An Endpoint Version of Uniform Sobolev Inequalities
AAN ENDPOINT VERSION OF UNIFORM SOBOLEVINEQUALITIES
TIANYI REN, YAKUN XI, CHENG ZHANG
Abstract.
We prove an endpoint version of the uniform Sobolev inequalitiesin Kenig-Ruiz-Sogge [8]. It was known that strong type inequalities no longerhold at the endpoints; however, we show that restricted weak type inequali-ties hold there, which imply the earlier classical result by real interpolation.The key ingredient in our proof is a type of interpolation first introduced byBourgain [2]. We also prove restricted weak type Stein-Tomas restriction in-equalities on some parts of the boundary of a pentagon, which completelycharacterizes the range of exponents for which the inequalities hold.
Keywords.
Uniform Sobolev inequalities, Stein-Tomas inequality, Bourgain’sinterpolation Introduction
In this paper, we consider the second order constant coefficient differential op-erator P p D q “ Q p D q ` n ÿ j “ a j BB x j ` b, where Q p ξ q denotes a nonsingular real quadratic form on R n , n ě ď k ď n , is given by Q p ξ q “ ´ ξ ´ ... ´ ξ k ` ξ k ` ` ... ` ξ n ,D “ ´ i pB{B x , ..., B{B x n q , and a , ..., a n , b are complex numbers. If k “ n , theoperator P p D q has principal part Q p D q “ ∆ and is called elliptic. Otherwise, it iscalled non-elliptic.Uniform Sobolev inequalities(1) } u } L q p R n q ď C } P p D q u } L p p R n q , u P W ,p p R n q , have been of interest to the study of unique continuation for partial differentialequations. Here the constant C should depend only on n and p . If P p D q “ ∆, (1)is just the classical Sobolev inequality. For more general elliptic operators, Kenig-Ruiz-Sogge [8, Theorem 2.2] characterized the optimal range of exponents p p, q q forwhich (1) holds. Indeed, they showed that (1) holds for elliptic P p D q if and only if p p, q q satisfies the two conditions(i) p ´ q “ n , (ii) min t| p ´ | , | q ´ |u ą n , (i.e., p { p, { q q lies on the open line segment joining α p n ` n , n ´ n q and β p n ` n , n ´ n q in Figure 1). As pointed out in [8, p.330], the chief technical difficulty in proving(1) comes from the first order terms of P p D q . Indeed, (1) follows from a localization a r X i v : . [ m a t h . A P ] J u l TIANYI REN, YAKUN XI, CHENG ZHANG argument and the uniform resolvent estimates:(2) } u } L q p R n q ď C }p ∆ ` z q u } L p p R n q , u P W ,p p R n q , z P C . Since strong type inequality like (1) and (2) no longer holds at the endpoints α and β , it is natural to ask whether restricted weak type inequality can be establishedat these endpoints. In this paper, we give a positive answer to this question when P p D q is elliptic. Theorem 1.
Let n ě . If p { p, { q q “ α or β , then for any z P C , the inequalityholds: (3) } u } L q, p R n q ď C }p ∆ ` z q u } L p, p R n q , where the constant C depends only on n and p . Theorem 2.
