Cylinder multipliers associated with a convex polygon

ylinder multipliers associated with a convex polygon Sunggeum Hong, Joonil Kim and han Woo Yang Abstract. In this paper we prove that sharp weak-type estimates for the maximal operators associated with a cylindric distance function associated with a convex polygon on H p (R 3 ) when 2/3 3( ). p Mathematics Subject lassification (2). Primary 42B5 ; Secondary 42B3. Keywords. ylinder multipliers, Minkowski functional, Polygon, Hardy spaces.. Introduction Let P be a convex polygon in R 2 which contains the origin in its interior. Let ρ be the associated Minkowski functional defined by We define a distance function m as ρ(ξ ) = inf{ɛ > : ɛ ξ P}, ξ. (.) m(ξ, ξ 3 ) = max{ρ(ξ ), ξ 3 } with ξ = (ξ, ξ 2 ). We note that ( m(ξ, ξ 3 ) 2 ) δ is supported on a bounded cylinder. Here t δ = t δ for t > and zero otherwise. For a Schwartz function f S(R 3 ) let f(ξ) = R 3 f(y) e i<y,ξ> dy denote by the Fourier transform of f. For δ > we consider T δ ɛ f(ξ) = ( m(ξ)2 ɛ 2 ) δ f(ξ), ξ = (ξ, ξ 3 ) R 2 R, and the maximal operators T δ f(x, x 3 ) = sup Tɛ δ f(x, x 3 ), (x, x 3 ) R 2 R. ɛ> This work was supported (in part) by research funds from hosun University, 27.

2 Sunggeum Hong, Joonil Kim and han Woo Yang The cylinder multipliers were previously studied by H. Luer in [4]. He considered the case where ρ(ξ ) = ξ with ξ R n. It is worth noticing that the the multiplier m with ρ(ξ ) = ξ has singularities on the boundary and along the cone ξ = ξ 3 and that the singularity inside the bounded cylinder yields the restriction on p for L p boundedness of the operator associated with the multiplier ( m 2 ) δ. He showed that if n 3 and p >, there is a restriction on p for which L p boundedness of T δ holds (see also [9]). Recently it is shown in [3] that the maximal operator associated with this cylindric distance function m(ξ, ξ 3 ) = max{ ξ, ξ 3 } with n = 2 satisfies a sharp weak type estimate on H p (R 3 ) when 4/5 δ c (p). In the cases of higher dimensional cases, n 3, one can show that a critical index in the above sense does not exist because the kernel F [( m 2 ) δ ] is not even integrable independently of δ where F [g] is the inverse Fourier transform of g (see [4, 9] for details). This type of restriction on p was also observed by P. Oswald in [5, 6]. He considered Marcinkiewicz-Riesz means associated with a non-smooth distance function m defined by m(ξ) = max{ ξ,..., ξ n } where ξ R n. We note that this distance function coincides with cylindric distance function when n = 2. He established sharp weak type estimates for the maximal operators associated with this multiplier when (n )/n δ M (p). However when the distance function does not have any singularity inside the boundary of the unit ball, this kind of phenomenon is not detected. Actually for the case m(ξ) = ξ Stein, Taibleson and Weiss in [8] finished drawing the full picture of the corresponding maximal operators, that is, when δ = δ s (p) := n(/p /2) /2 the maximal operators is of weak type (p, p) for the functions in H p (R n ), < p <. The purpose of this paper is to obtain a sharp weak type (p, p) estimate for the maximal operator T δ associated with a convex polygon on H p (R 3 ). This is a general version of Marcinkiewicz-Riesz means associated with non-smooth distance functions due to P. Oswald [6] on three dimensions. Here H p are the standard real Hardy space as defined in E. Stein [7]. Theorem.. The maximal operator T δ is bounded from H p (R 3 ) to L p (R 3 ); that is T δ f f Lp (R 3 ) H p (R 3 ) (.2) if and only if 2/3 3( p ). The constant does not depend on f. Theorem.2. If 2/3 3( p ), then T δ maps H p (R 3 ) boundedly into weak-l p (R 3 ), that is, { (x, x 3 ) R 2 R : T δ f(x, x 3 ) > α } α p f p H p (R 3 ), (.3) where the constant does not depend on α or f, and B denotes the Lebesgue measure of B.

ylinder multipliers associated with a convex polygon 3 Remark.3. (i) Let δ p = 3( p ) be the critical index. If δ δ p or p 2/3, one can find that T δ fails to be bounded on L p (R 3 ) in Section 4. We also show that T 3/2 is unbounded from H 2/3 (R 3 ) to weak-l 2/3 (R 3 ). Thus we note that the indicated ranges of parameters p and δ can not be improved. (ii) For the maximal operator T δ associated with a convex polyhedron on H p (R n ), n 4, we can show that (n )/n n( p ) conditions are sufficient for strong and weak type (p, p) estimates by extending the methods in Theorems. and.2. However, we do not know whether the indicated ranges of parameters p and δ are sharp. (iii) For δ > since the kernel function in Lemmas 2.2 and 2.3 of Section 2 is integrable, T δ is bounded on L p (R 3 ) for p. (iv) When p = and δ =, the operator is not bounded on L (R 3 ) but it can be proved that it is bounded from Hprod (R3 ) to L (R 3 ), where Hprod (R3 ) is the product Hardy space. We shall use A B if cb A B for some constants c, >. 2. Kernel estimates In this section we shall estimate the kernel of cylinder multipliers associated with a convex polygon. We shall consider Tɛ δ f(x, x 3 ) = (2π) 3 where K ɛ (x, x 3 ) = R R R 2 e i<x For each ɛ, the kernel K ɛ has the property R 2 K ɛ (x y, x 3 y 3 ) f(y, y 3 ) dy dy 3,ξ >ix 3ξ 3 ( m(ξ) 2 ɛ 2 ) δ dξ dξ 3. K ɛ (, ) = ɛ 3 K (ɛ, ɛ ). (2.) We initially decompose ( ρ(ξ ) 2) δ by triangulation, where ρ is the associated Minkowski functional defined by ρ(ξ ) = inf{ɛ > : ɛ ξ P}. Definition 2.. () For θ, θ 2 < 2π, (cos θ, sin θ ) (cos θ 2, sin θ 2 ) if and only if θ < θ 2. (2) For w = (w, w 2 ) R 2 \ {}, we define w = ( w, w 2 ) = w/ w. Let V [P] be the set of vertices of P. We write V [P] = {v i } L i= and v i = (v,i, v 2,i ). We may assume that ṽ i ṽ i for i =,..., L. Let i be a triangle with vertices {v i, v i, } when i =,..., L and {v L, v, } when i = L. Let ϕ (R 2 ) be supported in B(, ) with ϕ (ξ ) = for ξ B(, /2). Let φ(ξ) be the angle between the positive ξ 3 -axis and ξ. We let ω and ω 2 be smooth functions in φ such that ω and ω 2 are supported in {φ(ξ) π/2 ɛ } and {φ(ξ) π/2 ɛ },

