SQUARE FUNCTIONS AND MAXIMAL OPERATORS ASSOCIATED WITH RADIAL FOURIER MULTIPLIERS

Size: px

Start display at page:

Download "SQUARE FUNCTIONS AND MAXIMAL OPERATORS ASSOCIATED WITH RADIAL FOURIER MULTIPLIERS"

Erin Fitzgerald
5 years ago
Views:

1 SQUARE FUNCTIONS AND MAXIMAL OPERATORS ASSOCIATED WITH RADIAL FOURIER MULTIPLIERS SANGHYUK LEE KEITH M. ROGERS ANDREAS SEEGER Dedicated to Eli Stein We begin with an overview on square functions for spherical and Bochner Riesz means which were introduced by Eli Stein, and discuss their implications for radial multipliers and associated maximal functions. We then prove new endpoint estimates for these square functions, for the maximal Bochner Riesz operator, and for more general classes of radial Fourier multipliers. Overview Square functions. The classical Littlewood Paley functions on R d are defined by ( g[f] = t P tf ) / tdt 0 where (P t ) t>0 is an approximation of the identity defined by the dilates of a nice ernel (for example (P t ) may be the Poisson or the heat semigroup). Their significance in harmonic analysis, and many important variants and generalizations have been discussed in Stein s monographs [38], [39], [44], in the survey [45] by Stein and Wainger, and in the historical article [43]. Here we focus on L p -bounds for two square functions introduced by Stein, for which (P t ) is replaced by a family of operators with rougher ernels or multipliers. The first is generated by the generalized spherical means A β t f(x) = Γ(β) y ( y ) β f(x ty)dy defined a priori for Re β > 0. The definition can be extended to Re β 0 by analytic continuation; for β = 0 we recover the standard spherical means. In [4] Stein used (a variant of) the square function ( G β f = t Aβ t f ) / tdt 0 99 Mathematics Subject Classification. 4B5, 4B5. Key words and phrases. Square functions, Riesz means, spherical means, maximal Bochner Riesz operator, radial multipliers. Supported in part by NRF grant , ERC grant 77778, MINECO grants MTM00-658, SEV and NSF grant 006.

2 SANGHYUK LEE KEITH M. ROGERS ANDREAS SEEGER to prove L p -estimates for the maximal function sup t>0 A β /+ε t f, in particular he established pointwise convergence for the standard spherical means when p > d d and d 3; see also [45]. The second square function ( G α f = t Rα tf /, tdt) generated by the Bochner Riesz means Rt α f(x) = ( ξ ) α f(ξ)e i x,ξ (π) d t dξ, 0 ξ t was introduced in Stein s 958 paper [37] and used to control the maximal function sup t>0 R α /+ε t f for f L in order to prove almost everywhere convergence for Bochner Riesz means of both Fourier integrals and series (see also Chapter VII in [46]). Later, starting with the wor of Carbery [3], it was recognized that sharp L p bounds for G α with p > imply sharp L p -bounds for maximal functions associated with Bochner Riesz means and then also maximal functions associated with more general classes of radial Fourier multipliers ([4], [3]). In[45], Stein and Wainger posed the problem of investigating the relationships between various square functions. Addressing this problem, Sunouchi [48] (in one dimension) and Kaneo and Sunouchi [3] (in higher dimensions) used Plancherel s theorem to establish among other things the uniform pointwise equivalence () G α f(x) G β f(x), β = α d. In view of this remarable result we shall consider G α only. Implications for radial multipliers. We recall Stein s point of view for proving results for Fourier multipliers from Littlewood Paley theory. Suppose the convolution operator T is given by Tf = h f where h satisfies the assumptions of the Hörmander multiplier theorem. That is, if ϕ is a radial nontrivial C function with compact support away from the origin, and L α(r d ) is the usual Sobolev space, it is assumed that sup t>0 ϕh(t ) L α is finite for some α > d/. Under this assumption T is bounded on L p for 0 where g is a standard Littlewood Paley function and gλ is a tangential variant of g which does not depend on the specific multiplier. As gλ(α) [f] p f p for p < and α > d/, this proves the theorem since (under certain nondegeneracy assumptions on the generating ernel) one also has g[f] p f p for < p <.

3 SQUARE FUNCTIONS AND MAXIMAL OPERATORS 3 A similar point of view was later used for radial Fourier multipliers. Let m be a bounded function on R +, let ϕ C 0 (,), and let T m be defined by (3) Tm f(ξ) = m( ξ ) f(ξ). The wor of Carbery, Gasper and Trebels [5] yields an analogue of () for radial multipliers in which the gλ -function is replaced with a robust version of G α which has the same L p boundedness properties as G α. A variant of their argument, given by Carbery in [4], shows that one can wor with G α itself and so there is a pointwise estimate (4) g[t m f](x) Csup ϕ m(t ) L α (R)G α f(x) t>0 where again g is a suitable standard Littlewood Paley function. L p mapping properties of G α together with (4) have been used to prove essentially sharp estimates for radial convolution operators, with multipliers in localized Sobolev spaces. However it was not evident whether (4) could also be used to capture endpoint results, for radial multipliers in the same family of spaces. We shall address this point in below. Carbery [4] also obtained a related pointwise inequality for maximal functions, (5) sup T m(t ) f(x) C m exp L α (R) G α f(x), t>0 which for p yields effective L p bounds for maximal operators generated by radial Fourier multipliers from such bounds for G α ; see also Dappa and Trebels[3] for similar results. Only little is currently nown about maximal operators for radial Fourier multipliers in the range p < ; cf. Tao s wor [50], [5] for examples and for partial results in two dimensions. L p -bounds for G α. We now discuss necessary conditions and sufficient conditions on p (, ) for the validity of the inequality (6) G α f p f p ; here the notation A B is used for A CB with an unspecified constant. By (4) it is necessary for (6) that α > / since for L α (R) to be imbedded in L we need α > /. For α(p) = d( p )+. This is seen by writing ( (7) G α f = Kt α f dt ) / where Kα t t (ξ) = α ξ ( t ξ ) α t. + 0 Then, for a suitable Schwartz function η, with η vanishing near 0 and compactly supported in a narrow cone, and for t and large x in an open cone, we have (8) K α t η(x) = c α t d e it x tx d α +E t (x)

