TWO VERSIONS OF THE NIKODYM MAXIMAL FUNCTION ON THE HEISENBERG GROUP

Transcription

1 TWO VERSIOS OF THE IKODYM MAXIMAL FUCTIO O THE HEISEBERG GROUP JOOIL KIM Abstract The ikody axial function on the Euclidean plane R 2 is defined as the supreu over averages over rectangles of eccentricity ; its erator nor in L 2 is known to be O log ) We consider two variants, one on the standard Hensenberg group H and the other on the polarized Heisenberg group H pol The latter having logarithic L2 H )-erator nor behaves rather like the classical ikody axial function on the Euclidean plane, while the forer has the erator nor which grows essentially of order O /4 ) Finally we investigate the class of the ikody type axial erators having certain hoogeneous structures on R n whose L n R n ) erator nor is not logarithic Introduction For each integer 2, let R be the faily of all rectangles centered at the origin whose eccentricity the length of long side divided by the length of the short side) is Euclidean plane is defined by Then the classical ikody axial function on the M R fx) = ) sup f x + y dy, R R R R where x R andf is a locally integrable function on R 2 A Córdoba [2] proved that ) M R L 2 R 2 ) L 2 R 2 ) C + log ) a with a = 2 to obtain the result of the Bochner-Riesz eans on R 2 The sharp bound a = in ) was obtained by Ströberg [9] 2000 Matheatics Subject Classification Priary 42B20, 42B25

2 2 JOOIL KIM In this article we study the the classical ikody axial function on the Euclidean plane in the setting of the three diensional Heisenberg group We consider two realizations of the Heisenberg group First let H be the usual Heisenberg group identified with R 3 endowed with the group ultiplication x y = x + y, x 2 + y 2, x 3 + y 3 + ) 2 x y 2 x 2 y ) where we use coordinate x = x, x 2, x 3 ) ext let H p be the polarized Heisenberg group endowed with the group law ) x p y = x + y, x 2 + y 2, x 3 + y 3 + x y 2 Associated with the two cases of Heisenberg groups, we consider two axial averages over the all rectangles of eccentricity supported on the hyperplane Π = {x, x 2, 0) : x, x 2 R} We define the ikody-axial erator M Hisenberg group H by 2) M fx) = sup R R R R associated with the standard f x y, y 2, 0) ) dy dy 2 On the polarized Heisenberg group H p, the ikody-axial erator M p is defined by 3) M p fx) = sup f x p y, y 2, 0) ) dy dy 2 R R R R The ain purpose of this article is to prove the following results Theore There are positive constants c and C independent of such that 4) c log M p L 2 H p ) L2 H p ) Clog ) 3/2 Theore 2 There are positive constants c and C independent of such that 5) c /4 M L 2 H ) L 2 H ) C /4 log ) 3/2

3 IKODYM MAXIMAL FUCTIO 3 We shall ibed these two axial erators in the faily of erators associated to the hypersurfaces {x, x 2, αx x 2 )} where α R in the Heisenberg group H where the exceptional blow up in of Theore 2 occurs when α = 0 The L 2 estiates are based on the induction and group Fourier transfor We exploit the induction arguent by Wainger [0] to reduce the L 2 H ) estiation of the axial erators to that of certain square su erators The application of the group Fourier transfor for these square sus leads to the estiations of oscillatory integral erators on the real line It turns out that the phase functions of these integral erators are degenerate in the sense that their ixed second derivatives vanish in the case of H, but do not vanish in the case of the polarized Heisenberg group H p Reark It would be interesting to relate the behavior of M p and M to the results of curved ikody and Kakeya axial erators considered by Bourgain [], Minicozzi and Sogge [7] and Wisewell [] Reark 2 Consider the spherical axial erator S 2n on the 2n diensional hyperplane of the 2n + diensional Heisenberg group H n A Seeger and D Müller [7] prove that S 2n is bounded on L p H n ) when p > 2n 2n when n 2 The unresolved case n = cobined with the previous results of [4, 5] leads us to consider the ikody axial function on the plane of H Organization In section 2, we devel the induction arguent of [0] via the group Fourier transfor to reduce obtaining the upper bounds of Theores and 2 to the unifor L 2 estiations of a certain faily of one diensional oscillatory integral erators In section 3, we prove Theore 2 by showing the unifor L 2 estiation of the oscillatory integrals by T T ethods and Cotlar- Stein lea In sections 4, we show the lower bound in Theore 2 In section 5, we construct the faily of ikody type axial erators having certain dilation invariant structures on R n, whose L n R n ) erator nors are at least O β ) where β > 0 In section 6, we obtain the upper bound by using the siilar arguent of section 3

