∗ f k L , ∞ + k X j ∈ Z ( K l,j ∗ F − (1 − Ψ j ) l,j ) ∗ f k L , ∞ where we define H l,j := ( K l,j ∗ F − (Ψ j )) l,j . Also let the operator T l be defined as T l f := P j ∈ Z H l,j ∗ f .By Theorem 2.2, we conclude that k P j ∈ Z ( K l,j ∗F − (1 − Ψ j ) l,j ) ∗ f k L , ∞ . Bl − α +1 k f k L .Hence, in order to prove that S l is of weak type (1 , 1) (with the respective norm . d Bl − α +1 ),it is enough to prove the same for T l , which is the content of the next theorem. The proof wegive here essentially uses the same ideas as Hörmander’s original proof of the (Hörmander-)Mikhlin Multiplier Theorem (see [5], also [3]). COTLAR TYPE MAXIMAL FUNCTION ASSOCIATED WITH FOURIER MULTIPLIERS 7 Theorem 4.1. For all f ∈ L ( R d ) , we have k T l f k L , ∞ . Bl − α +1 k f k L for l ≥ and k T f k L , ∞ . B k f k L Proof. We prove the result for l > 0. The result for l = 0 follows similarly. Also, for thisproof, we can assume that B = 1. Let f ∈ L ( R d ) and fix σ > 0. Let f = f + f be the standard Calderón-Zygmund decomposition of f at the level l α − σ . More precisely,let { I k } k ∈ N be axis-parallel cubes with centres { a k } k ∈ N respectively such that l α − σ < | I k | − Z I k | f ( y ) | dy ≤ d l α − σ, | f ( x ) | ≤ l α − σ a . e . for x [ k ∈ N I k ,f ( x ) = ( f ( x ) − | I k | − R I k f ( y ) dy x ∈ I k , , otherwise. f ( x ) = ( | I k | − R I k f ( y ) dy x ∈ I k ,f ( x ) , otherwise.Now meas( { x : | X j ∈ Z H l,j ∗ f ( x ) | > σ } ) ≤ meas( { x : | X j ∈ Z H l,j ∗ f ( x ) | > σ/ } ) + meas( { x : | X j ∈ Z H l,j ∗ f ( x ) | > σ/ } )We estimate meas( { x : | P j ∈ Z H l,j ∗ f ( x ) | > σ/ } ) ≤ meas( { x : | X j ∈ Z H l,j ∗ f ( x ) | > σ/ } \ ( [ k ∈ N I k ) c ) + meas( [ k ∈ N I k ) . d meas( { x : | X j ∈ Z H l,j ∗ f ( x ) | > σ/ } \ ( [ k ∈ N I k ) c ) + l − α +1 k f k σ . (a)Now since the mean value of f over I k vanishes, we have Z ( S k ∈ N I k ) c | X j ∈ Z H l,j ∗ f ( x ) | dx ≤ X k ∈ N Z I k Z ( S k ∈ N I k ) c | X j ∈ Z H l,j ( x − y ) − H l,j ( x − a k ) | dx ! | f ( y ) | dy . d l − α +1 X k ∈ N Z I k | f ( y ) dy | . d l − α +1 k f k , (b)provided we prove that(c) Z ( S k ∈ N I k ) c | X j ∈ Z H l,j ( x − y ) − H l,j ( x − a k ) | dx . d l − α +1 , y ∈ I k . COTLAR TYPE MAXIMAL FUNCTION ASSOCIATED WITH FOURIER MULTIPLIERS 8 We postpone the proof of (c) in order to conclude the estimates. By (a), (b) and (c), weobtain(d)meas( { x : | X j ∈ Z H l,j ∗ f ( x ) | > σ/ } ) . d σ − l − α +1 k f k + σ − l − α +1 k f k . d σ − l − α +1 k f k . Set p = 2 l + 4. Then by Theorem 2.3, we have k S l f k . d l α − k f k L , and hence σ meas( { x : | X j ∈ Z H l,j ∗ f ( x ) | > σ/ } ) ≤ k X j ∈ Z H l,j ∗ f k . l − α +1) k f k = l − α +1) X k ∈ N | I k | − (cid:12)(cid:12)(cid:12) Z I k f ( x ) dx (cid:12)(cid:12)(cid:12) + Z ( S k ∈ N I k ) c | f ( x ) | dx ! . d l − α +1 σ n X k ∈ N (cid:12)(cid:12)(cid:12) Z I k f ( x ) dx (cid:12)(cid:12)(cid:12) + Z ( S k ∈ N I k ) c | f ( x ) | dx o which enables us to conclude(e) meas( { x : | X j ∈ Z H l,j ∗ f ( x ) | > σ/ } ) . l − α +1 k f k σ . Combining the estimates (d) and (e) yieldsmeas( { x : | X j ∈ Z H l,j ∗ f ( x ) | > σ } ) . d l − α +1 k f k σ . There remains the proof of (c). It is sufficient to prove that X j ∈ Z Z | x |≥ t | H l,j ( x − y ) − H l,j ( x ) | dx . l − α +1 ( | y | ≤ t, t > . Fix t > 0. Let m ∈ Z such that | t | ∼ m . Now for j > − m , Z | x |≥ t | H l,j ( x − y ) − H l,j ( x ) | dx ≤ Z | x |≥ t | H l,j ( x ) | dx ≤ Z | x |≥ m | K l,j ∗ F − (Ψ j )( x ) | l,j ( x ) dx. Now we observe that for the last term to be non-zero, 2 l − j ≥ m . Hence we have X j> − m Z | x |≥ t | H l,j ( x − y ) − H l,j ( x ) | dx ≤ X − m The following theorem now follows almost immediately using standard L p − interpolationtheory. Theorem 4.2. T l defines a bounded operator on L p for p ∈ (1 , ∞ ) , with k T l k L p . d max ( p, ( p − − Bl − α +1 for l > . We also have k T k L p . d max ( p, ( p − − B. Proof. The operator T l is bounded on L (by Theorem 2.3) and maps L to L (1 , ∞ ) (byTheorem 4.1), with norm Bl − α +1 (norm B for l = 0). Interpolating between the L and L spaces then yields the required norm for the operator acting on L p with p ∈ (1 , p ∈ (2 , ∞ ) as well. This concludes the proofof the theorem. (cid:3) Boundedness on L p for large p In this section, we investigate the properties of the truncated operator T ln f := P j ≤ n H l,j ∗ f (for n ∈ Z and l ≥ L p → L p bounds for the associated max-imal operator T l ∗ f = sup | T ln f | . Let M [ f ] denote the standard Hardy-Littlewood Maximalfunction.We wish to show that for l ≥ 0, the operator T l ∗ is bounded on L p for 1 < p < ∞ withthe corresponding operator norm . p Bl − α +1 ( . p B for l = 0). We will achieve this byproving a Cotlar type inequality T l ∗ f ( x ) . d − δ ) /q ( M ( | T l f | q )( x )) /q + 2 ldq l − α +1 (1 + δ − /q ) B ( m )( M ( | f | q )( x )) /q . for each T l ∗ and for a large enough exponent q (here 0 < δ ≤ / q = q l ∼ l will work. In particular, q l → ∞ as l → ∞ . Using this inequality, wecan conclude that T l ∗ is bounded (with norm . p Bl − α +1 ) for all p ∈ ( p l , ∞ ).The following lemma is the main step in establishing the Cotlar type inequality. Theideas used are similar to Lemma A.1 in [4]. Lemma 5.1. Fix ˜ x ∈ R d , n ∈ Z and q > . Let g ( y ) = f ( y ) B (˜ x, − n ) ( y ) and h = f − g .Then we have(i) | T ln g (˜ x ) | . ( B ( M [ f q ](˜ x )) /q l = 0 . l > . (ii) | T ln h (˜ x ) − T l h (˜ x ) | . ( l = 0 .B ld/q l − α +1 ( M [ f q ](˜ x )) /q l > . COTLAR TYPE MAXIMAL FUNCTION ASSOCIATED WITH FOURIER MULTIPLIERS 10 (iii) For | w − ˜ x | ≤ − n − , we have | T l