4 4 SANGHYUK LEE KEITH M. ROGERS ANDREAS SEEGER where E t are lower order error terms. This leads to ( / Kt α η dt) L p (R d ) = α > α(p). Note that the oscillation for large x in (8) plays no role here. Concerning positive results for p, the L -bound for α > / follows from Plancherel and was already observed in [37]. The case < p, α > d+ is covered by the Calderón Zygmund theory for vector-valued singular integrals, and analytic interpolation yields L p -boundedness for α(p), see [47], []. Thereis also an endpoint resultfor α = α(p), indeed one can use the arguments by Fefferman [4] for the wea type endpoint inequalities for Stein s gλ function to prove that G α(p) is of wea type (p,p) for < p < (Henry Dappa, personalcommunication, seealsosunouchi[48] for the case d = ). The range < p < is more interesting, since now the oscillation of the ernel Kt α plays a significant role, and, in dimensions d, the problem is closely related to the Fourier restriction and Bochner Riesz problems. A necessary condition for p > can be obtained by duality. Inequality (6) for p > implies that for all b L ([,]) and η as above ( (9) b(t)kt α /. ηdt b(t) dt) p [,] If we again split Kt α as in (8), and prove suitable upper bounds for the expression involving the error terms then we see that, for R, x R b( x ) x d +α p dx <, which leads to the necessary condition α > d( p ) = d( p ). It is conjectured that (6) holds for α(p) = max{d( p ), }. For d = this can be shown in several ways, and the estimate follows from Calderón Zygmund theory (one such proof is in [48]). The full conjecture for d = was proved by Carbery [3], and a variable coefficient generalization of his result was later obtained in [7]. The partial result for p > d+ d which relies on the Stein Tomas restriction theorem is in Christ [9] and in [30]. A better range (unifying the cases d = and d 3) was recently obtained by the authors [5]; that is, inequality (6) holds for α > d(/ /p) and d in the range +4/d < p <. This extends previous results on Bochner Riesz means by the first author [4] and relies on Tao s bilinear adjoint restriction theorem [5]. Motivated by a still open problem of Stein [4], the authors also proved a related weighted inequality in [5], namely for d, q < d+, [G α f(x)] w(x)dx f(x) W q w(x)dx, α > d q,

5 SQUARE FUNCTIONS AND MAXIMAL OPERATORS 5 where W q is an explicitly defined operator which is of wea type (q,q) and bounded on L r with q < r. This is an analogue of a result by Carbery and the third author in two dimensions[6] and extends a weighted inequality by Christ [9] in higher dimensions. One might expect that recent progress by Bourgain and Guth [] on the Bochner Riesz problem will lead to further improvements in the ranges of these results but this is currently open. By the equivalence () one can interpret the boundednessof G α as a regularity result for spherical means and then for solutions of the wave equation. By a somewhat finer analysis in conjunction with the use of the Fefferman Stein #-function [7] the authors obtained an L p (L ) endpoint result, local in time, in fact not just for the wave equation, but also for other dispersive equations. Namely if γ > 0, d, +4/d . Concerning endpoint estimates, many such results for Bochner Riesz multipliers and variants had been previously nown (cf. [0], [], [], [33], [34], [49]). For the Bochner Riesz means R λ t with the critical exponent λ(p) = d(/ /p) /, Tao [50] showed that if for some p > d/(d ) the L p boundedness holds for all λ > λ(p ), then one also has a bound maps L p, to L p, and L p L p,. In contrast no positive result for G d/ d/p seems to have been nown, even for the version with dilations restricted to (/, ). It should be emphasized that, despite the pointwise equivalence of the two square functions in (), the sharp regularity result (0) does not imply a corresponding endpoint bound for G d/ d/p (in fact the latter is not bounded on L p ). In this paper we will prove a sharp result for G d/ d/p in the restricted open range of the Stein Tomas adjoint restriction theorem, and obtain related results for maximal operators and Fourier multipliers. in the limiting case, for p < p <, namely R λ(p) t. Endpoint results Theorem.. Let d, (d+) d . This can be shown by the argument in (9) namely, if b L ([,]) then the function b( )( + ) d p + belongs to L p, but not necessarily to L p,r for r <. The space L p, has occured earlier in endpoint results related to other square functions, see [3], [35], [53]. The pointwise bound(5) and Theorem. yield a new boundfor maximal functions, in particular for multipliers in the Sobolev space L d/ d/p which

6 6 SANGHYUK LEE KEITH M. ROGERS ANDREAS SEEGER are compactly supported away from the origin. This Sobolev condition is too restrictive to give any endpoint bound for the maximal Bochner Riesz operator. However such a result can be deduced from a related result on maximal functions M m f(x) = sup F [m(t ) f ](x) t>0 with m compactly supported away from the origin. Our assumptions involve the Besov space Bα,q (which is L α when q = ) and thus the following result seems to be beyond the scope of a square function estimate when q. Theorem.. Let d, (d+) d < p <, α = d( p ) and p q. Assume that m is supported in (/,) and that m belongs to the Besov space B α,q. Then M m f L p C m B α,q f L p,q. We apply this to the Bochner Riesz maximal operator R λ R λ f(x) = sup Rt λ f(x). t>0 defined by Split ( t ) λ + = u λ (t) + m λ (t) where m λ is supported in (/,) and u λ C 0 (R). Then the maximal function M u λ f is pointwise controlled dominated by the Hardy Littlewood maximal function and thus bounded on L p for all p >. The function m λ belongs to the Besov space B λ+/, and Theorem. with q = yields a maximal version of (the dual of) Christ s endpoint estimate in []. Corollary.3. Let d, (d+) d < p <, and λ = d( p ). Then R λ f p C f L p,. We now consider operators T m with radial Fourier multipliers, as defined in (3), which do not necessarily decay at. The pointwise bounds (4), Theorem. and duality yield optimal L p L p, estimates in the range < p < (d+) d+3, for Hörmander type multipliers with localized L α conditions in the critical case α = d( p ). This demonstrates the effectiveness of Stein s point of view in () and (4). The following more general theorem is again beyond the scope of a square function estimate. We use dilation invariant assumptions involving localizationsofbesov spacesbα,q. Wenotethatin[33]ithadbeenleftopenwhether one could use endpoint Sobolev space or Besov spaces with q > in () below. Theorem.4. Let d, 0 Then T m maps L p to L p,q and L p,q to L p.

7 SQUARE FUNCTIONS AND MAXIMAL OPERATORS 7 It is not hard to see that the assumption () is independent of the choice of the particular cutoff ϕ. The result is sharp as T m does not map L p to L p,r for r < q. This can be seen by considering some test multipliers of Bochner Riesz type. Indeed, let Φ bearadial C function, with Φ (x) = for / x / and supported in {/ < x < } and similarly let χ be a radial C function compactly supported away from the origin and so that χ(ξ) = in a neighborhood of the unit sphere. Set (now with p < ) m(ξ) = χ(ξ) c j ( ξ η ) d( p ) + jd Φ ( j η)dη. j= We first remar that if we write m(ξ) = m ( ξ ), then m B α,q(r) if and onlyifm B α,q(r d )(hereweusethat m iscompactly supportedaway from the origin). Now considering the explicit formula for the ernel of Bochner Riesz means (cf. (4) below) it is easy to see that m B d/p d/,q (Rd ) if and only if {c j } j= belongs to lq ; moreover the necessary condition F [m] L p,q is satisfied if and only if {c j } belongs to l q. These considerations show the sharpness of Theorem.4 and also the sharpness of Theorem.. For the operator T m acting on the subspace L p rad, consisting of radial L p functions, the estimate corresponding to Theorem.4 has been nown to be true in the optimal range 0 p,q (R d ) if < p < d d+. This easily implies the L p rad Lp,q boundedness under assumption (), see [8]. Similarly, if in Theorem.4 we replace the range (, d+ d+3 ) with the smaller p-range (, d d+ ) (applicable only in dimension d 4) the result follows from the characterization of radial L p Fourier multipliers acting on general L p functions inarecent article by Heo, Nazarov andthethird author [9]. There it is proved that for p q, (3) T m L p Lp,q sup F [ φ( )m(t ) ] L p,q (R d ) t>0 if < p < d d+. The remainder of this paper is devoted to the proofs of the above theorems. They are mostly based on ideas in [9]. It remains an interesting open problem to extend the range of (3), in particular to prove such a result for some p > in dimensions two and three. Moreover it would be interesting to prove the above theorems beyond the Stein Tomas range.. Convolution with spherical measures In this section we prove an inequality for convolutions with spherical measures acting on functions with a large amount of cancellation. It can be

