A UNIFORM L p ESTIMATE OF BESSEL FUNCTIONS AND DISTRIBUTIONS SUPPORTED ON S n 1

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 5, Number 5, May 997, Pages 39 3 S ) A UNIFORM L ESTIMATE OF BESSEL FUNCTIONS AND DISTRIBUTIONS SUPPORTED ON S n KANGHUI GUO Communicated by Christoher D. Sogge) Abstract. A uniform L estimate of Bessel functions is obtained, which is used to get a characterization of the L measures on the unit shere of R n in terms of the mixed L norm of the Fourier transform of the measures.. Introduction For n, let SR n ) be the sace of Schwartz class functions. The dual sace of SR n ) is denoted by S R n ) and called the temered distributions. Let S n be the unit shere of R n.forµ S R n ), we list the following statements. A) µ is a finite measure on S n. B) µ, f =forfx) SR n ), vanishing on S n. C) µ is suorted on S n. It is easy to see that A) B) C). An examle of Schwartz [8] indicates that in general C) B), while the Hilbert transform distribution on a iece of S could be used to show that in general B) A). One motivation for us to study these roerties comes from the area of artial differential equations with constant coefficients. For examle, if ˆµ is a solution of the equation ux) +ux)=, then µ satisfies the roerty B). In [3], Hörmander roved that if ˆµ L R n ), n n n, then C) imlies µ =. For> n,wehave Theorem. [], Lemma, age 5). If ˆµ L R n ), <, n =, then C) imlies B). 3<<, n =3, n n << n n 3, n > 3, A natural question is: When does C) imly A)? First we notice that since S n is comact, the fact that ˆµ L R n )forsome imlies that all derivatives of ˆµ are also in the sace L R n ). Thus the Sobolev embedding lemma [9], Theorem.3.7, age 6) yields that ˆµ L R n ) and hence ˆµ L R n )for all >. Therefore if one seeks L conditions on ˆµ, then a smaller means a Received by the editors March, 995 and, in revised form, October, Mathematics Subject Classification. Primary 3A5. The author s research was suorted in art by the National Science Foundation, Grant DMS- 98. Some of the work was done while the author was attending the harmonic analysis worksho at Edinburgh, Scotland, June, c 997 American Mathematical Society License or coyright restrictions may aly to redistribution; see htt://

2 33 KANGHUI GUO stronger condition. To find a recise condition on ˆµ, let us study the measure µ on S n given by µ = gs) ds, wheregs) is a smooth function on S n and ds is the area measure of S n. In this secial case, it is well known [7] that ˆµξ) is essentially a constant multile of + ξ ) n, which imlies the following facts: ) ) lim inf n n )+ { lim su R R n ) ˆµ L R n n ) <, } ˆµξ) dξ <. ξ <R Previously, Agmon and Hörmander [], Theorems. and.) found that ) is true if and only if µ is a L density on S n. Their roof uses the Plancherel theorem and some smooth cut-off functions. For x R n, denote its olar coordinates by r, x ). Define the mixed norm saces L R + )L S n )) by { L R + )L S n )) = fr, x ), f L L ) ) ) } = f r, x ) dx r n dr S <. n We list the following statement for L L ): lim inf n ) 3) ˆµ L n n L ) <. n )+ The L L ) sace is very close to the sace L R n ). For instance, we have f L L ) C f L R n ) for, and f L L ) = C f L R n ) if f is radial. The main result of this article is Theorem., which is stated in terms of L L ) saces. In articular, it yields that if su µ S n, then 3) is equivalent to ). Since it is trivial to see that ) imlies 3), we conclude that if su µ S n,then) imlies ). The well known otimal L L restriction theorem says that the Fourier transform restriction oerator T is bounded from L R n ) to L S n ) for all n+) n+3. If one considers the L L q restriction roblem, then the otimal range of should be < n n+. This turns out to be a long-standing conjecture for n 3 [], age 388). In [], L. Vega roved the L L )R n ) L S n ) restriction theorem for in the otimal range < n n+. As a corollary of Theorem., we will recature L. Vega s result with the recise information for the constant in the inequality see Remark.). Working with the mixed norm saces enables us to substitute the Plancherel theorem with the Parseval identity for Fourier coefficients so that >inther direction is allowed, while in the revious work, = in all directions is imerative. For our aroach, the difficulty stems from the assumtion that > n n, since, when invoking the Hölder inequality in the Lorentz saces see [5]), one can only assert that if ˆµ L n n, the weak L n n sace), then ) is satisfied. The key ingredients in our roof are the L L ) version of Theorem. and the best ossible License or coyright restrictions may aly to redistribution; see htt://