Let n ą . If p { p, { q q “ α or β , then there exists a constant C ,depending only on n , such that whenever P p D q is a second order constant coefficientdifferential operator with principal part ∆ , we have (4) } u } L q, p R n q ď C } P p D q u } L p, p R n q . A few remarks are in order. First, the above two theorems imply the corre-sponding classical results of Kenig-Ruiz-Sogge [8] by real interpolation. Second, S.Guti´errez [7] obtained restricted weak type resolvent estimates for the Laplacianat points A p n ` n , p n ´ q n p n ` q q and B p n ` n ´ n p n ` q , n ´ n q in Figure 1. The estimates can-not be uniform, but depend on z , because the exponent pairs are not on the line p ´ q “ n . Finally, for non-elliptic P p D q , uniform restricted weak type estimateshave been established in a recent work of Jeong-Kwon-Lee[4], which completelycharacterizes the range of p p, q q for which (1) holds in the non-elliptic case.Theorem 2 follows from Theorem 1, a restricted weak type Stein-Tomas inequal-ity (see Section 3) and the localization argument in [8, p.335-337], after adaptingseveral classical results to Lorentz spaces. To our knowledge, (4) is still open when n “
3. The difficulty here is the failure of Littlewood-Paley inequality when anexponent becomes (see Proposition 1).The rest of the paper is organized as follows. In Section 2, we prove Theorem1 by an interpolation result first obtained by Bourgain [2] and a variant of Stein’soscillatory integral theorem due to Sogge [10]. The interpolation method of Bour-gain was first brought to our attention by Bak-Seeger[1]. In Section 3, by a similarargument, we prove restricted weak type Stein-Tomas restriction inequalities onsome parts of the boundary of a pentagon; this completely characterizes the rangeof exponents for which the inequalities hold. In Section 4, we prove Theorem 2 byestablishing several classical results in the setting of Lorentz spaces and carryingout the localization argument in Kenig-Ruiz-Sogge [8].2. Proof of Theorem 1
First of all, we need some reductions. It suffices to prove the theorem for oneendpoint, say α p n ` n , n ´ n q , because the other follows from duality. Furthermore,noting the gap condition p ´ q “ n on the exponents, we are able to reduce thetheorem to the case where z has unit length, | z | “
1, after a simple rescalingargument. Finally, by continuity, we may assume that Im z ‰ p´| ξ | ` z q ´ , whose inverse Fouriertransform is the fundamental solution of the operator “∆ ` z ” in our theorem. N ENDPOINT VERSION OF UNIFORM SOBOLEV INEQUALITIES 3
Figure 1.
The interpolation diagram for the resolvent estimatesTherefore, our theorem is a consequence of the following estimate for a multiplieroperator:(5) ›››! ˆ u p ξ q´| ξ | ` z ) q ››› L nn ´ , p R n q ď C } u } L nn ` , p R n q . This in turn, amounts to the inequality for a convolution operator(6) ››› u p x q ˚ ! ´| ξ | ` z ) q p x q ››› L nn ´ , p R n q ď C } u } L nn ` , p R n q . The proof of the theorem then transforms to the study of the kernel of this convo-lution operator: K p x q “ ! ´| ξ | ` z ) q p x q . The expression for this kernel is actuallyalready in literature, e.g. Gelfand-Shilov [6]: K p x q “ ´ z | x | ¯ n ´ K n ´ p a z | x | q , where(7) K ν p w q “ ż e ´ w cosh t cosh p νt q d t denotes the modified Bessel function. Along with the expression for K p x q , we willalso need the following facts about the Bessel function, all of which are containedin [8, p.339].First, a change of variable u “ e t in the expression (7) for K ν p w q immediatelyyields(8) | K ν p w q| ď C | w | ´| Re p ν q| , TIANYI REN, YAKUN XI, CHENG ZHANG for | w | ď p w q ą
0, where the constant C depends only on ν . Second,applying the formula (see [5, p.19])(9) Γ p ν ` q K ν p w q “ ´ π w ¯ e ´ w ż e ´ t t ν ´ ´ ` t w ¯ ν ´ d t, which is valid when Re ν ě