8 8 SANGHYUK LEE KEITH M. ROGERS ANDREAS SEEGER used to obtain results such as Theorems.4 for radial multipliers which are compactly supported away from the origin. To formulate this inequality let η be a Schwartz function on R d and let ψ be a radial C function with compact support in {x : x } and such that ψ(ξ) = u( ξ ) vanishes of order 0d at the origin. For j let I j = [ j, j+ ] and denote by σ r the surface measure on the sphere of radius r which is centered at the origin. Thus the norm of σ r as a measure is O(r d ). We recall the Bessel function formula (4) σ r (ξ) = r d J(r ξ ) with J(s) = c(d)s d Jd (s), which implies σ r (ξ) r d (+r ξ ) d. In view of the assumed cancellation of ψ, we have (5) ψ σ r = O(r (d )/ ). In what follows let ν be a probability measure on [,]. We will need to wor with functions with values in the Hilbert space H = L (R +, dr r ) and write ( F L p (L (H)) = F t (r, ) dr ) /dν(t) p. r j 0 Proposition.. Let p < (d+) d+3. Then ( ψ σ rt η F t,j (r, )drdν(t) jd Fj p /p. I j p L p (L (H))) The measure ν is used here to unify the proofs of Theorems. and.4. For our applications we are only interested in two such measures. For Theorems. and.4 we tae for ν the Dirac measure at t = (and consequently in this case we can set σ rt = σ r and eliminate all t-integrals in the proofs below). For the application to Theorem. we tae for ν the Lebesgue measure on [,]. We first give a proof for the L p bound of each term in the j-sum, which uses standard arguments ([4], [5]). Lemma.. Let p (d+) d+3. Then ψ σ rt F t (r, )drdν(t) jd/ F L p (L I j (H)). Proof. We use Plancherel s theorem and then the Stein Tomas restriction theorem [54]. With J as in (4) so that σ r (ξ) = r d J(r ξ ) and ψ(ξ) = j

9 SQUARE FUNCTIONS AND MAXIMAL OPERATORS 9 u( ξ ), we get from the restriction theorem ψ σ rt F t (r, )drdν(t) I j = c u(ρ) (rt) d J(rtρ) F t (r,ρξ )drdν(t) dσ(ξ )ρ d dρ S d I j u(ρ) ρ d d p (rt) d J(rtρ)F t (r, )drdν(t) dρ I j p ( u(ρ) ρ d d p r d J(rtρ)F t (r, )dr ) /dν(t) dρ. I j p In the last step we have used Minowsi s integral inequality. We claim that, for fixed x R d and t [,], (6) u(ρ) ρ d d p r I d J(rtρ)F t (r,x)dr dρ Ft (r,x) r d dr, j I j with the implicit constant uniform in x,t, and the lemma follows by substituting this in the previous display. To see (6) we first notice that for a radial H(w) = H ( w ) we have H (r)r d J(r ξ )dr = c d Ĥ(ξ). Thus, if we tae H x,t (w) = χ Ij ( w )F t ( w,x), the left-hand side of (6) is a constant multiple of ψ(ξ) ξ d p d Ĥx,t (tξ) dξ Ĥx,t (ξ) dξ = c H x,t (w) dw = c I F t (r,x) r d dr, j and we are done. In the inequality we used that ψ vanishes of high order at the origin. If we fix j and assume that F Q,t (r, ) is supported for all r in a cube Q of sidelength j then theexpression I j ψ σ rt F Q,t (r, )drdν(t) is supported in a similar slightly larger cube. From this it quicly follows that ψ σ rt F Q,t (r, )drdν(t) I j p ( jd/p F Q,t (r, ) dr ) /dν(t) p. r This estimate is however insufficient to prove Proposition. for p >. We shall also need the following orthogonality lemma. 0

10 0 SANGHYUK LEE KEITH M. ROGERS ANDREAS SEEGER Lemma.3. Let J,J (0, ) be intervals and let E, E be compact sets in R d with dist(e,e ) M. Suppose that for every r J i, the function x f i (r,x) is supported in E i. Then, for t,t [,], ψ σr t f (r, ),ψ σ r t f (r, ) dr dr J J [ ( ) /dy ] M d f i (r,y) r d dr. J i i= Proof. We follow [9] and apply Parseval s identity and polar coordinates in ξ. Then, ψ σr t f (r, ),ψ σ r t f (r, ) = c = c ψ(ξ) σ r t (ξ) σ r t (ξ) f (r,y )f (r,y )e i ξ,y y dy dy dξ u(ρ) (r t ) d J(r t ρ)(r t ) d J(r t ρ) f (r,y )f (r,y )J(ρ y y )dy dy ρ d dρ, so that the left-hand side of the desired inequality is equal to a constant multiple of (7) u(ρ) J (r t ) d J(r t ρ)f (r,y )dr (r t ) d J(r t ρ)f (r,y )dr J(ρ y y )dy dy ρ d dρ. J Now define two radial ernels by H y i i (w) = f i ( w,y i )χ Ji ( w ) so that the expression (7) can be written as a constant times (8) ψ(ξ) Ĥ y (tξ)ĥy (t ξ)j( ξ y y )dy dy dξ. Then, using the decay for Bessel functions and the M-separation assumption, J( ξ y y ) (+ρm) d, y i E i, i =,. By the Cauchy Schwarz inequality, the left-hand side of the desired inequality is thus bounded by [ ( ψ(ξ) i=, M d y i R d i=, (+ ξ M) d [ Ĥ y i i dy y R d ) Ĥy i /dy ] i (t i ξ) dξ ],