3 A UNIFORM L ESTIMATE OF BESSEL FUNCTIONS 33 uniform L estimate of the Bessel functions J v r) for all nonnegative real v, which is roved via the Fourier transform restriction theorem on S. We organize this aer as follows. Section sulies some background information. Section 3 is devoted to the estimates of Bessel functions. The main result is roved in Section, where a counter-examle is also given to show that for any fixed >, there exists a distribution µ S R ) with su µ S and ˆµ L L ) such that µ is not a finite measure on S.. Preliminaries We start with the dual form of the restriction theorem of the Fourier transform to the unit circle S in R. We refer the reader to [], age, for the original form. In the rest of this aer, the same C will stand for different uniform constants. Lemma.. Let µ = gs)ds be a finite measure on S. Then there is a constant C, indeendent of, suchthat ) ) ) ˆµξ) dξ C ) gs) q q 3 ds, R S + q =,<<. Proof. For <α<nand <s<, the roof of the Hardy-Littlewood-Sobolov inequality [], age 35) indicates the inequality 5) f y α L s R n ) Cn α) f Ls R n ), s =+n α n, where s = s s and C is indeendent of α or s). Then an alication of the Hölder inequality yields 6) fx)fy) x y α dx dy Cn α) f L s R n ) R n R n with <α<nand <s<, such that s =+n α n. Since S is comact, we may cut it into several ieces if necessary so that for each iece we can follow the roof of Theorem 5. in [], ages, to get ) r ˆµξ) r r 7) dξ A fx) r fy) r x y r dx dy, R R R where <r<andr = r r. Here the constant A is indeendent of r. Using 6) for n = to control the right-hand side of 7), we have ) r ) ˆµξ) r r dξ C r) fx) rs s 8) dx, R R where <r<,r = r r,and s =3 r. Let =r and q = rs; then an easy calculation shows that 3 + q =. Thus ) follows from 8) since there is a constant C, indeendent of r or ), such that r) r C ) for all <r< and hence for all <<. The roof of Lemma.3 is comlete The following lemma is called van der Corut s lemma. License or coyright restrictions may aly to redistribution; see htt://

4 33 KANGHUI GUO Lemma. [], age 33). Suose that φ is real-valued and smooth in a, b) and that φ k) x) for all x a, b). Then there exists a constant c k indeendent of φ, t, a and b such that for t> b e itφx) dx c kt k holds when i) k, orii) k =and φ x) is monotone. a For x R n,writex=rx, with r = x and x S n. We will need the following identity, which can be derived easily from Theorem 3., age 58 in []. Lemma.3. Let µ = Y x ) dx with Y x ) a sherical harmonic of degree m ). Then ˆµξ) =Ci m r n Jn +mr)y ξ ), where C is indeendent of m. Checking the roof of Theorem. in [] carefully, one sees that the same roof with a minor modification yields its L L ) version as follows. Lemma.. If then C) imlies B). ˆµ L L ), <,n=, 3<<, n =3, n n << n n 3, n > 3, 3. Estimates of Bessel functions Recall that for integers m, the Bessel functions J m r) are defined by J m r) = π e irsin θ mθ) dθ π π and we have J m r) = ) m J m r). Also for ositive real numbers v, the Bessel functions J v r) can be written as A v r) B v r), where A v r) = π 9) e irsin θ vθ) dθ, π π B v r) = sinvπ) ) e vt r sinht) dt. π We refer the readers to [3], age 76, for details. Lemma 3.. For <λ, we have ) J v r)r λ dr C λ, v, ) where C is indeendent of v, λ. J v r)r λ dr C λ v λ, v, License or coyright restrictions may aly to redistribution; see htt://

