here dθ denotes surface measure on the sphere. Then it is easy to see that for (Rd ) the principal value integral

Transcription

1 JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Number 1, January 1996 SINGULAR INTEGRAL OPERATORS WITH ROUGH CONVOLUTION KERNELS ANDREAS SEEGER 1. Introduction The purpose of this paper is to investigate the behavior on L 1 (R d ), d, of a class of singular convolution operators which are not within the scope of the standard Calderón-Zygmund theory. An important special case occurs if the convolution kernel K is homogeneous of degree d. Suppose that Ω L 1 (S d 1 )and (1.1) S d 1 Ω(θ)dθ =0; here dθ denotes surface measure on the sphere. Then it is easy to see that for f C0 (Rd ) the principal value integral (1.) T Ω f(x) =p.v. Ω(y/ y ) y d f(x y) dy exists for all x R d. Calderón and Zygmund [1] used the method of rotations to show that if Ω L 1 (S d 1 ) and if the even part of Ω belongs to the class L log L(S d 1 ), then T extends to a bounded operator on L p (R d ), 1 <p<. Proposition. Suppose that Ω L log L(S d 1 ) and suppose that the cancellation property (1.1) holds. Then T Ω extends to an operator of weak type (1, 1). In two dimensions this result was previously obtained by Christ and Rubio de Francia [3], and, under a slightly stronger hypothesis, by Hofmann [6]. In [], [3] a weak type (1, 1) inequality was also proved for the less singular maximal operator M Ω f(x) =supr d Ω(y/ y )f(x y) dy, r>0 y r in all dimensions, again under the assumption Ω L log L. It is conceivable that a variant of the arguments in [3] for the maximal operator could also work for the singular integral operator; in fact, in unpublished work, the authors of [3] obtained a weak type (1, 1) inequality in dimension d 7. However their arguments if applied to the singular integral operator lead to substantial technical difficulties and no Received by the editors August 5, Mathematics Subect Classification. Primary 4B0; Secondary 4B5. Research supported in part by a grant from the National Science Foundation. 95 c 1996 American Mathematical Society

2 96 ANDREAS SEEGER proof has been known for the higher-dimensional cases. In this paper we develop a different and conceptually simpler method, based on a microlocal decomposition of the kernel (cf. (.) below). Incidentally this method also gives a new proof of the weak type bounds for M Ω. The proposition is a special case of a more general theorem concerning translation invariant operators T with rough convolution kernels K S. We assume that K is locally integrable away from the origin, so that (1.3) Tf,g = g(x)f(y)k(x y)dy dx for all f,g C c (Rd ) with disoint supports. Clearly T extends to a bounded operator on L (R d ) if and only if the Fourier transform K belongs to L (R d ). Introducing polar coordinates x = rθ, r>0, θ S d 1, we shall assume a weak regularity condition for r K(rθ). However only size conditions will be imposed in the θ variable. In order to formulate our assumptions let (1.4) V R (θ) = R R K(rθ) r d 1 dr and (1.5) V (θ) =supv R (θ). R>0 Moreover, for a let (1.6) η(a) = sup R as S d 1 R We shall always assume the Dini-condition R K((r s)θ) K(rθ) r d 1 dr dθ. (1.7) η(a) da a <. Theorem. Suppose that T is as in (1.3) and that K L (R d ). Suppose that (1.7) holds and that V L log L(S d 1 ). Then T is bounded in L p, 1 <p< ;moreover, T is of weak type (1, 1). Note that for the operators in (1.) we have η(a) = O(a 1 )andv =c Ω. Therefore the Proposition follows from the Theorem. Remarks. (i) It may be more natural to impose an integrability condition on V R, uniformly in R, rather than on the maximal quantity V. Indeed the hypothesis V L log L can be replaced by sup V R (θ)(1 + (V R (θ)/ V R 1 ))dθ < R S d 1

