AN ENDPOINT VERSION OF UNIFORM SOBOLEV INEQUALITIES

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1 AN ENDPOINT VERSION OF UNIFORM SOBOLEV INEQUALITIES TIANYI REN, YAKUN XI, CHENG ZHANG Abstract. We prove an endpoint version of the uniform Sobolev inequalities in Kenig-Ruiz-Sogge [8]. It was known that strong type inequalities no longer hold at the endpoints. However, we show that restricted weak type inequalities hold there, which imply the earlier classical result by real interpolation. The key ingredient here is a type of interpolation first introduced by Bourgain []. We also prove restricted weak type Stein-Tomas restriction inequalities on some parts of the boundary of a pentagon, which completely characterizes the range of exponents for which the inequalities hold. Keywords. Uniform Sobolev inequalities, Stein-Tomas inequality, Bourgain s interpolation. Introduction In this paper, we consider the second order constant coefficient differential operator ņ B P pdq QpDq a j b, Bx j where Qpξq denotes a nonsingular real quadratic form on R n, n 3 which, for some k n, is given by j Qpξq ξ... ξ k ξ k... ξ n, D ipb{bx,..., B{Bx n q, and a,..., a n, b are complex numbers. If k n, the operator P pdq has principal part QpDq and is called elliptic. Otherwise, it is called non-elliptic. Uniform Sobolev inequalities () }u} Lq pr n q C}P pdqu} Lp pr n q, u P W,p pr n q, have been of interest to the study of unique continuation for partial differential equations. Here the constant C should depend only on n and p. If P pdq, () is just the classical Sobolev inequality. For more general elliptic operators, Kenig- Ruiz-Sogge [8, Theorem.] characterized the optimal range of exponents pp, qq for which () holds. Indeed, they showed that () holds for elliptic P pdq if and only if pp, qq satisfies the two conditions (i) p q n, (ii) mint p, q u, (i.e., p{p, {qq lies on the open line segment joining αp n, n 3 n q and βp, n q in Figure ). As pointed out in [8, p.330], the chief technical difficulty in proving () comes from the first order terms of P pdq. Indeed, () follows from a localization

2 TIANYI REN, YAKUN XI, CHENG ZHANG argument and the uniform resolvent estimates: () }u} Lq pr n q C}p zqu} Lp pr n q, u P W,p pr n q, z P C. Since strong type inequality like () and () no longer holds at the endpoints α and β, it is natural to ask whether restricted weak type inequality can be established at these endpoints. In this paper, we give a positive answer to this question when P pdq is elliptic. Theorem. Let n 3. If p{p, {qq α or β, then for any z P C, the inequality holds: (3) }u} L q,8 prn q C}p zqu} L p, pr n q, where the constant C depends only on n and p. Theorem. Let n 3. If p{p, {qq α or β, then there exists a constant C, depending only on n, such that whenever P pdq is a second order constant coefficient differential operator with principal part, we have (4) }u} L q,8 prn q C}P pdqu} L p, pr n q. A few remarks are in order. First, the above two theorems imply the corresponding classical results of Kenig-Ruiz-Sogge [8] by real interpolation. Second, S. Gutierrez [7] obtained restricted weak type resolvent estimates as in Theorem at points Ap n, pn q pn q q and Bp n 4n pn q, n q in Figure. The estimates cannot be uniform, but depend on z, because the exponent pairs are not on the line p q n. Finally, for non-elliptic P pdq, uniform restricted weak type estimates have been established in a recent work of Jeong-Kwon-Lee[4], which completely characterizes the range of pp, qq for which () holds in the non-elliptic case. Theorem follows from Theorem, a restricted weak type Stein-Tomas inequality (see Section 3) and the localization argument in [8, p ], after adapting several 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 an exponent becomes 8 (see Proposition ). The rest of the paper is organized as follows. In Section, we prove Theorem by an interpolation result first obtained by Bourgain [] and a variant of Stein s oscillatory integral theorem due to Sogge [0]. The interpolation method of Bourgain was first brought to our attention by Bak-Seeger[]. In Section 3, by a similar argument, we prove restricted weak type Stein-Tomas restriction inequalities on some parts of the boundary of a pentagon; this completely characterizes the range of exponents for which the inequalities hold. In Section 4, we prove Theorem by establishing several classical results in the setting of Lorentz spaces and carrying out the localization argument in Kenig-Ruiz-Sogge [8].. Proof of Theorem First of all, we need some reductions. It suffices to prove the theorem for one endpoint, say αp n, n 3 q, because the other follows from duality. Furthermore, noting the gap condition p q n on the exponents, we are able to reduce the theorem to the case where z has unit length, z, after a simple rescaling argument. Finally, by continuity, we may assume that Imz 0.