11 SQUARE FUNCTIONS AND MAXIMAL OPERATORS and by Plancherel s theorem this is M d M d and so we are done. [ ( ) /dy ] f i ( w,y) χ Ji ( w )dw w i R d [ ( ) /dy ] f i (r,y) r d dr, J i i=, i=, Proof of Proposition.. The case p = is trivial and we assume p > in what follows. For z Z d consider the cube q z of all x with z i x i < z i + for i =,...,d. Let ( γ j,z (f) = sup η(x y)f j,t (r,y)dy dr ) /dν(t), x q z r 0 and since η is a Schwartz function it is straightforward to verify that, for every j, ( ) /p ( (9) γ j,z (f) p F j,t (r, ) dr ) /dν(t) p, z Z d 0 r with the implicit constant independent of j. If γ j,z (f) 0 we set b j,z,t (r,x) = [γ j,z (f)] χ qz (x) η(x y)f j,t (r,y)dy and if γ j,z (f) = 0 we set b j,z,t = 0. Then ( (0) z Z d sup x q z 0 b j,z,t (r,x) dr r ) /dν(t). Let V j,z (x) = ψ σ rt b j,z,t (r,x)drdν(t). I j In view of (9) it suffices to show that for arbitrary functions z γ j,z on Z d we have, for < p < (d+) d+3, ( ) /p () γ j,z V j,z γ j,z p jd p j z Z d j z Z d where the implicit constant is independent of the specific choices of the b j,z,t (satisfying (0) with b j,z,t supported in q z ). Let µ d denote the measure on N Z d given by µ d (E) = j jd #{z Z d : (j,z) E}. Then () expresses the L p (Z d N,µ d ) L p (R d ) boundedness of an operator T. In the open p-range it suffices by real interpolation to show that T

12 SANGHYUK LEE KEITH M. ROGERS ANDREAS SEEGER maps L p, (Z d N,µ d ) to L p, (R d ). This amounts to checing the restricted wea-type inequality () meas ({ x : }) > λ λ p j j jd #(E j ) z E j V j,z where E j are finite subsets of Z d. Now for each (j,z) the term V j,z is supported on a ball of radius C j+ and therefore the entire sum is supported on a set of measure j jd #(E j ). Thus the estimate () holds for λ 0. Assume now that λ > 0. WedecomposeR d intodyadic halfopen cubesofsidelength j andlet Q j be the collection of these j -cubes. For each Q Q j let Q be the cube with same center as Q but sidelength j+5. Note that for z Q the term V j,z is supported in Q. Letting and Q j (λ) := {Q Q j : #(E j Q) > λ p } Ω = j we have the favorable estimate meas(ω) 5 j Q Q j (λ) λ p j jd #(E j ). Q Q j (λ) Q 5 j Q, jd Q Q j (λ) #(E j Q) λ p Thus the remaining estimates need only involve the good part of E j ; Ej λ = Q E j. Q Q j \Q j (λ) Note that every subsetof diameter C j, with C >, contains C d λ p points in Ej λ. Letting V j = V j,z, it remains to show that meas ({ x : j z E λ j V j (x) > λ }) λ p j jd #(E j ). This will follow from p (3) V j Cλd+ logλ jd #(E j ) j j and Tshebyshev s inequality since, for p < (d+) d+3 and λ >, λ p d+ logλ C p λ p.

13 SQUARE FUNCTIONS AND MAXIMAL OPERATORS 3 Proof of (3). Setting N(λ) = 0log λ, we treat the sums over j N(λ) and j > N(λ) separately. Using the Cauchy Schwarz inequality for the first sum, (4) V j log(λ) V j + V j j j N(λ) j>n(λ) + j>n(λ) N(λ)<<j 0 V j,v. Since the expression z E λ j QV j,z is supported in Q it follows easily from Lemma. (applied with the endpoint exponent (d+) d+3 ) that V j ( b j,z,t (r, ) dr ) /dν(t) r Q Q j I j jd z Q E λ j Since Q E λ j contains no more than λp points we have by (0) and thus (5) ( I j jd z Q E λ j b j,z,t (r, ) dr ) /dν(t) r j= (d+) d+3 (d+) d+3 jd( #(E λ j Q) )d+3 d+ jd #(E j Q)λ p d+ V j λ p d+ jd #E j. Thus we get the asserted bound (3) for the sum of the first two terms on the right-hand side of (4). It remains to estimate the mixed terms V j,v for N(λ) < < j 0. For fixed j, we let Ij, n = [ n, (n+)] I j with n Z, n j. Then with V,n j,z,t : = ψ σ rt b j,z,t (r, )dr we can write V,z : = ψ V j,v = Ij, n n j I σ rs b,z,s(r, )drdν(s) z Ej λ z Z (n,z,t) V,n j,z,t,v,z dν(t); here, in view of the support properties, we were able to restrict the z summation to the set Z (n,z,t) := {z E λ : z z nt C },.

14 4 SANGHYUK LEE KEITH M. ROGERS ANDREAS SEEGER with C a suitable constant. Observethat for z Z (n,z,t), with j 0, we have z z j since nt I j. By Lemma.3 (applied with the parameter M j ) we have for fixed z,z,t, (6) with V,n j,z,t,v,z j d h z,n j,,t (y)dy y z C y z C h z,n j,,t (y) = ( h z (y) = I n j, By our normalization assumption (0), (7) ( n b j,z,t (r,y) r d dr) /, ( ) /dν(s). b,z,s(r,y) r d dr I h z (y )dy h z,n j,,t (y) ) / dν(t) jd and h z (y ) d and, by the Cauchy Schwarz inequality, we also have (8) Altogether, using (6) and (7), V j,v d j z E λ j n n h z,n j,,t (y) dν(t) jd j. y z C h z,n j,,t (y)dyd/ #(Z (n,z,t))dν(t). Recall that for every cube Q of sidelength the set Z (n,z,t) Q contains at most λ p points. Moreover, for each z,n,t there are no more than O( (j )(d ) ) dyadiccubesof sidelength which intersect Z (z,n,t). Thus This and (8) yield, for j 0, d j V j,v z E λ j #(Z (n,z,t)) λ p (j )(d ). y z C n h z,n j,,t (y)dν(t)dyd/ λ p (j )(d ) d j #(Ej λ )jd j d λ p (j )(d ) λ p d jd #(Ej λ ). By summing a geometric series, we see that V j,v λ p N(λ)d jd #E j, j>n(λ) N(λ)<<j 0 j

15 SQUARE FUNCTIONS AND MAXIMAL OPERATORS 5 and by the choice of N(λ) = 0log λ, we have λ p N(λ)d λ p 5. This gives the desired estimate (indeed a better estimate) for the third term on the right-hand side of (4) and finishes the proof of (3). Lorentz space estimates. We will use the following interpolation lemma in which we allow any d > 0; this is the only place where d does not necessarily denote the dimension. Lemma.4. Let p 0 < p, d > 0, and, for j N, let S j be an operator acting on functions on a measure space (M,µ) with values in a Banach space B. Suppose that the inequality pi ( (9) S j g j M i jd ) gj p /pi i L p i(b) j holds for i = 0,. Then for p 0 < p < p, p = ϑ p 0 + ϑ p, and p q, jd/p L S j f j C p,qm θ p,q 0 M θ ( ) /q f j q Lp B j j j with q = interpreted as usual by taing a supremum. Proof. Let µ d denote the measure on N M given by µ d (E) = jd dµ. j x:(j,x) E By real interpolation of the assumptions (9) we have L (30) S j g j C p,qm θ {gj } p,q L p,q (µ d,b). j 0 M θ We may apply this with g j = jd/p f j and then our assertion follows from the inequality (3) { jd/p f j } L p,q (µ d,b) {fj } L p (l q (B)). The case for p = q is immediate. We also have µ d({ (j,x) : jd p f j (x) B > λ }) µ d({ (j,x) : jd p sup f (x) B > λ }) = jd dx λ p sup f (x) p B dx, j: jd < sup f (x) p B λ p which yields (3) for q =. By complex interpolation (with fixed p) we obtain (3) for p q. As an immediate consequence of Lemma.4 we obtain a Lorentz space version of Proposition. which is the main ingredient in the proof of Theorem..

