DISTRIBUTIONS SUPPORTED in a HYPERSURFACE and LOCAL H 1. Introduction the Aim of This Paper Is to Solve a Problem Which Arose In

Home , Bump function

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 10, October 1998, Pages 2933–2943 S 0002-9939(98)04437-2

DISTRIBUTIONS SUPPORTED IN A HYPERSURFACE AND LOCAL hp

GALIA DAFNI

(Communicated by Christopher D. Sogge)

Abstract. We give a necessary condition for a distribution with compact support in a hypersurface to be in the local Hardy space hp(Rn). We apply this condition to prove a result distinguishing two types of Hardy spaces of distributions on a smooth domain Ω Rn. ⊂

1. Introduction The aim of this paper is to solve a problem which arose in the study of various local Hardy spaces of distributions on a smooth bounded domain Ω Rn.In[CKS], p p ⊂ Chang, Krantz and Stein deﬁned the spaces hr (Ω) and hz(Ω), 0

1. p The space hr(Ω) consists of those distributions on Ω which are the restriction to Ω of distributions in hp(Rn), the local Hardy spaces defined by Goldberg (see p p [G].) The space hz(Ω) is a subspace of hr(Ω), consisting of those distributions on Ω which are the restriction to Ω of distributions in hp(Rn) which also vanish p outside Ω. Note that two such extensions of the same element of hz(Ω) differ by a distribution in hp(Rn) which is supported on ∂Ω. Thus in order to understand p p n hz(Ω), it is important to understand the nature of distributions in h (R )which are supported in a smooth hypersurface, or (by a choice of local coordinates) in a hyperplane. Some simple sufficient conditions can be stated for a distribution with compact support in a hyperplane to be in hp(Rn) (see for example [S], 5.18, which implies that f hp(Rn) if its order is strictly less than 1 n). Here we prove a necessary ∈ p − condition, namely that the order of the distribution “in the normal direction” must be strictly less than n( 1 1) (Theorem 1). We do this by means of a lower bound p − on the integral of the maximal function of such a distribution on planes parallel to

Received by the editors February 28, 1997. 1991 Mathematics Subject Classiﬁcation. Primary 42B30, 46F05. Key words and phrases. hp spaces, distributions.

c 1998 American Mathematical Society

2933

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the hyperplane (Lemma 1). It is this lemma that allows us to construct a counter- example showing that for p = n , k =0,1,2,..., hp(Ω) = hp(Ω) (Theorem 2). n+k r 6 z 2. hp Distributions supported in a hyperplane n n 1 n 1 For n 2, write R = R − R, and let Π = R − = xn =0 be our chosen hyperplane.≥ If f is a distribution× in Rn of order K with{ compact} support in Π, then K (1) f(φ)= fk(φk), =0 kX n 1 where each f is a distribution of compact support and order K k in R − ,and k − k φ (x0)=∂ φ(x0,x ) =0 k xn n |xn (see [H], Theorem 2.3.5). Let N be the largest integer such that fN is non-zero. We will call N the order of f in the normal direction. Then we have the following: Theorem 1. A distribution f in Rn with compact support in Π is in the local Hardy space hp(Rn), 0

α = α1 + ...+α N +1, | | n ≤ p where Np is the greatest integer in n(1/p 1). For a tempered distribution f, we deﬁne− x m(f)(x)=supf(ϕr), x ϕr x where the supremum is taken over all normalized bump functions ϕr supported in balls of radii r 1 containing x.Thenf hp(Rn) if and only if ≤ ∈ m(f) Lp(Rn). ∈ That this is equivalent to the deﬁnition given by Goldberg ([G]) can be seen from the remarks in [S], 5.17. In view of this characterization, the theorem will follow from the following Lemma 1. Let f be a distribution in Rn with compact support in Π, and let N be the order of f in the normal direction. Then for 0

p n 1 (n+N)p m(f) (x0,xn)dx0 Cxn − − n 1 ≥ ZR − for all suﬃciently small xn > 0.

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Note that when N n(1/p 1), n 1 (n + N)p 1 so the lemma shows p ≥ − − − ≤− that m(f) is not locally integrable near xn = 0. This proves the theorem. Proof of Lemma 1. Write N f(φ)= fk(φk), =0 kX as in equation 1, with fN nonzero. We may assume the support of f lies in the n 1 interior of the unit cube Q =[0,1] − .SincefNis nonzero, there is a smooth function φ such that fN (φ)=>0. We may also take the support of φ to lie inside Q. By changing if necessary, we normalize φ so that sup ∂γ φ 1. x0 γ N +1 | |≤ | |≤ p j Consider the dyadic cubes Qα, with j =0,1,2,... and α =(α1,... ,αn 1), j j j − 0 αi 2 1, where the side length of Qα is 2− and its closest corner to the ≤ ≤ − j j 0 origin lies at the point (α1/2 ,... ,αn 1/2 ). Thus Q = Q . − (0,... ,0) For each j , we associate with the cubes Qj a partition of unity ηj ,whereeach α { α} j j j ηα is a smooth function supported in the double of the cube Qα, i.e. the cube Qα with the same center but double the side length, and such that f j ηα =1 α X on Q. Furthermore, we may assume that, for every j and α, γ j jγ ∂ (η ) 2| | | x0 α |≤ for all derivatives of order γ Np+1. Set | |≤ j j(n 1) j φα = An,p2 − φηα,

where the constant An,p is chosen so that γ j j(n 1+ γ ) ∂ (φ ) 2 − | | | x0 α |≤ for all γ N + 1. Consider a smooth one-variable function ψ(t) with | |≤ p 1 ψ(t)= tN N! for t [ 1/2, 1/2], ψ(t)=0for t >1, and dmψ(t) 1form N +1.Let ∈ − || | t |≤ ≤ p j j ψj(t)=2 ψ(2 t), and j j Φα(x0,xn)=Cn,p φα(x0) ψj (xn),

where again we chose the constant Cn,p so that β j j (n+ β ) ∂ (Φ ) (2− √n)− | | | α |≤ j j j j for all β Np+ 1. Since Φα is supported in the cube Qα [ 2− , 2− ] of side | |≤j+1 j j× − length 2− , hence in a ball of radius 2− √n,thismakesΦα a normalized bump function. f Now note that j j j(N+1) j f(Φα)=fN((Φα)N )=Cn,p2 fN (φα).

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j j j Thus for x Qα [ 2− , 2− ], ∈ × − m(f) C 2j(N+1) f (φj ) . f ≥ n,p | N α | Recall that, for every j,

j j(n 1) j(n 1) fN (φα)=An,p2 − fN (φ)=An,p2 − . α X (j+1) j Therefore for 2− x 2− , ≤ n ≤ p p j(N+1) j j m(f) (x0,xn)dx0 Cn,p 2 fN (φα) Qα n 1 ≥ | | | | ZR − α X p j(N+1)p j(n 1) j C 2 2− − f (φ ) ≥ n,p N α α X j(N+1)p j(n 1) j(n 1)p p = Cn,p0 2 2− − 2 −

p j[n 1 (n+N)p] = Cn,p0 2− − − n 1 (n+N)p C x − − . ≥ n,p,N, n This proves the lemma.

We will now give an example to show that Theorem 1 does not hold with N replaced by K, the total order of the distribution. Example 1. For 0

Np+1 n n/p+Np+1 a(x0)(x1 (c )1) dx0 = C δ − j − j n j Z for some Cn > 0. Deﬁne linear functionals τ on (Rn)by j S

τj(ϕ)= aj(x0)ϕ(x0,0)dx0 n 1 ZR − n for every ϕ (R ). By expanding ϕ(x0, 0) in a Taylor expansion around cj,one can see that∈S

n n/p+Np+1 τj(ϕ) C ϕ Np+1 δj − Cn,p ϕ Np+1 | |≤ k kC ≤ k kC

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n since δj 0andn n/p+Np +1>0. This shows the τj are continuous on (R ), → − n S and we can deﬁne f 0(R )by ∈S

f= λjτj, X where the sequence λ is chosen to satisfy λ p < and λ /δ as j j j ∞ j j →∞ →∞ (for example take δ =2 j,λ =j 1 1/p). j − j − − P To see that the order of f is at least Np, take a sequence of smooth functions n ϕ (R ) with ϕ (x0, 0) supported in 2Q , j ∈S j j ∂αϕ 1 | j |≤ for α N 1, and | |≤ p− 1 Np+1 2 ϕj(x0, 0) = (x1 (cj)1) δj− (Np +1)! −

on Qj.Thus

n n/p+Np+1 2 f(ϕ)=λ a (x0)ϕ (x0,0)dx0 = C λ δ − − j j j j n,p j j →∞ ZQj

as j ,sincen n/p + Np +1 2 1. Finally,→∞ to see that− f hp(Rn), consider− ≤− the local grand maximal function m(f). ∈ n It suﬃces to show m(τj) Lp(Rn) C uniformly in j.Takex R.Notethatif x k k ≤ ∈ ϕt is a normalized bump function supported in a ball of radius t containing x,then x x τj(ϕt)= 0 only if t xn/2, and tϕt (y0, 0) is a normalized bump function in n 1 variables.6 Thus ≥ − 2 m( )( ) m n 1 ( )( ) τj x R − aj x0 , ≤ xn n 1 where m n 1 denotes the local grand maximal function on . Recalling that R − R − 2 n 1 m n 1 is bounded on ( ), we get that R − L R −

δj p p p m( ) ( ) m n 1 ( ) ( ) τj x dx C xn− R − aj x0 dx0 2Q [ δ ,δ ] ≤ δ 2Q Z j × − j j Z− j Z j 1 p p 1 p/2 Cδj − m(aj) 2 n 1 2Qj − ≤ k kL (R − )| | 1 p p 1 p/2 Cδj − aj 2 n 1 2Qj − ≤ k kL (R − )| | 1 p p n Cδ − δ − Q ≤ j j | j| = C.

Note that here we used the fact that p<1. x For x 2Qj [ δj,δj], if ϕt is a normalized bump function supported in a ball of radius6∈t containing× − x,thenasabove

x x n n/p+Np+1 τj(ϕt) Cϕt Np+1 δj − | |≤ k kC n (N +1) n n/p+Np+1 Ct− − p δ − ