Endpoint maximal and smoothing estimates for Schroedinger equations
aa r X i v : . [ m a t h . C A ] M a y ENDPOINT MAXIMAL AND SMOOTHING ESTIMATES FORSCHR ¨ODINGER EQUATIONS
KEITH M. ROGERS AND ANDREAS SEEGER
Abstract.
For α > i∂ t u + ( − ∆) α/ u = 0. We prove an endpoint L p inequality for the maximal functionsup t ∈ [0 , | u ( · , t ) | with initial values in L p -Sobolev spaces, for p ∈ (2 + 4 / ( d + 1) , ∞ ). Thisstrengthens the fixed time estimates due to Fefferman and Stein, and Miyachi. As anessential tool we establish sharp L p space-time estimates (local in time) for the samerange of p . Introduction
For α > L p estimates for solutions to the initial value problem (cid:26) i∂ t u + ( − ∆) α/ u = 0 u ( · ,
0) = f. The case α = 2 corresponds to the Schr¨odinger equation. We will not consider α = 1 whichcorresponds to the wave equation and exhibits different mathematical features.When f is a Schwartz function, the solution can be written as u ( x, t ) = U αt f ( x ), where(1.1) d U αt f ( ξ ) = e it | ξ | α b f ( ξ )with b f ( ξ ) = R f ( y ) e − i h y,ξ i dy as the definition of the Fourier transform. The sharp endpoint L p -Sobolev bounds for fixed t are due to Fefferman and Stein [11] and Miyachi [15]. Theirresult states that for any compact time interval I and any p ∈ (1 , ∞ ),sup t ∈ I (cid:13)(cid:13) U αt f (cid:13)(cid:13) L p ( R d ) C I,p,α k f k L pβ ( R d ) , βα = d (cid:12)(cid:12)(cid:12) − p (cid:12)(cid:12)(cid:12) ;this is sharp with respect to the regularity index β and can also be deduced from certainendpoint versions of the H¨ormander multiplier theorem ([1], [19]).We strengthen the fixed time estimates as follows. Theorem 1.1.
Let p ∈ (2 + d +1 , ∞ ) and α > . Then, for any compact time interval I , (1.2) (cid:13)(cid:13) sup t ∈ I | U αt f | (cid:13)(cid:13) L p ( R d ) C I,p,α k f k L pβ ( R d ) , βα = d (cid:16) − p (cid:17) . This implies pointwise convergence results; indeed we shall prove a little more, namelyif χ ∈ C ∞ c ( R ) then the function t χ ( t ) U αt f ( x ) belongs to the Besov space B p /p, ( R ), foralmost every x ∈ R d . In particular these functions are continuous (for almost every x ) andtherefore this implies almost everywhere convergence to the initial datum as t → Mathematics Subject Classification.
Key words and phrases.
Schr¨odinger equation, dispersive equations, pointwise convergence, maximalfunctions, smoothing, space time regularity.The first author was supported by MEC project MTM2007-60952 and UAM-CM project CCG07-UAM/ESP-1664. The second author was supported in part by NSF grant DMS 0652890.
Our maximal function result is closely related to certain space-time estimates whichimprove the regularity index. The first such bounds are due to Constantin and Saut [7],Sj¨olin [21], and Vega [27] who showed that better L regularity properties hold locally when α ∈ (1 , ∞ ); namely, if f ∈ L − ( α − / ( R d ) then u ∈ L ( R d +1 ). However, it is not possibleto replace the L -norms over compact sets by L -norms which are global in space. This isknown as the local smoothing phenomenon. For functions in L -Sobolev spaces the variouslocal and global problems for smoothing and for maximal operators have received a lot ofattention, starting with [4]. We do not have a contribution to the L -Sobolev problems butrather consider corresponding questions with initial data in L p -Sobolev spaces for p > p not close to 2.In [17] the first author considered L p regularity estimates which are global in space butinvolve an integration over a compact time interval I ,(1.3) (cid:16) Z I k U αt f k pp dt (cid:17) /p C I k f k L pβ ( R d ) . This question was motivated by the similar (although deeper) question for the wave equation( cf . [22], [28]). In [17], it was proven that (1.3) holds for α = 2 when p > / ( d + 1)with β/ > d (1 / − /p ) − /p . We remark that smoothing results of this type could alsobe deduced from square-function estimates related to Bochner-Riesz multipliers such as in[2], [6], [18] and [14] however these arguments do not apply when d = 1, and in dimensions d > p > /d .The L p smoothing result in [17] was obtained from an L p → L p estimate for the adjointFourier restriction (or ‘extension’) operator associated to the paraboloid, and the range p > d +1 corresponds to the known range of L q → L p bounds for the extension operator;see [9], [12] and [29] for the sharp bounds when d = 1, and [24] for the best known partialresults for d >
2. The reduction in [17] to the extension estimate used the explicit formula e it ∆ f ( x ) = 1(4 πit ) d/ Z e i | x − y | / t f ( y ) dy together with a ‘completing of the square’ trick; see [3] for a similar argument. Unfortu-nately this reasoning is not available when α = 2.We generalize to all α >
1, and establish the endpoint regularity result.
Theorem 1.2.
Let p ∈ (2 + d +1 , ∞ ) and α > . Then, for any compact time interval I , (cid:16) Z I k U αt f k pp dt (cid:17) /p C I,p,α k f k L pβ ( R d ) , βα = d (cid:16) − p (cid:17) − p . In Theorem 4.1 below we formulate a slightly improved version of this result which canalso be used to prove Theorem 1.1. We remark that for d = 1 our arguments also give theanalogous results for the range 0 < α < u t + u xxx = 0 . For f := u ( · ,
0) a Schwartz function, we can write u ( · , t ) = U t P + f + U − t P − f , where P + and P − are the projection operators with Fourier multipliers χ (0 , ∞ ) and χ ( −∞ , , respectively.Thus, for initial values in L pβ the solution of (1.4) satisfies the sharp bound k u k L p ( R × [ − T,T ]) C T k u ( · , k L pβ ( R ) , β = 3( p − p , < p < ∞ , AXIMAL AND SMOOTHING ESTIMATES FOR SCHR ¨ODINGER EQUATIONS 3 and if u ( · , ∈ L pε ( R ) for any ε > < p
4, then u ∈ L p ( R × [ − T, T ]).The proofs will be based on the bilinear adjoint restriction theorem for elliptic surfacesdue to Tao [24]. In §
3, having discussed the necessary conditions in §
2, we combine Tao’stheorem with a variation of a localization technique employed in [10] to prove L p estimatesfor some oscillatory integrals with elliptic phases; this yields the smoothing estimate forfunctions which are frequency supported in an annulus. In §
4, we extend to the general caseby decomposing the Fefferman-Stein sharp function; here we use a variant of an argumentin [19].