16 6 SANGHYUK LEE KEITH M. ROGERS ANDREAS SEEGER Corollary.5. Let < p < (d+) d+3 and p q. Then jd p ψ η σ rt F j,t (r, )drdν(t) j I j L p,q ( /qdν(t) F j,t H) q p. A preparatory result. For the proof of Theorems. and.4 we shall need a more technical variant of the corollary which is compatible with atomic decompositions. In what follows we let ν be Dirac measure at t = so that the integrals in t disappear. Let l and for z Z d let j R l z = {x : l z i x i < l (z i +), i =,...,d}; these sets form a grid of disjoint cubes with sidelength l covering R d. In the following proposition we use the conclusion of Proposition. as our hypothesis. Proposition.6. Suppose that, for some p (,), ( ψ σ r η F j (r, )dr jd F j p /p. I p L (H)) p j j l+ Let b j,z L (H) with b j,z L (H), let β j (z) C and define S j β j (x) = ( β j (z) ψ η σ r ( χ R l z b j,z (r, ) ) ) dr z I j Then, for < p < p and p q, jd/p L S j β j C p ld(/p /) ε(p))( ( β p,q j (z) q) p/q) /p, j l+ where ε(p) = (d )p ( p p ). j z Z d Proof. We argue as in [0], Prop. 3.. First note that ( p (3) S j β j ld(/p /) β j (z) ) p /p. j l+ j jd z Indeed, by hypothesis the left-hand side is dominated by a constant times ( jd p ) /p β j (z)χ R l z b j,z j ( z j jd z L p (H) j β j (z) p ) χ R l z b j,z p /p L p (H) and after using Hölder s inequality on each R l z and the L normalization of b j,z we obtain (3).

17 SQUARE FUNCTIONS AND MAXIMAL OPERATORS 7 There is a better L bound. Note that for r j the term ψ σ r b j,z (r, ) is supported on an annulus with radius j and width l. We use the Cauchy Schwarz inequality on this annulus and then (5) and estimate η j l+ψ β j (z) σ r (b j,z (r, )χ R l z )dr z I j β j (z) ψ σ r (b j,z (r, )χ R l z ) dr j l+ z I j β j (z) ( l j(d ) ) / ψ σ r (b j,z (r, )χ R l z ) dr j l+ z I j l/ j(d ) β j (z) b j,z (r, ) dr, j z I j and by Cauchy Schwarz on I j and the normalization assumption on b j,z we get (33) j β j j l+s l/ j jd β j (z). z NowLemma.4isusedtointerpolate(3) and(33)andtheassertionfollows. 3. Proof of Theorem. We start with a simple fact on Besov spaces, namely if ζ is a C function supported on a compact subinterval of (0, ) then (34) ζ( )g( ) B α,q (R d ) g B α,q (R), α > 0. To see this note that the corresponding inequality with Sobolev spaces L α, α = 0,,,... is true by direct computation, and then (34) follows by real interpolation. Next if F [m( ](x) = κ( x ) we can use polar coordinates to see that ( [ ] q/ ) /q; (35) m( ) B α,q (R d ) κ(r) r α+d dr I j j=0 here, as in, I j = [ j, j+ ] for j, and I 0 = (0,]. We shall first prove a dual version of a bound for a maximal operator where the dilations are restricted to [, ]. Proposition 3.. Let d, < p < (d+) d+3, α = d( p ), p q. Then, for m Bα,q with support in (/,), (36) T m(t ) f t dt m L p,q Bα,q f t dt. p

18 8 SANGHYUK LEE KEITH M. ROGERS ANDREAS SEEGER Proof. Let φ be a radial C -function so that φ is supported in {/8 ξ 8} and equal to one in {/4 ξ 4}. Then T m(t ) f t = T m(t ) (φ f t ) for t. Also T m(t ) f = 0 κ(r)t d σ rt φ f dr, t, where κ is bounded and smooth, and the right-hand side of (35) is finite with α = d/p d/. We may split φ = ψ η where ψ Cc with ψ vanishing of high order at the origin. It then suffices to show that (37) κ(r)t d ψ σ rt f t drdt L p,q ( ( κ(r) r d/pdr ) q/ ) /q f t dt I j r j= This estimate follows by applying Corollary.5. Tae ν to be Lebesgue measure on [,], use the tensor product F j,t (r,x) = jd/p χ Ij (r)κ(r)t d f t (x) and observe that F j L p (L (H)) can be estimated by the right-hand side of (37). We also need a standard orthogonality estimate, in Lorentz spaces. Lemma 3.. Let {β } Z a family of L -functions, satisfying (i) sup β L (R d ) <, (ii) sup ξ Z β (ξ) <. Then L ( ) /p, (38) β f f p,q p L < p <, p q, p,q and (39) ( ) /p β f p L p,q f L p,q, < p <, q p. Here the functions {f } are allowed to have values in a Hilbert space H (and f may have values in H ). Proof. By duality (38) and (39) are equivalent. To see (38) we define m d to be the product measure on R d Z of Lebesgue measure on R d and counting measure on Z. Define an operator P acting on functions (x,) f (x), letting F = {f }, by PF = β f. By assumption (i) P maps the space L (R d Z,m d ) to L (R d ) and by the almost orthogonality assumption (ii) it maps L (R d Z,m d ) to L (R d ). Hence by real interpolation P maps L p,q (R d Z,m d ) to L p,q (R d ) for all 0. Let E,m (F) = {x : f (x) H > m }. p

19 SQUARE FUNCTIONS AND MAXIMAL OPERATORS 9 If p q we have, by the triangle inequality in l q/p, ( F L p,q (m d ;H) mq meas(e,m (F)) m ( ( mq meas(e,m (F)) m q p )p q ) p q p ) q ( f p L p,q (H) )/p, where for q = we mae the usual modification. This proves (38). Proof of Theorem., conclusion. Now let (d+) d < p < and p q. Let φ be as above and define L by L f(ξ) = φ( ξ) f(ξ). We may then estimate ( M m f p sup T m( t )L f /p. p) p t Z For every Z, sup T m( t )L f p C m B L f t d/ d/p,q L p,q ; this follows for = 0 by duality from Proposition 3., and then for general by scaling. By Lemma 3. ( L f ) p /p L f L p,q p,q Z and combining the estimates we are done. 4. Proofs of Theorems. and.4 Many endpoint bounds for convolution operators on Lebesgue spaces can be obtained by interpolation involving a Hardy space estimate and an L estimate; this idea goes bac to [40], [7]. In some instances it has been advantageous to use Hardy space or BMO methods such as atomic decompositionsorthefefferman Stein #-maximal functiondirectly on L p toprove theorems which cannot immediately be obtained by interpolation (see for example endpoint questions treated in [3], [46], [5], [9], [9]). We formulate such a result suitable for application in the proofs of Theorems. and.4. In order to give a unified treatment we need to consider vector-valued operators. Let H, H be Hilbert spaces. We consider translation invariant operators mapping L (H ) to L (H ), with convolution ernels having values in the space L(H,H ) of bounded operators from H to H. On the Fourier transform side, the operators are given by Tf(ξ) = M(ξ) f(ξ) where f(ξ) H, Tf(ξ) H, with sup ξ M(ξ) L(H,H ) <. If S is an L (R d ) convolution operator with scalar ernel (and multiplier) and H is a Hilbert space then S extends to a bounded operator on L (R d,h), denoted temporarily by S Id H. If T is as before with L(H,H )-valued ernel then