4 Sunggeum Hong, Joonil Kim and han Woo Yang and ω ω 2, where ɛ is chosen to be so small that the following arguments hold true. We set ν(ξ) = ( m(ξ) 2 ) δ and decompose the multiplier ν as ν(ξ) = ν(ξ)ϕ (ξ ) ν(ξ)( ϕ (ξ )) = ν(ξ)ϕ (ξ ) ν(ξ)ϕ 2 (ξ ) = ν(ξ)ϕ (ξ )ω (ξ) ν(ξ)ϕ (ξ )ω 2 (ξ) ν(ξ)ϕ 2 (ξ )ω (ξ) ν(ξ)ϕ 2 (ξ )ω 2 (ξ), where ξ = (ξ, ξ 3 ) R 2 R. Let a i, b i (i =,..., L) be unit vectors satisfying ṽ i a i b i ṽ i when i =,..., L and ṽ L a i b i. For unit vectors u and w, we define seg(u, w) as a segment in S from u to w counterclockwise. Let η i be smooth functions in S supported in seg(a i, b i ) when i =,..., L and in seg(a L, b ) when i = L satisfying L η i on S. i= Now we decompose ν as L ν(ξ) = ν(ξ)ϕ (ξ )ω (ξ)η i (ξ / ξ ) = i= L i= L ν(ξ)ϕ (ξ )ω 2 (ξ)η i (ξ / ξ ) i= L ν(ξ)ϕ 2 (ξ )ω (ξ)η i (ξ / ξ ) i= M,up i (ξ) L i= M,down i (ξ) L ν(ξ)ϕ 2 (ξ )ω 2 (ξ)η i (ξ / ξ ) i= L i= M,up i (ξ) L i= M,down i (ξ). Let φ (R) be supported in [/2, 2] such that k=2 φ(2k s) = for s (, /4) and set φ k = φ(2 k ) (k = 2, 3,...) and φ (s) = χ (, ) k=2 φ(2k s). To simplify the notations we define l i (ξ) = α i ξ β i ξ 2. For the notational convenience we set K = K. We denote and K,up i K,up i K = = F [M,up i ], K,down i = F [M,down i ], = F [M,up i ], K,down i = F [M,down i ], L i= [ K,up i K,down i K,up i K,down i ],

ylinder multipliers associated with a convex polygon 5 where F [f] is denoted by the inverse Fourier transform of f. We let α i ξ β i ξ 2 = (i =,..., N) be equations of edges of P whose end points are v i and v i when i =,..., N and v N and v when i = N. We set l i (ξ) = α i ξ β i ξ 2, ( ) αi β A = i, α i β i l i, (x) = deta (β ix α i x 2 ), and l i,2 (x) = deta ( β ix α i x 2 ), l i,3 (x) = x 3. For the decays of the kernels, it is enough to estimate K,up i and K,up i because the arguments for K,down i and K,down i are similar to those for K,up i and K,up i. We denote K,up i = Ki and K,up i = Ki in Lemmas 2.2 and 2.3 for our convenience. We first consider the decays of Ki = F [M i ]. From now on since the arguments will be symmetric, it suffices to treat the case of l i (ξ), l i (ξ), ξ 3 >. Lemma 2.2. For each i there are estimates as follows : Ki (x) ( max a { l i,a (x) } l i,b (x) ± l i,c (x) ) 2 (2.2) where a b c 3. ( max a { l i,a (x) }) 2 ( min b { l i,b (x) }), Proof. Suppose that l i, (x) max{ l i,2 (x), l i,3 (x) }. If l i, (x) l i,2 (x) l i,3 (x), then we write M i (ξ) = [ ( l i (ξ) 2 ) δ ( l i (ξ) 2 ) δ ] ϕ (ξ )ω (ξ)η i (ξ / ξ ) (2.3) ( l i (ξ) 2 ) δ ϕ (ξ )ω (ξ)η i (ξ / ξ ) = M, i (ξ) M,2 i (ξ). For the estimates of M, i we need more decomposition as follows : M, i (ξ) = φ j (l i (ξ) l i (ξ))φ k (l i (ξ))φ l (l i (ξ) ξ 3 ) M, i (ξ) j,k,l = j,k,l M, i,j,k,l (ξ), and set K, i,j,k,l = F [M, i,j,k,l ]. By using a change of variables ξ = l i (ξ), ξ 2 = l i (ξ), and ξ 3 = ξ 3, we have K, i,j,k,l (x) = R e i<(a ) t x,ξ >ix 3ξ 3 M, R 2 i,j,k,l (ξ, ξ 3) dξ dξ 3,

6 Sunggeum Hong, Joonil Kim and han Woo Yang where ξ = (ξ, ξ 2). We make a change of variables ζ = ξ 2 ξ, ζ 2 = ξ 2, and ζ 3 = ξ 2 ξ 3 to write (x) (2.4) = e i{ li,(x)ζ (l i,(x)l i,2(x)l i,3(x))ζ 2 l i,3(x)ζ 3} φ j (ζ )φ k (ζ 2 )φ l (ζ 3 ) R 3 K, i,j,k,l Ω(ζ, ζ 3 )ζ (2ζ 2 ζ ) ( sζ 2 2 ( s)(ζ 2 ζ ) 2 ) δ ds dζ dζ 2 dζ 3, where Ω(ζ, ζ 3 ) = ϕ (A ζ ) ω (A ζ, ζ 3 ) η i (A ζ / A ζ ), ζ = (ζ 2 ζ, ζ 2 ) and ζ = (ζ, ζ 3 ). We note that ζ (2ζ 2 ζ ) 2 j 2 min{j,k}. The integration by parts yields K, i,j,k,l (x) 2 j 2 min{j,k} ( 2 j l i, (x) ) N ( 2 l (2.5) l i,3 (x) ) N2 ( 2 k l i, (x) l i,2 (x) l i,3 (x) ) N3 ( ) N ( ) N2 ( ) N3φj (ζ )φ k (ζ 2 )φ l (ζ 3 )Ω(ζ, ζ 3 ) dζ dζ 2 dζ 3. R ζ 3 ζ 2 ζ 3 We make use of the size of the support to obtain K, i,j,k,l (x) 2 2j 2 k 2 min{j,k} 2 l ( 2 j (2.6) l i, (x) ) N ( 2 k l i, (x) l i,2 (x) ± l i,3 (x) ) N2 ( 2 l l i,3 (x) ). N3 We now consider the summation over j, k, and l for (2.6). If we assume that min{j, k} = j, then j,k,l K, j,k,l (x) ( 2 j l i,(x) ( 2 l l i,3(x) ( 2 k l i,(x) 2 3j 2 l 2 k 2 j l i,(x) > 2 l l i,3(x) > 2 k l i,(x) > 2 j(3 N) l i, (x) N) 2 l( N2) l i,3 (x) N2) (2.7) 2 k( N3) l i, (x) N3) ( l i, (x) ) 3 ( l i, (x) l i,2 (x) ± l i,3 (x) )( l i,3 (x) ), where l i, (x) = l i, (x) l i,2 (x) ± l i,3 (x).

