Paraproducts for bilinear multipliers associated with convex sets
PPARAPRODUCTS FOR BILINEAR MULTIPLIERSASSOCIATED WITH CONVEX SETS
OLLI SAARI AND CHRISTOPH THIELE
Abstract.
We prove bounds in the local L range for exotic paraprod-ucts motivated by bilinear multipliers associated with convex sets. Oneresult assumes an exponential boundary curve. Another one assumes ahigher order lacunarity condition. Introduction
Given a bounded measurable function m in the plane, we define the associ-ated bilinear Fourier multiplier operator acting on a pair of one dimensionalSchwartz functions ( f, g ) by B m ( f, g )( x ) := (cid:90) (cid:90) R m ( ξ, η ) (cid:98) f ( ξ ) (cid:98) g ( η ) e πi ( ξ + η ) x dξdη. (1.1)The constant multiplier m = 1 reproduces the pointwise product that maps( f, g ) to f g , and it is a fundamental question to determine whether and forwhich m H¨older’s inequality, true for the pointwise product, extends to B m .We focus on bounds (cid:107) B m ( f, g ) (cid:107) p (cid:48) ≤ C p ,p ,m (cid:107) f (cid:107) p (cid:107) g (cid:107) p (1.2)in the open local L region2 < p , p , p < ∞ , p + 1 p + 1 p = 1 , (1.3)where p (cid:48) = p / ( p −
1) denotes the dual exponent.A well-understood example is the classical linear Mikhlin multiplier m in two dimensions. It leads to the theory of paraproducts, for which werefer to the textbooks [21, 23] and the original references therein. BeyondMikhlin multipliers, the bilinear Hilbert transforms are the most prominentexamples. Their essence is captured by choosing m to be the characteristicfunction of a half-plane. The bounds (1.2) and (1.3) for this case wereestablished in [13, 14]. They also hold with a constant independent of theslope and location of the associated half-plane [10, 29], and the range ofexponents can be extended beyond the local L range as in [8, 15, 16].Our focus here is on multipliers m which are characteristic functions ofconvex sets rather than just half-planes as in the case of the bilinear Hilberttransforms. Certain curved regions to be discussed shortly, most notablythe epigraph of a parabola, were considered in [22], and the bilinear discmultiplier has been studied in [11]. The bounds (1.2) and (1.3) are knownfor the disc. The results in [11] also imply the previously known uniform a r X i v : . [ m a t h . C A ] F e b OLLI SAARI AND CHRISTOPH THIELE local L bounds for the half-plane multipliers as can be seen by using theinvariance of the bounds under dilation and translation of the convex setand approximating half-planes by large discs. In [5], a lacunary polygon isdiscussed.A further list of examples of convex sets illuminating the increasinglycomplicated structure can be given as follows. We say a line has degeneratedirection if it is orthogonal to one of the three vectors (0 , ,
0) and (1 , { ( ξ, η ) : 0 ≤ ξ ≤ , γ ( ξ ) ≤ η } for someconvex function γ : [0 , → [0 ,
1] such that all tangent lines atpoints ( ξ, γ ( ξ )) with 0 < ξ < a and 2 a forsome 0 < a < C boundary curve such that the cur-vature of the boundary is nonzero at every boundary point with adegenerate tangent line.(4) A bounded convex set C such that every boundary point with adegenerate tangent line has tangent lines for every direction in aneighborhood of this degenerate direction.(5) The convex set { ( ξ, η ) : ξ ≤ , ξ ≤ η } .(6) The convex set { ( ξ, η ) : 0 ≤ ξ ≤ , γ ( ξ ) ≤ η } , where γ is a monotoneincreasing convex function mapping [0 ,
1] to [0 , − ] such that the set { γ (2 − j ) , j ∈ N } is multi-lacunary. See Definition 1.3 below for thedefinition of multi-lacunarity.(7) A general bounded convex set C .The first four examples are known to satisfy bounds (1.2) and (1.3).The first example is easily reduced to the bilinear Hilbert transforms. Thereduction works by iteratively cutting the multiplier by a line parallel to adegenerate direction, which results in two new multipliers, whose operatorscan be expressed through the original bilinear multiplier operator and pre-or post-composition with a linear multiplier such as the Riesz projection.The second example is a perturbation of the bilinear Hilbert transforms.Thanks to the comparable upper and lower bounds on the slope, the methodsfor the bilinear Hilbert transform can be adapted to this situation [22, 11].The third example includes the case of the parabola and the disc [22, 11].It requires an additional technique to put together infinitely many piecesas in the second example as well as an additional bound for a central piecenear each degenerate direction. These additional pieces are what we callexotic paraproducts. The techniques of [22, 11] still apply at this levelof generality. Likewise, the fourth example can be handled with similarmethods. Such multipliers are more general away from the critical directionsbut have strongly regulated behavior at the degenerate directions.In this paper, we study relaxation of the additional conditions near thedegenerate directions. Beginning with the proof of the first bounds for thebilinear Hilbert transform, the common approach to understand bilinearmultipliers associated with characteristic functions has been to decompose ARAPRODUCTS FOR CONVEX SETS 3 them into smoother multipliers, paraproducts, which are singular only at asingle boundary point instead of a one-dimensional set. As this boundarypoint comes closer to a degenerate tangent direction, the relevant paraprod-ucts undergo a deformation. At a boundary point with degenerate tangentdirection, one encounters an entirely exotic paraproduct, whose structure isclosely tied to the behaviour of the boundary in the vicinity of that point. Inthe present paper, in particular due to the local L range, it seems prudentto consider rougher exotic paraproducts which correspond to characteristicfunctions of certain staircase sets.One of the results in the present paper provides bounds for the exoticparaproduct associated with case five of the previous list. Theorem 1.1.
Let p , p , p be as in (1.3) . Define m ( ξ, η ) := (cid:88) j ∈ N [ − ( j +1) , − j ) ( ξ )1 [2 − j , ( η ) . Then the operator (1.1) satisfies the a priori estimate (1.2) . Here we are also able to complete the passage from a paraproduct estimateto a multiplier bound as in [22, 11] and reduce the bounds for case five tobounds for case two.