20 0 SANGHYUK LEE KEITH M. ROGERS ANDREAS SEEGER (S Id H )T = T(S Id H ). With a slight abuse of notation we shall continue to write S for either S Id H and S Id H. We need to formulate a hypothesis which will be used for convolution operators with multipliers compactly supported away from the origin. Hypothesis 4.. Let 0 and A > 0. We say that the ernel K satisfies Hyp(p,q,ε,A) if for every l 0 one can split the ernel into a short and long range contribution K = K sh l +K lg l so that the following properties hold: (i) K sh l is supported in {x : x l+0 }. (ii) sup ξ R d F[K sh l ](ξ) L(H,H ) A. (iii) For every family of L functions {a z } z Z d, with supp(a z ) Rz l and sup z a z L (H ), and for γ l p (Z d ) the inequality holds. z l (γ(z)a z) L p,q Al(d( p ) ε)( γ(z) p) /p K lg Theorem 4.. Given p (,), p q, ε > 0 and A > 0 suppose that K, Z are L(H,H )-valued ernels satisfying hypothesis Hyp(p,q,ε,A). Define the convolution operator T by T f(x) = d K ( (x y))f(y)dy Let η be a scalar Schwartz function with η supported in {ξ : /4 ξ 4} and let η = d η( ). Then the operator f Z η T f, initially defined on H valued Schwartz functions with compact Fourier support away from the origin, extends to an operator acting on all f L p (H ) so that the inequality η T f C pa f L L p,q (H ) p (H ) holds. The proof of Theorem 4. is by now quite standard, but for completeness we include it in Appendix A below. Given Theorem 4. we now show how it can be used to deduce Theorems. and.4 from the results in. Remar 4.3. We actually prove a slightly more general result: Assuming d that the estimate of Proposition. holds for some exponent p (, d+ ) then the conclusion of Theorem.4 holds for < p < p and the conclusion of Theorem. holds for p < p <. A similar remar also applies to Theorem.. z

21 SQUARE FUNCTIONS AND MAXIMAL OPERATORS Proof of Theorem.. With p as in Remar 4.3, by duality and changes of variables t = s it is enough to show that, for < p < p and α = d( p ), (40) Z F [ ξ ( s ξ ) α s + f s ] ds s ( L p, f s ds ) / p. s Let φ be such that φ is supported in {/4 ξ 4} with φ(ξ) = in {/3 ξ 3}. Let (4) J α (ρ) = ρ d α Jd +α(ρ) so that F[J α (t )](ξ) = c α t d ( ξ /t ) α + (see Chapter VII of [46]). In particular J 0 = J as in (4). Let φ = d φ( ). Then (40) follows from (4) φ Z J α (s y )f s ( y)dy ds s ( L p, f s ds s ) / p. The reduction of (40) to (4) involves incorporating irrelevant powers of s in the definition of f s and an application of standard estimates for vectorvalued singular integrals ([39]) to handle the contribution of ( ξ ) α + away from the unit sphere. We omit the details. We now split φ = η ψ ψ where η has the same support as φ and ψ is a radial C0 function supported in {x : x /0}, furthermore ψ vanishes to order 0d at the origin. If H = L ([,],dr) then we wish to apply Theorem 4. with the H valued ernel K K (independent of ) defined by (43) K(x),v = v(s) 0 J α (sr)ψ σ r (x)drds. We define the corresponding short range ernel K sh l by letting the r-integral in (43) extend over [0, l+ ] and the long range ernel K lg l by letting the r-integral extend over ( l+, ). Clearly the support condition (i) in Hypothesis 4. holds. Note that d/p d/ > /forp < d/(d+). Thustochec condition(ii)ofhypothesis 4. it suffices to verify that sup ξ R d ( l+ 0 J α (rs) ψ(ξ) σ r (ξ)dr / ds) Aα, α > /. Writing ψ(ξ) = u( ξ ), this reduces to (44) sup u(ρ) ( l+ J α (rs)j(rρ)r d dr / ds) Aα. ρ>0 0 We may tae the r-integral over [, l+ ] since the estimate for the contribution for r [0,] is immediate. We use the standard asymptotic expansions

22 SANGHYUK LEE KEITH M. ROGERS ANDREAS SEEGER for the modified Bessel-function J α, (45) J α (u) = u d α[ n=0 u n( ] c + n,α eiu +c n,α e iu) +O( u ), u andalsotheanalogous expansion forj = J 0. Ifweconsideronlytheleading terms in both asymptotic expansions we are led to bound sup ρ>0 u(ρ) ( ρ d l+ e ir(±s±ρ) r α dr / ds) Aα, α > /, which follows from Plancherel s theorem on R. The other terms with lower order or nonoscillatory error terms are similar or more straightforward. Note that we also use u(ρ) ρ 0d for ρ (0,). This establishes condition (ii) in Hypothesis 4.. Finally we verify condition (iii). Let {a z } z Z d be L (H ) functions with sup z a z L (H ), supported on l -cubes with disjoint interiors. We then need to show that (46) ψ ψ σ r γ(z) J α (sr)a z (s, )ds j l+ I j z L p, A l(d( p ) ε)( γ(z) p) /p. Setting ( c j,z = I j R z we may apply Proposition.6 for q = with β j (z) = jd/p γ(z)c j,z, J α (sr)a z (s,x)ds dr ) / r dx and b j,z (r,x) = χ Ij (r)c j,z z J α (sr)a z (s,x)ds if c j,z 0 and b j,z = 0 if c j,z = 0. We can then dominate the left-hand side of (46) by a constant times l(d p d ε(p))( ( β j (z) ) p/ ) /p with ε(p) > 0 for p < p. We are only left to show that for fixed z ( β j (z) ) / γ(z) j z where the implicit constant is uniform in z. This estimate follows from (47) jd/p J α (sr)a z (s,x)ds dr r a z (s,x) ds j l+ I j j