Notation.
Throughout, c and C will denote positive constants that may depend on thedimensions, exponents or indices of the Sobolev spaces, or the parameter α , but never onthe functions. Such constants are called admissible and their values may change from line toline. We shall mostly use the notation A . B if A CB for an admissible constant C . Wemay sometimes indicate the dependence on a specific parameter c by using the notation . c .We write A ≈ B if A . B and B . A .2. Necessary conditions
Let θ be a nonnegative and smooth function supported in { − < | ξ | < } and equalto 1 in { − / < | ξ | < / } . For large λ , we consider initial data f λ defined by b f λ ( ξ ) = e − i | ξ | α θ ( λ − ξ ) and note that, by a change of variables, f λ ( x ) = (cid:18) λ π (cid:19) d Z θ ( ξ ) e i ( h λx,ξ i− λ α | ξ | α ) dξ. Thus | f λ ( x ) | . λ d − dα , by the method of stationary phase (keeping in mind that α = 1). Onthe other hand, when | x | ≫ λ α − , by repeated integration by parts, there exists constants C N such that | f λ ( x ) | C N ( | x | λ − α ) − N for all N ∈ N . Combining the two bounds, we seethat k f λ k L pβ ( R d ) ≈ λ β k f λ k L p ( R d ) . λ d − dα + d ( α − p + β . Next we consider U αt f λ and compute | U αt f λ ( x ) | = (cid:12)(cid:12)(cid:12) (cid:18) λ π (cid:19) d Z R d θ ( ξ ) e i ( h λx,ξ i + λ α ( t − | ξ | α ) dξ (cid:12)(cid:12)(cid:12) , so when | x | (10 λ ) − and | t − | (10 λ α ) − , we have | U αt f λ ( x ) | > cλ d for some positiveconstant c . Thus, (cid:16) Z − (10 λ α ) − k U αt f λ k pp dt (cid:17) /p > Cλ d − d + αp . Comparing this with the upper bound for k f λ k L pβ ( R d ) , and letting λ → ∞ , we see that β/α > d (1 / − /p ) − /p is a necessary condition for (1.3) to hold when α = 1.Note that alternatively one can argue that by Sobolev embedding any improvement inthe smoothing would give a better fixed time estimate than the sharp known bounds in[11], [15], which is impossible.The range p > / ( d + 1) for the smoothing estimate in Theorem 1.2 is sharp for d = 1, and for d > p > /d , see [17].For Theorem 1.1 however our range may not be sharp even in one dimension. We cansay that the maximal estimate (1.2) cannot hold when p < /d . This follows from thenecessary condition β/α > / p which we now show, modifying a calculation in [8]. KEITH M. ROGERS AND ANDREAS SEEGER
Let χ be a nonnegative and smooth function supported in ( − ε, ε ) where ε will be smalldepending only on α . Let e = (1 , , . . . ,
0) and define g λ ( x ) = 1(2 π ) d Z χ ( λ α − | ξ + λe | ) e i h x,ξ i dξ. Then immediately k g λ k L pβ . λ β + d ( α − ( p − . Now U αt g λ ( x ) = 1(2 π ) d Z χ ( λ α − | ξ + λe | ) e i ( h x,ξ i + t | ξ | α ) dξ = 1(2 π ) d Z χ ( λ α − | h | ) e iφ λ ( x,t,h ) dh where φ λ ( x, t, h ) = tλ α | − e + h/λ | α + h x, − λe + h i . A Taylor expansion gives for | h | ≪ λφ λ ( x, t, h ) = tλ α − x λ + h x − tαλ α − e , h i + O ( λ α − h )where the implicit constants in the error term depend on α . The error term in the phaseis ≪ ε is sufficiently small).Let 0 < c ≪ α and let R be the rectangle where 0 x cλ α − , and | x i | λ ( α − / for i = 2 , . . . , d . We define t ( x ) = α − λ − α x for x ∈ R so that t ( x ) ∈ [0 ,
1] for x ∈ R ,and for x / ∈ R we may choose any (measurable) t ( x ) ∈ [0 , x ∈ R , we have | U αt ( x ) g λ ( x ) | > c λ − d ( α − / and thus (cid:13)(cid:13) sup t | U αt g λ | (cid:13)(cid:13) p > k U αt ( · ) g λ k p & λ α − p + ( α − d − p − ( α − d . Comparing with the upper bound for k g λ k L pβ leads to the condition β/α > / p .3. L p estimates for oscillatory integrals with elliptic phases In the sequel, we will rescale inequalities for U αt when acting on functions with compactfrequency support. This process will give rise to the operator S defined by(3.1) Sf ( x, t ) ≡ S φχ f ( x, t ) = 1(2 π ) d Z χ ( ξ ) e itφ ( ξ ) b f ( ξ ) e i h x,ξ i dξ where χ ∈ C ∞ ( U ) and φ is elliptic ; here a C ∞ function φ on an open set U in R d is calledelliptic if for every ξ ∈ U the Hessian φ ′′ is positive definite.We ask for L p ( R d ) → L p ( R d × [0 , λ ]) bounds for S . Note that for | t | χ ∈ C ∞ the function χe itφ is a Fourier multiplier of L p , 1 p ∞ , and consequently the questionis only nontrivial for large λ . Proposition 3.1.
Let p > d +1 , χ ∈ C ∞ ( U ) , and let φ be an elliptic phase on U . Then k Sf k L p ( R d × [ − λ,λ ]) . λ d (1 / − /p ) k f k L p ( R d ) . The key ingredient will be Tao’s bilinear estimate for the adjoint restriction operator[24] which applies to phases which are small perturbations of | ξ | /
2. We need to formulatemore specific assumptions on the phases allowed and follow [25]. Let N > d . We say φ : [ − , d → R is a phase of the class Φ( N, A ) if | ∂ α j x j φ ( x ) | A for all x ∈ [ − , d andall | α j | N , where j = 1 , . . . , d . To add an ellipticity condition we say that φ is of class AXIMAL AND SMOOTHING ESTIMATES FOR SCHR ¨ODINGER EQUATIONS 5 Φ ell ( ε, N, A ) if φ (0) = ∇ φ (0) = 0, and if for all x ∈ [ − , d the eigenvalues of the Hessian φ ′′ ( x ) lie in [1 − ε, ε ].We define the adjoint restriction operator E ≡ E φ by E h ( x, t ) = Z [ − , d e i ( h x,ξ i + tφ ( ξ )) h ( ξ ) dξ. so that Sf = (2 π ) − d E b f , where U = ( − , d . Now Tao’s theorem can be stated as follows.Suppose p > d +1 . Then there exists an N (depending on d and p ) and for A > ε = ε ( A, N, d, p ) > φ ∈ Φ( ε, N, A ): For all pairs of L functions h , h so that dist(supp ( h ) , supp ( h )) > c > (cid:13)(cid:13) E h E h (cid:13)(cid:13) p/ . c k h k k h k , p > d + 1 , holds. In what follows we fix N , A and ε for which Tao’s theorem applies. The constantsmay all depend on these parameters. Lemma 3.2.