ylinder multipliers associated with a convex polygon 7 We use l i, (x) l i, (x) l i,2 (x) ± l i,3 (x) to have K, (x) (2.8) ( l i, (x) ) 2 ( l i, (x) l i,2 (x) ± l i,3 (x) ) 2 ( l i,3 (x) ). If min{j, k} = k, then we obtain (2.8) by straightforward computations, replacing 2 k by 2 2k, and 2 k( N3) by 2 k(2 N3) in (2.7). On the other hand if l i, (x) l i,3 (x) l i,2 (x), we replace M, i by [ ( ξ 2 3 ) δ ( l i (ξ) 2 ) δ ] ϕ (ξ )ω (ξ)η i (ξ / ξ ), (2.9) and i,j,k,l (ξ) = φ j(ξ 3 l i (ξ))φ k (ξ 3 l i (ξ))φ l (ξ 3 ) M, i (ξ). M, If we make a change of variables and integration by parts as in (2.4) and (2.5) we have K, i,j,k,l (x) 2 2j 2 k 2 l 2 min{j,l} ( 2 j l i, (x) ) N ( 2 l l i, (x) l i,2 (x) ± l i,3 (x) ) N2 ( 2 k l i,2 (x) ). N3 If we sum over j, k, and l likewise (2.7), then K, (x) (2.) ( l i, (x) ) 2 ( l i, (x) l i,2 (x) ± l i,3 (x) ) 2 ( l i,2 (x) ). Thus, from (2.8) and (2.) we get K, (x) ( l i, (x) ) 2 ( min{ l i,2 (x), l i,3 (x) }) ( l i, (x) l i,2 (x) ± l i,3 (x) ) 2. Now it remains to treat the case M,2 i. We decompose M,2 i (ξ) = [( l i (ξ) 2) δ ϕ (ξ )ω (ξ)η i (ξ / ξ ) (2.) ϕ (ξ )ω (ξ)η i (ξ / ξ ) = M,3 i (ξ) M,4 i (ξ). To control the size of l i (ξ), l i (ξ) and ξ 3, we split M,3 i such as M,3 i (ξ) = φ j (l i (ξ)) φ k (l i (ξ)) φ l (ξ 3 )M,3 i (ξ) j,k,l = j,k,l M,3 i,j,k,l (ξ),

8 Sunggeum Hong, Joonil Kim and han Woo Yang and set K,3 i,j,k,l = F [M,3 i,j,k,l ]. Then by a change of variables ζ = l i (ξ), ζ 2 = l i (ξ), and ζ 3 = ξ 3, we have K,3 i,j,k,l (x) = e i<(a ) t x,ζ >ix 3ζ 3 M,3 i,j,k,l (ζ, ζ 3 ) dζ dζ 3 R R 2 = e i{li,(x)ζ l i,2(x)ζ 2l i,3(x)ζ 3} φ j (ζ ) φ k (ζ 2 ) φ l (ζ 3 ) ζ 2 R R 2 η i (A ζ / A ζ ) ϕ (A ζ )ω (A ζ, ζ 3 ) ( s ζ 2 ) δ ds dζ dζ 3, where ζ = (ζ, ζ 2 ). We integrate by parts to obtain K,3 i,j,k,l (x) 2 3j 2 k 2 l ( 2 j l i, (x) ) N ( 2 k l i,2 (x) ) N2 ( 2 l l i,3 (x) ) N3. After we sum over j, k and l, we have K,3 i (x) ( l i, (x) ) 3 ( l i,2 (x) ) ( l i,3 (x) ) χ { l i,(x) max{ l i,2(x), l i,3(x) }}. Finally we deal with M,4 i. By the choice of η i we obtain L L M,4 i (ξ) = η i (ξ / ξ )ϕ (ξ )ω (ξ) = ϕ (ξ )ω (ξ), i= i= which gives a fast decay after the inverse Fourier transform. The arguments for the remaining cases are the same with those of the previous case. If l i,2 (x) max{ l i,3 (x), l i, (x) }, we can switch the roles of l i (ξ) and l i (ξ) in (2.3), (2.9) and (2.). If l i,3 (x) max{ l i,2 (x), l i, (x) }, we can replace the roles of l i (ξ) and ξ 3 in (2.3), (2.9) and (2.).We thus leave the details to the interested readers. K,up i Now we turn to estimate Ki = F [M i ]. We recall that we denote K i =. We write M i (ξ) = ( m(ξ) 2 ) δ ϕ 2 (ξ )ω (ξ)η i (ξ / ξ ). For this we need further decompositions as follows : M i (ξ) = j,k,l φ j ( l i (ξ) ) φ k ( l i (ξ) ) φ l ( ξ 3 )M i (ξ) = j,k,l M i,j,k,l(ξ). We set K i,j,k,l = F [M i,j,k,l], and K i = j,k,l K i,j,k,l.

ylinder multipliers associated with a convex polygon 9 Likewise Lemma 2.2, without loss of generality it is enough to show the case of l i (ξ), l i (ξ), ξ 3 >. Lemma 2.3. For each i there are estimates as follows : Ki (x) (2.2) ( max a { l i,a (x) }) δ ( l i,b (x) )( l i,c (x) ) ( max a { l i,a (x) }) 2 ( min b { l i,b (x) }) { ( l i,b (x) l i,c (x) ) δ }, ( l i,a (x) l i,b (x) l i,c (x) ) δ b c where a b c 3. Proof. We split M i,j,k,l as three cases dependent on the size of j, k and l. ase (i) j k l, or k j l. ase (ii) j l k, or l j k. ase (iii) k l j, or l k j. We begin with ase (i). From this case we can consider four subcases, that is, j > k > l, k > j > l, j k, and j k, k l. Suppose first that j > k > l. Then by definition of m we have m(ξ) = l i (ξ). Now we write M i,j,k,l(ξ) = ( l i (ξ) 2 ) δ φ j ( l i (ξ))φ k ( l i (ξ))φ l ( ξ 3 ) B i (ξ), where B i (ξ) = ϕ 2 (ξ )ω (ξ)η i (ξ / ξ ). By change of variables ζ = l i (ξ), ζ 2 = l i (ξ) and ζ 3 = ξ 3, we have Ki,j,k,l(x) = ) t x,ζ >ix 3ζ 3 φ j ( ζ )( ζ) 2 δ R R 2 e i<(a φ k ( ζ 2 )φ l ( ζ 3 ) B i (A ζ, ζ 3 ) dζ dζ 3, where (A ) t x = (l i, (x) l i,2 (x)) and ζ = (ζ, ζ 2 ). If we integrate by parts with respect to ζ, ζ 2 and ζ 3, we obtain Ki,j,k,l(x) (2.3) ( 2 j l i, (x) ) N ( 2 k l i,2 (x) ) N2 ( 2 l l i,3 (x) ) N3 R R 2 ( ) N ( ) N2 ( ) N3 µi,j,k,l (ζ, ζ 3 ) ζ ζ 2 ζ 3 dζ dζ 3, where µ i,j,k,l (ζ, ζ 3 ) = ( ζ 2 ) δ φ j ( ζ )φ k ( ζ 2 )φ l ( ζ 3 )B i (A ζ, ζ 3 ). The integrand of (2.3) is bounded by 2 j(δ) 2 k 2 l ( 2 j l i, (x) ) N ( 2 k l i,2 (x) ) N2 ( 2 l. (2.4) N3 l i,3 (x) )

Sunggeum Hong, Joonil Kim and han Woo Yang If l i,a (x) max{ l i,b (x), l i,c (x) }, we use (2.4) directly when a =, apply the fact j > k when a = 2, and use the fact j > l in (2.4) when a = 3 to get Ki,j,k,l(x) ( max{ l i,a (x) }) δ ( l i,b (x) ) j,k,l where a b c 3. Suppose now that k > j > l, we have m(ξ) = l i (ξ), and ( l i,c (x) ), M i,j,k,l(ξ) = ( l i (ξ) 2 ) δ φ j ( l i (ξ))φ k ( l i (ξ))φ l ( ξ 3 ) B i (ξ). If we replace j by k in the previous arguments, we obtain the same results. Suppose now that j k and j, k > l. In this case, the size of l i (ξ) and l i (ξ) is comparable, we subtract the harmless term to gain the decays. If l i, (x) max{ l i,2 (x), l i,3 (x) }, we decompose M i,j,k,l(ξ) = [ ( l i (ξ) 2 ) δ ( l i (ξ) 2 ) δ ] φ j ( l i (ξ)) φ k ( l i (ξ)) φ l ( ξ 3 ) B i (ξ) ( l i (ξ) 2 ) δ φ j ( l i (ξ))φ k ( l i (ξ)) φ l ( ξ 3 ) B i (ξ) = M, i,j,k,l (ξ) M,2 i,j,k,l (ξ). Now dyadic parallelogram may intersect with the singular line l i (ξ) = l i (ξ), and thus we need to overcome the singularity. Hence, we decompose once more such as i,j,k,l (ξ) = φ σ ( l i(ξ) l i (ξ) ) M, i,j,k,l (ξ) M, = σ=k σ=k M, i,j,k,l,σ (ξ), and set K, i,j,k,l,σ = F [M, i,j,k,l,σ ], K,2 i,j,k,l = F [M,2 i,j,k,l ]. By using a change of variables ξ = l i (ξ), and ξ 2 = l i (ξ), and ξ 3 = ξ 3, we have K, i,j,k,l,σ (x) = R = R e i<(a ) t x,ξ >ix 3ξ 3 M, R 2 R 2 e i<(a (ξ 2 2 ξ 2 ) i,j,k,l,σ (ξ, ξ 3) dξ dξ 3 ) t x,ξ >ix 3ξ 3 φj ( ξ )φ k ( ξ 2)φ l ( ξ 3)φ σ ( ξ ) ( s ξ 2 2 ( s)ξ 2 ) δ ds B i (A ξ, ξ 3) dξ dξ 3. ξ 2

ylinder multipliers associated with a convex polygon We make a change of variables ζ = ξ /ξ 2, ζ 2 = ξ 2, and ζ 3 = ξ 3 to write K, i,j,k,l,σ (x) = R 3 e i{li,(x)ζ ζ 2 l i,2 (x)ζ 2 l i,3 (x)ζ 3 } φ j ( ζ ζ 2 )φ k ( ζ 2 ) φ l ( ζ 3 )φ σ ( ζ )ζ 2 2( ζ 2 ) (2.5) ( sζ 2 2 ( s) ζ 2 2 ζ 2 ) δ ds B i (A ζ, ζ 3 ) ζ 2 dζ dζ 2 dζ 3, where ζ = (ζ ζ 2, ζ 2 ). We note that ζ 2 2( ζ 2 ) 2 σ and ( sζ 2 2 ( s) ζ 2 2 ζ 2 ) δ ds 2 k(δ ). We integrate by parts and make use of the size of the support to obtain K, i,j,k,l,σ (x) 2 σ ( 2 σ l i, (x) ) N ( 2 l (2.6) l i,3 (x) ) N3 R3 2 k(δ ) ( ) N ( ) N2 ( ) N3φj ( 2 k ( ζ ζ 2 ) l i, (x)ζ l i,2 (x) ) N2 ζ ζ 2 ζ 3 φ l ( ζ 3 )φ k ( ζ 2 )φ σ ( ζ ) B i (A ζ, ζ 3 ) ζ2 3 dζ dζ 2 dζ 3 2 2σ 2 δk 2 l ( 2 σ l i, (x) ) N ( 2 k l i, (x) l i,2 (x) ) N2 ( 2 l l i,3 (x) ). N3 Also in view of (2.3) we can get K,2 i,j,k,l (x) 2 j(δ) ( 2 j l i, (x) ) N 2 k ( 2 k l i,2 (x) ) N 2 l ( 2 l l i,3 (x) ) N. If we sum over σ, k, and l, then K, i,j,k,l,σ (x) K,2 i,j,k,l (x) l σ k j,k,l j k ( l i, (x) ) 2 ( l i, (x) l i,2 (x) ) δ ( l i,3 (x) ) ( l i, (x) ) δ ( l i,2 (x) )( l i,3 (x) ) χ { l i,(x) max{ l i,2(x), l i,3(x) }. If l i,2 (x) max{ l i, (x), l i,3 (x) }, we write M i,j,k,l(ξ) = [ ( l i (ξ) 2 ) δ ( l i (ξ) 2 ) δ ] φ j ( l i (ξ)) φ k ( l i (ξ)) φ l ( ξ 3 ) B i (ξ) (2.7) ( l i (ξ) 2 ) δ φ j ( l i (ξ))φ k ( l i (ξ)) φ l ( ξ 3 ) B i (ξ) = M,3 i,j,k,l (ξ) M,4 i,j,k,l (ξ).

2 Sunggeum Hong, Joonil Kim and han Woo Yang Since dyadic parallelogram may intersect with the singular line l i (ξ) = l i (ξ), we need further dyadic decomposition. We write M,3 i,j,k,l (ξ) = = σ=k σ=k φ σ ( l i(ξ) l i (ξ) ) M,3 i,j,k,l (ξ) M,3 i,j,k,l,σ (ξ), and set K,3 i,j,k,l,σ = F [M,3 i,j,k,l,σ ], K,4 i,j,k,l = F [M,4 i,j,k,l ]. By using a change of variables and integrate by parts likewise (2.5) and (2.6), we have K,3 i,j,k,l,σ (x) (2.8) 2 2σ 2 δk 2 l ( 2 σ l i,2 (x) ) N ( 2 k l i, (x) l i,2 (x) ) N ( 2 l l i,3 (x) ) N, and thus l j k σ k K,3 i,j,k,l,σ (x) K,4 i,j,k,l (x) j,k,l ( l i,2 (x) ) 2 ( l i, (x) l i,2 (x) ) δ ( l i,3 (x) ) ( l i, (x) )( l i,2 (x) ) δ ( l i,3 (x) ) χ { l i,2(x) max{ l i,3(x), l i,(x) }. If l i,3 (x) > max{ l i, (x), l i,2 (x) }, in this case the subtraction of the multiplier likewise (2.8) does not help to get the desired estimate. Thus, we use the fact σ j, k > l in (2.5) and (2.8) to obtain K,3 i,j,k,l,σ (x) 2 σ ( 2 σ l i,a (x) ) N 2 δk ( 2 k l i, (x) l i,2 (x) ) N 2 2l ( 2 l l i,3 (x) ) N where a = or 2. Thus, this leads to l j k σ k K,3 i,j,k,l,σ (x) ( l i,3 (x) ) 2 ( l i, (x) l i,2 (x) ) δ ( min{ l i, (x), l i,2 (x) }).