5 A UNIFORM L ESTIMATE OF BESSEL FUNCTIONS 333 Proof. From the definition of J v r), it is obvious that J v r) Cfor all v and r. It follows that J v r)r λ dr C for <λ and v. Also from 9) and ), we have J v r)r λ dr C v for <λ and v. Thus the lemma will be roved if we can show that 3) J v r)r λ dr C λ, <λ, v, ) Jv r)r λ dr C λ v λ, <λ, v. The following formula can be found in [3], age 3, which we only need for v and<λ, even though it holds in other cases: 5) J v r)r λ dr = )λ Γλ)Γv λ + ) Γ λ + )) Γv + λ + ). The inequality 3) will follow from 5) and the fact that Γλ) C λ for < λ. The inequality ) will follow from 5) and the following inequality whose roof is an easy exercise in calculus: Γv λ + 6) ) Γv + λ + ) C v λ, <λ, v. The roof of the lemma is finished. Lemmas. and.3 and the fact that J m r) = ) m J m r) give the following uniform L estimate of Bessel functions J m r) for all integers m. This estimate seems to be well known in the community, but it is not available for the author in the literature. Lemma 3.. 7) J m r) rdr ) C ), where C is indeendent of >and all integers m. Remark. For each fixed m, Lemma 3. is trivial due to the fact that as r,j m r) is essentially C m r. What makes Lemma 3. interesting is that C m as m [], age 357). In the remainder of this section, we will rove that the same estimate 7) holds for J v r), uniformly for all v. Lemma 3.3. For v, we have 8) i) J v r) J v+ r) rdr 9) ii) J v r)+j v+ r) rdr where C is indeendent of >and v. ) ) C ), C ) License or coyright restrictions may aly to redistribution; see htt://

6 33 KANGHUI GUO Proof. First of all it is easy to check that 7) is true for B v r), uniformly for v. Thus it remains to verify 8) for A v r) A v+ r) and 9) for A v r)+a v+ r). Write v as m + α with α <. Let F α θ) = e i α)θ e i+α)θ and G α θ) =e αθ + e i+α)θ. Since F α π)=g α π)=f α π)=g α π) =, and both F α θ) andg α θ) are smooth, integrating by arts yields that for θ [ π, π], one has F α θ) = a ke ikθ and G α θ) = b ke ikθ,where a k Cand b k C. Here the bound C is indeendent of α. We have π A v r) A v+ r) = e irsin θ mθ) e i α)θ e i+α)θ ) dθ, π π A v r)+a v+ r) = π e irsin θ mθ) e iαθ + e i+α)θ ) dθ. π π Invoking Lemma 3. and Minkowski s integral inequality finishes the roof of 8) and 9). Lemma 3.. J v r) C v, r v, v, J v r) Cr 3, r, v. Proof. The estimates of this lemma are trivial for B v r), hence we only rove the lemma for A v r). We write A v r) asa v r)+a v r), where πa v r) = πa v r) = 3π π π 3π e irsin θ vθ) dθ + e irsin θ vθ) dθ + π π 3π π e ir sin θ vθ) dθ + e ir sin θ vθ) dθ. π 3π e ir sin θ uθ) dθ, When r v, we use Lemma. with k =forbotha v r)anda v r)toget ) A v r) C v, r v, v. When r, we use Lemma. with k = 3 to control A v r) andwithk=to control A v r). This together with ) comletes the roof of the lemma. Now we are in a osition to rove the main result in this section. Theorem 3.5. For >, we have ) i) J v r) ) rdr C ), v, ) ii) J v r) rdr C,δ v δ 3, v, <δ<, where C is indeendent of and v, while C,δ deends on and δ, but is indeendent of v. Proof. The estimate ) follows directly from ) and Lemma 3.. It remains to verify ). For each fixed v, ) is trivial, so we may assume v. We will use License or coyright restrictions may aly to redistribution; see htt://

7 A UNIFORM L ESTIMATE OF BESSEL FUNCTIONS 335 the following recurrence formulae [3], age 5): 3) ) J v r) J v+ r) =J vr), rj v r) vj vr) = rj v+ r). From 8) and 3), we see that J vr) satisfies the estimate ), uniformly for v. It follows from ) that v r J vr) J v+ r) satisfies the estimate ). This together with 9) yields that + v r )J vr) satisfies the estimate ). In articular, there exists a constant C indeendent of > and v such that which imlies 3v + v ) J v r) rdr r ) C ), 5) 3v ) J v r) rdr C ). Using the cutting of the integral as in the roof of Lemma 3. and invoking Lemma. with k =ork=, one can verify that there exists a constant C indeendent of >andv such that ) A v r) 6) rdr C ). 3v As in the roof of Lemma 3.3, 6) is trivial for B v r), v. It follows that 6) is true for J v r). Putting 5), 6) together comletes the roof of the theorem.. Proof of the main result Theorem.. Let µ be a temered distribution suorted on S n.then i) if µ is a L density on S n,then su n ) 7) ˆµ L L > n ) C µ L S n ); n n n ii) if lim inf n n )+ n ) ˆµ L L ) <, thenµis a L density on S n such that 8) µ L S n ) C lim inf n n+ )+ where C only deends on the dimension n. n ) ˆµ L n L ) Proof of i). Without loss of generality, we assume that µ L S n ) =. Wecan write µ = gx ) dx with gx )= c my m x ), where c m =andy m x )is some sherical harmonic of degree m with Y m =. From Lemma.3, we have ˆµr, x )=C i m c m r n Jn +mr)y mx ). License or coyright restrictions may aly to redistribution; see htt://