Corollary 1.2.
Let p , p , p be as in (1.3) . Let C be the convex set { ( ξ, η ) : ξ ≤ , ξ ≤ η < } . Then B m as in (1.1) with m = 1 C satisfies the a priori bounds (1.2) . The particular cut-offs at 0 and 1 are not important in this theorem.One also obtains bounds for similar convex sets with constraints of thetype a ξ ≤ η with a (cid:54) = 2 by applying translation and isotropic dilation tothe multiplier. Hence the number 2 has no fundamental importance in thecorollary.Our second result proves bounds for exotic paraproducts related to thesixth case of our list. Definition 1.3 (Multi-lacunarity) . Let b be a non-negative integer. Wecall a finite set X of real numbers (0 , b )-lacunary, if it consists of a singleelement.Let d be a non-negative integer and assume we have already defined ( d, b )-lacunarity. We call a finite set X of real numbers ( d + 1 , b )-lacunary, if itcan be partitioned into two sets L and O such that L is ( d, b )-lacunary andfor any pair of different points ξ, ξ (cid:48) in O we havedist( ξ, ξ (cid:48) ) ≥ − b dist( ξ, L ) . Theorem 1.4.
Let b, d ≥ be integers and let p , p , p be as in (1.3) .Assume that we have sequences ( η j ) j ∈ N , ( ζ j ) j ∈ N , ( ξ j ) j ∈ N OLLI SAARI AND CHRISTOPH THIELE such that for all j − j ≤ η j ≤ ζ j < − j , (1.4)0 ≤ ξ j + 2 − j ≤ ξ j − . (1.5) Assume the image X of the sequence ( ξ j ) be ( d, b ) -lacunary. Then the mul-tiplier operator B m as in (1.1) with m ( ξ, η ) = (cid:88) j (0 ,ξ j ) ( ξ )1 ( η j ,ζ j ) ( η ) (1.6) satisfies the estimate (1.2) with constant C p ,p ,m = C p ,p ,b,d that depends on m only through b and d . In this case, however, we are not able to complete the program as theconvex set lacks the regularity relevant for techniques in [22, 11] to work.Moreover, the bounds (1.2) and (1.3) for the general case seven appear en-tirely beyond the techniques known to us. We are without any bias towardsvalidity or invalidity of these bounds.Standard paraproducts come with exponentially growing sequences ξ j , η j and ζ j , all other configurations with an increasing sequence ξ j and aninterlaced pair of increasing sequences ( η j , ζ j ) may be considered exotic.We emphasize that the point here rests in the growth conditions on thesesequences, the characteristic functions in place of smoother versions foundelsewhere in the literature being a minor modification in the consideredrange of exponents.Previous research on various assumptions on the sequences can be foundin [25, 24, 18, 19, 20], partially in connection with bilinear Hilbert transformson curves, but none of these references appears to go beyond the case d = 1of multi-lacunarity. The multi-lacunarity assumption also appears in similarcontext elsewhere in harmonic analysis. For example, it provides a sharpcondition on sets of directions for which the directional maximal operator isunbounded [28, 1]. These are deep facts building on a long history of relatedresults. Directional Hilbert transforms in the plane with multi-lacunarityassumptions are considered in [6]. Multi-lacunary sets of directions in higherdimensions appear in [3, 26, 7]. Acknowledgement.
The authors were funded by the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation) under Germany’s Ex-cellence Strategy – EXC-2047/1 – 390685813 as well as SFB 1060. Part ofthe research was carried out while the authors were visiting the Oberwol-fach Research Institute for Mathematics, the workshop Real and HarmonicAnalysis. 2.
Proof of Theorem 1.1
Throughout this section, we fix p , p , p as in (1.3). We work through asequence of lemmata, at the end of which we are ready to prove Theorem ARAPRODUCTS FOR CONVEX SETS 5 V r norm for a function h on Z or R by (cid:107) h (cid:107) V r := sup x | h ( x ) | + sup N, x Let ρ, φ be Schwartz functions supported in [ − , . Let r ∈ R and α ∈ { , } . Let (cid:15) > be small enough so that (cid:12)(cid:12)(cid:12)(cid:12) − p (cid:12)(cid:12)(cid:12)(cid:12) < 12 + (cid:15) . Define the operator B ( f, g )( x ) := (cid:90) R (cid:98) f ( ξ ) (cid:98) g ( η ) e πi ( ξx + ηr ) (cid:88) j ∈ N − αj ρ ( ξ + j ) φ (2 j η ) dξdη and the averages Ag ( r )( j ) := (cid:90) (cid:98) g ( η ) φ (2 j η ) e πiηr dη. Then there is a constant C ρ,φ,p ,(cid:15) such that for all Schwartz functions f and g , we have (cid:107) B ( f, g ) (cid:107) p ≤ C ρ,φ,p ,(cid:15) (cid:107) f (cid:107) p (cid:107) Ag ( r ) (cid:107) V (cid:15) . (2.2) Proof. We decompose ρ into a Fourier series on an interval of length four ρ ( ξ ) = 1 [ − , ( ξ ) (cid:88) k ∈ Z / (cid:98) ρ k e πikξ . Splitting B correspondingly and using the rapid decay of (cid:98) ρ k , we see it sufficesto prove bounds analogous to (2.2) on (cid:90) R (cid:98) f ( ξ ) (cid:98) g ( η ) e πi ( ξx + ηr ) (cid:88) j ∈ N − αj [ − , ( ξ + j ) e πik ( ξ + j ) φ (2 j η ) dξdη, uniformly in k ∈ Z / j into four sums depending on the congruence classof j modulo four, and we notice that the factor e πikj is constant in