ylinder multipliers associated with a convex polygon 3 Suppose that j k and k l. If l i, (x) max{ l i,2 (x), l i,3 (x) }, we decompose M i,j,k,l(ξ) = [ ( m(ξ) 2 ) δ ( l i (ξ) 2 ) δ ] (2.9) φ j ( l i (ξ)) φ k ( l i (ξ)) φ l ( ξ 3 ) B i (ξ) ( l i (ξ) 2 ) δ φ j ( l i (ξ))φ k ( l i (ξ)) φ l ( ξ 3 ) B i (ξ) = M,5 i,j,k,l (ξ) M,6 i,j,k,l (ξ). If l i, (x) l i,2 (x) l i,3 (x), we take m(ξ) = l i (ξ) in (2.9). Then since dyadic parallelogram may intersect with two singular lines l i (ξ) = l i (ξ) and l i (ξ) = ξ 3, we need more dyadic decomposition such as We set M,5 i,j,k,l (ξ) = = σ=j τ=l σ=j τ=l φ σ ( l i(ξ) l i (ξ) ) φ τ ( ξ 3 l i (ξ) ) M,5 i,j,k,l (ξ) M,5 i,j,k,l,σ,τ (ξ). K,5 i,j,k,l,σ,τ = F [M,5 i,j,k,l,σ,τ ], If we follow the same way as (2.5) and 2.6, we have K,5 i,j,k,l,σ,τ (x) 2 2σ ( 2 σ l i, (x) ) N K,6 i,j,k,l = F [M,6 i,j,k,l ]. 2 δk 2 τ ( 2 k l i, (x) l i,2 (x) l i,3 (x) ) N ( 2 τ l i,3 (x) ) N. If l i, (x) l i,3 (x) l i,2 (x), we take m(ξ) = ξ 3 in (2.9). Then we have M,5 i,j,k,l (ξ) = Similarly as before, = σ=j τ=k σ=j τ=k φ σ ( l i(ξ) ) φ τ ( l i(ξ) ) M,5 i,j,k,l ξ 3 ξ (ξ) 3 M,5 i,j,k,l,σ,τ (ξ). K,5 i,j,k,l,σ,τ (x) 2 2σ ( 2 σ l i, (x) ) N 2 δl ( 2 l l i, (x) l i,2 (x) l i,3 (x) ) N 2 τ ( 2 τ l i,2 (x) ) N,

4 Sunggeum Hong, Joonil Kim and han Woo Yang Thus the summation leads to [ j k, j l k, σ j, τ l l, σ j, τ k ] K,5 i,j,k,l,σ,τ (x) K,6 i,j,k,l (x) j,k,l ( l i, (x) ) 2 ( l i, (x) l i,2 (x) l i,3 (x) ) δ ( min{ l i,2 (x), l i,3 (x) }) ( l i, (x) ) δ ( l i,2 (x) )( l i,3 (x) ) χ { l i,(x) max{ l i,2(x), l i,3(x) }. To treat the remaining cases l i,2 (x) max{ l i,3 (x), l i, (x) } and l i,3 (x) max{ l i,2 (x), l i, (x) }, if we apply the similar arguments, we have the desired estimates such that Ki,j,k,l(x) ( max{ l i,2 (x), l i,3 (x) }) 2 ( l i, (x) l i,2 (x) l i,3 (x) ) δ ( min b3 { l i,b (x) }) ( l i, (x) )( max{ l i,2 (x), l i,3 (x) }) δ ( min{ l i,2 (x), l i,3 (x) }). Lastly since the arguments for ase (ii) and ase (iii) are symmetric to those for ase (i), and we omit here. This completes the proof. Remark 2.4. The estimates for the derivatives of the kernel can be obtained by the fact that γ =N (K i )(γ) = γ =N Ψ(γ) K i and γ =N (K i )(γ) = γ =N Ψ(γ) 2 Ki for some Schwartz functions Ψ, Ψ 2 S(R 3 ). Thus, it is easily seen that the decays of the derivatives of the kernels are the same with those of kernels in Lemmas 2.2 and 2.3 with different constants, respectively. 3. Hardy spaces In this section we shall prove Theorems. and.2. We begin with the definition of Hardy spaces. Definition 3.. Let < p and s be an integer that satisfies s 3( p ). Let Q be a cube in R 3. We say that a is a (p, s)-atom associated with Q if a is supported on Q R 3 and satisfies (i) a L (R 3 ) Q /p ; (ii) a(x) x β dx =, R 3 where β = (β, β 2, β 3 ) is a pair of non-negative integers satisfying β β β 2 β 3 s, and x β = x β xβ2 2 xβ3 3.

ylinder multipliers associated with a convex polygon 5 If {a j } is a collection of (p, s)-atoms and {c j } is a sequence of complex numbers with j= c j p <, then the series f = j= c ja j converges in the sense of distributions, and its sum belongs to H p with the quasinorm (see [7]) [8]. ( f H p = inf c j p) /p. j= cjaj=f To prove Theorem.2 we shall need a lemma by Stein, Taibleson and Weiss Lemma 3.2. Suppose α > } α p for j =, 2, 3,. If j= c j p, then { } x : c j f j (x) > α 2 p p α p. Proof. See Lemma.8 in [8]. j= We shall use the following elementary lemma to obtain weak type estimates in the proof of Proposition 3.4. Set Ω = { x : x x 2 x 3 > 2 }, Ω 2 = { x : x x 2 x 3 > 2, x x x 2 ± x 3 > }. Lemma 3.3. Let a, b, c and p < be positive real numbers. (a) If a b c = 3 p, a > p, b < p, and c < p, then { x Ω : x a x 2 b x 3 c > α/ } α p. (b) If a b c > 3 p, a = p, b > p, and c > p, then { x Ω 2 : x a x x 2 ± x 3 b x 3 c > α/ } α p. Proof. We consider (a). If a b c = 3 p, a > p and b, c < p, then (a c)/b > 2. The weak type set in the right-hand side of (a) is bounded by {(x,x 2,x 3) :2< x 2 α b x 3 c b x a b,2< x 3 x, x >α abc } By applying Fubini s theorem, we obtain the weak type estimates x 2 dx α b x {x :2< x α abc } α 3 abc = α p, since x x 3 > 2. {x : x > α abc } dx 2 dx 3 dx. ac b dx