8 336 KANGHUI GUO Let > n n ) n so that >. We aly i) of Theorem 3.5 to obtain J n ) n n ) n ) n ) n +m r)rdr C n C n n ) n n ) ) n n ). Moreover an easy calculation shows ) n n ) r +n n n ) ) n ) n ) n dr C n ) n n ). n Therefore, the following inequalities are true: ) c m J n +mr) r n ) r n dr ) C c m J n +m r) r n ) r n dr C C C c m c m [ c m C C n n n n J n +m r)r n ) r n dr n J n +m r)r n ) r n n ) r n ) J n [ ) n ) ] n n ) n ) n +m r)rdr r n dr ] n n ) r +n n n ) ) n ) n ) n dr n ) n ) n n ) n where, in the first inequality, we used the Jensen convex inequality for the discrete integral since c m =. Since Y m =, which imlies Y m C for all m, Lemma.3 imlies that n ) J n +mr) r are uniformly bounded by a constant B for all r and m. This gives ) c m J n +mr) r n ) r n dr B c m r n dr B. License or coyright restrictions may aly to redistribution; see htt://

9 A UNIFORM L ESTIMATE OF BESSEL FUNCTIONS 337 Thus the definition of L L ) yields ˆµ L L ) = = + c m J n +mr) ) C n ) + B n ) c m J n +mr) r n ) r n dr ) r n ) r n dr where C only deends on the dimension n. Since lim n n ) =, 7) follows from the last inequality. Proof of ii). Since S n is comact, we see that µ, f is well defined for f C R n )andthatˆµis a C function in R n. In articular for each r, ˆµr, ξ )isa L function on S n. Under the assumtion of ii), we can aly Lemma. to see that µ, f =iffx) C R n ), vanishing on S n. Thus we can write µ, f = µ, f S n for every f C R n ), where f S n is the restriction of f on S n.in articular this imlies that ˆµr, ξ )= µ, e irξ x and for any sherical harmonic Y ξ ) of degree m we have µ, Y ξ ) = µ, P ξ), wherepξ)= ξ m Y ξ ξ )isa solid sherical harmonic of degree m. Nowwehave n ˆµr, ξ )Y ξ ) dξ S = µ, e irξ x Y ξ ) dξ S n = µ, x Y ξ ) dξ S n e irξ = µ, Ci m r n J n +mr)y x ) = Ci m µ, Y x ) r n J n +mr). Let Y mj ξ ), j =,,...,m d, be all of the sherical harmonics of degree m with Y mj =. Letc mj = µ, Y mj,c m = m d c mj ) and Y m ξ )= m d c mj Y mj ξ ) c m rovided c m. With the above notation, we have shown that ˆµr, ξ )=C i m c m r n Jn +my mξ ). Given > n n,letλ = n + n ) n )sothat n λ ++ n ) n + n ) =n n and that <λ C n n ) with C indeendent of. We may assume λ by restricting close to n n. License or coyright restrictions may aly to redistribution; see htt://