h (˜ x ) − T l h ( w ) | . ( B ( M [ f q ](˜ x )) /q l = 0 .B ld/q l − α +1 ( M [ f q ](˜ x )) /q l > . Proof. We may assume without loss of generality that B = 1. To prove (i), we consider for j ≤ n | H l,j ∗ g (˜ x ) | ≤ Z | ˜ x − y |≤ − n | H l,j (˜ x − y ) g ( y ) | dy ≤ k K l,j ∗ F − (Ψ j )(˜ x − y ) k L q ′ Z | ˜ x − y |≤ − n | g ( y ) | q | l,j (˜ x − y ) | q dy ! /q . We note that l,j (˜ x − y ) is supported around ˜ x in either a dyadic annulus of radius ∼ − j + l for l > − j for l = 0. As j ≤ n , the second term above is non-zero onlywhen l = 0. In this case, we estimate | T ,j ∗ g (˜ x ) | . jd/q (log 2) − α − nd/q nd Z | ˜ x − y |≤ − n | g ( y ) | q dy ! /q (using Proposition 3 . . ( j − n ) d/q ( M [ g q ](˜ x )) /q . Summing up in j < n , the assertion now follows as | g | ≤ | f | .For (ii), we observe that | T ln h (˜ x ) − T l (˜ x ) | ≤ P j>n | H l,j ∗ h (˜ x ) | . For j > n , we then have | H l,j ∗ h (˜ x ) | ≤ Z | ˜ x − y |≥ − n | H l,j (˜ x − y ) h ( y ) | dy ≤ k K l,j ∗ F − (Ψ j )(˜ x − y ) k L q ′ Z | ˜ x − y |≥ − n | h ( y ) | q | l,j ((˜ x − y )) | q dy ! /q . Again, using the support property of l,j , we observe that the second term above is non-zeroonly when l > j < l + n . For each j ∈ ( n, l + n ), we estimate | H l,j ∗ h (˜ x ) |≤ jd/q (log (2 + 2 l )) − α Z | ˜ x − y |≥ − n | h ( y ) | q | l,j (2 j (˜ x − y )) | q dy ! /q (using Proposition 3 . . jd/q l − α Z | ˜ x − y |∼ − l + j | h ( y ) | q dy ! /q ≤ jd/q l − α ( − j + l ) d/q ( − j + l ) d Z | ˜ x − y |≤ − l + j | h ( y ) | q dy ! /q ≤ l − α ld/q ( M [ h q ](˜ x )) /q . Summing up in n < j < n + l and noting that | h | ≤ | f | , we get X j>n | H l,j ∗ h (˜ x ) | . ld/q l − α +1 ( M [ f q ](˜ x )) /q . Now for (iii), we consider the terms H l,j ∗ h (˜ x ) − H l,j ∗ h ( w ) separately for j ≤ n and j > n .The sum P j>n | H l,j ∗ h (˜ x ) | was already dealt with in (ii) and like before, only matters for COTLAR TYPE MAXIMAL FUNCTION ASSOCIATED WITH FOURIER MULTIPLIERS 11 l > 0. Since | w − ˜ x | ≤ − n − we have | w − y | ≈ | ˜ x − y | for | ˜ x − y | ≥ − n and for non-zero l ,the previous calculation leads to X j>n | H l,j ∗ h ( w ) | . ( l = 0 .B ld/q l − α +1 ( M [ f q ](˜ x )) /q l > . It remains to consider the terms for j ≤ n . We write H l,j ∗ h (˜ x ) − H l,j ∗ h ( w ) = Z Z | ˜ x − y |≥ − n D ˜ x − w, ∇ ( K l,j ∗F − (Ψ j ))( w + s (˜ x − w ) − y ) E h ( y ) dyds. Since | w − ˜ x | ≤ n − we can replace | w + s (˜ x − w ) − y | in the integrand with | ˜ x − y | . AlsoProposition 3.1 yields |∇ ( K l,j ∗ F − (Ψ j ))(˜ x − y ) | ≤ j | ( K l,j ∗ F − (Ψ j ))(˜ x − y ) | . Thus we have | H l,j ∗ h (˜ x ) − H l,j ∗ h ( w ) | . j | ˜ x − w | Z | ˜ x − y |≥ − n | ( K l,j ∗ F − (Ψ j ))(˜ x − y ) h ( y ) | dy ≤ j − n − k ( K l,j ∗ F − (Ψ j ))(˜ x − y ) k L q ′ Z | ˜ x − y |≥ − n | h ( y ) | q | l,j ((˜ x − y )) | q dy ! /q . Proposition 3.1 now gives us | H l,j ∗ h (˜ x ) − H l,j ∗ h ( w ) | . j − n − jd/q (log (2 + 2 l )) − α Z | ˜ x − y |∼ − j + l | h ( y ) | q dy ! /q . j − n − jd/q (log (2 + 2 l )) − α ( − j + l ) d/q ( M [ h q ](˜ x )) /q . ( ld/q j − n − ( M [ f q ](˜ x )) /q l = 0 . ld/q j − n − l − α ( M [ f q ](˜ x )) /q l > . Summing in j ≤ n leads to X j ≤ n | H l,j ∗ h (˜ x ) − H l,j ∗ h ( w ) | . ( ld/q ( M [ f q ](˜ x )) /q l = 0 . ld/q l − α ( M [ f q ](˜ x )) /q l > . (cid:3) We can now prove the main result of this section. Proposition 5.2. Let α > , q > and B ( m ) be as in (6). Let f be a Schwartz function.Then for almost every x and for < δ ≤ / , we have T l ∗ f ( x ) . − δ ) /q ( M ( | T l f | q )( x )) /q + C d A l (1 + δ − /q )( M ( | f | q )( x )) /q where A l = ( B l = 0 B ld/q l − α +1 l > . COTLAR TYPE MAXIMAL FUNCTION ASSOCIATED WITH FOURIER MULTIPLIERS 12 Proof. The proof is essentially the same as that of an analogous result in [4], which is in turna modification of the argument for the standard Cotlar inequality regarding the truncationof singular integrals (see [8], sec 1.7).Fix ˜ x ∈ R d and n ∈ Z and define g , h and q as in the previous lemma. For w (to bechosen later) with | w − ˜ x | ≤ − n − we can write T ln f (˜ x ) = T ln g (˜ x ) + ( T ln − T l ) h (˜ x ) + T l h (˜ x )= T ln g (˜ x ) + ( T ln − T l ) h (˜ x ) + T l h (˜ x ) − T l h ( w ) + T l f ( w ) − T l g ( w ) . (14)By Lemma 5.1, we have | T ln g (˜ x ) | + | ( T ln − T l ) h (˜ x ) | + | T l h (˜ x ) − T l h ( w ) | . A l ( M [ f q ](˜ x )) /q . All that remains in this case is to consider the term T l f ( w ) − T l g ( w ) for w in a substantialsubset of B (˜ x, − n − ). By Theorem 4.1 we have that for all f ∈ L q ( R d ) and all λ > { x : | T l f ( x ) | > λ } ≤ A ql λ − q k f k qq . Now let δ ∈ (0 , / 2) and consider the setΩ n (˜ x, δ ) = { w : | w − ˜ x | < − n − , | T l g ( w ) | > d/q δ − /q A l ( M [ f q ](˜ x )) /q } . In (14) we can estimate the term | T l g ( w ) | by A l d/q δ − /q ( M [ f q ](˜ x )) /q whenever w ∈ B (˜ x, − n − ) \ Ω n (˜ x, δ ). Hence we obtain(15) | T ln f (˜ x ) | . inf w ∈ B (˜ x, − n − \ Ω n (˜ x,δ ) | T l f ( w ) | + C d A l (1 + δ − /q )( M [ f q ](˜ x )) /q . By the weak type inequality for T l we havemeas(Ω n (˜ x, δ )) ≤ A ql k g k qq d δ − A ql M [ f q ](˜ x ) = δ d M [ f q ](˜ x ) Z | x − y |≤ − n | f ( y ) | q dy ≤ δ − d meas( B (˜ x, − n ) = δ meas( B (˜ x, − n − ) . Hence meas( B (˜ x, − n − ) \ Ω n (˜ x, δ )) ≥ (1 − δ )meas( B (˜ x, − n − ) and thusinf w ∈ B (˜ x, − n − \ Ω n (˜ x,δ )) | T l f ( w ) | ≤ B (˜ x, − n − \ Ω n (˜ x, δ )) Z B (˜ x, − n − ) | T l f ( w ) | q ! /q ≤ − δ )meas( B (˜ x, − n − )) Z B (˜ x, − n − ) | T l f ( w ) | q ! /q . We obtain | T ln f (˜ x ) | . − δ ) /q ( M ( | T