3 AN ENDPOINT VERSION OF UNIFORM SOBOLEV INEQUALITIES 3 Figure. The interpolation diagram for the resolvent estimates This last reduction enables us to study p ξ zq, whose inverse Fourier transform is the fundamental solution of the operator z in our theorem. Therefore, our theorem is a consequence of the following estimate for a multiplier operator:! ûpξq ) q L (5) ξ z,8 C}u} n 3 pr n q L n., pr n q This in turn, amounts to the inequality for a convolution operator! ) q (6) upxq pxq ξ z,8 C}u} L n 3 pr n q L n., pr n q The proof of the theorem then transforms to the study of the kernel of this convolution operator: Kpxq ξ z! ) q pxq. The expression for this kernel is actually already in literature, e.g. Gelfand-Shilov [6]: z n 4 Kpxq x K n pa z x q, where (7) K ν pwq» 8 0 e wcosht coshpνtqdt denotes the modified Bessel function. Along with the expression for Kpxq, we will also need the following facts about the Bessel function, all of which are contained in [8, p.339]. First, a change of variable u e t in the expression (7) for K ν pwq immediately yields (8) K ν pwq C w Repνq,

4 4 TIANYI REN, YAKUN XI, CHENG ZHANG for w and Repwq 0, where the constant C depends only on ν. Second, applying the formula (see [5, p.9]) π» 8 (9) Γpν qk νpwq e w e t t ν t ν dt, w w which is valid when Reν 0, we obtain the behavior of the Bessel function for large w : (0) K ν pwq Ce Repwq w whenever w and Repwq 0. Finally, formula (9) in fact tells us that () K ν pwq a ν pwqw e w for Repwq 0, where the function a ν pwq enjoys the decaying property B αaν () pwq C α w α. Bw With these preparations, we embark on the task of proving the estimate (6) for the convolution operator Kpxq. The idea is to treat the part of Kpxq inside the unit ball and the part outside separately, hence we break Kpxq K pxq K pxq, where K pxq is defined to equal Kpxq when x, and equal 0 elsewhere. By estimate (8), considering the expression for Kpxq, we easily obtain that K pxq C x pn q. Then the desired, indeed strong, inequality (3) }upxq K pxq} L n 3 pr n q C}u} L n pr n q follows from Hardy-Littlewood-Sobolev inequality whenever the dimension n 3, noticing that the point αp n, n 3 q is on the line p q n. If n 3 however, we cannot apply Hardy-Littlewood Sobolev inequality, because one exponent n 3 is 8 then. Nevertheless, the restricted weak type estimate (4) }upxq K pxq} L 8 pr3 q }u} L 3, pr 3 q }K } L 3,8 pr3 q C}u} L 3, pr 3 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 pxq, the part of Kpxq away from the origin. Applying (0) yields the estimate K pxq C x n e x cosp argzq. Because of the exponential term, which may have the desired decaying property, we separate the case where argz P r π, π s, from the case where argz R r π, π s. For the former situation, as just mentioned, the effect of the exponential decay yields the strong estimate (5) }upxq K pxq} L n 3 pr n q C}u} L n pr n q, which follows from Young s inequality. The difficult situation is the latter one, and this is where Bourgain s interpolation comes into play. As pointed out before, this interpolation technique first appeared in [] when Bourgain was proving an endpoint bound for the spherical maximal function, and we first noticed it in []. There is also an elaboration on 0