4 4 JOOIL KIM otation As usual, the notation A B for two scalar expressions A, B will ean A CB for soe positive constant C independent of A, B and A B will ean A B and B A 2 Induction and Group Fourier transfor In this section we change the L 2 estiates for the axial averages on the three diensional Heisenberg group to the estiate for a certain class of one diensional oscillatory integrals This shall be established by using the group Fourier transfor and adapting the induction arguent of [0] 2 Reductions In proving Theores and 2, it suffices to assue that the angle between the long side of the rectangle R R and the x -axis is restricted on [0, π/4] Let D = {,, } For such rectangle, we can find a parallelogra Rk, r) with soe k D and r > 0, { Rk, r) = y, y 2 + k ) } 2) y : r < y < r, r/ < y 2 < r/ satisfying the following engulfing prerty 22) P k, r/0) R P k, 0r) Thus we now define P as the faily of all parallelogra of the for 2), P = {Rk, r) : k D, r > 0} Using the group isoorphis I : H H p defined by Ix, x 2, x 3 ) = x, x 2, x 3 + x x 2 /2) where Ix) p Iy) = Ix y), M p fx) sup f II x, x 2, x 3 )) p II y, y 2, 0)) ) dy dy 2 R P R = sup R P R where f I x) = fix)) R R fi I x, x 2, x 3 ) y, y 2, y y 2 /2) ) dy dy 2 Thus, in order to prove Theore, we work with the ikody axial erator M p associated with the hyper-surface Π =

5 {y, y 2, y y 2 /2)}: M p fx) = IKODYM MAXIMAL FUCTIO 5 sup f x y, y 2, y y 2 /2)) dy dy 2 R P R R Fro 2), the support of the above integral {y, y 2, y y 2 /2) : y, y 2 ) R} with R = Rk, r) is written as { y, y 2 + k y, 2 y2 + k ) } y ) y : r < y < r, r/ < y 2 < r/ Due to the group ultiplication, y, y 2 + k y, y2 + 2 k ) y ) y Therefore we have, = 0, y 2, 0) y, k y, k 2 y2 ) 23) where M α fx) = sup M 2 fx) = sup r>0 M p k D,r>0 r 2r /2 fx) M M 2 fx) r 2r r ) )) f x, x 2, x 3 0, t, 0 dt r f x, x 2, x 3 ) t, k t, α k t2)) dt, The erator M 2 is bounded bounded in L p H ) for all < p since r M 2 fx) = sup f I x, x 2 + t, x 3 x x 2 /2 ) dt r>0 2r r which is the Hardy-Littlewood axial function with respect to the second variable Hence, in proving Theore, we shall prove that for α = /2, 24) M α L 2 H ) L 2 H ) Clog ) 3/2 By change of variable t = t, we rewrite 25) Mfx) α r = sup k D,r>0 2r f x r t, k t, α k t2 ) ) dt where we note that y, y 2, y 3 ) = y, y 2, y 3 ), which is the group inverse of y, y 2, y 3 ) in H In proving 24), it suffices to consider the case that the support of integral is restricted to [0, r] in 25) because of siilarity Let us

6 6 JOOIL KIM choose a positive sooth function ϕ supported [/2, 4] and ϕt) on [, 2] Put ϕ j t) = ϕ t/2 j) /2 j For each j Z, k D and α R, we define a easure µ α j,k, by µ α j,k,f) = f t, k t, α k t2 ) ϕ j t)dt Fix k D and r > 0 where 2 < r 2 for soe Z Then, r r 0 f x t, k t, α k ) ) t2 dt 2 j 2 j 2 2 j 2 f x t, k t, α k ) ) ) t2 dt j j= sup f µ α j,k, x, x 2, x 3 ) = j Z,k D sup µ α j,k, [f] 3 x, x 2, x 3 ) j Z,k D where is a convolution of the Heisenberg group H and [f] 3 x, x 2, x 3 ) = fx, x 2, x 3 ) Here the last line follows fro the identity 26) [f] 3 [g] 3 x, x 2, x 3 ) = [g f] 3 x, x 2, x 3 ) Thus, we let M α fx, x 2, x 3 ) = sup µ α j,k, fx, x 2, x 3 ), j Z,k D and prove the following general result in Section 3 Prosition For all α 0 where α = /2 corresponds to Theore ), 27) M α f L 2 H ) Clog ) 3/2 f L 2 H )