0, we obtain the behavior of the Bessel function forlarge | w | :(10) | K ν p w q| ď Ce ´ Re p w q | w | ´ whenever | w | ě p w q ą
0. Finally, formula (9) in fact tells us that(11) K ν p w q “ a ν p w q w ´ e ´ w for Re p w q ą
0, where the function a ν p w q enjoys the decaying property(12) ˇˇˇ´ BB w ¯ α a ν p w q ˇˇˇ ď C α | w | ´ α . With these preparations, we embark on the task of proving the estimate (6) forthe convolution operator K p x q . The idea is to treat the part of K p x q inside theunit ball and the part outside separately, hence we break K p x q “ K p x q ` K p x q ,where K p x q is defined to equal K p x q when | x | ď
1, and equal 0 elsewhere. Byestimate (8), considering the expression for K p x q , we easily obtain that | K p x q| ď C | x | ´p n ´ q . Then the desired, indeed strong, inequality(13) } u p x q ˚ K p x q} L nn ´ p R n q ď C } u } L nn ` p R n q follows from Hardy-Littlewood-Sobolev inequality whenever the dimension n ą α p n ` n , n ´ n q is on the line p ´ q “ n . If n “ nn ´ is then. Nevertheless, the restricted weak type estimate(14) } u p x q ˚ K p x q} L p R q ď } u } L , p R q } K } L , p R q ď C } u } L , p R q still holds, by the Holder’s inequality for Lorentz spaces. See for instance, [9,Theorem 3.6].After that, we turn to our analysis of K p x q , the part of K p x q away from theorigin. Applying (10) yields the estimate | K p x q| ď C | x | ´ n ´ e ´| x | cos p arg z q . Because of the exponential term, which may have the desired decaying property, weseparate the case where arg z P r´ π , π s , from the case where arg z R r´ π , π s . Forthe former situation, as just mentioned, the effect of the exponential decay yieldsthe strong estimate(15) } u p x q ˚ K p x q} L nn ´ p R n q ď C } u } L nn ` p R n q , which follows from Young’s inequality.The difficult situation is the latter one, and this is where Bourgain’s interpo-lation comes into play. As pointed out before, this interpolation technique firstappeared in [2] when Bourgain was proving an endpoint bound for the sphericalmaximal function, and we first noticed it in [1]. There is also an elaboration on N ENDPOINT VERSION OF UNIFORM SOBOLEV INEQUALITIES 5 the abstract theory, developed for fairly general normed vector spaces, in Carbery-Seeger-Wainger-Wright [3]. We are not going into such abstractness and generality,but would rather state the result in our specific setting, that of L p spaces. Lemma 1.
Suppose that an operator T between function spaces is the sum of theoperators T j : T “ ÿ j “ T j . If for ď p , p , q , q ď 8 , there exist β , β ą and M , M ą such that each T j satisfies } T j } L p Ñ L q ď M ´ jβ , and } T j } L p Ñ L q ď M jβ , then we have restricted weak type estimate for the operator T between two interme-diate spaces: } T f } L q, ď C p β , β q M ´ θ M θ } f } L p, , where θ “ β β ` β , p “ ´ θp ` θp , q “ ´ θq ` θq . We return to the proof of the main theorem, dealing with the second situationwhere arg z R r´ π , π s . By the expression (11) for the Bessel function, K p x q “ | x | ´ n ´ e ´ i | x | sin p arg z q e i n ´ arg z e ´| x | cos arg z a n ´ p| x | e i arg z q , where a n ´ p w q satisfies the decaying property (12).We dyadically decompose the kernel K p x q . Fix a smooth function η p x q that hassupport in t x : | x | ă u and is equal to 1 for | x | ă . Denote δ p x q “ η p x q ´ η p x q .Then let β p x q “ η p x q , and for each j ą
1, let β j p x q “ δ p ´ j x q . It is easy to verifythat ř j “ β j p x q “ j ě
0, consider the operator T j given by the kernel K ,j p x q “ β j p x q K p x q , i.e. T j u “ u ˚ K ,j . We need invoke the following variant of Stein’s oscillatoryintegral theorem. Lemma 2.