5 AN 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. Suppose that an operator T between function spaces is the sum of the operators T j : 8 T T j. j If for p, p, q, q 8, there exist β, β 0 and M, M 0 such that each T j satisfies }T j } L p ÑL q M jβ, and }T j } L p ÑL q M jβ, then we have restricted weak type estimate for the operator T between two intermediate spaces: }T f} L q,8 Cpβ, β qm θ M θ }f} L p,, where θ β β β, p θ p θ p, q θ q We return to the proof of the main theorem, dealing with the second situation where argz R r π, π s. By the expression () for the Bessel function, K pxq x n e i x sinp argzq e i n 3 4 argz e x cos argz a n p x e i argz q, where a n pwq satisfies the decaying property (). We dyadically decompose the kernel K pxq. Fix a smooth function ηpxq that has support in tx : x u and is equal to for x. Denote δpxq ηpxq ηpxq. Then let β 0 pxq ηpxq, and for each j, let β j pxq δp j xq. It is easy to verify that 8 j0 β jpxq. For each j 0, consider the operator T j given by the kernel K,j pxq β j pxqk pxq, i.e. T j u u K,j. We need invoke the following variant of Stein s oscillatory integral theorem. Lemma. Let n 3. Suppose that p, q n n p ; in other words, the pair of exponents pp, qq lies on the closed line segment joining Ep, n pn q q and F p, 0q. Then, given a kernel of the form Lpxq δpxqbpxqe iλ x x n, θ q. where λ 0, δpxq is a smooth function supported in tx P R n : 4 bpxq P C 8 pr n q, and pp B Bx qα bqpxq C α x α, we have the inequality }L f} Lq pr n q C λ n q }f} Lp pr n q, x u, where the constant C depends only on the function δpxq and finitely many of the C α above.

6 6 TIANYI REN, YAKUN XI, CHENG ZHANG This oscillatory integral theorem, proved by Sogge [0], follows from Stein s oscillatory integral theorem [3]. See also [8, p.34]. It holds for pairs of exponents lying on the closed line segment EF in Figure. Because of this, we are tempted to interpolate between the point P p n, 0q on the p axis and the point Qp n n, n 3n q, which is the intersection of the line αβ and the line EF. See Figure. When n 3 however, αp n, n 3 q goes down to the r axis and coincides with P, so interpolating 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 3: At P, since K,j pxq C x n by (0) and K,j pxq is supported in tx : j x j u, Young s inequality yields At Q, we seek to prove (6) }K,j pxq upxq} }K,j pxq upxq} L 8 prn q C j n 3 }u} L n pr n q. C n j }u} L n 3n pr n q L n n pr n q Changing the scale, replacing x with j x, we would be done if we could show. where } K,j pxq upxq} L n 3n pr n q C n 3n j }u} L K,j pxq δpxq x n e i j x sinp argzq bp j xq, n n pr n q, bpxq e i n 3 4 argz e x cos argz a n p x e i argz q, and δpxq is as mentioned in the dyadic decomposition. The kernel K,j is easily seen to fall within the hypotheses of the above oscillatory integral theorem, remembering that a n pwq satisfies the decaying property (). Hence we obtain the inequality (6) by applying Lemma, with the constant C independent of j. The estimates at P and Q for the operator T j enables us to utilize Bourgain s interpolation, resulting in the desired restricted weak type estimate for the operator T, which is the sum of the T j. Specifically, θ n n 3 n n 3n ; n n, n 3n n p θq n, 0 θ, n 3. Noticing that the last pair of exponents is precisely the endpoint α, we have by Lemma (7) }K pxq upxq} L n 3,8 pr n q C}u} L n, pr n q, where, independent of z P C, the constant C depends exclusively on the dimension n. (7) together with (3), (4) and (5) gives the conclusion of Theorem whenever n 3. Case n 3: We interpolate instead between the two points Op0, 0q and F p, 0q, bearing in mind that the oscillatory integral theorem Lemma holds for the latter pair too. At O, Young s inequality gives }K,j pxq upxq} L 8 prn q C j }u} L 8 prn q,