6 Sunggeum Hong, Joonil Kim and han Woo Yang For (b) we assume that a b c > 3 p, a = p, b > p, and c > p. By change of variables x 2 = x ± x ± x 2 we obtain { (x, x 2, x 3 ) : x a x 2 b x 3 c > α/ } since b a > and c a >. α p α p, x 2 b a x3 c a dx 2 dx 3 To prove Theorem.2, we shall need uniform weak type estimates for T δ with a (p, N)-atom (N 3( p )). Proposition 3.4. Suppose f is a (p, N)-atom (N 3( p )) on R3. Suppose that 2/3 3( p ). Then there exists a constant = (p) such that {(x, x 3 ) R 2 R : T δ f(x, x 3 ) > α} α p (3.) for all α >. Proof. Let f be supported in a cube Q of diameter centered at the origin. We first consider the case (x, x 3 ) Q which is the cube centered at the origin with diameter 4. In view of (2.) and Lemmas 2.2 and 2.3 we can easily see that K ɛ is integrable and its L norm is independent of ɛ. Thus we have T δ ɛ f(x, x 3 ) K ɛ f K ɛ Q /p. Therefore T δ f(x, x 3 ) = sup Tɛ δ f(x, x 3 ) Q /p ɛ> for all (x, x 3 ) Q, and which implies that for α > { (x, x 3 ) Q : T δ f(x, x 3 ) > α/ } α p. (3.2) Hence it suffices to show that for α > { (x, x 3 ) R 3 \ Q : T δ f(x, x 3 ) > α/ } α p. We note that Tɛ δ = L i= T ɛ,i δ. We denote T ɛ,i δ f = T ɛ,i f T ɛ,i f, where T ɛ,i f = Kɛ,i f and T ɛ,i f = K ɛ,i f. For the notational convenience we set K,i = K i and K,i = K i. We fix ɛ and use the fact that f is supported in Q to write T δ ɛ,if(x, x 3 ) T ɛ,if(x, x 3 ) T ɛ,if(x, x 3 ) ɛ 3 Q f(y, y 3 ) K i (ɛ(x y ), ɛ(x 3 y 3 )) dy dy 3 ɛ 3 Q f(y, y 3 ) K i (ɛ(x y ), ɛ(x 3 y 3 )) dy dy 3.

ylinder multipliers associated with a convex polygon 7 We set x = l i, (x), x 2 = l i,2 (x), x 3 = l i,3 (x), x = x x 2 x 3, y = l i, (y), y 2 = l i,2 (y), and y 3 = l i,3 (y) in (2.2) and (2.2). By the kernel estimates in Lemmas 2.2, 2.3 and (2.), we write and Tɛ,if(x) [ ɛ Q 3 f(y) ( ɛ max j { x j y j }) δ ( ɛ x k y k )( ɛ x l y l ) ( ɛ max j { x j y j }) 2 ( ɛ min k { x k y k }) { ] ( ɛ x j y j x k y k ) δ } ( ɛ x y ) δ dy dy 2 dy 3 j k = T A ɛ,if(x) T B ɛ,if(x) T ɛ,if(x), Tɛ,if(x) [ ɛ Q 3 f(y) ( ɛ max j { x j y j }) 2 ( ɛ min k { x k y k }) ] ( ɛ max j { x j y j } (x k y k ) ± (x l y l ) ) 2 dy dy 2 dy 3, where j k l 3. From (x, x 3 ) (Q ) c, we consider the following three cases : ase (i) x > 2, or x 2 > 2, or x 3 > 2, ase (ii) x 2, or x 2 2, or x 3 2, ase (iii) x > 2, x 2 > 2, and x 3 > 2. We first treat ase (i). Suppose that x > 2, x 2 2, and x 3 2. Then we have T A ɛ,if(x) ɛ (3 δ 2) x δ χ { x 4}. (3.3) From (x y ) (x 2 y 2 ) (x 3 y 3 ) x, x y x 2 y 2 x, and x y x 3 y 3 x, we have and Tɛ,if(x) B Tɛ,if(x) (3.4) ɛ (3 2 δ ) x 2δ ɛ(3 2 δ ) x 2 χ { x 4}, T ɛ,if(x) ɛ (3 2 2) x 4 χ { x 4}. (3.5) For the remaining case x 2 > 2, or x 3 > 2, we switch x and x 2, x and x 3, respectively in (3.3) through (3.5) to obtain Tɛ,if(x) δ max j { x j } δ χ { x k 4, x l 4} χ { xj 4, j3}, (3.6)

8 Sunggeum Hong, Joonil Kim and han Woo Yang where j k l 3. We now treat ase (ii). Suppose that x 2, x 2 > 2, and x 3 > 2. Then we have Tɛ,if(x) A ɛ (3 δ ) max{ x 2, x 3 } δ min{ x 2, x 3 } χ { x2 4, x 3 4}. (3.7) From (x y )(x 2 y 2 )(x 3 y 3 ) (x 2 y 2 )(x 3 y 3 ), x y x 2 y 2 x 2, and x y x 3 y 3 x 3, we have T B ɛ,if(x) T ɛ,if(x) {ɛ (3 2 δ ) max{ x 2, x 3 } 2 x 3 x 2 δ (3.8) Likewise ɛ (3 2 δ ) max{ x 2, x 3 } 2 χ { x 2 x 3 2} ɛ (3 2 δ ) max{ x 2, x 3 } 2δ χ {min{ x 2, x 3 }4} ɛ (3 2 δ ) max{ x 2, x 3 } 2 min{ x 2, x 3 } δ χ { x2 4, x 3 4}}. T ɛ,if(x) {ɛ (3 2 2 ) max{ x 2, x 3 } 2 x 3 x 2 2 (3.9) ɛ (3 2 2 ) max{ x 2, x 3 } 2 χ { x 2 x 3 2} ɛ (3 2 2 ) max{ x 2, x 3 } 4 χ {min{ x 2, x 3 }4} χ { x2 4, x 3 4}}. Thus, (3.7) through (3.9), and symmetric arguments for the remaining two cases x 2, or x 3 2 lead to Tɛ,if(x) δ χ { xj 4, j3 } (3.) { max j { x j } min{2,δ} min k { x k } min{δ,} max j { x j } 2 x j x k min{2,δ} } max j { x j } 2 χ { x j x k 2} χ { xj >2, x k >2, min l { x l }2}, where j k l 3.

ylinder multipliers associated with a convex polygon 9 Finally, we consider ase (iii). Similarly (3.6) and (3.), we have Tɛ,if(x) δ { max j { x j } δ x k x l max j { x j } 2 min k { x k } j l x j x l δ max j { x j } 2 min k { x k } χ { x j x l 2, j l} max j { x j } 2 min k { x k } ( x min{2,δ} χ { x 2} (3.) )} χ{ xj >2}. where j k l 3 and x = max j { x j } x k ± x l. Then we apply hebyshev s inequality for (3.6), (3.), and use Lemma 3.3 and hebyshev s inequality for (3.) to obtain { (x, x 3 ) R 3 \ Q : sup Tɛ,if(x δ, x 3 ) > α/ } α p. (3.2) ɛ Now we consider the complementary case we fix ɛ <. Let P i,n be the N-th order Taylor polynomial of the function y 3 ɛ 3 K i (ɛ(x y ), ɛ(x 3 y 3 )) expanded about the origin. Then by using the moment conditions on f in Definition 3..(ii) and integrating with respect to y 3 first, we write Tɛ,if(x δ, x 3 ) = f(y, y 3 ) [ɛ 3 K i (ɛ(x y ), ɛ(x 3 y 3 )) P i,n (y 3 )] dy 3 dy. Q By using the integral version of the mean value theorem, we obtain ɛ 3 K i (ɛ(x y ), ɛ(x 3 y 3 )) P i,n (y 3 ) ɛ 3 ɛ (N) [,] N N K,i y N 3 (ɛ(x y ), ɛ(x 3 ûy 3 )) du du N where û = N i= u i. Since we gain ɛ (N) and the kernel K,i has the same decay after taking derivatives, we use the same argument as above to obtain (3.6), (3.), (3.) and Lemma 3.3 for ɛ <, which imply { (x, x 3 ) R 3 \ Q : T δ ɛ,if(x, x 3 ) > α/ } α p. (3.3) Together with (3.2), and (3.3), we see that { (x, x 3 ) (Q ) c : T δ ɛ,if(x, x 3 ) > α/ } α p. For the summation over i, we apply Lemma 3.2. Since L i= ( ) p =, we (L) /p thus get { (x, x 3 ) (Q ) c : T δ f(x, x 3 ) > α/ } { (x, x 3 ) (Q ) c : α p. L i= sup ɛ> Tɛ,i δ } f(x, x 3 ) α > (L ) /p (L ) /p