Let p > d +1 , let B , B ⊂ [ − , d be balls so that dist( B , B ) > c , andlet φ ∈ Φ ell ( ε, N, A ) . Then for f , g with supp b f ⊂ B , supp b f ⊂ B , (cid:13)(cid:13) Sf Sg (cid:13)(cid:13) L p/ ( R d × [0 ,λ ]) . c,p λ d (1 − /p ) k f k L p ( R d ) k g k L p ( R d ) . Proof.
Let C = 10(1+max ξ ∈ [ − , d |∇ φ ( ξ ) | ), and let η , η ∈ C ∞ be supported in ( − , d sothat η ( ξ ) = 1 on B and η ( ξ ) = 1 on B . Moreover assume that η and η are supportedon slightly larger concentric balls e B , e B with the property that dist( e B , e B ) > c/
2. Wealso set P i f = F − [ η i b f ] , i = 1 , . Let K it = F − [ e itφ η i χ ], for i = 1 ,
2, so that S i f ( x, t ) := SP i f ( x, t ) = K it ∗ f ( x ) . Then
Sf Sg = S f S g . We first note that for all t ∈ [ − λ, λ ](3.3) | K it ( x ) | . | x | − N , if | x | > C λ This follows by a straightforward N -fold integration by parts, which uses the inequality |∇ ξ ( h x, ξ i + tφ ( ξ )) | > | x | / | x | > C λ , | t | λ .Now let Q ( λ ) be a tiling of R d by cubes of sidelength λ , and for each Q ∈ Q ( λ ) let Q ∗ denote the enlarged cube with sidelength 2 C λ , with the same center as Q . For each cubewe split each function into a part supported in Q ∗ and a part supported in its complement.Thus we can write (cid:13)(cid:13) Sf Sg (cid:13)(cid:13) p/ L p/ ( R d × [0 ,λ ]) = I + II + III + IV KEITH M. ROGERS AND ANDREAS SEEGER where I = X Q ∈Q ( λ ) (cid:13)(cid:13) S [ f χ Q ∗ ] S [ gχ Q ∗ ] (cid:13)(cid:13) p/ L p/ ( Q × [0 ,λ ]) ,II = X Q ∈Q ( λ ) (cid:13)(cid:13) S [ f χ Q ∗ ] S [ gχ R d \ Q ∗ ] (cid:13)(cid:13) p/ L p/ ( Q × [0 ,λ ]) ,III = X Q ∈Q ( λ ) (cid:13)(cid:13) S [ f χ R d \ Q ∗ ] S [ gχ Q ∗ ] (cid:13)(cid:13) p/ L p/ ( Q × [0 ,λ ]) ,IV = X Q ∈Q ( λ ) (cid:13)(cid:13) S [ f χ R d \ Q ∗ ] S [ gχ R d \ Q ∗ ] (cid:13)(cid:13) p/ L p/ ( Q × [0 ,λ ]) . The first term gives the main contribution and is estimated using Tao’s theorem, i.e. (3.2).One obtains, | I | X Q ∈Q ( λ ) (cid:13)(cid:13) S P [ f χ Q ∗ ] S P [ gχ Q ∗ ] (cid:13)(cid:13) p/ L p/ ( R d × R ) . c X Q (cid:13)(cid:13) P [ f χ Q ∗ ] (cid:13)(cid:13) p/ (cid:13)(cid:13) P [ gχ Q ∗ ] (cid:13)(cid:13) p/ . X Q (cid:13)(cid:13) f χ Q ∗ (cid:13)(cid:13) p/ (cid:13)(cid:13) gχ Q ∗ (cid:13)(cid:13) p/ . (cid:16) X Q k f χ Q ∗ k p (cid:17) / (cid:16) X Q k gχ Q ∗ k p (cid:17) / . By H¨older’s inequality, (cid:16) X Q k f χ Q ∗ k p (cid:17) /p . (cid:16) X Q | Q ∗ | p/ − k f χ Q ∗ k pp (cid:17) /p . λ d (1 / − /p ) k f k p , and we have the same estimate for g . Thus I /p . c λ d (1 − /p ) k f k p k g k p which is the desiredbound for the main term.The corresponding estimates for II , III , IV are straightforward as we use (3.3) for theterms supported in R d \ Q ∗ . We examine II and begin with | II | X Q ∈Q ( λ ) (cid:13)(cid:13) S [ f χ Q ∗ ] (cid:13)(cid:13) p/ L p ( Q × [0 ,λ ]) (cid:13)(cid:13) S [ gχ R d \ Q ∗ ] (cid:13)(cid:13) p/ L p ( Q × [0 ,λ ]) (cid:16) X Q ∈Q ( λ ) (cid:13)(cid:13) S [ f χ Q ∗ ] (cid:13)(cid:13) pL p ( Q × [0 ,λ ]) (cid:17) / (cid:16) X Q ∈Q ( λ ) (cid:13)(cid:13) S [ gχ R d \ Q ∗ ] (cid:13)(cid:13) pL p ( Q × [0 ,λ ]) (cid:17) / . (3.4)We use the trivial bound k S f ( · , t ) k p . (1 + | t | ) d k f k p for f replaced with f χ Q ∗ , so that thefirst factor in (3.4) is bounded by ( Cλ d +1 k f k p ) p/ . By (3.3) we get (cid:16) X Q ∈Q ( λ ) (cid:13)(cid:13) S [ gχ R d \ Q ∗ ] (cid:13)(cid:13) pL p ( Q × [0 ,λ ]) (cid:17) /p . (cid:16) Z λ − λ Z x ∈ R d h Z | z | > λ | z | − N | g ( x − z ) | dz i p dxdt (cid:17) /p . λ d +1 − N k g k p . Hence | II | /p . c λ d +1) − N k f k p k g k p . As N > d this estimate is negligible. Becauseof symmetry III is estimated by the same term. For the estimation of IV we proceedin the same way but use (3.3) for both terms, the result is the (again negligible) bound | IV | /p . λ d +1 − N ) k f k p k g k p . (cid:3) We now formulate an analogous result for functions with smaller frequency support andsmaller separation.
AXIMAL AND SMOOTHING ESTIMATES FOR SCHR ¨ODINGER EQUATIONS 7
Lemma 3.3.