7 AN ENDPOINT VERSION OF UNIFORM SOBOLEV INEQUALITIES 7 while a change of scale argument as in the above case along with the oscillatory integral theorem Lemma shows immediately that at F, }K,j pxq upxq} L 8 prn q C j }u} L pr n q. Then we compute θ 3, p0, 0q θ p, 0q p θq p 3, 0q. Again, the last pair of exponents is our target pair. That concludes our proof of the Theorem. 3. Endpoint Version of Stein-Tomas Fourier Restriction Theorem The original Stein-Tomas restriction theorem [3], [5] states that if the dimension n 3, one has the inequality» (8) ˆfpξqe πixx,ξy dσpξq C}f} L Lq pr n q p pr n q, S n at the point p{p, {qq p n 3, n q, which is the midpoint of AB in Figure. Sogge [0] extended this result in showing that the same inequality holds for pairs of exponents pp, qq off the line of duality satisfying p n, q n n p. They constitute the half open line segment connecting F p, 0q and Ap n, pn q pn q q (exclusive). Therefore by duality and interpolation, the Stein-Tomas restriction inequality is true for pairs of exponents in the interior of the pentagon in Figure. Furthermore, Bak-Seeger [, Proposition.]) established restricted weak type inequalities» (9) ˆfpξqe πixx,ξy dσpξq S n L q,8 prn q C}f} L p, pr n q at the vertices A and B. Then by real interpolation, strong type inequalities as (8) hold on the open segment AB. In addition, strong type inequalities trivially hold on 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. It is clear that strong Stein-Tomas can not hold on these two segments. Indeed, in [6], radial functions belonging to L n pr n q are constructed that have infinite Fourier transforms on S n. Moreover, neither strong type nor restricted weak type inequality holds outside of the pentagon in Figure. In fact, if there were a restricted weak type inequality somewhere outside this pentagon, then by real interpolation, we would either get a strong inequality on the line of duality q p, or get a strong inequality somewhere on AC or BD. This is a contradiction, remembering that the range p n 3 is sharp for a strong restriction estimate on the line of duality (see [4, p.387,..]). In this section, we show that restricted weak type inequality as (9) holds on the closed segments AC, BD, by an argument similar to the proof of Theorem. With this result and the discussion above, we completely characterize the range of pp, qq for which either strong Stein-Tomas or restricted weak type Stein-Tomas holds. with p n 3

8 8 TIANYI REN, YAKUN XI, CHENG ZHANG Figure. The interpolation diagram for the restriction estimates Theorem 3. Let n 3. If p and q n, then» S n n ˆfpξqe πixx,ξy dσpξq and pn q pn q q 8, or p L q,8 prn q C}f} L p, pr n q pn q n 4n Proof. Our result follows from an analysis of the convolution operator whose kernel is the Fourier transform of the Lebesgue measure on the unit sphere, like in the classical case. This kernel is well-known to have the expression Kpxq π x n J n pπ x q, where J ν pwq is the Bessel function, see for instance, [4, p ]. Later, we will need the following fact about J ν pwq for ν m positive, integral or half integral, and w r real, positive and greater than, which is also well-known: it takes the form (0) J m prq r e ir a prq, where the functions a prq, r are smooth and satisfy the decay property dk dr k a prq C k r k. This expression can be found in [4, p.338]; see also [, Theorem..]. Again, dyadically decompose Kpxq, letting β j pxq, j 0 be as in the decomposition before and K j pxq β j pxqkpxq. We first treat the case of the vertex at C, because it is exceptional. For this, we wish to apply Bourgain s interpolation to the point Op0, 0q and the point F p, 0q. There is no need to worry about the part K 0 pxq of Kpxq near the origin, since Kpxq

9 AN 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 0, at O, Young s inequality gives }K j pxq fpxq} L 8 prn q C n j }f} L 8 prn q, while at F, still applying Young s inequality }K j pxq fpxq} L 8 prn q C n j }f} L pr n q. With the following interpolation computation, θ n n n n, p, 0q p θq p0, 0q θ p n, 0q, we obtain the restricted weak type Stein-Tomas inequality at Cp n, 0q:» ˆfpξqe πixx,ξy dσpξq L 8 C}f} prn q L S n n, pr n q. Duality then produces the same restricted weak type inequality for the pair of exponents Dp, n q. However here, we cannot apply real interpolation to the points A and C, nor can we apply it to B and D, since they are along a vertical or horizontal line, which violates a hypothesis of the real interpolation theorem. Nevertheless, we can proceed as in the proof of Theorem to obtain a restricted weak 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 k n, we interpolate between the point Rpk, 0q and the point S p n n n k, p kqq, which is the intersection of the line p q k and the line EF. At R for each j 0, n jpnp kq }K j pxq fpxq} L 8 prn q C q }f} L k pr n q, while at S for j 0, the familiar change of scale argument in which we replace x with j x, together with Lemma produces }K j pxq fpxq} Ls pr n q C jp n k q }f} Lr pr n q, where r and s denote the exponents corresponding to S. Finally we verify n θ k n p kq, n n k, n n p kq p θq pk, 0qθ, k n, which gives us a restricted weak type inequality (9) at every point on the line segment AC, as we hoped. Duality then produces the same results for the dual line segment BD. 4. Proof of Theorem 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 several classical results to Lorentz spaces.