10 338 KANGHUI GUO We have [ n ) n [C λ ) c + m> c + m> C c m m C m C n ) n C C n n n n J m+ n )] c m m + n λ c m m + n λ r)r λ dr )] c m J r)r n ) r n ) r m+ n λ n + r n r + n c m J r)r n ) m+ n m r n ) r λ n + r n ) r + n n ) dr ) r n dr ) ) S n ˆµr, x ) dx n ) ˆµ n L L ). n ) dr Here the second inequality comes from Lemma 3. and, in the fourth inequality, we used Jensen s convex inequality for integrals and the fact that n + r n ) dr = n n ). Thus we have c + m> ) c m m + n λ C where C only deends on the dimension n. Since λ k we get c m C lim inf n m n )+ which roves 8). n ) ˆµ L n L ), for any k n n )+, n ) ˆµ L L n ), Remark.. For f SR n ), the dual form of 7) is the following otimal mixed norm restriction result: Tf L S n ) C n ) 9) f L L n ), < n n+, where is given by = and C is indeendent of. We now turn our attention to the counter-examle. We start with a lemma that is just the Fourier series version of Theorem 7.6.6, age 9 in []. License or coyright restrictions may aly to redistribution; see htt://

11 A UNIFORM L ESTIMATE OF BESSEL FUNCTIONS 339 Lemma.3. For a small ε > and any >, there exists µ S R) with su µ [ε, π ε], not a finite measure, such that ˆµm) <, where c m = µ, e imx. Proof. Checking the roof in [] for the case n =, one can see that everything could be adoted excet the fact that the Fourier transform of e itξ is Ct x it e t, which was used to show that the Fourier transform of e itξ is uniformly bounded by Ct. The author is not aware of the exact formula for the Fourier coefficients of the function e itξ on [, π]. Fortunately, by invoking van de Corut s lemma for k =,wecanhavethesameestimatethatthefouriercoefficientofe itξ on [, π] is uniformly bounded by Ct. Examle.. For any >, there exists a temered distribution µ suorted on S, not a finite measure, such that ˆµ L L ). Proof. Since >, we have 7 ) +8 <. Choose α such that 7 ) +8 <α<. From 7 ) +8 <α,wehave α α < 3. Let l = α α ;thenl>since>>. From the revious inequality, we have Let l be the dual of l; then 3 l>. α l = α). From α<,weseethat α) >. Now from Lemma.3, we can find a temered distribution µ S R) with su [ε, π ε], not a finite measure, such that c m α) <. Notice that the same µ can be viewed as a temered distribution in R with su µ S, but not a finite measure on S.Since 3 l>, ii) of Theorem 3.5 yields l J m r)rdr) <. License or coyright restrictions may aly to redistribution; see htt://

12 3 KANGHUI GUO The following inequalities will finish the roof of the examle: ) ˆµ L L ) = C c m Jm r) rdr =C C C C <. c m α c m α Jmr) c m α) c m α ) c m α l l J m r) dr ) ) rdr c m α Jm r) ) ) l l J m r)rdr ) rdr Acknowledgement We thank Professor Carl Herz for his insightful suggestions and encouragement during the rearation of this work. References. S. Agmon and L. Hörmander, Asymtotic roerties of solutions of differential equations with simle characteristics, Journal D Analyse Mathematique 3 976), 38. MR 57:6776. K. Guo, On the -aroximation roerty for hyersurfaces of R n,math.proc.camb.phil. Soc ), MR 9f:3 3. L. Hörmander, Lower bounds at infinity for solutions of differential equations with constant coefficients, Israel J. Math ), 3 6. MR 9:553. L. Hörmander, The analysis of linear artial differential oerators I, Sringer-Verlag, R. A. Hunt, On the L, q) saces, Enseign. Math. 966), MR 36:69 6. C. E. Kenig, A. Ruiz, and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential oerators, Duke Mathematical Journal ), MR 88d: W. Littman, Fourier transforms of surface-carried measures and differentiability of surface average, Bull. Amer. Math. Soc ), MR 7: L. Schwartz, Sur une roriete de synthese sectrale dans les groues non comact, C. R. Acad. Sci. Paris. 7 98), 6. MR :9e 9. C. D. Sogge, Fourier integrals in classical analysis, Cambridge University Press, 993. MR 9e:3578. E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean saces, Princeton University Press, 97. MR 6:. E. M. Stein, Harmonic Analysis, Princeton University Press, 993. MR 95c:. L. Vega, Restriction theorems and the Schrödinger multilier on the torus, Partial differential equations with minimal smoothness and alications B. Dahlberg et al., eds.), IMA Vol. Math. Al., vol., Sringer-Verlag, New York, 99, 99. MR 93e:5 3. G. N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, 958. MR 6:6a Deartment of Mathematics, Southwest Missouri State University, Sringfield, Missouri 658 address: kag6f@cnas.smsu.edu License or coyright restrictions may aly to redistribution; see htt://