2 Sunggeum Hong, Joonil Kim and han Woo Yang Thus, together with (3.2) it follows that (3.) holds for the cube Q of diameter. Now we suppose that f is a (p, N)-atom (N 3( p )), supported in a cube Q of diameter σ centered at (x Q, x 3Q). By translation invariance we can assume (x Q, x 3Q) = (, ). Let h(x, x 3 ) = σ 3/p f(σx, σx 3 ). Then h is an atom supported in the cube Q centered at (, ) and we write Tɛ δ f(x, x 3 ) = σ 3/p h(σ (x y ), σ (x 3 y 3 )) K ɛ (y, y 3 ) dy dy 3 R R 2 = σ 3/p Tɛσ δ h(σ x, σ x 3 ), which implies sup ɛ> Tɛ δ f(x, x 3 ) = σ 3/p sup Tɛ δ h(σ x, σ x 3 ). ɛ> We therefore have { (x, x 3 ) R 3 : T δ f(x, x 3 ) > α/ } = { (x, x 3 ) R 3 : T δ h(σ x, σ x 3 ) > σ 3/p α/ } (σ 3/p α) p σ 3 = α p. This completes the proof. We are now ready to prove Theorem.2 which is an immediate consequence of Proposition 3.4 and Lemma 3.2. Proof of Theorem.2. Let f = j= c jf j H p (R 3 ) where f j s are (p, N)-atoms and j= c j p <. Since the kernel (see Lemmas 2.2, 2.3) is integrable and the coefficient {c j } is l convergent, it is easy to see that Tɛ δ f is well defined and can be written Tɛ δ f = c j Tɛ δ f j. By Proposition 3.4, we therefore have j= {(x, x 3 ) : T δ f j (x, x 3 ) α} α p. Lastly, if we apply Lemma 3.2 we obtain the desired estimates (.3). 4. The proof of Theorem. In this section we prove Theorem. and the results in Theorems. and.2 cannot be improved in the sense that there exists a function f in H p space such that if δ δ p = 3( p ), then T δ f L p = and if p 2/3, then T δ f L p =. To construct those functions we shall need the following lemma.

ylinder multipliers associated with a convex polygon 2 Lemma 4.. If f S(R 3 ) and R 3 x β f(x) dx = for all β N p with N p > 3( p ), then f is in Hp (R 3 ), Φ t f(x), and want to show that Mf Lp (R 3 ). If x, it is easy to see that x (Mf(x))p dx f p. We now consider the case x >. When t x /, R t 3 y) Φ((x ) f(y) dy 3 t N ( x ) N. When t > x /, we use the moment condition and f S(R 3 ), we obtain [ Φ t (x y) f(y) dy = Φt (x y) (Φ t ) (α) (x) yα ] f(y) dy. R α! 3 α N p Thus, by the mean value theorem for integral Φ t (x y) f(y) dy R 3 R 3 α =N p (Φ t ) (α) (x θy) yα f(y) dθ dy α! ( x ) (Np3) R 3 y Np f(y) dy. We note that (Φ t ) (α) (y) t 3 α x 3 α on the last inequality above. Since ( N p 3)p > 3, Mf(x) = sup t> Φ t f(x) is in L p (R 3 ). Proof of Theorem.. We first show the sufficiency. Let f be a (p, N)-atom (N 3( p )) on R3 and supported in a cube Q. Due to the translation invariance and maximality of T δ as above we may assume that Q is centered at the origin with diameter. In view of atomic decomposition, it suffices to show that there exists a independent of f such that sup Tɛ δ f L p (R 3 ) Q /p. ɛ> By the integrability of the kernel, it is easy to see that sup Tɛ δ f p L ɛ> p (Q ) Q Q dx dx 3. For the complementary case we take the p-th power on both sides in (3.6), (3.), and (3.) and integrate them if 2/3 3( p ). Then we can obtain the L p boundedness (.2) as desired. We now show the necessity. Let φ be a smooth function in R 3 supported in a neighborhood of a vertex of P. Let B be an open ball centered at a vertex v i of P and v j / U if j i and let φ be a nonnegative smooth cut-off function supported

22 Sunggeum Hong, Joonil Kim and han Woo Yang in B and whose values in a neighborhood of v i are identically. We choose suitable cut-off functions φ β supported in R 3 \ P and constants c β such that R 3 [ φ (x) β N p c β φβ (x)] x β dx =, where β = (β, β 2, β 3 ) is a pair of non-negative integers satisfying β β β 2 β 3 s, and x β = x β xβ2 2 xβ3 3. We set f(x) = φ (x) c β φβ (x). β N p In view of Lemma 4., f is in H p (R 3 ). From the support conditions it is easy to see that T δ f(ξ) = ( m2 (ξ)) δ φ (ξ). We therefore have T δ f(x) = e i<x,ξ> ( m 2 (ξ)) δ φ (ξ)dξ. R 3 We may assume that φ is of the form φ (ξ) = φ(l i (ξ))φ(l i (ξ))φ(ξ 3 ), where φ is a smooth cut-off function which is supported in a small neighborhood of and identically near. We then have T δ f(x) = e i<x,ξ> ( m 2 (ξ)) δ φ(l i (ξ))φ(l i (ξ))φ(ξ 3 ) dξ dξ 2 dξ 3 R 3 = e i[li,(x)l i(ξ)l i,2(x)l i(ξ)l i,3(x)ξ 3] ( m 2 (ξ)) δ R 3 φ(l i (ξ))φ(l i (ξ))φ(ξ 3 ) dξ dξ 2 dξ 3. By setting ξ = l i (ξ), ξ 2 = l i (ξ), ξ 3 = ξ 3, x = l i, (x), and x 2 = l i,2 (x), we then have = T δ f(x ) ξ ξ ξ 2 ξ 2 ξ 3 ξ 3 e i(x ξ x 2 ξ 2 x 3 ξ 3 ) ( ξ 2 ) δ φ(ξ )φ(ξ 2)φ(ξ 3) dξ 2 dξ 3 dξ = U V W. e i(x ξ x 2 ξ 2 x 3 ξ 3 ) ( ξ 2 2 ) δ φ(ξ )φ(ξ 2)φ(ξ 3) dξ 3 dξ dξ 2 e i(x ξ x 2 ξ 2 x 3 ξ 3 ) ( ξ 3 2 ) δ φ(ξ )φ(ξ 2)φ(ξ 3) dξ dξ 2 dξ 3