3 SINGULAR INTEGRAL OPERATORS 97 for some nondecreasing function : [1, ) (0, ) satisfying Typical choices for are 1 da a (a) <. (t) =log 1+ε ( + t), or (t) = log( + t) log( + log 1+ε ( + t)), etc. (ii) Without the assumption Ω L log L(S d 1 )eventhel boundedness of T Ω may fail. This was pointed out by Calderón and Zygmund [1]. However if Ω L 1 (S d 1 ) is odd, then T Ω is bounded on L p,1<p< (see [1]). Presently it is not known whether a weak type (1, 1) inequality holds in this case. In we shall give the main estimates needed to prove the Theorem. The formal proof is contained in 3. The following notation is used: For a set E R d we denote the Lebesgue measure of E by E. For a set A S d 1 we also write A = dθ. The Fourier A transform of f is denoted by f, the inverse Fourier transform of f is denoted by F 1 [f]. Given two quantities a and b we write a b or b a if there is a positive constant C, depending only on the dimension, such that a Cb. Wewritea bif a b and a b.. Main estimates Let {H } be a family of functions with supp H {x: x + }. We assume that the H are differentiable in the radial variable and that the estimates (.1) sup sup r d+l ( 0 l N r )l H (rθ) M N hold uniformly in θ and r. Convolution kernels of this type come up in a dyadic decomposition of the kernel of the operator defined in (1.), if Ω L (S d 1 ). We shall be interested in estimates for H Q b Q where each b Q is a building block in a Calderón-Zygmund decomposition, supported in a cube Q, and where the sidelength L(Q) of Q is small compared to the diameter of supp H ;saybya factor of s. For lete s ={e s ν }be a collection of unit vectors with mutual distance > s 10 d 1 such that for each θ S d 1 there is an e s ν with θ es ν s 1. It is easy to see that we may construct disoint measurable sets Eν s S d 1 with e s ν Es ν,diam(es ν ) s and ν Es ν = Sd 1. Then clearly card(e s ) s(d 1). Let H s ν (x) =H (x)χ E s ν (x/ x ).

4 98 ANDREAS SEEGER A further decomposition will be based on the observation that the Fourier transform Ĥν s is concentrated near the hyperplane perpendicular to es ν. Fix a parameter κ, such that 0 <κ<1. Let ψ C0 (R) besupportedin[ 4,4] such that ψ(t) =1fort [,]. Define Pν s by P ν s (ξ) =ψ( s(1 κ) ξ,e s ν / ξ ). Our basic splitting is (.) H =Γ s +(H Γ s ) where Γ s = ν P s ν H s ν. Lemma.1. Let Q be a collection of cubes Q with disoint interiors. Define L(Q) =mif m 1 < sidelength(q) m and let Q m = {Q Q : L(Q) =m}.for each Q let f Q be an integrable function supported in Q satisfying f Q (x) dx α Q. Let F m = Q Q m f Q. Then for Γ s F s C[M 0] s(1 κ) α Q f Q 1. In our application of Lemma.1 the functions f Q will be the basic building blocks which arise in a Calderón-Zygmund decomposition at height cα. Note however that for this part no cancellation condition for f Q is assumed. Lemma.. Let Q be a cube of sidelength s and let b Q be integrable and supported in Q; moreover, suppose that b Q =0. Then for N d+1 and 0 ε 1 (H Γ s ) b Q 1 C N [ M0 sε + M N s(d+(ε κ)n)] b Q 1 where C N does not depend on or Q. It is important to keep track of how the estimates depend on M N since we shall apply the lemmas in a situation where this norm is large and depends on s itself. The bounds in Lemma. are not best possible, but this is irrelevant for our purpose. ProofofLemma.1. We use an orthogonality argument based on the following observation. Given, each ξ 0iscontainedinatmostC s(d +κ) of the sets supp P ν s where C only depends on d. In fact by homogeneity it suffices to check this for ξ S d 1.Ifξ supp P ν s S d 1 and ξ is the hyperplane perpendicular to ξ, then (.3) dist(e s ν,ξ ) c s(1 κ).

5 SINGULAR INTEGRAL OPERATORS 99 Since the mutual distance of the e s ν is bounded below by c s there are at most c s(d +κ) of the e s ν satisfying (.3). This implies the observation. We apply Plancherel s theorem, the Cauchy-Schwarz inequality and then Plancherel s theorem again to obtain (.4) = Γ s F s =(π)d/ For fixed ν write Hν s F s = ν C s(d +κ) ν = C s(d +κ) ν = + P s ν Ĥν s F s (π) d/ Ĥν s F s H s ν F s. H s ν Hs ν F s (x)f s (x)dx 1 = i= H s ν Hs iν F i s (x) F s (x) dx where H ν s (x) =Hs ν ( x). Next observe that H ν s Hs iν is for i supported in a rectangle Rs ν centered at 0 with d 1 short sides of length s+10 and one long side of length +10,the long side being parallel to e s ν. Since the measure of Eν s is bounded by C s(d 1) we have Hiν s 1 M 0 s(d 1) for all i and consequently H s ν Hs iν H s ν H s iν 1 M 0 d s(d 1). Therefore, since the cubes Q are disoint, H ν s Hs ν F s(x) + H ν s Hiν s F i s(x) i< [M 0 ] d s(d 1) [M 0 ] d s(d 1) i [M 0 ] d s(d 1) α i x+r s ν i F i s (y) dy L(Q)=i s Q (x+r s ν ) L(Q)=i s Q (x+r s ν ) [M 0 ] d s(d 1) α x +R s ν [M 0 ] s(d ) α f Q (x) dx Q

6 100 ANDREAS SEEGER for all x R d. This finally implies that Hν s F s [M 0] α s(d ) card(e s ) ν F s 1 [M 0 ] α s(d 1) F s 1 [M 0 ] α s(d 1) Q f Q 1 and the asserted inequality follows from (.4). ProofofLemma.. Let β C (R d \{0}) be supported in {ξ :1/ ξ } such that k β ( k ξ) = 1 for all ξ 0. LetL k be defined by L k (ξ) =β( k ξ). Consider the multipliers m sk ν (ξ) =β( k ξ)(1 P s ν (ξ))ĥs ν (ξ); then H Γ s = ν k F 1 [m sk ν ] L k. Since diam(q) s and L k 1 k we obtain using the cancellation of b Q (.5) F 1 [m sk ν ] L k b Q 1 F 1 [m sk ν ] 1 min{1, k+ s } b Q 1. Let l k sν be the invertible linear transformation with l k sνe s ν = k s(1 κ) e s ν and l k sνy = k y if y, e s ν = 0. It is straightforward to check that α ξ [ L k (1 P s ν )(lk sν )] C α for all multi-indices α. Therefore L k (1 P s ν )isanl 1 Fourier multiplier with norm independently of k, s and ν. Consequently (.6) F 1 [m sk ν ] 1 H s ν 1 s(d 1) M 0. In order to get a better bound for large k we estimate Ĥs ν and its derivatives using integration by parts. Note that 1 P ν s (ξ) =0if ξ,e s ν s(1 κ) ξ. Therefore if θ Eν s and if ξ supp(1 P ν s)and ξ k,then θ, ξ s(1 κ) k.now Ĥν s(ξ) = χ E s ν (θ) H (rθ)e ir θ,ξ r d 1 dr dθ = χ E s ν (θ)(i θ, ξ ) N r N H (rθ)e ir θ,ξ r d 1 dr dθ. Hence we obtain the size estimate m sk ν (ξ) C NM N E s ν [s(1 κ) k)]n,