7 IKODYM MAXIMAL FUCTIO 7 22 Group Fourier transfor Let BL 2 R)) be the space of all bounded erators on L 2 R) The group Fourier transfor of f L H ) L 2 H ) is defined as an erator-valued function fro R \ {0} to BL 2 R)) such that λ R \ {0} fλ) BL 2 R)), given by [ ] 28) fλ)h x) = F 2,3 fx y, λx + y)/2, λ)hy) dy R where F 2,3 is the Fourier transfor with respect to second and third variables ote that fλ) BL 2 R)) is well defined since the integral kernel is square integrable with respect to x, y variables We introduce the criteria of the L 2 H n ) boundedess of convolution type erators on the Heisenberg group Prosition 2 Let G be a convolution erator defined by Gf = K f for f SH ) and where the convolution kernel K is a tepered distribution in S H ) Then G L 2 H ) L 2 H ) sup λ R\{0} Kλ) L 2 R ) L 2 R ) Proof For the proof of Prosition, we refer Chapter of [8] 23 on vanishing Hessian vs Vanishing Hessian We write µ α j,k,y, y 2, y 3 ) = ϕ j y )δ y 2 k ) y δ y 3 + α k ) y2, where δ is the Dirac ass By applying Prosition 2 and the forula 28), one is led to estiate the one diensional oscillatory integral erator, 29) where [ µ α j,k, λ)h ] x) = = [F 2,3 µ j,k,] α x y, λx + y)/2, λ)hy)dy exp 2πiλ k ) P αx, y) ϕ j x y)hy)dy, P α x, y) = 2 x y)x + y) + αx y)2 ote that the ixed second derivative of the oscillatory function P α for α 0 does not vanish as 20) [P α ] xyx, y) = 2α 0

8 8 JOOIL KIM This non-vanishing ixed second derivative condition enables us to apply integration by parts for the kernel of the erator [ ] µ α j,k, λ) µ α j,k, λ) to prove Prosition But the vanishing second derivative condition α = 0 in 20) yields the lower bound of Theore 2 which is proved in Sections 4 24 Induction Arguent on the Scale of The proof of 27) is based on the following induction arguent Assue that 2) M α L 2 H ) L 2 H ) Clog ) 3/2, where C does not depend on Under this assuption we show that 22) M α L 2 H ) L 2 H ) Clog ) 3/2 This iplies that 2) holds for every positive integer of the for 2 l by induction on l, which gives the desired result for every positive integer ow we prove 22) under the assuption of 2) For any j Z, sup µ α j,k, fx, x 2, x 3 ) sup µ α j,2k, µ α j,2k,) fx, x 2, x 3 ) k D k D 23) ote that + sup k D µ α j,2k, fx, x 2, x 3 ) 24) µ α j,2k, = µ α j,k, for all j Z, k D Hence we take the suprea over j Z on both sides of 23) to obtain that 25) M α f L 2 H ) Gf L 2 H ) + M α f L 2 H ), where G α fx, x 2, x 3 ) = j Z, k D µ α j,2k, µ α j,2k,) fx, x 2, x 3 ) 2 ) /2 In proving 22) it suffices to show that for α 0, 26) G α f L 2 H ) C log ) /2 f L 2 H ),

9 IKODYM MAXIMAL FUCTIO 9 since 26) 25) and 2) with C > 0C lead us to obtain that M α L 2 H ) L 2 H ) C log ) /2 + Clog ) 3/2 Clog)) 3/2 For the estiate of 26), we now use the group Fourier transfor For each k {,, }, j Z and fixed λ, α 0, put T α,λ j,k, = µ α j,2k, λ) µ α j,2k, λ) where µ α j,k, λ) is the group Fourier transfor which is expressed in 29) In proving 26), by Prosition, it suffices to prove that there exists a constant C > 0 independent of λ such that for α 0, 27) G α λ)h L 2 R ) Clog ) /2 h L 2 R ), where [G α λ)h] x) = j Z, k D T α,λ j,k, hx) 2 ) /2 3 Proof of Theore In this section we prove 27) in order to prove Theore 3 Decoposition into local and global parts Take a function ψ Cc [ 2, 2]) such that 0 ψ and ψu) = for u /2 Put ηu) = ψu) ψ2u) We choose the integer a satisfying 3) 2 a < 2 0 α + ) 2 a ote that a > 0 For fixed λ and α, let Sj,k, loc hx) = exp iλ k ) P αx, y) ϕ j x y)ψ y )hy)dy 2j+a Sj,k,hx) = exp iλ k ) y ) 32) P αx, y) ϕ j x y)η gy)dy, 2 where > j + a Then, µ α j,k, λ) = Sloc j,k, + =j+a+ S j,k,hx)

10 0 JOOIL KIM ote that we oit 2π on the oscillatory part of 32) Also we do not put λ and α in S loc j,k, and S j,k, for the notational siplicity We set T loc j,k, = S loc j,2k, S loc j,2k, T glo j,k, = =j+a+ ) S j,2k, Sj,2k, In proving 27), we show that there is a constant C > 0 independent of λ, T loc j,k,h ) /2 2 33) Clog ) /2 h L 2 R ), j Z, k D L 2 R ) and 34) j Z, k D ) /2 j,k, h 2 T glo L 2 R ) Clog ) /2 h L 2 R ) 32 Local part estiate 33) It suffices to show that for each fixed nonzero α and λ, 35) since fro 35) T loc j Z, k D j,k, T loc j,k,h 2 ) /2 2 j Z, k D k 22j λ L 2 R ) k 22j λ log ) h 2 L 2 R ) /2 { } in, 22j λ, { in, 2} ) 22j λ h 2 L 2 R ) Proof of 35) Our proof is based on T T ethod In order to estiate the L 2 nor of T loc j,k, = S loc j,2k, S loc j,2k,,

11 IKODYM MAXIMAL FUCTIO [ we copute the integral kernel Sx, z) of the erator Sj,k, loc S loc : j,k,] Sx, z) = exp iλ k Pα x, y) P α z, y) )) ϕ j x y)ϕ j z y)ψ y 2 j+a )2 dy where we recall P α x, y) = 2 x y)x + y) + αx y)2 The derivative of the phase function with respect to y variable is 36) [P α ] y x, y) [P α] y z, y) = 2α x z), which enables us to apply integration by parts to obtain that Sx, z) C α 2 j k 2j λx z) 2 + By Schur s test, S loc j,k, S = [ loc j,k, S loc j,k,] sup z Sx, z) dx k 22j λ /2 Hence 37) T loc j,k, S loc j,2k, + S loc j,2k, k 22j λ /2 In order to gain 22j λ in 35), we need to use the ean value theore as well as integration by parts We denote by T x, z) the integral kernel of Tj,k, loc where T x, y) = exp iλ 2k ) Pα x, y) P α z, y) )) exp iλ ) exp iλ P αz, y) ) ϕ j x y)ϕ j z y)ψ y 2 j+a )2 dy [T loc j,k, ] P αx, y) ) )

12 2 JOOIL KIM By using the ean value theore and the support condition of the above integral such that y 2 j and x 2 j, ) iλ exp P 38) αx, y) ) iλ exp y P αx, y) ) 22j λ 2j λ ote that we can replace P α x, y) by P α z, y) in 38) We apply integration by parts, T x, y) = [ exp iλ y exp iλ P αz, y) ) by using 36) and 38) to obtain that Hence, 39) T x, z) T loc j,k, which cobined with 37) yields 35) P αx, y) ) ) ) ] ϕj x y)ϕ j z y)ψy/2 j+a 2 λ ) 2k Pα x, y) P α z, y) ) dy 2 j λ 22j 2 k 2j λx z) 2 + /2 k 22j λ 22j λ, 33 Global part estiate 34) For > j + a, we let Then we can write [ T j,k, h ] x) = T j,k, = S j,2k, S j,2k,) ) ) 2k exp iλ P α x, y) exp iλ ) ) ) P α x, y) ϕ j x y)η y 2 )hy)dy Fro the support condition x y 2 j and y 2 > 2 j+a where a > 0, we can observe that x 2 if y 2

13 IKODYM MAXIMAL FUCTIO 3 Therefore, in proving 34) it suffices to show that for each fixed Z, 30) j Z, k D T j,k,h 2 ) /2 L 2 R ) log ) /2 h L 2 R ) The size of derivative of phase function on the coposition kernel of S j,k, [S j,k, ] is the sae as 36) to yield 3) S j,k, k 22j λ /2 so that Tj,k, k 22j λ /2, which gives the sae bound as 37) But when we use the ean value theore on the region y 2 2 j+a, 32) y iλ exp exp ) P αx, y) ) iλ P αx, y) ) 2j+ λ 2 λ appearing 2 instead of 2 j in 38) and this leads us to obtain only 33) T j,k, /2 k 22j λ 2j+ λ which is different fro 39) Fro this observation we notice that, for the global case y 2 j+a, the direct application of the above T T estiate does not lead us to have the unifor bound of 39) independent of the size of y In order to overcoe this difficulty, we shall apply the Cotlar-Stein alost orthogonality lea By duality, the estiate 30) follows fro 34) [T j,k Z j,k,] Tj,k, log

14 4 JOOIL KIM where we regard Tj,k, = 0 for k / D In proving 34), it suffices to prove that 35) j,k [T j,k,] Tj,k,[Tj,k 2,] T k k 2 j Z j,k 2, 2 2+j λ since the Cotlar-Stein lea with 35) yields that [T j,k,] Tj,k, [T j,k,] Tj,k, k { in, 4} 2+j λ log { in 2+j λ j Z log In proving 35), it suffices to show that 36) and 37) [T j,k,] T j,k,[t j,k 2,] T j,k 2, [T j,k,] T j,k,[t j,k 2,] T j,k 2, k k 2 k k 2 2+j λ We now conclude the proof by showing 36) and 37) }, 2+j λ 2 2+j λ j λ Proof of 36) By 32), the integral kernel Sx, z) of Sj,k, [S j,k 2, ] is exp iλ k P αx, y) k 2 P αz, y) ) ) y ) 2 ϕ j x y)ϕ j z y)η dy 2 The y-derivative of phase function in the oscillatory ter is λ 38) k k 2 )y 2α k x y) k 2 z y) )) We assue k k 2 without loss of generality We now show 36) by distinguishing two cases Case k k 2 )2 0k α + )2 j For this case λ k k 2 )y is the,

15 IKODYM MAXIMAL FUCTIO 5 doinating factor in 38), which enables us to apply integration by parts We obtain that 39) and this iplies that 320) S j,k,[s j,k 2,] T j,k,[t j,k 2 ] Since [T j,k,] T j,k,[t j,k 2,] T j,k 2, T we obtain 36) fro 320) for Case k k 2 k k 2 j,k, Case 2 k k 2 )2 < 0k α + )2 j By 3) 2 2+j λ 2 2+j λ T j,k,[t k k 2 2 j+a k k 2 2 < 0k a+j j,k 2,] T j,k2,, Thus it follows k k 2 ) < k /2 5 Therefore we have 32) k 2 2 j k 2 j k k α + ) Hence by 32) and 3) Sj,2k, k 22j λ Sj,2k 2, k 2 22j λ to obtain 36) for Case 2 /2 /2 k k 2 k k 2 /2 2+j λ /2 2+j λ, Proof of 37) We write the integral kernel of T glo j,k [T glo j,k 2 ] as exp iλ k P αx, y) k 2 P αz, y) ) ) exp ) iλ P α x, y) ) ) exp ) iλ P α z, y) ) ) ϕ j x y)ϕ j z y)η y 2 )dy Again we distinguish two cases Case k k 2 )2 0k α + )2 j For this case λ k k 2 )y is the

16 6 JOOIL KIM doinating factor in the phase function This cobined with 32) leads us to apply integration by parts and the ean value theore to obtain that 2 Tj,k,[Tj,k 2,] k k j λ 322) 2+j λ We also cobine322) with the ean value estiate Tj,k, + Tj,k 2, 2+j λ to obtain 37) Case 2 k k 2 )2 < 0k α + )2 j By 32) and 33), /2 Tj,k, k 22j λ /2 2+j λ k k 2 2+j λ 2+j λ to obtain 37) Define the erator by 4 Proof of lower bound of Theore 2 fx, x 2, x 3 ) = R sup R P R f x, x 2, x 3 ) y, y 2, 0)) dy dy 2 where P is the subfaily of P whose side lengths are fixed as and Obviously L 2 H ) L 2 H ) M L 2 H ) L 2 H ) We show that there exists c > 0 such that 4) c /4 L 2 H ) L 2 H ) Let S k fx, x 2, x 3 ) be ) k f x + y, x 2 + y + y 2, x 3 + ) ) ) k x 2 y + y 2 x 2 y dy R where R = {y, y 2 ) : y <, y 2 < /}

17 By 22), we observe that for f 0 IKODYM MAXIMAL FUCTIO 7 fx) sup S k fx) where x = x, x 2, x 3 ) k D We let fx, x 2, x 3 ) = fx, x 2, x 3 x x 2 /2) and check that ) k S k fx) = f x + y, x 2 + y + y 2, x 3 + ) 2 x x 2 + P x, y, y 2 ) dy where R P x, y, y 2 ) = 2 ) k ) x + y ) 2 x 2 + y y 2 Thus it suffices to prove that there exists a function f L 2 H ) such that 42) sup k D Sk f 2 L 2 H ) C /2 f 2 L 2 H ) where S k fx, x 2, x 3 ) is ) k f x + y, x 2 + y + y 2, x R ow we shall show 42) Let us define )) k ) x + y ) 2 x 2 + y y 2 dy 43) fx, x 2, x 3 ) = χ [/,/ ] x )χ [ 0,0] x 2 )χ [ 00/,00/] x 3 ) where χ B is a characteristic function supported on the set B R Let { U = x, x 2, x 3 ) : 2 x, 0 x 2, 6 x 3 } 8 Then we show that for each x = x, x 2, x 3 ) U 44) This iplies that Therefore we have 45) k D depending on x such that S k fx) sup k sup k D Sk f 2 L 2 H ) = S k fx) on U U sup k D Sk fx) 2 dx 32

18 8 JOOIL KIM Fro 43) 46) f 2 L 2 H ) = 8000 Hence 42) follows fro 45) and 46) ow it suffices to show 44) Proof of 44) Let V = { x, x 3 ) : x 2, x 6 3 8} Then we need to observe that for each x, x 3 ) V, k = kx, x 3 ) D such that x 3 47) k 2 x2 < 5, which iplies that on the region x +y /, y 2 / and x, x 3 ) V x 3 + ) k x x + y ) 2) 48) + y y 2 20 Obviously we see that for any k D, y < and y 2 /, x 2 + k y ) 49) + y 2 0 for 0 x 2 Thus by 48) and 49), we check the support condition of 43) to obtain that for any x, x 2, x 3 ) U with k in 47) f x + y, x 2 + k y ) + y 2, x3 + k x x + y ) 2) ) ) + y y 2 =, on the region R x = {y R : x + y } This cobined with the easure estiate of the set R x yields 44) 5 onlogarithic L n R n ) bound for curved surface By using the previous proof we can show that L 3 R 3 ) nor of the following ikody type axial erator K 3 is at least β with β > 0 where Kfx 3, x 2, x 3 ) = sup f x y, y 2, 0, )) dy dy 2 P P P P

19 IKODYM MAXIMAL FUCTIO 9 where {x y, y 2, 0, ) : y, y 2 ) P } is the curved iage of soe parallelogra whose side lengths are and Here 5)x y = x + y, x 2 + y 2, x 3 + y 3 + 2x 2 y 2 2x x 2 y + x y y 2 x 2 y 2 )) We shall obtain this lower bound β in R n for any n 3 Define a binary eration on R n by x y = x + y,, x n + y n, x n + y n + Qx, x 2, y, y 2 )) where x = x,, x n ), y = y,, y n ) and Qx, x 2, y, y 2 ) is [ ) )] ) x l y l y 2 x l y l x 2 l l l l= ote that 5) corresponds to = 3 For each n 3, we define the axial erator l=2 Kfx) n = sup f x y, y 2, 0,, 0)) dy dy 2 P P P P Then we can obtain that Theore 3 There exists c n > 0 and β n > ɛ for arbitrary sall ɛ > 0 such n that K n L n R n ) L n R n ) c n βn for large Proof of Theore 3 Let fx) = fx,, x n + x x 2 ), then f x y, y 2 + k )) y, 0,, 0 where = f x + y, x 2 + y 2 + k ) y ), x 3,, x n, x n + V x, x 2, y, y 2 ) V x, x 2, y, y 2 ) = x x 2 + k x + y ) x ) + ) x l y l y 2 l l=

20 20 JOOIL KIM Thus we see that K n fx) sup k K k fx) where K k fx) = where Set R f x + y, x 2 + y 2 + k ) y ), x 3,, x n, x n + V x, x 2, y, y 2 ) dy R = {y, y 2 ) : y <, y 2 < /} fx) = χ [ / /,/ / ]x )χ [ 0,0] x 2 ) χ [ 0,0] x n )χ [ 00 /,00 /]x n ) where χ B is a characteristic function supported on the set B R Let { U = x,, x 3 ) : 2 x, 0 x 2,, x n, 0 x n + x x 2 } 2 Then for each x U 52) k D depending on x such that S k fx) / Proof of 52) For each x U, 53) k = kx) D such that x n + x x 2 k x < 0, which iplies that on the region x + y / /, y 2 / and x U ) x n + x x 2 + k x + x + y ) ) + x l y l y 2 00 l Thus we check that for any x U with k in 53) f x + y, x 2 + y 2 + k ) y ), x 3,, x n, x n + V x, x 2, y, y 2 ) =, l= on the region R x = {y R : x + y } / easure estiate of the set R x yields 52) This cobined with the

21 IKODYM MAXIMAL FUCTIO 2 By 52), Therefore, 54) sup S k fx) k sup k D S k f n L n R n ) = Fro the definition of f, 55) Thus the L n R n ) nor is at least U sup k D on U / Sk fx) n dx f n L n R n ) = 20n / n/ 2 + +/ n/)/n 20 /n 2 +)/n We finish the proof of Theore 3 by choosing β n = + / n/)/n for sufficiently large n 6 Proof of upper bound of theore 2 We are able to assue that the region of integral is restricted to [0, r] in 2) as we did in Theore For each j Z, k D, we define the easure ν j,k,, ν j,k, f) = f y, y 2, 0) 2 ϕ y ) j 2 j 2 ψ y j 2 j 2 + k )) y dy dy 2 By using 22) and 26), M fx, x 2, x 3 ) sup j Z, k D ν j,k, fx, x 2, x 3 ) By applying the sae induction arguent as in Section 22, it suffices to show that j Z, k D νj,2k, ν j,2k, ) f 2 ) /2 L2H) ) /4 log ) /2 f L 2 H )

22 22 JOOIL KIM By using forula 28), [ ν j,k, λ)h] x) = exp iλ k x2 y 2 ) ) ) λ2 ψ j x + y) ϕ j x y)hy)dy Let U j,k, λ) = ν j,2k, λ) ν j,2k, λ) By Prosition, it suffices to prove that for soe C independent of λ, Uj,k, λ)h ) /2 6) 2 C /4 log ) /2 h L 2 R ) j Z, k D L2R) Fix λ and choose l Z such that We let for each Z, 2 l /2 < 2 l+ 62) U j,k, = S j,2k, S j,2k, where S j,k,hx) = exp iλ k x2 y 2 ) ) ) λ2 ψ j x + y) y ) ϕ j x y)η hy)dy 2 We split U j,k, λ) = Uj,k, loc + U j,k, ed + U glo j,k, Uj,k, loc = 63) U ed j,k, = < j l U glo j,k, = such that U j,k, j l j 0 j 0 < U j,k, U j,k, ote each of the above three erators is defined according to the size of y in the above, naely U loc j,k, whose kernel supported on y 2j l, U ed j,k, on

23 IKODYM MAXIMAL FUCTIO 23 2 j l y 2 j 0, and U glo j,k, on 2j 0 y Estiate of Uj,k, loc We show that U loc j,k,h ) /2 2 64) j Z, k D L 2 R ) /4 h L 2 R ) ote U loc j,k, hx) = On the support of integral exp iλ 2k x2 y 2 ) ) exp iλ 2k x2 y 2 ) ) ) ) ψ λ2 j x + y) y ) ϕ j x y)ψ hy)dy 2 j l 65) x + y x y 2 j because y 2 j Since ψ is a Schwartz function, ) { } ψ λ2 j x + y) 66) C in λ2, 2j By using the ean value theore, exp iλ 2k x2 y 2 ) ) exp iλ 2k x2 y 2 ) ) { } λ2 2j 67) in, Fro 65)-67) cobined with the support condition y 2 j l, { } U j,k, loc λ2 C2 l 2j in, λ2 2j This yields 64) because 2 l /2 Estiate of U ed j,k, j Z, k D We show that U ed j,k,h 2 ) /2 L 2 R ) /4 log ) /2 h L 2 R ) For this estiate it suffices to prove that for fixed with j l 2 j 8, U j,k, h ) /2 68) 2 /4 h L 2 R ), j Z, k D L2R)

24 24 JOOIL KIM since l log This can be obtained fro the dual estiate [ ] U j,k, U 69) j,k, /2 j Z, k D L 2 R ) In proving 69), it suffices to prove that [U 60) j,k,] Uj,k,[Uj,k 2,] Uj,k 2, { k k 2 22j λ in, 2} 22j λ since the Cotlar-Stein lea with 60) yields that U j,k,[uj,k,] U j,k,[u j,k,] j,k j Z k ) { in k 22j λ j Z /2 k= /2, 22j λ We now show that 60) ote Uj,k,hx) = exp iλ 2k x2 y 2 ) ) exp iλ 2k x2 y 2 ) ) ) ) ψ λ2 j x + y) y ) 6) ϕ j x y)η hy)dy 2 /2} Let Kx, y) be the integral kernel of the above erator By using 67) cobined with the support condition y 2, we obtain the Hilbert Schidt nor, 62) { } U j,k, λ2 Kx, y) 2 dydx C2 j)/2 2j in, ext we show that 63) U j,k,[uj,k 2,] k k 2 2j We see that 60) follows fro 62) and 63) 22j λ

25 IKODYM MAXIMAL FUCTIO 25 Proof of 63) The integral kernel Sx, z) of the erator Sj,k, [S j,k 2, ] is Sx, z) = exp iλ k x2 y 2 ) k 2 z2 y 2 ) ) ) 64) Θx, y, z)dy where Θx, y, z) ) ) = ψ λ2 j x + y) λ2 ψ j z + y) y ) 2 ϕ j x y)ϕ j z y)η 2 Derivative of the oscillatory ter with respect to y variable is given by 65) 2 λ k k 2 )y ote that y 66) and 67) ψ )) λ 2 j x + y) = λ 2j ) ) λ 2 ψ j x + y) x + y y y )) η 2 2 By the support condition that y 2 2 j 0 and 2 j x y 2 j+, we observe that x + y 2 j in 66) Thus the derivative of the aplitude Θx, y, z) is doinated by 68) Θx, y, z) y C 2 2j+ By using 65) and 68), we apply integration by parts in 64) to obtain that 69) Sx, z) λ k k 2 2 2j+ This yields that S j,k,[sj,k 2,] λ k k 2 2 j+ which cobined with 62) iplies 63),

26 26 JOOIL KIM Estiate of U glo 620) We split again where j,k, j Z, k D We show that ) /2 j,k, h 2 U glo U glo j,k, U glo, j,k, = L 2 R ) = U glo, j,k, + U glo,2 j,k, j+0 U glo,2 j,k, = U j,k, j 0<<j+0 h L 2 R ) U j,k, For the proof of 620), we show that 62) and 622) [U glo, j,k, ] U glo, j,k, j Z,k D [U glo,2 j,k, ] U glo,2 j,k, j Z,k D Proof of 62) For the case y 2 > 2 j+0 in 6), it suffices to replace U glo, j,k, in 62) by only one piece U j,k,, For this, it suffices to show that 623) [Uj,k,] Uj,k, j Z,k D [U j,k,] Uj,k,[Uj,k 2,] Uj,k 2, k k 2 2j+ λ 5/2 { in, 3} 2j+ λ

27 IKODYM MAXIMAL FUCTIO 27 since Prosition 2 with 623) yields that U j,k,[uj,k,] U j,k,[u j,k,] j,k j Z k { 5/4 in 2j+ λ, 2j+ λ j Z /4} By using the support condition y 2 > 2 j+0 in 6), exp iλ 2k x2 y 2 ) ) exp iλ 2k x2 y 2 ) ) { } λ2 j+ C in, So, we get 624) ext we show that 625) { } U j,k, λ2 j+ C in, U j,k,[uj,k 2,] k k 2 5 2j+ λ If 625) is true, then by interpolation 625) and 624), U j,k,[uj,k 2,] 5/2 { } k k 2 2j+ λ λ2 j+ 626) in, We see that 623) follows fro 624) and 626) In proving 625), fro 62) it suffices to show that 627) S j,k,[sj,k 2,] k k 2 5 2j+ λ The integral kernel Sx, z) of the erator S j,k, [S j,k 2, ] is in 64) Fro the observation that x + y z + y 2 > 2 j+0 in 66) and 67), we obtain that 628) Θx, y, z) y C 2 3j

28 28 JOOIL KIM By using 65) and 628) with the support condition y 2 > 2 j+0, we apply integration by parts on the integral 64) to obtain that 5 Sx, z) C λ k k 2 2 j+ 2 j which yields that 627) The proof of 62) is finished Proof of 622) ote that our difficulty here coes fro the fact that for the case j 0 < < j + 0, x + y can vanish in 66), which prevent us fro having 628) Instead, we have soe bigger bound, Θx, y, z) y C { 629) 2 ax 2j 2, λ 2j j However, we can overcoe this obstacle by the following observation integral kernel Kx, y) of the erator [ ] [U glo, j,k, ] U glo, j,k, h y) = } Kx, y)hy)dy is supported on the set { x, y) : 2 j 5 < x < 2 j+5, 2 j 5 < y < 2 j+5} The Thus it suffices to prove for fixed j and where j 0 < < j + 0, [U j,k,] Uj,k, k D For proving this we show that [U 630) j,k,] Uj,k,[Uj,k 2,] Uj,k 2, { k k 2 ax 22j λ 5/2 since Prosition 2 with 630) yields that U j,k,[uj,k,] { in k5/4 22j λ k k, } { k k 2 5/2 in, 3} 22j λ 5/4, 22j λ /4}

29 IKODYM MAXIMAL FUCTIO 29 For the proof of 630), we replace 628) by 629) and apply the sae estiation of 623) with the support condition y 2 j instead of 2 References [] J Bourgain, L p -estiates for oscillatory integrals in several variables Geo Funct Anal 99), [2] A Córdoba, The Kakeya axial function and the spherical suation ultipliers, Aer J Math ), -22 [3] G B Folland, Haronic Analysis in Phase Space, Annals of Math Study 22, Princeton UnivPress 989) [4] J Ki, L p -estiates for singular integrals and axial erators associated with flat curves on the Heisenberg group Duke Math J ), [5] J Ki, Multi-paraeter Marcinkiewicz ultiplier theore for the Weyl functional calculus To appear in Math Z [6] W Minicozzi, C Sogge, eagative results for ikody axial functions and related oscillatory integrals in curved space 4 997) [7] D Müller, A Seeger, Singular spherical axial erators on a class of two step nilpotent Lie groups Israel J Math ), [8] E M Stein, Haronic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton press 993) [9] J-O Stroberg, Maxial functions associated to rectangles with uniforly distributed directions, Ann of Math ), [0] S Wainger, Applications of Fourier transfor to averages over lower diensional sets, Pro Sypos Pure Math 35) Aer Math Soc Providence, RI, 979) [] L Wisewell,Kakeya sets of curves Geo Funct Anal ), Departent of Matheatics, Chung-Ang University, Seoul , Korea E-ail address: jiki@cauackr