23 SQUARE FUNCTIONS AND MAXIMAL OPERATORS 3 and integration over x R z. To see (47) we use again the asymptotics (45). The estimate for the oscillatory terms (with n = 0,) becomes jd/p e ±isr s d α n a z (s,x)ds dr j l+ I j r α n d and since α = d/p d/ it suffices to show e ±isr v(s)a z (s,x)dt dr a z (s,x) ds a z (s,x) ds, with sup s v(s) C. But this is an immediate consequence of Plancherel s theorem. Lastly, ifin(47) weputtheerrortermo((sr) α d )forj α (sr) the resulting expression can be easily estimated by [ r 3 ] dr a z (s,x) ds a z (s,x) ds. l This concludes the proof of (47), and thus the proof of Theorem.. Proof of Theorem.4. We apply Theorem 4. with H = H = C. It is easy to see that it suffices to show that, for α = d(/p /), F [ m ( ) η( ) f ] L sup m p,q B α,q f p, Z where m are functions in Bα,q(R) supported in (/,) and η is a radial Schwartz function with η supported in the annulus {/4 < ξ < 4}. Now write F [m ( )](x) = κ ( x ). Using polar coordinates and (34) we see that (48) ([ j κ (r) r d/pdr I j r ] q/ ) /q m B α,q, α = d(/p /), and of course sup 0<r κ (r) <. With ψ as in the proof of Theorem. it suffices to show that the ernels [ Kl = K,sh l +K,lg l+ ] l = + κ (r)ψ ψ σ r dr l+ 0 satisfy the assumptions of Hypothesis 4., uniformly in. Note that by (5) [ ] F κ (r)ψ ψ σ r dr (ξ) κ (r) r d dr I j I j j(d p d+)( κ (r) r d/pdr I j r ) / and since p < d d+ we may sum in j to deduce that sup K,sh l <.

24 4 SANGHYUK LEE KEITH M. ROGERS ANDREAS SEEGER We turn to the ernels K,lg l and again show using Proposition.6 that they suffice condition (iii) in Hypothesis 4.. Define ( β,j (z) = γ(z) κ (r) r d/pdr ) / I j r and b,j,z (r,x) = [β,j (z)] jd/p χ Ij (r)κ (r)a z (x) if β,j (z) 0 (and b,j,z = 0 otherwise). Then ( b,j,z L (H) = b j,z (r,x) dr ) / r dx. Now K,lg l z γ(z)a z = z 0 j l+ β,j (z) ψ ψ σ r b,j,z (r, )dr I j and by Proposition.6 we have K,lg l p a z l(d p d ε(p))( ( β,j (z) q) p/q) /p, z with ε(p) > 0 for p < p. Finally, by (48) ( ( β,j (z) q) p/q) /p ( γ(z) p) /p m B, d(/p /),q z which completes the proof. j Appendix A. Proof of Theorem 4. By normalization we may assume that Hypothesis Hyp(p, q, ε, A) holds witha =. WeuseatomicdecompositionsinL p whichareconstructedfrom square functions, based on the ideas by Chang and Fefferman [8]. A convenient and useful form is given by an l -valued version of Peetre s maximal square function (cf. [8], [55]), ( ) /, Sf(x) = sup L f(x+y) H y 00d wherel f = φ f, with φ = d φ( ), and φ is a radial Schwartz function with φ supported in {ξ : /5 < ξ < 5}. Then Sf p C p f L p (H ), < p <. We closely follow the argument in [0]. Choose φ by splitting the function η in the statement of Theorem 4. as z z η = ψ φ j

25 SQUARE FUNCTIONS AND MAXIMAL OPERATORS 5 where ψ is a radial C0 -function with support in {x : x < /4} whose Fouriertransformvanishestoorder0dattheorigin. Wesetψ = d ψ( ), then η = ψ φ and we have η T f = ψ T L f. For Z, we tile R d by the dyadic cubes of sidelength and write L(Q) = if the sidelength of a dyadic cube Q is. For each n Z, let Ω n = {x : Sf(x) > n }. Let Q n be the set of all dyadic cubes of sidelength which have the property that Q Ω n Q / but Q Ω n+ < Q /. Let Ω n = {x : Mχ Ωn (x) > 00 d } with M the Hardy Littlewood maximal operator. The set Ω n is open, contains Ω n and satisfies Ω n Ω n. Let W n be the set of all dyadic cubes W for which the 50-fold dilate of W is contained in Ω n and W is maximal with respect to this property. The collection {W} forms a Whitneytype decomposition of Ω n. The interiors of the Whitney cubes are disjoint. For each W W n we denote by W the tenfold dilate of W; the dilates {W : W W n } have still bounded overlap. Note that each Q Q n is contained in a unique W W n. For W W n, set a,w,n = (L f)χ Q, Q Q n Q W and for any dyadic cube W define a,w = n:w W n a,w,n. The functions a,w,n can be considered as atoms, but without the usual normalization. For fixed n one has (49) a,w,n L (H ) n meas(ω n ). W W n Indeed (arguing as in [8]) the left-hand side is equal to L f(x) H dx Q W n Q W n Q Q n Q Q Q n sup Q (Ω n\ω n+ ) y d Sf(x) meas(ω n ) (n+). Ω n\ω n+ L f(x+y) H dx

26 6 SANGHYUK LEE KEITH M. ROGERS ANDREAS SEEGER Let T lg,l, Tsh,l, be the convolution operator with ernels d K,lg l ( ) and d K,sh l ( ), respectively. The desired estimate will follow once we establish the short range inequality (50) ψ T sh Sf p. L p (H ) l 0 W nw n L(W)= +l,l a,w and for fixed l 0 the long range inequality (5) ψ T lg L p,q (H ) lε Sf p. W nw n L(W)= +l,l a,w Proof of (50). We prove that for < r < and for fixed n Z (5) ψ T sh r C L r r nr meas(ω n ). (H ) l 0 W W n L(W)= +l,l a,w,n By real interpolation (cf. Lemma. in [9]) it follows that the stronger estimate ψ T,l sh a,w,n p np meas(ω n ). L p (H ) n n l 0 W W n L(W)= +l holds and this implies (50) since n np meas(ω n ) Sf p p. Since the expression inside the norm in (5) is supported in Ω n we see that the left-hand side of (5) is dominated by (53) meas(ω n ) r/ ψ T sh r. L (H ) l 0 W W n L(W)= +l,l a,w,n The convolution operators with ernel ψ are almost orthogonal and thus we can dominate the left-hand side of (53) by a constant times (54) meas(ω n ) r/( l 0 W W n L(W)= +l T,l sh a,w,n r/. L (H )) Now, for each W with L(W) = +l, the function T,l sha,w,n is supported in the expanded cube W. The cubes W with W Ω j have bounded overlap, and therefore the expression (54) is (55) meas(ω n ) r/( T sh r/. L (H )) Now we have for fixed W T sh,l a,w,n l 0 W W n L(W)= +l,l a,w,n L (H ) a,w,n L (H ).