10 0 TIANYI REN, YAKUN XI, CHENG ZHANG. Littlewood-Paley inequality. For t P R, let χptq be the characteristic function of the set tt : t P r, su, and let χ k pξ n q χp k ξ n q. Suppose g is any function in S pr n q for simplicity. Proposition. For any p 8 and q 8, there exist constants C, C, depending only on p, q and n, such that the inequalities below hold 8 C }g} L p,q pr n q tχ k pξ n qĝpξqu q L C }g} p,q L p,q pr pr n q n q. Proof. The upper bound follows directly from the usual Littlewood-Paley inequality and real interpolation. The lower bound can be obtained by imitating the duality argument in [, p.05].. Minkowski s Inequality. Proposition. 8 F k pxq L s,8 prn q C 8 }F k pxq} L s,8 pr n q, for any s, where the C depends only on s and n; 8 }F k pxq} L r, pr n q C 8 F k pxq L, r, pr n q for any r, 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 p 8, q 8. The second inequality results from a standard duality argument. 3. Stein-Tomas Inequality. Proposition 3. If n 3, then» ˆfpξqe πixx,ξy dσpξq,8 C}f} L n 3 pr n q L S n Proof. This is a special case of Theorem 3. n, pr n q 4. Hörmander s Multipliers Theorem. Proposition 4. Suppose that m P L 8 pr n q satisfies, for some integer s n, sup λ n }λ α D α βp {λqmp q} L pr n q 8, λ 0 0 α s whenever β P C 8 0 prn zt0uq. Then for p 8 and q 8, the inequality holds }T m f} L p,q pr n q C p,q }f} L p,q pr n q, where the T m is the multiplier operator with multiplier mpxq: T m f tmpξq ˆfpξqu q. Proof. This is a consequence of Hörmander s multiplier theorem and real interpolation.

11 AN ENDPOINT VERSION OF UNIFORM SOBOLEV INEQUALITIES Having the above four propositions at hand, we are able to prove Theorem, following the localization argument in [8, p ]. One thing worthy of mentioning is that in Theorem, we exclude the case n 3, in contrast with the corresponding result in Kenig-Ruiz-Sogge [8]. This is due to the fact that the exponent n 3 is 8 when n 3 and thus Littlewood-Paley inequality fails. Below is a sketch of this localization argument. By a reduction process similar to that at the beginning of the proof of Theorem (see [8, p.335]), it suffices to prove for Theorem the following special case () }u} B )!,8 L n3 pr n q C ɛ iβ u Bx, n L n, pr n q where ɛ 0, β 0. Theorem is then a consequence of the estimate for a multiplier operator below:! ) ˆfpξq q L () ξ iɛpξ n βq,8 C}f} n 3 pr n q L n., pr n q Denote the multiplier in () by mpξq. Also, for t P R, let χptq be the characteristic function of the set tt : t P r, su, and set χ k pξ n q χp k pξ n βq. For convenience, denote χ k pξ n qmpξq as m k pξq. It then suffices to prove a similar estimate for the multiplier m k pξq: (3) }tm k pξq ˆfpξqu q } L n 3,8 pr n q C}f} L n, pr n q. n Indeed, noting that n 3, if we had estimate (3), we may apply the second part of Proposition, the first part of Proposition, the second part of Proposition, and the first part of Proposition, in that order, to obtain }tmpξq ˆfpξqu q } 8,8 L n 3 pr n q C C C C C}f}, L n, pr n q tm k pξq ˆfpξqu q L n 3,8 pr n q }tm k pξq ˆfpξqu q } L n 3,8 pr n q }tχ k pξ n q ˆfpξqu q } L n, pr n q tχ k pξ n q ˆfpξqu q L n, pr n q which is the result we are seeking. To prove inequality (3), we first apply the special case we just proved, (3) in Theorem to z iɛ k and obtain! χ k pξ n q (4) ˆfpξq ) q L ξ iɛ k,8 C}f} n 3 pr n q L n., pr n q By taking difference, it then remains only to demonstrate the inequality! χ k pξ n qriɛpξ n β k qs ˆfpξq ) q L p ξ iɛpξ n βqqp ξ iɛ k q,8 n 3 (5) pr n q C}f} L n, pr n q.