j in each OLLI SAARI AND CHRISTOPH THIELE of the four sums. Using further that the translation x → x + k leaves the L p norm invariant, we see it suffices to prove bounds analogous to (2.2) on (cid:90) R (cid:98) f ( ξ ) (cid:98) g ( η ) e πi ( ξx + ηr ) (cid:88) j ∈ N , j ≡ j mod 4 − αj [ − , ( ξ + j ) φ (2 j η ) dξdη for a fixed parameter j .We identify this expression as a linear multiplier of the form (2.1) appliedto f . The multiplier symbol is n ( ξ ) = (cid:88) j ∈ N , j ≡ j mod 4 − αj [ − , ( ξ + j ) Ag ( r )( j ) , and it thus suffices to show (cid:107) n (cid:107) M p ≤ C ρ,φ,p ,(cid:15) (cid:107) Ag ( r ) (cid:107) V (cid:15) . (2.3)We apply the following well-known control of the multiplier norm by vari-ation norms proven by Coifman, Rubio de Francia and Semmes [4]. Theorem 2.2. Let n be a measurable function on R , then (cid:107) n (cid:107) M p ≤ C p,r (cid:107) n (cid:107) V r , provided < p < ∞ and | / − /p | ≤ /r . For α = 0, inequality (2.3) follows if one identifies n as a step functionconstant on intervals of length 4, taking precisely the value Ag ( r )( j ) in the j -th interval, counted in natural order, and taking the value 0 outside theunion of these intervals.For α = 1, we observe that for any sequence a ( k ) tending to zero, thevariation norm of the sequence b ( k ) = 2 − k a ( k ) is bounded by a constanttimes the supremum norm of a , which in turn is controlled by the variationnorm of a . This fact following from a plain application of the triangleinequality completes the proof of Lemma 2.1. (cid:3) The next lemma passes to a localized version of actual bilinear multipli-ers. The parameter r in the exponent disappears, but we introduce a newlocalization parameter s . In what follows, we use the translation operatordefined by T s h ( x ) = h ( x − s ) . Lemma 2.3. Let χ, ρ, φ be Schwartz functions supported in [ − , and B ( f, g )( x ) := T s χ ( x ) (cid:90) R (cid:98) f ( ξ ) (cid:98) g ( η ) e πi ( ξ + η ) x (cid:88) j ∈ N ρ ( ξ + j ) φ (2 j η ) dξdη. (2.4) Given the averages Ag ( r )( j ) and a parameter (cid:15) as in Lemma 2.1, there is aconstant C χ,ρ,φ,p ,(cid:15) so that for all Schwartz functions f and g we have (cid:107) B ( f, g ) (cid:107) p ≤ C χ,ρ,φ,p ,(cid:15) (cid:107) f (cid:107) p M ( (cid:107) Ag (cid:107) V (cid:15) )( s ) . ARAPRODUCTS FOR CONVEX SETS 7 Proof. We apply the support assumption on χ and the fundamental theoremof calculus as well as the product rule to write (2.4) as (cid:90) xs − T s χ (cid:48) ( r ) (cid:90) R (cid:98) f ( ξ ) (cid:98) g ( η ) e πi ( ξx + ηr ) (cid:88) j ∈ N ρ ( ξ + j ) φ (2 j η ) dξdηdr (2.5)+ (cid:90) xs − T s χ ( r ) (cid:90) R (cid:98) f ( ξ ) (cid:98) g ( η )2 πiηe πi ( ξx + ηr ) (cid:88) j ∈ N ρ ( ξ + j ) φ (2 j η ) dξdη. (2.6)We first discuss the term (2.5). We estimate the integral in r with the L norm to obtain the bound (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) T s χ (cid:48) ( r ) (cid:90) R (cid:98) f ( ξ ) (cid:98) g ( η ) e πi ( ξx + ηr ) (cid:88) j ∈ N ρ ( ξ + j ) φ (2 j η ) dξdη (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( r ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( x ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) T s χ (cid:48) ( r ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:90) R (cid:98) f ( ξ ) (cid:98) g ( η ) e πi ( ξx + ηr ) (cid:88) j ∈ N ρ ( ξ + j ) φ (2 j η ) dξdη (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( r ) for the L p ( x ) norm of (2.5). We used Minkowski’s inequality and the condi-tion 1 < p here. Next we estimate the inner norm with Lemma 2.1. Setting α = 0 and (cid:15) > C ρ,φ,p ,(cid:15) (cid:13)(cid:13)(cid:13) T s χ (cid:48) ( r ) (cid:107) f (cid:107) p (cid:107) Ag ( r )) (cid:107) V (cid:15) (cid:13)(cid:13)(cid:13) L ( r ) ≤ C ρ,φ,χ,p ,(cid:15) (cid:107) f (cid:107) p M ( (cid:107) Ag (cid:107) V (cid:15) )( s ) . The last inequality follows by recognizing a smooth average over the supportof T s χ (cid:48) , which is near s , and dominating it by the Hardy–Littlewood maximalfunction. This completes the bound for first term (2.5).We rewrite the second term (2.6) as2 πi (cid:90) xs − T s χ ( r ) (cid:90) R (cid:98) f ( ξ ) (cid:98) g ( η ) e πi ( ξx + ηr ) (cid:88) j< j ρ ( ξ − j ) ˜ φ (2 − j η ) dξdη, where the new Schwartz function ˜ φ ( η ) = ηφ ( η ) depends on φ only. We canproceed exactly as with the first term (2.5), but now applying Lemma 2.1with α = 1 and the Schwartz function ˜ φ instead of α = 0 and the Schwartzfunction φ . This results in the desired bound and hence completes the proofof Lemma 2.3. (cid:3) Finally, we can pass to a standard bilinear multiplier estimate. Thisentails getting rid of the localization present in the previous estimate. Themultiplier below is a smooth model of the one in Theorem 1.1. Lemma 2.4. Let ρ and φ be Schwartz functions supported in [ − , and B ( f, g )( x ) := (cid:90) R (cid:98) f ( ξ ) (cid:98) g ( η ) e πi ( ξ + η ) x (cid:88) j ∈ N