Let p > d +1 and λ / > j > . Let Q , Q ⊂ [ − , d be cubes of side j λ − / , so that dist( Q , Q ) > c j λ − / and let φ ∈ Φ ell ( ε, N, A ) . Then for all f and g such that supp b f ⊂ Q , supp b f ⊂ Q , (cid:13)(cid:13) Sf Sg (cid:13)(cid:13) L p/ ( R d × [0 ,λ ]) . c j ( d − d +1 p ) λ p k f k L p ( R d ) k g k L p ( R d ) . Proof.
By finite partitions and the triangle inequality, we may suppose that Q and Q areballs of radius 2 j λ − / . We reduce matters to the statement in Lemma 3.2 by scaling. Let ξ be the midpoint of the interval connecting the center of the balls. We change variables ξ = ξ + δη where δ = 2 j λ − / . Then a short computation shows that S φ f ( x, t ) = e i ( h x,ξ i + tφ ( ξ )) S ψ f ∗ ( δ ( x + t ∇ φ ( ξ )) , δ t ) where f ∗ ( y ) = f ( δ − y ) e iδ − h y,ξ i , and the phase ψ is given by ψ ( η ) = 12 Z h φ ′′ ( ξ + sδη ) η, η i ds. The same consideration is applied to S φ g . Note that ψ is elliptic (with estimates uniformin ξ and δ ) and the frequency supports of f ∗ and g ∗ are now separated, independently of δ , j and λ . Thus we can apply Lemma 3.2 to obtain k S φ f S φ g k L p/ ( R d × [0 ,λ ]) = δ − ( d +2) / ( p/ k S ψ f ∗ S ψ g ∗ k L p/ ( R d × [0 ,λδ ]) . δ − (2 d +4) /p ( λδ ) d (1 − /p ) k f ∗ k p k g ∗ k p . δ d − d +1) /p λ d (1 − /p ) k f k p k g k p . As δ = 2 j λ − / the assertion follows. (cid:3) We will also require the following lemma for when we have no frequency separation.
Lemma 3.4.
Let p > , let Q ⊂ [ − , d be a cube of side λ − / , and let φ ∈ Φ( N, A ) .Then for all f such that supp b f ⊂ Q , k Sf ( · , t ) k L p ( R d ) . k f k L p ( R d ) , | t | λ. Proof.
Let ξ B be the center of the cube Q , and let χ ∈ C ∞ so that χ ( ξ ) = 1 for | ξ | √ d . Itsuffices to show that χ ( λ / ( ξ − ξ B )) e itφ ( ξ ) is a Fourier multiplier of L p for all | t | λ , withbounds uniform in t . By modulation, translation and dilation invariance of the multipliernorm it suffices to check that h ( · , t ) defined by h ( η, t ) = χ ( η ) e it ( φ ( λ − / η + ξ B ) − φ ( ξ B ) −h λ − / η, ∇ φ ( ξ B ) i ) , is a Fourier multiplier of L p , uniformly in | t | λ . However this follows since ∂ αη h ( η, t ) = O (1) for | t | λ as one can easily check. (cid:3) Proof of Proposition 3.1.
By a partition of unity and a compactness argument it sufficesto show that for every ξ ∈ U there is a neighborhood U ( ξ ) so that the statement ofthe theorem holds with χ replaced by χ ∈ C ∞ supported in U ( ξ ). Now let H be the(symmetric) positive definite squareroot of φ ′′ ( ξ ) and let ψ ( η ) = ε − (cid:0) φ ( ξ + ε H − η ) − φ ( ξ ) − ε hH − η, ∇ φ ( ξ ) i (cid:1) . Then it suffices to show that S ψ (defined with the amplitude χ ( ξ + ε H − η )) satisfiesthe asserted estimates, with a dependence on ε . If ε is chosen sufficiently small then we KEITH M. ROGERS AND ANDREAS SEEGER have reduced matters to a phase function in Φ ell ( ε, N, A ) with parameters for which Tao’stheorem and therefore Lemma 3.3 applies.We now return to our original notation and work with a phase function φ but assumenow that φ ∈ Φ ell ( ε, N, A ); we may also assume that the amplitude function χ is smoothand supported in [ − (2 d ) − , (2 d ) − ] − d . We make a decomposition of the product Sf Sg in terms of bilinear operators, localizing the frequency variables in terms of nearness to thediagonal in ( ξ, η )-space; this is similar to arguments in [13], [20] and [25].Let χ be a radial C ∞ ( R d ) function so that χ ( ω ) = 1 for | ω | d / and so that supp χ is contained in { ω : | ω | < d / } . Fix λ > ( ξ, η ) = χ ( λ / ( ξ − η ))Θ j ( ξ, η ) = χ ( λ / − j ( ξ − η )) − χ (2 λ / − j ( ξ − η )) , j > , so that Θ is supported where | ξ − η | d / λ − / and, Θ j is supported in the region4 d / j λ − / | ξ − η | d / j λ − / . We may then decompose
Sf Sg = X j > B j [ f, g ]where B j [ f, g ]( x, t ) = 1(2 π ) d Z Z e i h x,ξ + η i e it ( φ ( ξ )+ φ ( η )) Θ j ( ξ, η ) b f ( ξ ) b g ( η ) dξdη Only values of j > j λ / will be relevant, as otherwise B j is identically zero. Wewill prove the estimate(3.5) (cid:13)(cid:13) B j [ f, g ] (cid:13)(cid:13) p/ . ( j ( d − d +1 p ) λ p k f k p k g k p , d +3) d +1 < p , j ( d − p ) λ d − d − p k f k p k g k p , < p < ∞ , and use this to bound k Sf k L p ( R d × [0 ,λ ]) = k ( Sf ) k / L p/ ( R d × [0 ,λ ]) (cid:16) X j log ( λ / ) kB j [ f, f ] k p/ (cid:17) / , and then sum a geometric series.In order to prove (3.5), we decompose B j into pieces on which we may apply Lemma 3.3.Let ϑ ∈ C ∞ ( R d ) a function supported in [ − / , / d , equal to 1 on [ − / , / d , andsatisfying X n ∈ Z d ϑ ( ξ − n ) = 1for all ξ ∈ R d . For j > n ∈ Z d , define β j,n ( ξ ) = ϑ ( λ / − j ξ − n )and, for ( n, n ′ ) ∈ Z d × Z d , ϑ j,n,n ′ ( ξ, η ) = Θ j ( ξ, η ) β j,n ( ξ ) β j,n ′ ( η ) . Observe that β j,n , β j,n ′ are supported in cubes Q j,n , Q j,n ′ which have sidelengths slightlylarger than λ − / j , and that are centered at the points ξ j,n = λ − / j n and ξ j,n ′ = λ − / j n ′ , respectively. AXIMAL AND SMOOTHING ESTIMATES FOR SCHR ¨ODINGER EQUATIONS 9