l f | q )(˜ x )) /q + A l (1 + δ − /q )( M ( | f | q )(˜ x )) /q uniformly in n , which implies Proposition 5.2. (cid:3) The above proposition, in conjunction with Theorem 4.2 immediately leads to the fol-lowing: Theorem 5.3. T l ∗ is bounded on L p for p ∈ ( l + 2 , ∞ ) . COTLAR TYPE MAXIMAL FUNCTION ASSOCIATED WITH FOURIER MULTIPLIERS 13 Proof. Fix δ = 1 / q = l + 2. We note that both f → T l f and f → ( M [ f q ]) /q arebounded operators on L p for p ∈ ( l + 2 , ∞ ), with operator norms bounded by pBl − α +1 ( pB for l = 0) and p/ ( p − l − 2) respectively (upto multiplication by a dimensional constant).Hence, by Proposition 5.2, we obtain k T l ∗ k L p . d ( B, l = 0 l − α +1 p p − l − B, l > . (cid:3) Weak (1,1) Boundedness of the Maximal Operator In this section, we will prove that each of the pieces T l ∗ is of weak type (1 , . d Bl − α +1 ( . d B for l = 0). Combining this result with Theorem 5.3 andinterpolating, we will obtain bounds on the operator norm of T l ∗ on L p for all p ∈ (1 , ∞ ),with a decay of l − α +2 . This will allow us to achieve our final goal of summing up the piecestogether to get bounds on the operator S ∗ .The proof we give here is essentially the same as the one for Theorem 4.1, except forone notable difference. In order to obtain a weak bound for the "good" function in theCalderón-Zygmund decomposition, we use the bounds on L p l (as given by Theorem 5.3,with p l = 4 l ) in place of those on L , noting that the operator norm in the former case is . Bl − α +2 (for l > l in the weak (1,1) norm of T l ∗ goesup by 1. Instead of repeating the entire argument, we sketch an outline. Theorem 6.1. T l ∗ is weak (1,1) bounded with k T l ∗ k L → L , ∞ . ( B, l = 0 ,Bl − α +2 , l > . Proof. Let n ∈ Z and f ∈ L ( R d ). Fix σ > 0. We sketch the proof for l > l = 0 proceeds in almost the same way). Also, we might assume B = 1. As before, we makea Calderón-Zygmund decomposition of f at the level l α − σ . For the "good" function f , weuse the bound k T ln k L Pl . Bl − α +2 on L p l with p l = 4 l for l > k T n k L . B on L . Weobtain for l > k T ln f k L , ∞ . Bl − α +2 k f k L (and the corresponding result for l = 0). The argument for the "bad" part f proceeds theexact way as in proof of Theorem 5.3, with the index n playing no real role, and we get for l > k T ln f k L , ∞ . Bl − α +1 k f k L (and the corresponding result for l = 0). The result then follows by combining the twoestimates followed by taking a supremum over n . (cid:3) Theorems 5.3 and 4.1 together via L p interpolation lead to Theorem 6.2. T l ∗ is bounded on L p for p ∈ (1 , ∞ ) , with k T l ∗ k L p . d B max ( p, ( p − − ) , l = 0 , p ∈ (1 , ∞ ) B max ( p, ( p − − ) l − α +2 , l > , p ∈ (1 , l ) Bpl − α +1 , l > , p ∈ [4 l, ∞ ) . COTLAR TYPE MAXIMAL FUNCTION ASSOCIATED