SINGULAR INTEGRAL OPERATORS WITH ROUGH CONVOLUTION KERNELS. Andreas Seeger University of Wisconsin-Madison

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1 SINGULA INTEGAL OPEATOS WITH OUGH CONVOLUTION KENELS Andreas Seeger University of Wisconsin-Madison. Introduction The purpose of this paper is to investigate the behavior on L ( 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 (S d ) and (.) Ω(θ)dθ = 0; S d here dθ denotes surface measure on the sphere. Then it is easy to see that for f C0 ( d ) the principal value integral (.) T Ω f(x) = p.v. Ω(y/ y ) y d f(x y)dy exists for all x d. Calderón and Zygmund [] used the method of rotations to show that if Ω L (S d ) and if the even part of Ω belongs to the class LlogL(S d ) then T extends to a bounded operator on L p ( d ), < p <. Proposition. Suppose that Ω LlogL(S d ) and suppose that the cancellation property (.) holds. Then T Ω extends to an operator of weak type (,). In two dimensions this result was previously obtained by Christ and ubio de Francia [3], and, under a slightly stronger hypothesis, by Hofmann [6]. In [], [3] a weak type (, ) 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 LlogL. 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of the arguments in [3] for the maximal operator could also work for the singular integral operator; in fact, in unpublished work, the authors of [3] obtained a weaktype(,)inequality indimension d 7. Howevertheirarguments- ifapplied to the singular integral operator - lead to substantial technical difficulties and no esearch supported in part by a grant from the National Science Foundation. Typeset by AMS-TEX

2 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 (.3) Tf,g = g(x)f(y)k(x y)dydx for all f,g C c (d ) with disoint supports. Clearly T extends to a bounded operator on L ( d ) if and only if the Fourier transform K belongs to L ( d ). Introducing polar coordinates x = rθ, r > 0, θ S d 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 (.4) V (θ) = K(rθ) r d dr and (.5) V(θ) = supv (θ). >0 Moreover, for a let (.6) η(a) = sup as S d We shall always assume the Dini-condition K((r s)θ) K(rθ) r d drdθ. (.7) η(a) da a <. Theorem. Suppose that T is as in (.3) and that K L ( d ). Suppose that (.7) holds and that V LlogL(S d ). Then T is bounded in L p, < p < ; moreover T is of weak type (,). Note that for the operators in (.) we have η(a) = O(a ) and V = c Ω. Therefore the Proposition follows from the Theorem. emarks. (i) It may be more natural to impose an integrability condition on V, uniformly in, rather than on the maximal quantity V. Indeed the hypothesis V LlogL can be replaced by sup S d V (θ)(+ (V (θ)/ V )dθ <

3 for some nondecreasing function : [, ) (0, ) satisfying da a (a) <. Typicalchoicesfor are (t) = log +ǫ (+t), or (t) = log(+t)log(+log +ǫ (+ t)) etc. (ii) Without the assumption Ω LlogL(S d ) even the L boundedness of T Ω may fail. This was pointed out by Calderón and Zygmund []. However if Ω L (S d ) is odd then T Ω is bounded on L p, < p <, see []. Presently it is not known whether a weak type (,) 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 d we denote the Lebesgue measure of E by E. For a set A S d we also write A = Adθ. The Fourier transform of f is denoted by f, the inverse Fourier transform of f is denoted by F [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. We write a b if 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 (.) sup 0 l N supr d+l ( 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 (.), if Ω L (S d ). 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 ; say by a factor of s. For s > 3 let E s = {e s ν} be a collection of unit vectors with mutual distance > s 0 d such that for each θ S d there is an e s ν with θ e s ν s. It is easy to see that we may construct disoint measurable sets E s ν Sd with e s ν E s ν, diam(e s ν) s and ν E s ν = S d. Then clearly card(e s ) s(d ). Let H s ν (x) = H (x)χ E s ν (x/ x ). 3

4 A further decomposition will be based on the observation that the Fourier transform Ĥ s ν is concentrated near the hyperplane perpendicular to es ν. Fix a parameter κ, such that 0 < κ <. Let ψ C 0 () be supported in [ 4,4] such that ψ(t) = for t [,]. Define P s ν by Our basic splitting is P s ν (ξ) = ψ(s( κ) ξ,e s ν / ξ ). (.) H = Γ s +(H Γ s ) where Γ s = ν P s ν H s ν. Lemma.. Let Q be a collection of cubes Q with disoint interiors. Define L(Q) = m if m < 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 s > 3 Γ s F s C[M 0] s( κ) α Q f Q. InourapplicationofLemma.thefunctionsf Q willbethebasicbuildingblocks 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+, 0 ε (H Γ s ) b Q C N [ M0 sε +M N s(d+(ε κ)n)] b Q 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. Proof of Lemma.. We use an orthogonality argument based on the following observation. Given s > 3, each ξ 0 is contained in at most C s(d +κ) of the sets supp P s ν where C only depends on d. In fact by homogeneity it suffices to check this for ξ S d. If ξ supp P s ν Sd and ξ is the hyperplane perpendicular to ξ then (.3) dist(e s ν,ξ ) c s( κ). 4