7 SINGULAR INTEGRAL OPERATORS 101 uniformly in ξ. A similar calculation applies to the derivatives of m sk ν. Differentiating Ĥs ν yields additional factors of r in the above integral and differentiating β( k ) P ν s yields additional factors of s(1 κ) k. Since Eν s C s(d 1) we see that α ξ [m sk ν( k )] C α M N s(d 1) ( s(1 κ) + +k ) α N(+k s(1 κ)) for all multi-indices α with α N. Therefore if N N 1 >d/ (.7) F 1 [m sk ν] 1 Finally by (.5), (.6) (.8) k +s(1 ε) and by (.7) with N 1 = d, N d +1 (.9) k> +s(1 ε) ξ α [m sk ν ( k )] α N 1 M N s(d 1) N(+k s(1 κ)) ( (+k)n1 + s(1 κ)n1 ). F 1 [m sk ν ] L k b Q 1 M 0 s(d 1+ε) b Q 1 F 1 [m sk ν ] L k b Q 1 M N s(d 1) s(κ ε)n[ sd(1 ε) + sd(1 κ)] b Q 1. If we sum over ν and note that card(e s ) s(d 1), then (.8) and (.9) imply the statement of the lemma. 3. Proof of the Theorem Clearly the L p boundedness for 1 <p< follows from the weak type (1, 1) estimate and the assumed L boundedness, by a duality argument and the Marcinkiewicz interpolation theorem (see [7]). Therefore given λ>0wehavetoverify the inequality (3.1) where A = K + {x R d : Tf(x) >λ} Aλ 1 f 1 ; η(a) da a + S d 1 V (θ)(1 + log + (V (θ)/ V 1 )) dθ; here log + s =logsif s 1andlog + s=0where0 s<1. Then by assumption A<. Given f L 1 (R d ) we shall use the Calderón-Zygmund decomposition of f at height α = λ/a (see Stein [7]). We decompose f = g + b = g + Q b Q

8 10 ANDREAS SEEGER where g α, g 1 f 1,eachb Q is supported in a dyadic cube Q with sidelength L(Q) and the cubes Q have disoint interiors. Moreover b Q 1 α Q and Q Q α 1 f 1. For each Q let Q be the dilate of Q with same center and L(Q )=L(Q) + 10, and let E = Q.Thenalso E α 1 f 1 =Aλ 1 f 1. Finally, for each Q, themeanvalueofb Q vanishes: b Q = 0. We shall use a variant of Calderón-Zygmund theory due to Fefferman [5] and modified by Christ []. As in standard Calderón-Zygmund theory we have the estimate for the good function g Tg T L L g A g 1 g Aλ g 1 and by Tshebyshev s inequality {x R d : Tg(x) >λ/} 4λ Tg 4Aλ 1 g 1 Aλ 1 f 1. (3.) Therefore the proof of the Theorem is reduced to the estimate {x R d \ E : Tb(x) >λ/} Aλ 1 f 1. Note that the expressions Tb Q (x) are well defined for almost all x R d \ E since we assume that K is locally integrable away from the origin. We now introduce a dyadic decomposition of the kernel. Let β C0 (R +)beas in the previous section (supp β (1/, ), k β ( k t) = 1 for all t>0). Define For m Z let K (x) =β ( x )K(x). B m = L(Q)=m Then observe that the support of the functions K B s is contained in E if s 3. Therefore, in order to verify (3.), it suffices to prove that (3.3) b Q. {x R d : K B s (x) >λ/} Aλ 1 f 1. We now decompose the kernels K in the spherical variables according to the size of V ; moreover, we introduce a regularization in the radial variable. Let δ = [100(d +)] 1 and let D s = {θ S d 1 : V (θ) δs V L 1 (S d 1 )}. Let φ C 0 (R) such that φ(t)dt = 1 and such that φ(t) =0if t 10.Then (3.4) K = H s + Rs + Ss

9 SINGULAR INTEGRAL OPERATORS 103 where [ ] S s (rθ) =β ( r) K(rθ) K(ρθ) δs φ( δs (r ρ))dρ, R(rθ) s =β ( r)χ S d 1 \D s(θ) K(ρθ) δs φ( δs (r ρ))dρ, H s (r, θ) =β ( r)χ D s(θ) K(ρθ) δs φ( δs (r ρ))dρ. Observe that H s vanishes if x / [, + ]andthatfor r +, θ D s, r r d+l ( r )l H s (r, θ) Cl δs(l+1) K(ρθ) ρ d 1 dρ C l δs(l+) V L 1 (S d 1 ). r/ That is, for fixed andforalln, the family {H s } satisfies the assumption (.1) with (3.5) M N = C N V L1 (S d 1 ) δs(n+) where C N does not depend on V or s. We now decompose H s =Γ s +(H s Γ s ) exactly as in (.), except that this time the operator H itself depends on s. The decomposition (.) depended on a parameter 0 <κ<1; we may now choose κ =1/. We have split K B s (x) =I(x)+II(x) where I(x) = Γ s B s (x) and II(x) = [H s Γ s + R s + S s ] B s (x). Now by Tshebyshev s inequality {x R d : K B s (x) >λ/} (3.6) {x R d : I(x) >λ/4} + {x R d : II(x) >λ/4} 16λ I +4λ 1 II 1. By Lemma.1 and (3.5) with N =0wehave [ I ] Γ s B s [( (3.7) V L 1 (S d 1 ) s(4δ+κ 1) α ) 1/ ] b Q 1 Q A α Q b Q 1 Aλ f 1 ;