WITH FOURIER MULTIPLIERS 14 Proof. The result for l = 0 is a clear outcome of L p interpolation and the estimates provedearlier. For l > p ∈ [4 l, ∞ ), it follows easily from Theorem 5.3 and the observationthat p / ( p − l − . p for p ≥ l . For l > p ∈ (1 , l ), it is an outcome of interpolatingbetween the weak (1,1) estimate in Theorem 6.1 and the one contained in Theorem 5.3 for p = 4 l , again making the observation that p / ( p − l − . p = 4 l . (cid:3) Finally, we give the proof of Theorem 1.1. Proof. It is enough to prove the theorem for S n for a fixed n ∈ Z , as the result then followsby taking the supremum over n ∈ Z . For p ∈ (1 , ∞ ) and a Schwartz function f , we have k S n f k L p ≤ X l ≥ k S ln f k L p ≤ X l ≥ k T ln f k L p + X l ≥ k X j ≤ n K l,j ∗ F − (1 − Ψ j ) ∗ f k L p . Using Theorem 6.2 for the first sum and Theorem 2.2 for the second one (and noting thatsumming up for j ≤ n instead of j ∈ Z does not affect the proof), we get k S n f k L p . B max ( p, ( p − − ) X l ≥ ( l − α +2 + 2 − l ) . p B. For weak (1,1) boundedness, we argue in a similar way, only using Theorem 6.1 this timein place of Theorem 6.2. (cid:3) References [1] Anthony Carbery, Variants of the Calderón-Zygmund theory for L p -spaces , Revista MatemáticaIberoamericana (1986), no. 4, 381–396.[2] Michael Christ, Loukas Grafakos, Petr Honzík, and Andreas Seeger, Maximal functions associated withFourier multipliers of Mikhlin-Hörmander type , Mathematische Zeitschrift (2005), no. 1, 223–240.[3] Loukas Grafakos, Classical Fourier analysis , Vol. 2, Springer, 2008.[4] Shaoming Guo, Joris Roos, Andreas Seeger, and Po-Lam Yung, A maximal function for families ofHilbert transforms along homogeneous curves , arXiv preprint arXiv:1902.00096 (2019).[5] Lars Hörmander, Estimates for translation invariant operators in L p spaces , Acta Mathematica (1960Dec), no. 1, 93–140.[6] Andreas Seeger, Some inequalities for singular convolution operators in L p -spaces , Transactions of theAmerican Mathematical Society (1988), no. 1, 259–272.[7] Elias M Stein, Singular integrals and differentiability properties of functions (pms-30) , Vol. 30, Princetonuniversity press, 2016.[8] Elias M Stein and Timothy S Murphy, Harmonic analysis: real-variable methods, orthogonality, andoscillatory integrals , Vol. 3, Princeton University Press, 1993.[9] Elias M Stein and Antoni Zygmund, Boundedness of translation invariant operators on Hölder spacesand L p -spaces , Annals of Mathematics (1967), 337–349.[10] Charles McCarthy Walter Littman and Nestor Riviere, L p -multiplier theorems , Studia Mathematica (1968), 193–217.[11] Thomas H Wolff, Lectures on harmonic analysis , Vol. 29, American Mathematical Soc., 2003. Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr, Madison,WI-53706, USA E-mail address ::