Let n ě . Suppose that ď p ď , q “ n ` n ´ p ; in other words, the pairof exponents p p, q q lies on the closed line segment joining E p , n ´ p n ` q q and F p , q .Then, given a kernel of the form L p x q “ δ p x q b p x q e iλ | x | | x | ´ n ´ , where λ ‰ , δ p x q is a smooth function supported in t x P R n : ď | x | ď u , b p x q P C p R n q , and |pp BB x q α b qp x q| ď C α | x | ´| α | , we have the inequality } L ˚ f } L q p R n q ď C | λ | ´ nq } f } L p p R n q , where the constant C depends only on the function δ p x q and finitely many of the C α above. TIANYI REN, YAKUN XI, CHENG ZHANG
This oscillatory integral theorem, proved by Sogge [10], follows from Stein’s oscil-latory integral theorem [13]. See also [8, p.341]. It holds for pairs of exponents lyingon the closed line segment EF in Figure 1. Because of this, we are tempted to inter-polate between the point P p n , q on the p axis and the point Q p n ` n ` n , n ´ n ` n q ,which is the intersection of the line αβ and the line EF . See Figure 1. When n “ α p n ` n , n ´ n q goes down to the r axis and coincides with P , so interpo-lating between P and Q cannot produce a restricted weak type inequality at α .Fortunately, we are able to remedy it by interpolating between two other points. Case n ą : At P , since K ,j p x q ď C | x | ´ n ´ by (10) and K ,j p x q is supportedin t x : 2 j ´ ď | x | ď j u , Young’s inequality yields } K ,j p x q ˚ u p x q} L p R n q ď C j n ´ } u } L n p R n q . At Q , we seek to prove(16) } K ,j p x q ˚ u p x q} L n n ´ n ` p R n q ď C ´ n j } u } L n n ` n ` p R n q . Changing the scale, replacing x with 2 j x , we would be done if we could show } ˜ K ,j p x q ˚ u p x q} L n n ´ n ` p R n q ď C ´ n ´ n ` n j } u } L n n ` n ` p R n q , where ˜ K ,j p x q “ δ p x q| x | ´ n ´ e ´ i j | x | sin p arg z q b p j x q ,b p x q “ e i n ´ arg z e ´| x | cos arg z a n ´ p| x | e i arg z q , and δ p x q is as mentioned in the dyadic decomposition. The kernel ˜ K ,j is easily seento fall within the hypotheses of the above oscillatory integral theorem, rememberingthat a n ´ p w q satisfies the decaying property (12). Hence we obtain the inequality(16) by applying Lemma 2, with the constant C independent of j .The estimates at P and Q for the operator T j enables us to utilize Bourgain’sinterpolation, resulting in the desired restricted weak type estimate for the operator T , which is the sum of the T j . Specifically, θ “ n n ` n ´ “ n ´ n ` ´ n ` n ` n , n ´ n ` n ¯ ¨ p ´ θ q ` ´ n , ¯ ¨ θ “ ´ n ` n , n ´ n ¯ . Noticing that the last pair of exponents is precisely the endpoint α , we have byLemma 1(17) } K p x q ˚ u p x q} L nn ´ , p R n q ď C } u } L nn ` , p R n q , where, independent of z P C , the constant C depends exclusively on the dimension n . (17) together with (13) and (15) gives the conclusion of Theorem 1 whenever n ą Case n “ : We interpolate instead between the two points O p , q and F p , q ,bearing in mind that the oscillatory integral theorem Lemma 2 holds for the latterpair too. At O , Young’s inequality gives } K ,j p x q ˚ u p x q} L p R n q ď C j } u } L p R n q , N ENDPOINT VERSION OF UNIFORM SOBOLEV INEQUALITIES 7 while a change of scale argument as in the above case along with the oscillatoryintegral theorem Lemma 2 shows immediately that at F , } K ,j p x q ˚ u p x q} L p R n q ď C ´ j } u } L p R n q . Then we compute θ “ ` “ , p , q ¨ θ ` p , q ¨ p ´ θ q “ p , q . Again, the last pair of exponents is our target pair. That concludes our proof ofthe Theorem 1. (cid:3) Endpoint Version of Stein-Tomas Fourier Restriction Theorem
The original Stein-Tomas restriction theorem [13], [15] states that if the dimen-sion n ě
3, one has the inequality(18) ››› ż S n ´ ˆ f p ξ q e πi x x,ξ y d σ p ξ q ››› L q p R n q ď C } f } L p p R n q , at the point p { p, { q q “ p n ` n ` , n ´ n ` q , which is the midpoint of AB in Figure2. Sogge [10] extended this result in showing that the same inequality holds forpairs of exponents p p, q q off the line of duality satisfying 1 ď p ă nn ` , q “ n ` n ´ p .They constitute the half open line segment connecting F p , q and A p n ` n , p n ´ q n p n ` q q (exclusive). Therefore by duality and interpolation, the Stein-Tomas restrictioninequality is true for pairs of exponents in the interior of the pentagon in Figure2. Furthermore, Bak-Seeger [1, Proposition 2.1]) established restricted weak typeinequalities(19) ››› ż S n ´ ˆ f p ξ q e πi x x,ξ y d σ p ξ q ››› L q, p R n q ď C } f } L p, p R n q at the vertices A and B . Then by real interpolation, strong type inequalities as (18)hold on the open segment AB . In addition, strong type inequalities trivially holdon the half open segments CF and DF (excluding C and D ) by Young’s inequality.However, no results seem to have been established on AC and BD before. Itis clear that strong Stein-Tomas can not hold on these two segments. Indeed,in [16], radial functions belonging to L nn ` p R n q are constructed that have infiniteFourier transforms on S n ´ . Moreover, neither strong type nor restricted weaktype inequality holds outside of the pentagon in Figure 2. In fact, if there werea restricted weak type inequality somewhere outside this pentagon, then by realinterpolation, we would either get a strong inequality on the line of duality q “ p with p ą n ` n ` , or get a strong inequality somewhere on AC or BD . This is acontradiction, remembering that the range 1 ď p ď n ` n ` is sharp for a strongrestriction estimate on the line of duality (see [14, p.387, 2.1.1]).In this section, we show that restricted weak type inequality as (19) holds onthe closed segments AC and BD , by an argument similar to the proof of Theorem1. With this result and the discussion above, we completely characterize the rangeof p p, q q for which either strong Stein-Tomas or restricted weak type Stein-Tomasholds. TIANYI REN, YAKUN XI, CHENG ZHANG
Figure 2.