5 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/ ν C s(d +κ) ν = C s(d +κ) ν P s ν Ĥν F s s (π) d/ Ĥν F s s Hν s F s. For fixed ν write = H s ν F s where H s ν (x) = Hs ν ( x). = + = = i= H s ν Hs ν F s (x)f s (x)dx H s ν Hs iν F i s(x)f s (x)dx Next observe that H ν s Hs iν is for i supported in a rectangle s ν centered at 0 with d short sides of length s+0 and one long side of length +0, the long side being parallel to e s ν. Since the measure of Es ν is bounded by C s(d ) we have Hiν s M 0 s(d ) for all i and consequently H s ν Hs iν H s ν H s iν M 0 d s(d ). 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 ) [M 0 ] d s(d ) i [M 0 ] d s(d ) α i x+ s ν i F i s (y) dy L(Q)=i s Q (x+ s ν ) L(Q)=i s Q (x+ s ν ) [M 0 ] d s(d ) α x+ s ν [M 0 ] s(d ) α 5 f Q (x) dx Q

6 for all x d. This finally implies that Hν s F s [M 0] α s(d ) card(e s ) ν F s [M 0 ] α s(d ) F s [M 0 ] α s(d ) Q f Q and the asserted inequality follows from (.4). Proof of Lemma.. Let β C ( d \{0}) be supported in {ξ : / ξ } such that k β ( k ξ) = for all ξ 0. Let L k be defined by L k (ξ) = β( k ξ). Consider the multipliers m sk ν (ξ) = β( k ξ)( P s ν (ξ))ĥs ν (ξ); then H Γ s = ν k F [m sk ν ] L k. Since diam(q) s and L k k we obtain using the cancellation of b Q (.5) F [m sk ν] L k b Q F [m sk ν] min{, k+ s } b Q. Let l k sν be the invertible linear transformation with lk sν es ν = k s( κ) e s ν and lk sν y = k y if y,e s ν = 0. It is straightforward to check that α ξ [ L k ( P s ν )(lk sν )] C α for all multiindices α. Therefore L k ( P s ν ) is an L Fourier multiplier with norm independently of k, s and ν. Consequently (.6) F [m sk ν ] H s ν s(d ) 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 P ν s(ξ) = 0 if ξ,es ν s( κ) ξ. Therefore if θ Eν s and if ξ supp ( P ν s) and ξ k then θ,ξ s( κ) k. Now Ĥ s ν (ξ) = = Hence we obtain the size estimate χ E s ν (θ) H (rθ)e ir θ,ξ r d drdθ χ E s ν (θ)(i θ,ξ ) N r N H (rθ)e ir θ,ξ r d dr dθ. m sk ν (ξ) C NM N E s ν [s( κ) k)]n, 6

7 uniformly in ξ. A similar calculation applies to the derivatives of m sk ν. Differentiating Ĥ s ν yields additional factors of r in the above integral and differentiating β( k ) P s ν yields additional factors of s( κ) k. Since E s ν C s(d ) we see that α ξ [m sk ν( k )] C α M N s(d ) ( s( κ) + +k ) α N(+k s( κ)) for all multiindices α with α N. Therefore if N N > d/ (.7) F [m sk ν ] Finally by (.5), (.6) (.8) k +s( ε) α N α ξ [msk and by (.7) with N = d, N d+ (.9) k> +s( ε) ν ( k )] M N s(d ) N(+k s( κ)) ( (+k)n + s( κ)n ). F [m sk ν ] L k b Q M 0 s(d +ε) b Q F [m sk ν ] L k b Q M N s(d ) s(κ ε)n[ sd( ε) + sd( κ)] b Q. If we sum over ν and note that card(e s ) s(d ) then (.8) and (.9) imply the statement of the Lemma. 3. Proof of the theorem Clearly the L p boundedness for < p < follows from the weak-type (,) estimate and the assumed L boundedness, by a duality argument and the Marcinkiewicz interpolation theorem (see [7]). Therefore given λ > 0 we have to verify the inequality (3.) {x d : Tf(x) > λ} Aλ f where A = K + η(a) da a + S d V(θ)(+log + (V(θ)/ V ))dθ here log + s = logs if s and log + s = 0 where 0 s <. Then by assumption A <. Given f L ( 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 7