Now let ∆ = { ( n, n ′ ) ∈ Z d × Z d : | n − n ′ | d / } , ∆ = { ( n, n ′ ) ∈ Z d × Z d : 2 d / | n − n ′ | d / } . Then if ϑ ,n,n ′ is not identically zero then we necessarily have ( n, n ′ ) ∈ ∆ and if, for j > ϑ j,n,n ′ is not identically zero then we necessarily have ( n, n ′ ) ∈ ∆. Thesestatements follow by the definitions of our cutoff functions. Moreover,dist( Q j,n , Q j,n ′ ) d / j λ − / if ( n, n ′ ) ∈ ∆ , and 2 − d / j λ − / dist( Q j,n , Q j,n ′ ) d / j λ − / if j > n, n ′ ) ∈ ∆ . For the application of Lemma 3.3 it is convenient to eliminate the cutoff Θ j but still keepthe separation of the supports of β j,n and β j,n ′ . Set, for j > e B j [ f, g ]( x, t ) = 1(2 π ) d Z Z e i h x,ξ + η i e it ( φ ( ξ )+ φ ( η )) X n,n ′ ∈ ∆ β j,n ( ξ ) β j,n ′ ( η ) b f ( ξ ) b g ( η ) dξdη and define e B [ f, g ] similarly by letting the ( n, n ′ ) sum run over ∆ . The reduction of theestimate for B j to the estimate for e B j is straightforward; by an averaging argument. Indeed,let χ = χ − χ (2 · ) and use the Fourier inversion formulaΘ j ( ξ, η ) = 1(2 π ) d Z b χ ( y ) e iλ / − j h ξ − η,y i dy, j > B j [ f, g ] = 1(2 π ) d Z b χ ( y ) e B j [ f − y , g y ] dy where f − y ( x ) = f ( x + λ / − j y ) and g y ( x ) = g ( x − λ / − j y ). A similar formula holds for j = 0, only then χ is replaced with χ . Thus in order to finish the argument it is enoughto show that k e B j [ f, g ] k p/ is dominated by the right hand side of (3.5).Define convolution operators P j,n by [ P j,n f = β j,n b f . Note that for fixed j , each ξ iscontained in only a bounded number of the sets Q j,n + Q j,n ′ . This implies, by interpolationof ℓ ( L ) with trivial ℓ ( L ) or ℓ ∞ ( L ∞ ) bounds that, for j > p > (cid:13)(cid:13) e B j [ f, g ] (cid:13)(cid:13) L p/ ( R d × [0 ,λ ]) . max { , ( λ / − j ) d (1 − /p ) } (cid:16) X n,n ′ ∈ ∆ (cid:13)(cid:13) SP j,n f SP j,n ′ g (cid:13)(cid:13) p/ L p/ ( R d × [0 ,λ ]) (cid:17) /p . The analogous formula for j = 0 holds if we replace ∆ by ∆ . Notice that for all j ,(3.7) (cid:16) X n k P j,n f k pp (cid:17) /p . k f k p , p > . Now if j = 0 we use Lemma 3.4 to estimate k SP ,n f ( · , t ) SP ,n ′ g ( · , t ) (cid:13)(cid:13) L p/ ( R d ) . k SP ,n f ( · , t ) k p k SP ,n ′ g ( · , t ) k p . k P ,n f k p k P ,n ′ g k p ; hence, after integrating in t , (cid:13)(cid:13) e B [ f, g ] (cid:13)(cid:13) L p/ ( R d × [0 ,λ ]) . max { , λ d (1 / − /p ) } λ /p (cid:16) X n,n ′ ∈ ∆ k P ,n f k p/ p k P ,n ′ g k p/ p (cid:17) /p . max { , λ d (1 / − /p ) } λ /p (cid:16) X n k P ,n f k pp (cid:17) /p (cid:16) X n ′ k P ,n ′ g k pp (cid:17) /p . The asserted bound for j = 0 follows from (3.7).Next for j > p > d +1 , and estimate (cid:13)(cid:13) SP j,n f SP j,n ′ g (cid:13)(cid:13) L p/ ( R d × [0 ,λ ]) . j ( d − d +1 p ) λ /p k P j,n f k p k P j,n ′ g k p . Therefore by (3.6) (cid:13)(cid:13)(cid:13) e B j [ f, g ] (cid:13)(cid:13)(cid:13) L p/ ( R d × [0 ,λ ]) . max { , ( λ / − j ) d (1 − /p ) } j ( d − d +1 p ) λ /p (cid:16) X n k P j,n f k pp (cid:17) /p (cid:16) X n ′ k P j,n ′ g k pp (cid:17) /p and again the asserted bound for k e B j [ f, g ] k p/ follows from (3.7). (cid:3) Estimates for exp( it ( − ∆) α/ )We now prove the endpoint estimates of Theorems 1.1 and 1.2. First we remark that byvarious scaling and symmetry arguments we may assume that I = [0 , χ , χ ∈ C ∞ ( R ) supported in ( − ,
2) and (1 / , χ + X k > χ (2 − k · ) = 1 . We define the operators T αk ≡ T k by \ T f ( · , t )( ξ ) = χ ( | ξ | ) e it | ξ | α b f ( ξ ) , \ T k f ( · , t )( ξ ) = χ (2 − k | ξ | ) e it | ξ | α b f ( ξ ) , k > , so that U αt = P k > T k ( · , t ).Our main result is the following inequality for vector-valued functions { f k } ∞ k =0 ∈ ℓ p ( L p ). Theorem 4.1.
Let p ∈ (2 + d +1 , ∞ ) , α = 1 , d = 1 or α > , d > and β = αd ( − p ) − αp .Then (4.1) (cid:13)(cid:13)(cid:13) X k > (cid:16) Z | − kβ T k f k ( · , t ) | p dt (cid:17) /p (cid:13)(cid:13)(cid:13) L p ( R d ) . (cid:16) X k > k f k k pp (cid:17) /p . The proof will be given in §
5. We now discuss the implications to Theorem 1.1 and 1.2,in fact strengthened versions involving Triebel-Lizorkin spaces F pα,q and Besov spaces B pα,q .Here the norms on these spaces are given by the L p ( ℓ q ) and ℓ q ( L p ) norms (resp.) ofthe sequence { kα L k f } ∞ k =0 , with the usual inhomogeneous dyadic frequency composition I = P k > L k . See [26]. The following corollary is an immediate consequence of Theorem4.1, by Minkowski’s inequality and Fubini’s theorem. AXIMAL AND SMOOTHING ESTIMATES FOR SCHR ¨ODINGER EQUATIONS 11
Corollary 4.2.
Let p , α , β be as in Theorem 4.1. Then (cid:16) Z (cid:13)(cid:13) U αt f (cid:13)(cid:13) pF p , ( R d ) dt (cid:17) /p . k f k B pβ,p ( R d ) . This implies Theorem 1.2 since for p > B pβ,p ≡ F pβ,p contains the Sobolevspace L pβ ≡ F pβ, , via the embedding ℓ ֒ → ℓ p followed by the Littlewood-Paley inequality,and by the same reasoning F p , is imbedded in L p ≡ F p , . We remark that a similar sharpinequality for the wave equation is proved in [16], in sufficiently high dimensions.Another consequence of Theorem 4.1 is Corollary 4.3.
Let p , α , be as in Theorem 4.1. Let t ϑ ( t ) be smooth and compactlysupported. Then (4.2) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13) ϑ ( · ) U α ( · ) g (cid:13)(cid:13) B p /p, ( R ) (cid:13)(cid:13)(cid:13) L p ( R d ) . k g k B pγ,p ( R d ) , γ = αd (1 / − /p ) . Theorem 1.1 is an immediate consequence of Corollary 4.3 since the Besov space B p /p, ( R )is continuously embedded in the space C of continuous bounded functions which vanishat infinity.To see how Corollary 4.3 follows from Theorem 4.1 we introduce dyadic frequency cutoffsin the t variable. We decompose the identity as I = P j =0 L j where d L j f ( τ ) = e χ j ( τ ) b f ( τ )where e χ j = e χ (2 − j | · | ) for j >
1, with a suitable e χ ∈ C ∞ supported in (1 / ,
2) and e χ issmooth and vanishes for | τ | >
2. Now we apply L j to ϑT k g . If 2 j − αk / ∈ (2 − , ), then weapply an integration by parts in s to terms of the form Z Z χ (2 − j | τ | ) χ (2 − k | ξ | ) b g ( ξ ) e i ( h x,ξ i + tτ ) Z ϑ ( s ) e is ( | ξ | α − τ ) ds dξdτ. One finds that for this range the contribution of L j [ ϑT k g ] is negligible; namely (cid:16) Z R Z R d |L j [ ϑT k g ]( x, s ) | p dxds (cid:17) /p . C N min { − αkN , − jN }k g k p if 2 j − αk / ∈ (2 − , ) . Thus a localization in ξ where | ξ | ≈ k corresponds to a localization in τ where | τ | ≈ kα .We combine this with Theorem 4.1 applied to f k = 2 kβ + k/p F − [ χ (2 − k | · | ) b g ] and obtain (cid:13)(cid:13)(cid:13) X j > j/p (cid:13)(cid:13) L j [ ϑU α ( · ) g ] k L p ( R ) ,dt (cid:13)(cid:13)(cid:13) L p ( R d ) . (cid:16) X k > kγp (cid:13)(cid:13) F − [ χ k b g ] (cid:13)(cid:13) pL p ( R d ) (cid:17) /p which is (4.2). 5. Proof of Theorem 4.1
The localization of the multiplier near the origin T is easily handled as kF − [ χ ( | · | ) e it |·| α ] k L C uniformly for t ∈ [0 , F − [ χ ( | · | )] ∈ L , it suffices to show that for φ supported in (1 / , L norm of F − [ χ ( e it |·| α − φ (2 k | · | )] is O (2 − αk ) for k >
0. Butby scaling this follows from showing that the L norm of F − [ χ (2 − k · )( e it − αk |·| α − φ ( | · | )]is O (2 − αk ) which follows from the standard Bernstein criterion. Now, by scaling and Proposition 3.1 with λ ≈ αk , U = { ξ : 1 / < | ξ | < } and φ ( ξ ) = | ξ | α , we have already proven the estimates(5.1) k T k f k L p ( R d × [0 , . kβ k f k L p ( R d ) , β > β ( p ) := αd (cid:16) − p (cid:17) − αp for k > p > d +1 .It suffices thus to show that if (5.1) holds for all k > p > q , then (4.1) holds forall p ∈ ( q, ∞ ). Due to our restriction on (5.1) we let q = 2 + d +1 and fix 2 + d +1 < r < p .We can make the additional assumption that the k sum on the left hand side is extendedover a finite set (with the constant in the inequality independent of this assumption); thegeneral case then follows by the monotone convergence theorem.For later reference we state a Sobolev inequality which is proved linking frequency de-compositions in ξ and τ and Young’s inequality (just as in the argument used in § (cid:13)(cid:13) k T k f k L pt [0 , (cid:13)(cid:13) L rx . αk ( r − p ) (cid:13)(cid:13) k T k f k L rt [0 , (cid:13)(cid:13) L rx . holds for r p ∞ (including the endpoint). Alternatively one can also apply thefundamental theorem of calculus to | T k f ( x, · ) | r (see e.g. [23]) to get (5.2) for p = ∞ andthe general inequality follows by convexity.The main ingredient in the proof of (4.1) (besides (5.1)) will be the Fefferman-Steinsharp function [11] and their inequality k F k p . k F k p , where p ∈ (1 , ∞ ) and a priori F ∈ L p . We apply this to P k> − kβ ( p ) k T k f k ( x, · ) k L pt [0 , and by (5.1) this function is a priori in L p as the sum in k is assumed to be finite. Thusit will suffice to prove that (cid:13)(cid:13)(cid:13) sup x ∈ Q \ Z Q (cid:12)(cid:12)(cid:12) X k> − kβ ( p ) k T k f k ( y, · ) k L pt [0 , − \ Z Q X k> − kβ ( p ) k T k f k ( z, · ) k L pt [0 , dz (cid:12)(cid:12)(cid:12) dy (cid:13)(cid:13)(cid:13) L px is dominated by C ( P k> k f k k pp ) /p . Here the supremum is taken over all cubes containing x , and the slashed integral denotes the average | Q | − R Q . By the triangle inequality theprevious bound follows from (cid:13)(cid:13)(cid:13) sup x ∈ Q \ Z Q X k> \ Z Q − kβ ( p ) k T k f k ( y, · ) − T k f k ( z, · ) k L pt [0 , dzdy (cid:13)(cid:13)(cid:13) L px . (cid:16) X k k f k k pp (cid:17) /p . Denoting the sidelength of Q by ℓ ( Q ), we observe that, by Minkowski’s inequality, thiswould follow from the inequalities(5.3) (cid:13)(cid:13)(cid:13) sup x ∈ Q \ Z Q X k ℓ ( Q ) \ Z Q − kβ ( p ) k T k f k ( y, · ) − T k f k ( z, · ) k L pt [0 , dzdy (cid:13)(cid:13)(cid:13) L px . (cid:16) X k k f k k pp (cid:17) /p , (5.4) (cid:13)(cid:13)(cid:13) sup x ∈ Q \ Z Q X k ℓ ( Q ) > αk − kβ ( p ) k T k f k ( y, · ) k L pt [0 , dy (cid:13)(cid:13)(cid:13) L px . (cid:16) X k k f k k pp (cid:17) /p . and(5.5) (cid:13)(cid:13)(cid:13) sup x ∈ Q \ Z Q X αk > k ℓ ( Q ) > − kβ ( p ) k T k f k ( y, · ) k L pt [0 , dy (cid:13)(cid:13)(cid:13) L px . (cid:16) X k k f k k pp (cid:17) /p . AXIMAL AND SMOOTHING ESTIMATES FOR SCHR ¨ODINGER EQUATIONS 13
First we handle (5.3) and (5.4) by standard estimates and then prove the more interestinginequality (5.5).
Proof of (5.3) . It is enough to consider cubes Q of diameter ≈ j with x, y, z ∈ Q and j + k
0. Let H k = F − [ e χ (2 − k | · | )], where e χ is smooth, equal to one on (1 / , / , |∇ H k ( w ) | . k kd (1 + 2 k | w | ) N with large N > d . Thus T k f k ( y, t ) − T k f k ( z, t ) = Z h H k ( y − w ) − H k ( z − w ) i T k f k ( w, t ) dw = Z Z (cid:10) ( y − z ) , ∇ H k ( z + s ( y − z ) − w ) T k f k ( w, t ) (cid:11) ds dw which is controlled by a constant multiple of2 j + k Z kd (1 + 2 k | x − w | ) N | T k f k ( w, t ) | dw. Thus, using the embedding ℓ p ֒ → ℓ ∞ , the right hand side of (5.3) is bounded by (cid:13)(cid:13)(cid:13)(cid:16) X j (cid:12)(cid:12)(cid:12) X
1) and let C ( α ) = α α − if α ∈ (1 , ∞ ), and define B k ( α ) = { x : | x | C ( α )2 k ( α − } . Integration by parts yields favorable bounds in the complement of this ball. Observe that (cid:12)(cid:12) ∇ ξ (cid:0) k h x, ξ i + 2 αk t | ξ | α (cid:1)(cid:12)(cid:12) > c α k | x | if x / ∈ B k ( α ) , t ∈ [0 , , and we obtain(5.6) | K tk ( x ) | C N kd (1 + 2 k | x | ) − N if x / ∈ B k ( α ) , t ∈ [0 , . Consequently the main contribution of K tk ( x ) comes when | x | C ( α )2 k ( α − . We prove the estimate (5.4) by interpolation between (cid:13)(cid:13)(cid:13) sup x ∈ Q \ Z Q X k ℓ ( Q ) > αk − kβ ( p ) k T k f k ( y, · ) k L pt [0 , dy (cid:13)(cid:13)(cid:13) ∞ . sup k k f k k ∞ and (cid:13)(cid:13)(cid:13) sup x ∈ Q \ Z Q X k ℓ ( Q ) > αk − kβ ( p ) k T k f k ( y, · ) k L pt [0 , dy (cid:13)(cid:13)(cid:13) r . (cid:16) X k k f k k rr (cid:17) /r , where 2 + d +1 < r < p .Now, as β ( p ) > β ( r ) + α ( r − p ), the L r bound is proven by applying H¨older in k , followedby the inequality (cid:13)(cid:13) sup x ∈ Q \ Z Q (cid:16) X k − k (cid:0) β ( r )+ α ( r − p ) (cid:1) r k T k f k ( y, · ) k rL pt [0 , (cid:17) /r dy (cid:13)(cid:13) r . (cid:16) X k k f k k rr (cid:17) /r . This is a consequence of the L r –boundedness of the Hardy–Littlewood maximal operator,the interchange of the spatial integral and the sum, an application of (5.2), followed byFubini and the estimate (5.1) (for the admissible exponent r > / ( d + 1)).To prove the L ∞ bound, we let Q ∗ be a cube with the same center as Q satisfying ℓ ( Q ∗ ) = 10 dC ( α ) ℓ ( Q ). By Minkowski’s inequality it will suffice to prove that(5.7) \ Z Q X k ℓ ( Q ) > αk − kβ ( p ) k T k [ f k χ Q ∗ ]( y, · ) k L pt [0 , dy . sup k k f k k ∞ and(5.8) \ Z Q X k ℓ ( Q ) > αk − kβ ( p ) k T k [ f k χ R d \ Q ∗ ]( y, · ) k L pt [0 , dy . sup k k f k k ∞ uniformly in Q .To prove (5.7), again we apply H¨older a number of times and (5.2); \ Z Q X k − kβ ( p ) k T k [ f k χ Q ∗ ]( y, · ) k L pt [0 , dy . | Q | − /r X k − k ( β ( p ) − α ( r − p )) (cid:16) Z k T k [ f k χ Q ∗ ]( y, · ) k rL rt [0 , dy (cid:17) /r . sup k | Q | − /r − kβ ( r ) (cid:16) Z k T k [ f k χ Q ∗ ]( y, · ) k rL rt [0 , dy (cid:17) /r . sup k | Q | − /r (cid:16) Z | f k χ Q ∗ | r dx (cid:17) /r . sup k k f k k ∞ , where the third inequality holds again by the L r version of (5.1). AXIMAL AND SMOOTHING ESTIMATES FOR SCHR ¨ODINGER EQUATIONS 15
For (5.8), we note that as ℓ ( Q ) > k ( α − , and the function is supported in the comple-ment of Q ∗ we can use the rapid decay in formula (5.6). We have that \ Z Q X k ℓ ( Q ) > αk − kβ ( p ) k T k [ f k χ R d \ Q ∗ ]( y, · ) k L pt [0 , dy . sup k \ Z Q (cid:13)(cid:13)(cid:13)(cid:13)Z kd (1 + 2 k | y − z | ) d | f k ( z ) | dz (cid:13)(cid:13)(cid:13)(cid:13) L pt [0 , dy . sup k (cid:13)(cid:13)(cid:13)(cid:13)Z kd (1 + 2 k | · − z | ) d | f k ( z ) | dz (cid:13)(cid:13)(cid:13)(cid:13) ∞ . sup k k f k k ∞ . This concludes the proof of (5.4)
Proof of (5.5) . We let ζ j ( x ) = ( d j ) − d if | x | d j and ζ j ( x ) = 0 if | x | > d j . Replacingcubes by dyadic balls we see that (5.5) follows from(5.9) (cid:13)(cid:13)(cid:13) sup j ζ j ∗ X k + j> α − k > j − kβ ( p ) k T k f k k L pt [0 , (cid:13)(cid:13)(cid:13) L px . (cid:16) X k k f k k pp (cid:17) /p . Now, for fixed k we cover R d by a grid R α − k consisting of cubes of sidelength 2 k ( α − . Foreach R ∈ R α − k let R ∗ be the cube with same center as R and sidelength C ( α )2 k ( α − d where C ( α ) is as in the proof of (5.4)For R ∈ R α − k we let f Rk = χ R f k . We may then split the left hand side of (5.9) as I + II where I = (cid:13)(cid:13)(cid:13) sup j ζ j ∗ h X k + j> α − k > j − kβ ( p ) k X R ∈R α − k χ R ∗ T k f Rk k L pt [0 , i(cid:13)(cid:13)(cid:13) L px and II is the analogous expression where χ R ∗ is replaced with χ R d \ R ∗ .By Hardy–Littlewood, Minkowski, Fubini, (5.6), and Young’s inequality, we dominate II . X k > − kβ ( p ) (cid:13)(cid:13)(cid:13) X R ∈R α − k χ R d \ R ∗ T k f Rk (cid:13)(cid:13)(cid:13) L p ( R d × [0 , . X k > − kβ ( p ) (cid:16) Z Z h Z kd (1 + 2 k | x − y | ) d X R ∈R α − k | f Rk ( y ) | dy i p dxdt (cid:17) /p . X k > − kβ ( p ) (cid:13)(cid:13)(cid:13) X R ∈R α − k f Rk (cid:13)(cid:13)(cid:13) p . sup k k f k k p . (cid:16) X k k f k k pp (cid:17) /p . Concerning the main term I we use the imbedding ℓ p ֒ → ℓ ∞ , interchange a sum and anintegral, and apply Minkowski’s inequality, so that I . (cid:16) X j (cid:13)(cid:13)(cid:13) ζ j ∗ h X k + j> α − k > j − kβ ( p ) X R ∈R α − k χ R ∗ k T k f Rk k L pt [0 , i(cid:13)(cid:13)(cid:13) pL px (cid:17) /p . Now for R ∈ R α − k , R ∗ has sidelength greater than 2 j , so for fixed k the functions ζ j ∗ χ R ∗ have bounded overlap, uniformly in k . Setting n = k + j > I . X n> I n where I n = (cid:16) X j Acknowledgement. The first author thanks Gustavo Garrig´os, Manuel Portilheiro andAna Vargas for helpful conversations, and Luis Vega for bringing the L p –conjecture for theSchr¨odinger maximal operator to his attention. The authors also thank Jong-Guk Bak andSanghyuk Lee for a correction concerning the case α < References [1] A. Baernstein and E.T. Sawyer, Embedding and multiplier theorems for H p ( R n ) , Mem. Amer. Math.Soc. 53 (1985), no. 318.[2] A. Carbery, The boundedness of the maximal Bochner-Riesz operator on L ( R ) , Duke Math. J. 50(1983), no. 2, 409–416.[3] , Restriction implies Bochner-Riesz for paraboloids, Math. Proc. Cambridge Philos. Soc., 111(1992), 525–529.[4] L. Carleson, Some analytic problems related to statistical mechanics. Euclidean harmonic analysis (Proc.Sem., Univ. Maryland, College Park, Md., 1979), pp. 5–45, Lecture Notes in Math., 779, Springer, Berlin,1980. AXIMAL AND SMOOTHING ESTIMATES FOR SCHR ¨ODINGER EQUATIONS 17 [5] L. Carleson and P. Sj¨olin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44(1972), 287–299.[6] M. Christ, On almost everywhere convergence of Bochner-Riesz means in higher dimensions, Proc.Amer. Math. Soc. 95 (1985), 16–20.[7] P. Constantin, J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1(1988), 413–439.[8] B.E.J. Dahlberg and C.E. Kenig, A note on the almost everywhere behavior of solutions to theSchr¨odinger equation, Harmonic analysis, Lecture Notes in Math. 908 (1982), 205–209.[9] C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 1970 9–36.[10] , A note on spherical summation multipliers, Israel J. Math., 15, (1973), 44-52.[11] C. Fefferman and E.M. Stein, H p spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193.[12] L. H¨ormander, Oscillatory integrals and multipliers on F L p , Ark. Mat. 11 (1973), 1-11.[13] S. Lee, Improved bounds for Bochner-Riesz and maximal Bochner-Riesz operators, Duke Math. J. 122(2004), 205–232.[14] S. Lee and A. Seeger, manuscript.[15] A. Miyachi, On some singular Fourier multipliers, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28 (1981),267–315.[16] F. Nazarov and A. Seeger, Radial Fourier multipliers in high dimensions, preprint.[17] K.M. Rogers, A local smoothing estimate for the Schr¨odinger equation, Adv. Math. 219 (2008), no. 6,2105–2122.[18] A. Seeger, On quasiradial Fourier multipliers and their maximal functions, J. Reine Angew. Math. 370(1986), 61–73.[19] , Remarks on singular convolution operators, Studia Math. 97 (1990), 91–114.[20] , Endpoint inequalities for Bochner-Riesz multipliers in the plane, Pacific J. Math. 174 (1996),543–553.[21] P. Sj¨olin, Regularity of solutions to the Schr¨odinger equation, Duke Math. J., 55 (1987), 699-715.[22] C.D. Sogge, Propagation of singularities and maximal functions in the plane, Invent. Math. 104 (1991),no. 2, 349–376.[23] E. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, PrincetonUniversity Press, (1993).[24] T. Tao, A Sharp bilinear restriction estimate for paraboloids, Geom. Funct. Anal. 13 (2003), 1359–1384.[25] T. Tao, A. Vargas and L. Vega. A bilinear approach to the restriction and Kakeya conjectures, J. Amer.Math. Soc. (1998), 967–1000.[26] H. Triebel, Theory of function spaces. Monographs in Mathematics, 78. Birkh¨auser Verlag, Basel, 1983.[27] L. Vega, Schr¨odinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102(1988), no. 4, 874–878.[28] T. Wolff, Local smoothing type estimates on L p for large p , Geom. Funct. Anal. 10 (2000), no. 5,1237–1288.[29] A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50(1974), 189–201. Keith Rogers, Instituto de Ciencias Matematicas CSIC-UAM-UC3M-UCM, 28006 Madrid,Spain E-mail address : [email protected] Andreas Seeger, Department of Mathematics, University of Wisconsin, 480 Lincoln Drive,Madison, WI, 53706, USA E-mail address ::