The interpolation diagram for the restriction estimates
Theorem 3.
Let n ě . If p “ nn ` and n p n ` qp n ´ q ď q ď 8 , or ď p ď n p n ` q n ` n ´ and q “ nn ´ , then ››› ż S n ´ ˆ f p ξ q e πi x x,ξ y d σ p ξ q ››› L q, p R n q ď C } f } L p, p R n q Proof.
Our result follows from an analysis of the convolution operator whose kernelis the Fourier transform of the Lebesgue measure on the unit sphere, like in theclassical case. This kernel is well-known to have the expression K p x q “ π | x | ´ n ´ J n ´ p π | x |q , where J ν p w q is the Bessel function, see for instance, [14, p.347-348]. Later, we willneed the following fact about J ν p w q for ν “ m positive, integral or half integral,and w “ r real, positive and greater than 1, which is also well-known: it takes theform(20) J m p r q “ ÿ ˘ r ´ e ˘ ir a ˘ p r q , where the functions a ˘ p r q , r ą ˇˇˇ d k d r k a ˘ p r q ˇˇˇ ď C k r ´ k . This expression can be found in [14, p.338]; see also [11, Theorem 1.2.1]. Again,dyadically decompose K p x q , letting β j p x q , j ě K j p x q “ β j p x q K p x q .We first treat the case of the vertex at C , because it is exceptional. For this, wewish to apply Bourgain’s interpolation to the point O p , q and the point F p , q .There is no need to worry about the part K p x q of K p x q near the origin, since K p x q N ENDPOINT VERSION OF UNIFORM SOBOLEV INEQUALITIES 9 is the Fourier transform of a compactly supported distribution and is thus smooth.Away from the origin, i.e., for j ą
0, at O , Young’s inequality gives } K j p x q ˚ f p x q} L p R n q ď C n ` j } f } L p R n q , while at F , still applying Young’s inequality } K j p x q ˚ f p x q} L p R n q ď C ´ n ´ j } f } L p R n q . With the following interpolation computation, θ “ n ´ n ´ ` n ` “ n ´ n , p , q ¨ p ´ θ q ` p , q ¨ θ “ p n ` n , q , we obtain the restricted weak type Stein-Tomas inequality at C p n ` n , q : ››› ż S n ´ ˆ f p ξ q e πi x x,ξ y d σ p ξ q ››› L p R n q ď C } f } L nn ` , p R n q . Duality then produces the same restricted weak type inequality for the pair ofexponents D p , n ´ n q . However here, we cannot apply real interpolation to thepoints A and C , nor can we apply it to B and D , since they are along a vertical orhorizontal line, which violates a hypothesis of the real interpolation theorem.Nevertheless, we can proceed as in the proof of Theorem 1 to obtain a restrictedweak type Stein-Tomas inequality at every point on the line segment joining A and C and its dual line segment joining B and D . Indeed, for each n ` ă k ă n ` n , weinterpolate between the point R p k, q and the point S “ p n ´ n ` n ` n k, n ´ n p ´ k qq ,which is the intersection of the line p ´ q “ k and the line EF . At R for each j ą } K j p x q ˚ f p x q} L p R n q ď C j p n p ´ k q´ n ´ q } f } L k p R n q , while at S for j ą
0, the familiar change of scale argument in which we replace x with 2 j x , together with Lemma 2 produces } K j p x q ˚ f p x q} L s p R n q ď C j p´ n ` k ` q } f } L r p R n q , where r and s denote the exponents corresponding to S . Finally we verify θ “ n ` k ´ n ´ p ´ k q , ´ n ´ n ` n ` n k, n ´ n p ´ k q ¯ p ´ θ q ` p k, q θ “ ´ n ` n , ´ k ´ n ´ n ¯ , which gives us a restricted weak type inequality (19) at every point on the linesegment AC , as we hoped. Duality then produces the same results for the dual linesegment BD . (cid:3) Proof of Theorem 2
As noted in the introduction, the procedure is essentially the same as in [8],with a restricted weak type Stein-Tomas inequality and the adaptations of severalclassical results to Lorentz spaces.
1. Littlewood-Paley inequality . For t P R , let χ p t q be the characteristic functionof the set t t : | t | P r , su , and let χ k p ξ n q “ χ p k ξ n q . Suppose g is any function in S p R n q for simplicity. Proposition 1.
For any ă p ă 8 and ď q ď 8 , there exist constants C , C ,depending only on p , q and n , such that the inequalities below hold C } g } L p,q p R n q ď ›››´ ÿ k “´8 |t χ k p ξ n q ˆ g p ξ qu q | ¯ ››› L p,q p R n q ď C } g } L p,q p R n q . Proof.
The upper bound follows directly from the usual Littlewood-Paley inequalityand real interpolation. The lower bound can be obtained by imitating the dualityargument in [12, p.105]. (cid:3)
2. Minkowski’s Inequality . Proposition 2. ›››´ ÿ k “´8 | F k p x q| ¯ ››› L s, p R n q ď C ´ ÿ k “´8 } F k p x q} L s, p R n q ¯ , for any s ą , where the C depends only on s and n ; ´ ÿ k “´8 } F k p x q} L r, p R n q ¯ ď C ›››´ ÿ k “´8 | F k p x q| ¯ ››› L r, p R n q , for any ă r ă , where the C depends only on r and n .Proof. The first inequality follows easily from Minkowski’s inequality, recalling that L p,q is a Banach space when 1 ă p ă 8 , 1 ď q ď 8 . The second inequality resultsfrom a standard duality argument. (cid:3)
3. Stein-Tomas Inequality . Proposition 3. If n ě , then ››› ż S n ´ ˆ f p ξ q e πi x x,ξ y d σ p ξ q ››› L nn ´ , p R n q ď C } f } L nn ` , p R n q Proof.
This is a special case of Theorem 3. (cid:3)
4. H¨ormander’s Multipliers Theorem . Proposition 4.
Suppose that m P L p R n q satisfies, for some integer s ą n , ÿ ď| α |ď s sup λ ą λ ´ n } λ | α | D α β p¨{ λ q m p¨q} L p R n q ă 8 , whenever β P C p R n zt uq . Then for ă p ă 8 and ď q ď 8 , the inequalityholds } T m f } L p,q p R n q ď C p,q } f } L p,q p R n q , where the T m is the multiplier operator with multiplier m p x q : T m f “ t m p ξ q ˆ f p ξ qu q . Proof.