27 By (49) we have SQUARE FUNCTIONS AND MAXIMAL OPERATORS 7 l 0 W W n L(W)= +l a,w,n W W n a,w,n n meas(ω n ). Since meas(ω n) meas(ω n ) it follows that the right-hand side of (55) is dominated by a constant times meas(ω n ) nr which then yields (5) and finishes the proof of the short range estimate. Proof of (5). We use the first estimate in Lemma 3., with β = ψ, and the H valued functions F = W: T lg,l a,w. We then see that (5) follows from (56) W: L(W)= +l T lg,l a,w L(W)= +l p lεp L p,q (H ) n meas(ω n ) np. By rescaling and assumption (iii) in Definition 4. we have for every T lg [ ],l a,w L p,q (H ) W: L(W)= +l lε (l )d( p )( /p. a,w p L (H )) W: L(W)= +l Thus in order to finish the proof we need the inequality (57) (l )d( p ) a,w p L (H ) np meas(ω n ). n W: L(W)= +l For fixed and fixed W, the functions a,w,n, n Z live on disjoint sets (since the dyadic cubes of sidelength are disjoint and each such cube is in exactly one family Q n ). Therefore ( ) / a,w L (H ) a,w,n L (H ) and thus we can bound the left-hand side of (57) by (l )d( p ) a,w,n p L (H ) n n n Z W W n: L(W)= +l ( Z W W n: L(W)= +l meas(ω n) p/( n ) p/ ( meas(w) W W n a,w,n L (H ) W W n: L(W)= +l ) p/; a,w,n L (H ) ) p/

28 8 SANGHYUK LEE KEITH M. ROGERS ANDREAS SEEGER here we used the disjointness of Whitney cubes in W n. By (49) the last displayed expression is bounded by C n meas(ω n) p/ ( n meas(ω n )) p/ n np meas(ω n ) which gives (57). References [] J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. (99), no., [] J. Bourgain, L. Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal. (0), no. 6, [3] A. Carbery, The boundedness of the maximal Bochner Riesz operator on L 4 (R ), Due Math. J. 50 (983), [4], Radial Fourier multipliers and associated maximal functions, Recent progress in Fourier analysis (El Escorial, 983), 49 56, North-Holland Math. Stud.,, North- Holland, Amsterdam, 985. [5] A. Carbery, G. Gasper, W. Trebels, Radial Fourier multipliers of L p (R ), Proc. Nat. Acad. Sci. U.S.A. 8 (984), no. 0, Phys. Sci., [6] A. Carbery, A. Seeger, Weighted inequalities for Bochner Riesz means in the plane, Q. J. Math. 5 (000), no., [7] L. Carleson, P. Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (97), [8] S.Y.A. Chang, R. Fefferman, A continuous version of duality of H and BMO on the bidisc, Annals of Math. (980), [9] M. Christ, On almost everywhere convergence of Bochner Riesz means in higher dimensions, Proc. Amer. Math. Soc. 95 (985), 6-0. [0], Wea type (,) bounds for rough operators, Ann. of Math. () 8 (988), no., 9 4. [], Wea type endpoint bounds for Bochner Riesz multipliers, Rev. Mat. Iberoamericana 3 (987), no., 5 3. [] M. Christ, C.D. Sogge, The wea type L convergence of eigenfunction expansions for pseudodifferential operators, Invent. Math. 94 (988), no., [3] H. Dappa, W. Trebels, On maximal functions generated by Fourier multipliers, Ar. Mat. 3 (985), no., [4] C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 4 (970), [5], A note on spherical summation multipliers. Israel J. Math. 5 (973), [6] C. Fefferman, E.M. Stein, Some maximal inequalities, Amer. J. Math. 93 (97), [7], H p spaces of several variables, Acta Math. 9 (97), no. 3-4, [8] G. Garrigós, A. Seeger, Characterizations of Hanel multipliers, Math. Ann. 34 (008), no., [9] Y. Heo, F. Nazarov, A. Seeger, Radial Fourier multipliers in high dimensions, Acta Math. 06 (0), no., [0], On radial and conical Fourier multipliers, J. Geom. Anal. (0), no., [] L. Hörmander, Estimates for translation invariant operators in L p spaces, Acta Math. 04 (960), [] S. Igari, S. Kuratsubo, A sufficient condition for L p -multipliers, Pacific J. Math. 38 (97),

29 SQUARE FUNCTIONS AND MAXIMAL OPERATORS 9 [3] M. Kaneo, G. Sunouchi, On the Littlewood Paley and Marciniewicz functions in higher dimensions, Tôhou Math. J. () 37 (985), no. 3, [4] S. Lee, Improved bounds for Bochner Riesz and maximal Bochner Riesz operators, Due Math. J. (004), [5] S. Lee, K.M. Rogers, A. Seeger, Improved bounds for Stein s square functions, Proc. London Math. Soc. 04 (0), no. 6, [6] S. Lee, A. Seeger, Lebesgue space estimates for a class of Fourier integral operators associated with wave propagation, Math. Nach., to appear. [7] G. Mocenhaupt, A. Seeger, C D. Sogge, Local smoothing of Fourier integral operators and Carleson-Sjölin estimates. J. Amer. Math. Soc. 6 (993), no., [8] J. Peetre, On spaces of Triebel-Lizorin type, Ar. Mat. 3 (975), [9] M. Pramani, K.M. Rogers, A. Seeger, A Calderón Zygmund estimate with applications to generalized Radon transforms and Fourier integral operators. Studia Math. 0 (0), no., 5. [30] A. Seeger, On quasiradial Fourier multipliers and their maximal functions, J. Reine Angew. Math. 370 (986), [3], Estimates near L for Fourier multipliers and maximal functions, Arch. Math. (Basel) 53 (989), no., [3], Remars on singular convolution operators, Studia Math. 97 (990), 9 4. [33], Endpoint estimates for multiplier transformations on compact manifolds, Indiana Univ. Math. J. 40 (99), [34], Endpoint inequalities for Bochner Riesz multipliers in the plane, Pacific J. Math. 74 (996), [35] A. Seeger, T. Tao, Sharp Lorentz space estimates for rough operators. Math. Ann. 30 (00), no., [36] A. Seeger, T. Tao, J. Wright, Endpoint mapping properties of spherical maximal operators. J. Inst. Math. Jussieu (003), no., [37] E.M. Stein, Localization and summability of multiple Fourier series, Acta Math. 00 (958), [38], Topics in harmonic analysis related to the Littlewood Paley theory. Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Toyo Press, Toyo 970. [39], Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J [40], L p boundedness of certain convolution operators. Bull. Amer. Math. Soc. 77 (97), [4], Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (976), no. 7, [4], Some problems in harmonic analysis, in Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 978), Part, 3 0, Amer. Math. Soc., Providence, R.I. [43], The development of square functions in the wor of A. Zygmund, Bull. Amer. Math. Soc. (N.S.) 7 (98), no., [44], Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Princeton University Press, Princeton, NJ, 993. [45] E.M. Stein, S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (978), no. 6, [46] E. M. Stein, G. Weiss, Introduction to Fourier analysis in Euclidean spaces, Princeton Univ. Press, 97. [47] G. Sunouchi, On the Littlewood Paley function g of multiple Fourier integrals and Hanel multiplier transformations, Tôhou Math. J. () 9 (967),

ON MAXIMAL FUNCTIONS FOR MIKHLIN-HÖRMANDER MULTIPLIERS. 1. Introduction

ON MAXIMAL FUNCTIONS FOR MIKHLIN-HÖRMANDER MULTIPLIERS LOUKAS GRAFAKOS, PETR HONZÍK, ANDREAS SEEGER Abstract. Given Mihlin-Hörmander multipliers m i, i = 1,..., N, with uniform estimates we prove an optimal