8 where g α, g f, each b Q is supported in a dyadic cube Q with sidelength L(Q) and the cubes Q have disoint interiors. Moreover b Q α Q and Q Q α f. For each Q let Q be the dilate of Q with same center and L(Q ) = L(Q)+0, and let E = Q. Then also E α f = Aλ f. Finally, for each Q, the mean value of b 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 g Aλ g and by Tshebyshev s inequality (3.) {x d : Tg(x) > λ/} 4λ Tg 4Aλ g Aλ f. Therefore the proof of the theorem is reduced to the estimate {x d \E : Tb(x) > λ/} Aλ f. Note that the expressions Tb Q (x) are well defined for almost all x 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 C 0 ( + ) be as in the previous section (supp β (/,), k β ( k t) = for all t > 0). Define K (x) = β ( x )K(x). For m Z let B m = L(Q)=m b Q. 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) {x d : K B s (x) > λ/} Aλ f. 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 = [00(d+)] and let D s = {θ S d : V(θ) δs V L (S d )}. Let φ C 0 () such that φ(t)dt = and such that φ(t) = 0 if t 0. Then (3.4) K = H s +s +Ss 8

9 where [ ] S s (rθ) = β ( r) K(rθ) K(ρθ) δs φ( δs (r ρ))dρ s (rθ) = β ( r)χ S d \D s(θ) K(ρθ) δs φ( δs (r ρ))dρ H(r,θ) s = β ( r)χ D s(θ) K(ρθ) δs φ( δs (r ρ))dρ. Observethat H s vanishes if x / [, + ] and that for r +, θ D s, r d+l ( r r )l H(r,θ) s C l δs(l+) K(ρθ) ρ d dρ C l δs(l+) V L (S d ). r/ That is, for fixed s > 3 and for all N, the family {H s } satisfies the assumption (.) with (3.5) M N = C N V L (S d ) δs(n+) where C N does not depend on V or s. We now decompose H s = Γs +(Hs Γs ) exactly as in (.), except that this time the operator H itself depends on s. The decomposition (.) depended on a parameter 0 < κ < ; we may now choose κ = /. We have split K B s (x) = I(x)+II(x) where and II(x) = I(x) = Γ s B s(x) [H s Γ s + s +S] B s s (x). Now by Tshebyshev s inequality {x d : K B s (x) > λ/} {x d : I(x) > λ/4} + {x d : II(x) > λ/4} (3.6) 6λ I + 4λ II. By Lemma. and (3.5) with N = 0 we have [ I ] Γ s B s [( V L (S d ) s(4δ+κ ) α Q (3.7) A α b Q Aλ f ; Q b Q ) / ] 9

10 here we could sum the geometrical series since 4δ +κ < /4. Next we apply Lemma. with N = 5(d+), ε = /4 and obtain (H s Γ s ) B s A (N+)δs ( εs + s(d (κ ε)n) ) (3.8) A f ; B s nowwehaveused that(n+)δ+d (κ ε)n < (d+)/4and(n+)δ ε < /8. It remains to estimate the sums involving s and Ss. Observe that s S d \D s + K(rθ) r d drdθ V(θ)> δs V V(θ)dθ and therefore s B s B s V(θ)dθ V(θ)> δs V b Q V(θ)card({s N : δs V(θ) / V })dθ Q V(θ)(+log + (V(θ)/ V ))dθ b Q Q (3.9) A f. Finally S s sup V and for s > 0/δ + S s K((r ρ)θ) K(rθ) r d drdθ δs φ( δs ρ) dρ η( δs 3 ). Therefore S s B s B s sup V L (S d ) + (3.0) 0<s<0/δ [ sup V L (S d ) + A f. Now by (3.8), (3.9) and (3.0) η(a) da ] b Q a Q s>0/δ B s η( δs 3 ) (3.) II A f and the desired weak type inequality (3.3) follows from (3.6), (3.7) and (3.). 0

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 s in (3.4). Let C = sup V and, for Z D s D s = {θ Sd : V (θ)+v (θ)+v +(θ) +δs C}. In the present setting we define H s and s as before but with Ds replaced by. The estimate (3.9) is changed to s B s B s f sup f f σ= σ= ( δs /C) sup da a (a) sup V +σ(θ)> δs C V +σ(θ)> δs C sup s σ= V +σ(θ)dθ V +σ(θ)dθ V (θ) (V (θ)/c)dθ. V +σ(θ) (V +σ(θ)/c)dθ The other estimates remain essentially unchanged; in various instances one replaces V by sup V. eferences. A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (956), M. Christ, Weak type (,) bounds for rough operators, Annals of Math. 8 (988), M. Christ and J.-L. ubio de Francia, Weak type (,) bounds for rough operators, II, Invent. Math. 93 (988), M. Christ and C. D. Sogge, The weak type L convergence of eigenfunction expansions for pseudo-differential operators, Inv. Math. 94 (988), C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 4 (970), S. Hofmann, Weak (, ) boundedness of singular integrals with nonsmooth kernel, Proc. Amer. Math. Soc. 03 (988), E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 97. University of Wisconsin, Madison, WI address: seeger@math.wisc.edu