ρ ( ξ + j ) φ (2 j η ) dξdη. OLLI SAARI AND CHRISTOPH THIELE There is a constant C φ,ρ,p ,p such that for any Schwartz functions f and g (cid:107) B ( f, g ) (cid:107) p (cid:48) ≤ C φ,ρ,p ,p (cid:107) f (cid:107) p (cid:107) g (cid:107) p . Proof. It suffices to prove the dual estimate (cid:90) R h ( x ) B ( f, g )( x ) dx ≤ C φ,ρ,p ,p (cid:107) f (cid:107) p (cid:107) g (cid:107) p (cid:107) h (cid:107) p for all Schwartz functions h . Fixing a suitable normalized non-negativeSchwartz function χ supported on [ − , (cid:90) R h ( x ) B ( f, g )( x ) dx = (cid:90) R h ( x ) T s χ ( x ) B ( f T s + t χ, g )( x ) dsdtdx, (2.7)which can be done because (cid:82) R T s χ ( x ) ds is a constant function as is theintegral over s of T s χ ( x ). Next we seek an estimate for (cid:107) T s χB ( f T s + t χ, g ) (cid:107) p (2.8)for arbitrary real t .For | t | ≤ 4, we use Lemma 2.3 and an epsilon depending only on p toobtain the upper bound (cid:107) T s χB ( f T s + t χ, g ) (cid:107) p ≤ C χ,ρ,φ,p (cid:107) f T s + t χ (cid:107) p M ( (cid:107) Ag (cid:107) V (cid:15) )( s ) . For | t | > 4, we write the Fourier expansion B ( f T s + t χ, g )( x )= (cid:90) R (cid:98) f ( ζ ) (cid:98) χ ( ξ − ζ ) (cid:98) g ( η ) e πi [( ξ + η ) x − ( s + t )( ξ − ζ )] (cid:88) j ∈ N ρ ( ξ + j ) φ (2 j η ) dξdηdζ. (2.9)Integrating by parts twice, we can write the ξ integral in (2.9) as (cid:88) β =1 x − s − t ) (cid:90) R (cid:98) χ β ( ξ − ζ ) e πi [( ξ + η ) x − ( t + s )( ξ − ζ )] (cid:88) j< ρ β ( ξ − j ) dξ, where the three pairs of Schwartz functions ( χ β , ρ β ), β = 1 , , 3, are deter-mined by the product rule. Set˜ χ ( x ) = (cid:88) β =1 | χ β ( x ) | + | χ ( x ) | . Note that | x − s − t | is comparable to | t | when x ∈ [ s − , s + 2], which is theessential domain of integration in (2.7). Inserting this into (2.9) and usingLemma 2.3, we obtain for (2.8) the upper bound C χ,ρ,φ,p t − (cid:107) f T s + t ˜ χ (cid:107) p M ( (cid:107) Ag (cid:107) V (cid:15) )( s ) . Combining the two cases of small and large t , we obtain for (2.8) the bound C χ,ρ,φ,p (1 + t ) − (cid:107) f T s + t ˜ χ (cid:107) p M ( (cid:107) Ag (cid:107) V (cid:15) )( s ) . ARAPRODUCTS FOR CONVEX SETS 9 Turning back to (2.7), we apply H¨older’s inequality on the integral in x and a trivial bound on a factor (cid:107) T s χ (cid:107) p to estimate (2.7) by C χ,p (cid:90) R (cid:107) hT s χ (cid:107) p (cid:107) T s χB ( f T s + t χ, g ) (cid:107) p dsdt ≤ C χ,ρ,φ,p ,p (cid:90) R (cid:107) hT s χ (cid:107) p (1 + t ) − (cid:107) f T s + t ˜ χ (cid:107) p M ( (cid:107) Ag (cid:107) V (cid:15) )( s ) dsdt ≤ C χ,ρ,φ,p ,p (cid:90) R (cid:107) h (cid:107) p (1 + t ) − (cid:107) f (cid:107) p (cid:107)M ( (cid:107) Ag (cid:107) V (cid:15) ) (cid:107) p dt. In the last line, we have applied H¨older’s inequality and noted (cid:90) R (cid:107) hT s χ (cid:107) p p ds = (cid:90) R | h ( x ) T s χ ( x ) | p dxds = C χ,p (cid:107) h (cid:107) p p . The same computation also applies to the factor with L p norm. The integralin t is trivial, and hence we obtain for (2.7) the upper bound C χ,ρ,φ,p ,p (cid:107) h (cid:107) p (cid:107) f (cid:107) p (cid:107)M ( (cid:107) Ag (cid:107) V (cid:15) ) (cid:107) p . It remains to observe (cid:107)M ( (cid:107) Ag (cid:107) V (cid:15) ) (cid:107) p ≤ C φ,p (cid:107) g (cid:107) p . This follows from the Hardy–Littlewood maximal theorem and a well-knownvariational bound stated in the following theorem, whose general formulationwe quote from Jones, Seeger and Wright [12]. Theorem 2.5. Let < p < ∞ and < r < ∞ . Let φ be a Schwartzfunction and define φ t ( x ) = 2 − t φ (2 − t x ) . Define the averaging operators Ah ( x )( t ) = (cid:90) R h ( x − y ) φ t ( y ) dy. Then we have the bound (cid:107)(cid:107) Ah (cid:107) V r (cid:107) L p ≤ C p,r (cid:107) h (cid:107) p . The theorem goes back to L´epingle in the martingale setting [17], and itwas introduced for applications in harmonic analysis by Bourgain [2]. How-ever, the precise formulation we use is from [12]. Note that the convolutionoperator appearing in this Theorem can be written equivalently as a mul-tiplier corresponding to our definition of A . This completes the proof ofLemma 2.4. (cid:3) We are now ready to prove Theorem 1.1. What remains is to pass fromthe smooth model in Lemma 2.4 to the rough paraproduct defined usingcharacteristic functions of intervals instead of smooth Schwartz bumps. Proof of Theorem 1.1. We regroup the multiplier operator of Theorem 1.1as B m ( f, g ) = (cid:88) j ∈ N M ( −∞ , − j − f M [2 − j − , − j ) g. Pick a smooth function ρ supported in [ − , 1) such that (cid:88) k ∈ Z T − k ρ ( k ) = (cid:88) k ∈ Z ρ ( ξ + k ) = 1 . We compare B m with B ˜ m ( f, g ) = (cid:88) j ∈ N (cid:88) k>j M T − k ρ f M [2 − j − , − j ) g. The difference satisfies( B ˜ m − B m )( f, g ) = (cid:88) j ∈ N M T − j − ρ M [ − j − , − j ] f M [2 − j − , − j ) g, and we have the estimate (cid:107) ( B ˜ m − B m )( f, g ) (cid:107) p (cid:48) ≤ (cid:107) ( (cid:88) j ∈ N | M T − j − ρ M [ − j − , − j ] f | ) / (cid:107) p (cid:107) ( (cid:88) j ∈ N | M [2 − j − , − j ) g | ) / (cid:107) p . The second factor is bounded by C p (cid:107) g (cid:107) p by the following well-knownsquare function estimate due to Rubio de Francia [27]. Theorem 2.6. Let I be a collection of pairwise disjoint intervals. Then,for < p < ∞ , there is a constant C p such that for all Schwartz functions f we have (cid:107) ( (cid:88) I ∈I | M I f ( x ) | ) / (cid:107) p ≤ C p (cid:107) f (cid:107) p . To estimate the first factor, we note that the operator M T − j − ρ is dom-inated pointwise by a constant multiple of the Hardy–Littlewood maximalfunction. Using the Fefferman–Stein maximal inequality [9] as well as The-orem 2.6, we can estimate (cid:107) ( (cid:88) j ( M ( T − j − ρ M [ − j − , − j ] f ) ) / (cid:107) p ≤ C ρ,p (cid:107) ( (cid:88) j ( M M [ − ( j +1) , − j ] f ) ) / (cid:107) p ≤ C ρ,p (cid:107) ( (cid:88) j ( M [ − ( j +1) , − j ] f ) ) / (cid:107) p ≤ C ρ,p (cid:107) f (cid:107) p . It then remains to estimate B ˜ m . We cut the intervals [2 − j − , − j ) intotwo equally long halves. For β ∈ { , } , define m β ( ξ, η ) = (cid:88) k ∈ N T − k ρ ( ξ ) (cid:88) j ∈ N ,j 1] + [ − β/ , β/ − / , / 4] + [ − β/ , β/ . Hence the multiplier operator corresponding to (2.10) becomes (cid:88) k ∈ N ( M T − k ρ f )˜ g β − (cid:88) k ∈ N ( M T − k ρ f )( φ β,k ∗ ˜ g β ) . (2.12)The second term in (2.12) is bounded by (cid:107) (cid:88) k ∈ N ( M T − k ρ f )( φ β,k ∗ ˜ g β ) (cid:107) p (cid:48) ≤ C p ,p (cid:107) f (cid:107) p (cid:107) ˜ g β (cid:107) p ≤ C p ,p (cid:107) f (cid:107) p (cid:107) g (cid:107) p , the first inequality following from Lemma 2.4 and the second inequalitybeing a consequence of the following Theorem 2.7 below by Coifman, Rubiode Francia and Semmes [4]. Theorem 2.7. Let n be a bounded function on the real line. If sup j ∈ Z (cid:13)(cid:13) n [2 j , j +2 ) ∪ [ − j , − j +2 ) (cid:13)(cid:13) V ≤ A, then for each < p < ∞ there is a constant C p such that (cid:107) n (cid:107) M p ≤ C p A. This is the sharp version of the H¨ormander–Mikhlin multiplier theorem,a stronger version of Theorem 2.2 above. The assumption on the totalvariation is obviously satisfied by the multiplier in (2.11). It only jumps auniformly bounded number of times in each of the test intervals, each of thejumps being of height one. Hence we have the desired bound for the secondterm in (2.12).For the first term in (2.12), we have the bound (cid:107) (cid:88) k ∈ N ( M T − k ρ f )˜ g β (cid:107) p (cid:48) ≤ (cid:107) (cid:88) k ∈ N ( M T − k ρ f ) (cid:107) p (cid:107) ˜ g (cid:107) p ≤ C ρ,p (cid:107) f (cid:107) p (cid:107) g (cid:107) p where we again used Theorem 2.7 for the second factor. The first factor isestimated by Theorem 2.2 as the corresponding multiplier is locally constantexcept for in [ − , 1] where it is a smooth function of total variation one. (cid:3) Proof of Corollary 1.2. We write the multiplier in the corollary as m = (cid:88) j ∈ N [ − ( j +1) , − j ) ( ξ )1 [2 − j , ( η ) + (cid:88) j ∈ N m j , (2.13)where m j is the characteristic function of the set { ( ξ, η ) : − ( j + 1) ≤ ξ ≤ − j, ξ ≤ η < − j } . (2.14)The bilinear multiplier corresponding to the first summand in (2.13) isbounded by Theorem 1.1. To estimate the second summand, we pair the bilinear multiplier with anarbitrary function h ∈ L p and write (cid:90) R (cid:88) j ∈ N B m j ( f, g )( x ) h ( x ) dx = (cid:88) j ∈ N (cid:90) R B m j ( M [ − ( j +1) , − j ) f, M [2 − ( j +1) , − j ) g )( x ) × M [ j, ( j +1))+[ − − j , − − ( j +1) ) h ( x ) dx. Here we have inserted an additional multiplier to restrict the frequencysupport of h . This is possible as a bilinear multiplier applied to a pairof functions with frequency supports in the intervals I and J , results in afunction that has frequency support in − I − J .We identify each piece B m j with a multiplier of the form correspondingto the second case in the list of examples of convex sets in the introduction,that is, we notice the slope of the curved boundary line of (2.14) is boundedabove and below by comparable numbers. Each B m j is hence individuallybounded, and we can estimate the display above by (cid:88) j ∈ N (cid:107) M [ − ( j +1) , − j ) f (cid:107) p (cid:107) M [2 − ( j +1) , − j ) g (cid:107) p (cid:107) M [ j, ( j +1))+[ − − j , − − ( j +1) ) h (cid:107) p Applying H¨older’s inequality, we estimate this by (cid:107) ( (cid:88) j ∈ N | M [ − ( j +1) , − j ) f | p ) /p (cid:107) p × (cid:107) ( (cid:88) j ∈ N | M [2 − ( j +1) , − j ) g | p ) /p (cid:107) p × (cid:107) ( (cid:88) j ∈ N | M [ j, ( j +1))+[ − − j , − − ( j +1) ) h | p ) /p (cid:107) p . Because p > − ( j + 1) , − j ) are disjoint, we can useTheorem 2.6 and estimate the first factor by (cid:107) ( (cid:88) j ∈ N | M [ − ( j +1) , − j ) f | p ) /p (cid:107) p ≤ (cid:107) ( (cid:88) j ∈ N | M [ − ( j +1) , − j ) f | ) / (cid:107) p ≤ C p (cid:107) f (cid:107) p . Similarly, we can estimate the factors corresponding to g and h . Theintervals [2 − ( j +1) , − j ) relevant to g are also disjoint, and the intervals[ j, ( j + 1)) + [ − − j , − − ( j +1) ) are disjoint for j even or j odd separately.This concludes the the desired bound for the second term in (2.13) andhence the proof of Corollary 1.2. (cid:3) Multi-lacunary paraproducts In this section, we prove Theorem 1.4. We begin by noting the naturaldecomposition of multi-lacunary sets. ARAPRODUCTS FOR CONVEX SETS 13 Lemma 3.1. If X is ( d, b ) -lacunary, then there exists a partition X = O ∪ · · · ∪ O d such that for every i < d the set X i := O ∪ · · · ∪ O i is ( i, b ) -lacunary and any two points ξ, ξ (cid:48) in O i +1 satisfy dist( ξ, ξ (cid:48) ) ≥ − b dist( ξ, X i ) . Proof. We successively decompose the limit sets in the definition of ( d, b )-lacunarity. (cid:3) Reductions. Consider the assumptions of Theorem 1.4. We first break upthe multiplier by decomposing the interval [2 − j , − j ) into four equally longintervals and intersecting these intervals with [ ξ j , ζ j ). In other words, forintegers 0 ≤ β ≤ 3, we choose numbers η ( β ) j and ζ ( β ) j such that1 [ η j ,ζ j ) = (cid:88) m =0 [ η ( β ) j ,ζ ( β ) j ) and 2 − j + β − j ≤ η ( β ) j ≤ ζ ( β ) j ≤ − j + ( β + 1)2 − j (3.1)Then the multiplier (1.6) breaks up as a sum of four analogous expressionsand it suffices to show the desired bound for each of the summands. Wefix β and suppress the dependency from the superscript for the rest of theproof.We also modify the sequence ξ j . For each j ∈ N , let ξ (cid:48) j be the largestinteger multiple of 2 − j smaller than or equal to ξ j . Thanks to (1.5), thesequence ξ (cid:48) satisfies 0 ≤ ξ (cid:48) j + 2 − j ≤ ξ (cid:48) j − Moreover, distance between ξ (cid:48) j and ξ (cid:48) j (cid:48) for two indices j and j (cid:48) is comparableto the distance of ξ j and ξ j (cid:48) with upper and lower factor at most 2. Hencethe sequence ξ (cid:48) j is ( d, b − ξ (cid:48) j , ξ j ) arepairwise disjoint. Lemma 3.2. The multiplier B m (cid:48) with m (cid:48) ( ξ, η ) = (cid:88) n [ ξ (cid:48) j ,ξ j ) ( ξ )1 [ η j ,ζ j ) ( η ) satisfies (1.2) with constant depending only on p , p .Proof. Recall that we denote by M I the linear Fourier multiplier for theinterval I . We write (cid:107) B m (cid:48) ( f, g ) (cid:107) p (cid:48) = (cid:107) (cid:88) j ( M [ ξ (cid:48) j ,ξ j ) f )( M [ η j ,ζ j ) g ) (cid:107) p (cid:48) ≤ (cid:107) ( (cid:88) j | M [ ξ (cid:48) j ,ξ j ) f | ) / (cid:107) p (cid:107) ( (cid:88) j | M [ η j ,ζ j ) g | ) / (cid:107) p ≤ C p ,p (cid:107) f (cid:107) p (cid:107) g (cid:107) p , where we applied Cauchy-Schwarz and H¨older to pass to the second line andTheorem 2.6 to bound the individual factors. This proves the lemma. (cid:3) Because of the lemma, it suffices to estimate the multiplier (1.6) with ξ j replaced by ξ (cid:48) j . We shall do so and omit the prime in the forthcomingnotation. Let X be the multi-lacunary image of the sequence ξ j . Let Y bethe collection of dyadic intervals I such that 3 I contains a point of X . Let Z be the collection of maximal dyadic intervals I such that 3 I does not containany point of X . By maximality, the intervals of Z are pairwise disjoint. Let Y j be the collection of all dyadic intervals in Y that have length 2 − j andare contained in [0 , ξ j ). Let Z j be the collection of intervals in Z which arecontained in [0 , ξ j ) but not in any interval of Y j . Lemma 3.3. The intervals in Y j ∪ Z j have length at least − j and partition [0 , ξ j ) .Proof. The intervals in Y j have length 2 − j by definition. Let I be an intervalin Z j of length at most 2 − j . Let J be the dyadic interval of length 2 − j containing I . Then J is contained in [0 , ξ j ), because ξ j is an integer multipleof 2 − j and J contains I ⊂ [0 , ξ j ). The interval J is not in Y j , since I is notcontained in any interval of Y j . Hence J must be contained in an interval J (cid:48) ∈ Z . As the intervals in Z are pairwise disjoint, I cannot be strictlycontained in J (cid:48) , and hence I has to be equal to J (cid:48) . Consequently, | I | = 2 − j .This proves the statement about the length of the intervals.We turn to the claim about partitioning. The intervals in Y j are pairwisedisjoint, because they are dyadic intervals of equal length. The intervalsin Z j are pairwise disjoint, because they are maximal. By construction, nointerval of Z j can be contained in any interval of Y j . Conversely, no intervalin Y j can be contained in any interval of Z j . Indeed, 3 I with I ∈ Z j containsno point of X , but 3 J for J ∈ Y j does contain a point of X . Hence J (cid:42) I .Hence the intervals in Y j ∪ Z j are all pairwise disjoint.To prove that the intervals form a cover, let ξ be any point in [0 , ξ j ). Let I be the dyadic interval of length 2 − j containing ξ . It is contained in [0 , ξ j ).If I is in Y j , then ξ is covered by intervals in Y j ∪ Z j . If it is not in Y j , thenit is contained in an interval J in Z . The interval J does not contain ξ j bydefinition of Z . Hence it is contained in [0 , ξ j ). It is not contained in anyinterval of Y j , and hence it is in Z j . Again, ξ is covered by Y j ∪ Z j . Thisproves the partition statement. (cid:3) Using Lemma 3.3, we may split B m ( f, g ) as (cid:88) j (cid:88) I ∈ Z j ( M I f )( M ( η j ,ζ j ) g ) + (cid:88) j (cid:88) I ∈ Y j ( M I f )( M ( η j ,ζ j ) g ) . (3.2)It suffices to estimate the terms separately. Recall that we assume each ξ j to be an integer multiple of 2 − j and each ( η j , ζ j ) to actually be ( η ( β ) j , ζ ( β ) j )as defined in (3.1). ARAPRODUCTS FOR CONVEX SETS 15 First term. Regrouping the sum in the first term and pairing with a du-alizing function, the quantity to be estimated becomes (cid:90) (cid:88) I ∈ Z (cid:88) j : I ∈ Z j ( M I f )( M ( η j ,ζ j ) g )( x ) h ( x ) dx = (cid:90) (cid:88) I ∈ Z ( M I f ) (cid:88) j : I ∈ Z j ( M ( η j ,ζ j ) g )( x ) M − I − [0 , − | I | ] h ( x ) dx. (3.3)Here we used that the Fourier support of the product of M I f and M ( η j ,ζ j ) g is contained in I + ( η j , ζ j ) ⊂ I + [0 , − j ) ⊂ I + [0 , − | I | ) , the first inclusion following from (3.1) and the second one from Lemma 3.3.We may therefore apply the adjoint of the Fourier restriction to this intervalto the dualizing function h without changing the value of the duality pairing.The intervals I + [0 , − | I | ) have bounded overlap as I runs through Z .To see this, partition each such interval as disjoint union of I and J I , where J I is the dyadic interval to the right of I and has length 2 − | I | . Then 3 J I is contained in 3 I and thus does not contain any point of X . Hence J I iscontained in an interval of Z with which it shares the left endpoint. As theintervals in Z are pairwise disjoint and each of them can only contain oneinterval J I as above, the intervals J I are pairwise disjoint.Consider an interval I of Z and assume it has nonzero contribution to(3.3). Then there are integers j I < j I such that I is contained in [0 , ξ j )precisely if j < j I , and it is not contained in any interval of Y j precisely if j > j I . Hence we can notice (cid:88) j : I ∈ Z j M ( ηj,ζj ) g = (cid:88) j I Second term. We turn to the second term in (3.2). We decompose Y asthe union Y = (cid:91) i Y ( i ) where Y ( i ) contains the intervals I of Y such that 3 I contains a point of O ( i ) but no point of any O ( i (cid:48) ) with i (cid:48) < i . As the parameter i only ranges form0 to d , it suffices to consider the Y ( i ) separately and prove a bound for (cid:88) j (cid:88) I ∈ Y j ∩ Y ( i ) ( M I f )( M ( η j ,ζ j ) g ) . (3.5)Let W ( i ) be the maximal intervals in Y ( i ) . In the next lemma, we singleout two facts that we will need later. The proof is similar, but not identical,to the argument that proved an analogous statement for Z . Lemma 3.4. Let I ∈ W ( i ) . If there is J ∈ W ( i ) with J ∩ ( I + [0 , − | I | )) (cid:54) = ∅ , then | J | ≥ − | I | . In particular, the intervals I + [0 , − | I | ) with I ∈ W ( i ) have bounded overlap.Proof. Take I ∈ W ( i ) and write I + [0 , − | I | ] = I ∪ J I as a disjoint union.Assume J I intersects another interval J ∈ W ( i ) . It suffices to show that J I is contained in J . Suppose J is contained in J I . Then 3 J I contains 3 J andthus a point from O ( i ) . On the other hand, 3 J I is contained in 3 I and thusdoes not contain any point from any O ( i (cid:48) ) with i (cid:48) < i . Hence J I is containedin an interval of W ( i ) . As the intervals of W ( i ) are pairwise disjoint, thisinterval must be J and hence J is equal to J I . In particular, a right neighborof I ∈ W ( i ) has at least one quarter of the length of I . (cid:3) To estimate (3.5), we sort the intervals by their containment in maximalintervals, pair with a dualizing function, and realize the restriction of theFourier support of the dualizing function with a multiplier as (cid:90) (cid:88) j (cid:88) J ∈ W ( i ) (cid:88) I ⊂ JI ∈ Y j ∩ Y ( i ) M I f ( x ) M [ η j ,ζ j ) g ( x ) M − J − [0 , − | J | ) ( h )( x ) dx. (3.6)We break the innermost sum up by considering separately the cases | I | > − b − | J | and | I | ≤ − b − | J | . ARAPRODUCTS FOR CONVEX SETS 17 The sum with | I | > − b − | J | is estimated by (cid:107) ( (cid:88) j (cid:88) J ∈ W ( i ) ( (cid:88) I ⊂ JI ∈ Y j ∩ Y ( i ) | I | > − b − | J | M I f ) ) / (cid:107) p × (cid:107) ( (cid:88) j | M ( η j ,ζ j ) g | ) / (cid:107) p (cid:107) ( (cid:88) J ∈ W ( i ) | M J +2 − | J | h | ) / (cid:107) p . The three factors are estimated by Theorem 2.6. It is clear that the intervals( η j , ζ j ) are pairwise disjoint. By Lemma 3.4, J + 2 − | J | have boundedoverlap. For the first factor, we use the disjointness of the various intervalsin Y j to write it as (cid:107) ( (cid:88) J ∈ W ( i ) b +3 (cid:88) k =0 (cid:88) I ⊂ JI ∈ Y ( i ) | I | =2 − k | J | | M I f | ) / (cid:107) p As the various intervals J are disjoint, we obtain the estimate C p ( b + 4) (cid:107) f (cid:107) p by Theorem 2.6. This completes the bound of the sum over | I | > − b − | J | in (3.6).We turn to the sum in (3.6) over | I | ≤ − b − | J | . Let V ( i ) be the collectionof dyadic intervals I such that there exists J ∈ W ( i ) with I ⊂ J and 2 b +4 | I | = | J | and at least one I (cid:48) ⊂ I with I (cid:48) ∈ Y j ∩ Y ( i ) . Given a pair of neighboringintervals in W ( i ) , the right interval is at least one quarter as wide as leftinterval by Lemma 3.4. The same property is inherited by the refined family V ( i ) . Accordingly, it suffices to estimate (cid:90) (cid:88) j (cid:88) J ∈ V ( i ) (cid:88) I ⊂ JI ∈ Y j ∩ Y ( i ) M I f ( x ) M [ η j ,ζ j ) g ( x ) M − J − [0 , − | J | ) h ( x ) dx, (3.7)where the Fourier support of h is now given in terms of an interval in V ( i ) .For each J ∈ V ( i ) , there is exactly one point of O ( i ) contained in 7 J .Existence of at least one such a point follows by our requirement that J contains an interval from Y j ∩ Y ( i ) . To show that there cannot be more thanone, suppose there were two such points ξ, ξ (cid:48) . By the multi-lacunarity, therewould then exist a point ξ (cid:48)(cid:48) of O ( i (cid:48) ) with i (cid:48) < i and | ξ (cid:48)(cid:48) − ξ | ≤ b | ξ − ξ (cid:48) | ≤ · b | J | ≤ | I (cid:48) | where I (cid:48) is the interval in W ( i ) with I (cid:48) ⊃ J . This would imply ξ (cid:48)(cid:48) ∈ I (cid:48) ,which contradicts the fact I (cid:48) ∈ Y ( i ) . Hence there is at most one point ξ J ∈ J ∩ O ( i ) , and this point is the unique point from O ( i ) withing thewhole 7 J .Let φ be a Schwartz function whose Fourier transform is supported on[ − − , − ] and equal to one in [ − − , − ]. Write (cid:98) φ ξ,j ( η ) = (cid:98) φ (2 j ( η − ξ )). We compare (3.7) with (cid:90) (cid:88) j (cid:88) J ∈ V ( i ) ( φ ξ J ,j ∗ f )( x ) M ( η j ,ζ j ) g ( x ) M − J − [0 , − | J | ) ( h )( x ) dx, (3.8)where we have replaced M I by a convolution with φ ξ J ,j , and the sum over I has been removed. We first prove bounds on (3.8), and then we show howto pass from the original expression to (3.8).We note that we may now further restrict the Fourier transform of h tothe interval I J,j = − [ ξ J − − − j , ξ J + 2 − − j ] − [ η j , ζ j ) . Because ξ J is a point unique to 7 J , because the intervals J are disjoint, andbecause the sequences [ η j , ζ j ) are subject to the condition (1.4), we see thatthe intervals I J,j form a disjoint family as J varies and disjoint and lacunaryfamily as j varies. We thus estimate (3.8) by (cid:107) ( (cid:88) J ∈ V ( i ) sup j | φ ξ J ,j ∗ ( M J f ) | ) / (cid:107) p (cid:107) ( (cid:88) j | M ( η j ,ζ j ) g | ) / (cid:107) p × (cid:107) ( (cid:88) J ∈ V ( i ) (cid:88) j | M I J,j h | ) / (cid:107) p The last two factors we estimate with Theorem 2.6. In the first factor, webound the convolution product by the Hardy–Littlewood maximal functionand apply the Fefferman–Stein maximal inequality as well as Theorem 2.6.This concludes the proof of the bound for (3.8).It remains to bound the difference of (3.7) and (3.8). Define w j,J = (cid:88) I ⊂ JI ∈ Y j ∩ Y ( i ) I − (cid:91) φ ξ J ,j J . Let ˜ ψ j,J be a Schwartz function equal to one in [ ξ J − − j , ξ J + 2 − j ] andzero outside [ ξ J − − j , ξ J + 2 − j ]. Defining ψ j,J = ˜ ψ j,J − (cid:91) φ ξ J ,j , we then seethat ψ j,J is a Schwartz function supported in [ ξ J − − j , ξ J +2 − j ], vanishingin [ − − − j , − − j ] and satisfying w j,J = 1 J ψ j,J (cid:88) I ⊂ JI ∈ Y j ∩ Y ( i ) I \ ( ξ J − − − j ,ξ J +2 − − j ) . (3.9)Indeed, ξ J is the unique point of (cid:83) i (cid:48) ≤ i O ( i ) in 7 J . If there is I ∈ Y j ∩ Y ( i ) with I ⊂ J and ∂I \ ∂J (cid:51) ξ J , then I (cid:48) with | I (cid:48) | = | I | and I (cid:48) ∩ I = { ξ J } is alsoin Y j ∩ Y ( i ) and contained in J . Hence ξ J / ∈ ∂ (cid:91) I ⊂ JI ∈ Y j ∩ Y ( i ) I \ ∂J. ARAPRODUCTS FOR CONVEX SETS 19 On the other hand, the union above is contained in [ ξ J − − j , ξ J + 2 − j ],and so the equation (3.9) is justified.As we know the bounds for (3.8), it suffices to bound (cid:107) ( (cid:88) J ∈ V ( i ) (cid:88) j | (cid:88) I ⊂ JI ∈ Y j ∩ Y ( i ) M I M J f − φ ξ J ,j ∗ ( M J f ) | ) / (cid:107) p × (cid:107) ( (cid:88) j | M ( η j ,ζ j ) g | ) / (cid:107) p (cid:107) ( (cid:88) J ∈ V ( i ) | M ( − J − [0 , − | J | )) h | ) / (cid:107) p . Here the last two factors are readily estimated by Theorem 2.6. We focuson the first factor. By the observation (3.9), we can bound the multiplieroperator associated with frequency symbol ψ j,J by the Hardy–Littlewoodmaximal function and apply the Fefferman–Stein inequality to control thefirst factor by (cid:107) ( (cid:88) J ∈ V ( i ) (cid:88) j | (cid:88) I ⊂ JI ∈ Y j ∩ Y ( i ) M I \ ( ξ J − − − j ,ξ J +2 − − j ) M J f | ) / (cid:107) p . 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