This is a consequence of H¨ormander’s multiplier theorem and real interpo-lation. (cid:3)
N ENDPOINT VERSION OF UNIFORM SOBOLEV INEQUALITIES 11
Having the above four propositions at hand, we are able to prove Theorem 2,following the localization argument in [8, p.335-337]. Below is a sketch of thisargument.By a reduction process similar to that at the beginning of the proof of Theorem1 (see [8, p.335]), it suffices to prove for Theorem 2 the following special case(21) } u } L nn “ , p R n q ď C ›››! ∆ ` ` (cid:15) ´ BB x n ` iβ ¯) u ››› L nn ` , p R n q , where (cid:15) ‰ β ‰
0. Theorem 2 is then a consequence of the estimate for a multiplieroperator below:(22) ›››! ˆ f p ξ q´| ξ | ` ` i(cid:15) p ξ n ` β q ) q ››› L nn ´ , p R n q ď C } f } L nn ` , p R n q . Denote the multiplier in (22) by m p ξ q . Also, for t P R , let χ p t q be the characteristicfunction of the set t t : | t | P r , su , and set χ k p ξ n q “ χ p k p ξ n ` β q . For convenience,denote χ k p ξ n q m p ξ q as m k p ξ q . It then suffices to prove a similar estimate for themultiplier m k p ξ q :(23) }t m k p ξ q ˆ f p ξ qu q } L nn ´ , p R n q ď C } f } L nn ` , p R n q . Indeed, noting that nn ` ă ă nn ´ , if we had estimate (23), we may apply thesecond part of Proposition 1, the first part of Proposition 2, the second part ofProposition 2, and the first part of Proposition 1, in that order, to obtain }t m p ξ q ˆ f p ξ qu q } L nn ´ , p R n q ď C ›››´ ÿ k “´8 |t m k p ξ q ˆ f p ξ qu q | ¯ ››› L nn ´ , p R n q ď C ´ ÿ k “´8 }t m k p ξ q ˆ f p ξ qu q } L nn ´ , p R n q ¯ ď C ´ ÿ k “´8 }t χ k p ξ n q ˆ f p ξ qu q } L nn ` , p R n q ¯ ď C ›››´ ÿ k “´8 |t χ k p ξ n q ˆ f p ξ qu q | ¯ ››› L nn ` , p R n q ď C } f } L nn ` , p R n q , which is the result we are seeking.To prove inequality (23), we first apply the special case we just proved, (3) inTheorem 1 to z “ ` i(cid:15) ´ k and obtain(24) ›››! χ k p ξ n q ˆ f p ξ q´| ξ | ` ` i(cid:15) ´ k ) q ››› L nn ´ , p R n q ď C } f } L nn ` , p R n q . By taking difference, it then remains only to demonstrate the inequality ›››! χ k p ξ n qr i(cid:15) p ξ n ` β ´ ´ k qs ˆ f p ξ qp´| ξ | ` ` i(cid:15) p ξ n ` β qqp´| ξ | ` ` i(cid:15) ´ k q ) q ››› L nn ´ , p R n q ď C } f } L nn ` , p R n q . (25)Now if we use polar coordinates ξ “ ρω , we will get, after applying Minkowski’sinequality for the Lorentz space L nn ´ p R n q and Proposition 3, the following string of inequalities ›››! χ k p ξ n qr i(cid:15) p ξ n ` β ´ ´ k qs ˆ f p ξ qp´| ξ | ` ` i(cid:15) p ξ n ` β qqp´| ξ | ` ` i(cid:15) ´ k q ) q ››› L nn ´ , p R n q ď ż ››› ż S n ´ (cid:15) ˆ f p ρω q χ k p ξ n qp ξ n ` β ´ ´ k q e iρ x ω,x y p´ ρ ` ` i(cid:15) p ξ n ` β qqp´ ρ ` ` i(cid:15) ´ k q d ω ››› L nn ´ , p R n q ρ n ´ d ρ ď C ż ρ ›››! (cid:15) ˆ f p ξ q χ k p ξ n qp ξ n ` β ´ ´ k qp´ ρ ` ` i(cid:15) p ξ n ` β qqp´ ρ ` ` i(cid:15) ´ k q ) q ››› L nn ` , p R n q d ρ. Finally, by Proposition 4, this last expression is majorized by C } f } L nn ` , p R n q ż | (cid:15) ´ k ρ |p ρ ´ q ` p (cid:15) ´ k q d ρ, which is dominated by C } f } L nn ` , p R n q . This concludes the proof of Theorem 2. (cid:3) One thing worthy of mentioning is that in Theorem 2, we exclude the case n “ nn ´ is when n “ Acknowledgement . The authors would like to express their gratitude to theiradvisor, Professor Christopher D. Sogge, for bringing this research topic and Bour-gain’s interpolation to their attention, and also for the invaluable guidance andsuggestions he provided.
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