12 TIANYI REN, YAKUN XI, CHENG ZHANG Now if we use polar coordinates ξ ρω, we will get, after applying Minkowski s inequality for the Lorentz space L n 3 pr n q and Proposition 3, the following string of inequalities! χ k pξ n qriɛpξ n β k qs ˆfpξq ) q L p ξ iɛpξ n βqqp ξ iɛ k q,8 n 3 pr n q» 8»Sn ɛ ˆfpρωqχ k pξ n qpξ n β k qe iρxω,xy 0 p ρ iɛpξ n βqqp ρ iɛ k q dω L,8 n 3 pr n q ρn dρ» 8! ɛ C ρ ˆfpξqχ k pξ n qpξ n β k ) q q L p ρ iɛpξ n βqqp ρ iɛ k q dρ. n, pr n q 0 Finally, by Proposition 4, this last expression is majorized by C}f} L n, pr n q» 8 0 ɛ k ρ pρ q pɛ k q dρ, which is dominated by C}f}. This concludes the proof of Theorem. L n, pr n q Acknowledgement. The authors would like to express their gratitude to their advisor, Professor Christopher D. Sogge, for bringing this research topic and Bourgain s interpolation to their attention, and also for the invaluable guidance and suggestions he provided. References [] J. G. Bak and A. Seeger. Extensions of the Stein-Tomas theorem. Math. Res. Lett., 8(4):767 78, 0. [] J. Bourgain. Esitmations de certaines functions maximales. C.R. Acad. Sci. Paris, 30:499 50, 985. [3] A. Carbery, A. Seeger, S. Wainger, and J. Wright. Classes of singular integral operators along variable lines. J. Geom. Anal., 9: , 999. [4] Y. Kwon E. Jeong and S. Lee. Uniform sobolev inequalities for second order non-elliptic differential operators. Advances in Mathematics, 30:33 350, 06. [5] A. Erdelyi. Higher transcendental functions, Vol. II. Bateman Manuscript Project. Krieger Publishing Company, Florida, 98. [6] I. M. Gelfand and G. E. Shilov. Generalized Function, volume I. Academic Press, New York, 964. [7] S. Gutiérrez. Non trivial l q solutions to the Ginzburg-Landau equation. Math. Ann., 38(- ): 5, 004. [8] C. E. Kenig, A. Ruiz, and C. D. Sogge. Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J., 55:39 347, 987. [9] R. O Neil. Convolution operators and Lpp, qq spaces. Duke Math. J., 30:9 4, 963. [0] C. D. Sogge. Oscillatory integrals and spherical harmonics. Duke Math. J., 53:43 65, 986. [] C. D. Sogge. Fourier integrals in classical analysis, volume 05 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 993. [] E. M. Stein. Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, N.J., 970. [3] E. M. Stein. Beijing lectures in harmonic analysis, volume of Ann. of Math. Stud., chapter Oscillatory integrals in Fourier analysis. Princeton University Press, Princeton, N.J., 986. [4] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton, N.J., 993. [5] P. Tomas. Resriction theorems for the Fourier transform. Proc. Symp. Pure Math., 35: 4, 979.

13 AN ENDPOINT VERSION OF UNIFORM SOBOLEV INEQUALITIES 3 [6] N. J. Vilenkin. Special functions and the theory of group representations, volume of Translations of Mathematical Monographs. American Mathematical Society, Providence, R.I., address: tyren@math.jhu.edu Department of Mathematics, Johns Hopkins University, Baltimore, MD 8, USA address: yxi4@math.rochester.edu Department of Mathematics, University of Rochester, Rochester, NY 467, USA address: czhang67@math.jhu.edu Department of Mathematics, Johns Hopkins University, Baltimore, MD 8, USA