10 104 ANDREAS SEEGER here we could sum the geometrical series since 4δ + κ 1 < 1/4. Next we apply Lemma. with N =5(d+1),ε=1/4andobtain (3.8) (H s Γ s 1 ) B s A (N+)δs ( εs + s(d (κ ε)n) ) A f 1 ; B s 1 now we have used that (N +)δ+d (κ ε)n < (d+1)/4and(n+)δ ε < 1/8. It remains to estimate the sums involving R s and Ss.Observethat R s 1 S d 1 \D s +1 1 K(rθ) r d 1 dr dθ V (θ)> δs V 1 V (θ)dθ and therefore R s B s B s 1 V (θ)dθ 1 V (θ)> δs V 1 b Q 1 V (θ)card({s N: δs V(θ) / V 1 })dθ (3.9) Q V (θ)(1 + log + (V (θ)/ V 1 ))dθ b Q 1 Q A f 1. Finally S s 1 sup R V R 1 and for s>10/δ +1 S s 1 K((r ρ)θ) K(rθ) r d 1 dr dθ δs φ( δs ρ) dρ 1 η( δs 3 ). Therefore (3.10) S s 1 B s B s 1 sup V R L 1 (S d 1 ) 0<s<10/δ R + B s 1 η( δs 3 ) s>10/δ [ sup V R L 1 (S d 1 ) + R A f 1. η(a) da ] b Q 1 a Q Now by (3.8), (3.9) and (3.10) (3.11) II 1 A f 1

11 SINGULAR INTEGRAL OPERATORS 105 and the desired weak type inequality (3.3) follows from equations (3.6), (3.7) and (3.11). We conclude by proving the remark following the statement of the Theorem. We have to change the definitions of the functions H s and Rs in (3.4). Let C = sup R V R 1 and for Z D s D s = {θ Sd 1 : V 1(θ)+V (θ)+v +1(θ) +δs C}. In the present setting we define H s and Rs as before but with Ds replaced by. The estimate (3.9) is changed to R s B s 1 1 B s 1 V +σ(θ) dθ σ= 1 V +σ (θ)> δs C 1 f 1 sup V +σ(θ) dθ σ= 1 V +σ (θ)> δs C 1 1 f 1 ( δs /C) sup sup V +σ (θ) (V +σ (θ)/c)dθ s σ= 1 da f 1 a (a) sup V R (θ) (V R (θ)/c)dθ. R 1 The other estimates remain essentially unchanged; in various instances one replaces V 1 by sup R V R 1. References 1. A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), MR 18:894a. M. Christ, Weak type (1, 1) bounds for rough operators, Ann. of Math. () 18 (1988), MR 89m: M. Christ and J.-L. Rubio de Francia, Weak type (1, 1) bounds for rough operators, II, Invent. Math. 93 (1988), MR 90d: M. Christ and C. D. Sogge, TheweaktypeL 1 convergence of eigenfunction expansions for pseudo-differential operators, Invent. Math. 94 (1988), MR 89: C. Fefferman, Inequalities for strongly singular convolution operators, ActaMath. 14 (1970), MR 41: S. Hofmann, Weak (1, 1) boundedness of singular integrals with nonsmooth kernel, Proc. Amer. Math. Soc. 103 (1988), MR 89f: E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, MR 44:780 Department of Mathematics, University of Wisconsin, Madison, Wisconsin address: seeger@math.wisc.edu