ylinder multipliers associated with a convex polygon 23 Integrating by parts about ξ 2 and ξ 3 variables in U we obtain and ξ = eix 2 ξ ix 2 ξ = eix 3 ξ ix 3 e ix 2 ξ 2 φ(ξ 2 ) dξ 2 φ(ξ ) eix 2 ξ dφ(ξ ) (ix 2 )2 dξ (ix 2 )2 e ix 3 ξ 3 φ(ξ 3 ) dξ 3 φ(ξ ) eix 3 ξ dφ(ξ ) (ix 3 )2 dξ (ix 3 )2 ξ ξ e ix 2 ξ 2 d2 φ(ξ 2) (dξ 2 )2 dξ 2, e ix 3 ξ 3 d2 φ(ξ 3) (dξ 3 )2 dξ 3, because φ is a smooth function in R 3 supported in a neighborhood of a vertex of P. Thus we see that the first terms eix 2 ξ ix φ(ξ ) and eix 3 ξ 2 ix φ(ξ ) are leading terms in the above integrals, respectively. With those terms we rewrite the integral about ξ to have U = (ix 2 )(ix 3 ) ( ) e i(x x 2 x 3 )ξ ξ 2 δ φ(ξ ) 3 dξ. If we apply the asymptotic expansion in [] (p.46-5) in U, we obtain where e i(x x 2 x 3 )ξ ( ξ 2 ) δ φ(ξ ) 3 dξ = A N (x, x 3) O((x x 2 x 3) N ), = A N (x, x 3) N µ= Γ(µ δ ) [φ(ξ µ! ) 3 ] (µ) ()(x x 2 x 3) µ δ e i(x x 2 x 3 ), and φ(ξ ) 3 are -functions. Then we write U = x 2 x 3 (x x 2 x 3 )δ j x 2 x 3 (x µ= x 2 x 3 )µδ = U a U b. Since U b has a nice decay U b x 2 x 3 (x x 2, x 3 )δ

24 Sunggeum Hong, Joonil Kim and han Woo Yang we have that U a is a leading term, where U a = B. (4.) l i,2 (x)l i,3 (x)(l i, (x) l i,2 (x) l i,3 (x)) δ If we apply the same argument as above for V and W, the leading terms are B 2, (4.2) l i,3 (x)l i, (x)(l i, (x) l i,2 (x) l i,3 (x)) δ B 3 l i, (x)l i,2 (x)(l i, (x) l i,2 (x) l i,3 (x)) δ, respectively. Thus from (4.) and (4.2) we obtain that T δ f(x) l i, (x) l i,2 (x) l i,3 (x) δ j k3 3 j= l i,j(x) l i, (x) l i,2 (x) l i,3 (x) δ. l i,j (x)l i,k (x) If δ 3( p ) and 2, j3} j= l dx i,j(x) p l i, (x) l i,2 (x) l i,3 (x) δp l i, (x) p l i,2 (x) p l i,3 (x) (δ)p dx { l i,3(x) > l i,2(x) > l i,(x) >2} 2 ( p) 2 If 2, j3} 3 j= l dx i,j(x) p l i, (x) l i,2 (x) l i,3 (x) δp { l i,(x) l i,2(x) l i,3(x), l i,3(x) 2,/2< l i,(x)l i,2(x)l i,3(x) <2} 4 ( p) 2 l i,3 (x) 3p dl i,3 (x) =. dx l i,3 (x) 3p Finally, we consider the negative result of Theorem.2. Let p = 2/3 and δ δ p = 3(/p ) on the region D = {x : l i, (x) l i,2 (x) l i,3 (x), l i,3 (x) 2} in (4.3).

ylinder multipliers associated with a convex polygon 25 If p = 2/3 and δ > δ p, we put δ = δ p 2ɛ for some ɛ >. By a change of variable l i, (x) l i,2 (x) l i,3 (x) = l i (x), we have {x : T δ f(x) > α} α 2 3 α 2 3. { x D : l i,(x) 2 l i,(x)l i,2(x)l i,3(x) δ l i,3(x) >α} { x D : l i,(x) 2 3 l i(x) ( 2 3 ɛ ) l i,3(x) ( 2 3 ɛ ) >α} 2 2 dx dx l i (x) 3 2 ( 2 3 ɛ) l i,3 (x) 3 2 ( 2 3 ɛ) dl i (x) dl i,3 (x) On the other hand if p = 2/3 and δ = 3/2, then l i, (x) 2 l i, (x) l i,2 (x) l i,3 (x) δ l i,3 (x) l i, (x)l i,2 (x)l i,3 (x) 3 2 on D. Thus, it follows that {x : T δ f(x) > α} dx = { x D : l i,(x)l i,2(x)l i,3(x) 3 2 >α} {x : l i,(x) l i,2(x), l i,2(x) l i,3(x), 2 l i,3(x) α 2 9 } {x : l i,(x) l i,2(x), l i,2(x) α 3 l i,3(x) 2, l i,3(x) > α 2 9 } {2< l i,3(x) α 2 9 } α 2 3 ( ln(/α)). l i,3 (x) dl i,3 (x) α 2 3 dx { l i,3(x) >α 2 9 } dx l i,3 (x) dl i,3(x) References [] A. Erdélyi, Asymptotic expansions, Dover, New York, 956. [2] S. Hong, J. Kim and. W. Yang, Risez means associated with convex polygons, J. Math. Anal. Appl. 33 (27), 377 395. [3] S. Hong, P. Taylor and. W. Yang, Weak type estimates for maximal operators with a cylindric distance function, Math. Z. 253 (26), 24. [4] H. Luers, On Riesz means with respect to a cylinder distance function, Anal. Math. 4 (988), 75 84. [5] P. Osvalćd, Marcinkiewicz means of double Fourier integrals in H p, p, Moscow Univ. Math. Bull. 38 (983), 65 73. [6] P. Oswald, On Marcinkiewicz-Riesz summability of Fourier integrals in Hardy spaces, Math. Nachr. 33 (987), 73 87. [7] E. M. Stein, Harmonic analysis : Real variable method, orthogonality and oscillatory integrals, Princeton Univ. Press, 993. [8] E. M. Stein, M. H. Taibleson and G. Weiss, Weak type estimates for maximal operators on certain H p classes, Rend. irc. Mat. Palermo(2) suppl. (98), 8 97.

26 Sunggeum Hong, Joonil Kim and han Woo Yang [9] P. Taylor, Bochner-Riesz means with respect to a rough distance function, Trans. Amer. Math. Soc. 359 (27), 43 432. Sunggeum Hong Department of Mathematics hosun University Gwangju 5-759, Korea e-mail: skhong@mail.chosun.ac.kr Joonil Kim Department of Mathematics hung-ang University Seoul 56-756, Korea e-mail: jikim@cau.ac.kr han Woo Yang Department of Mathematics Korea University Seoul 36-7, Korea e-mail: cw yang@korea.ac.kr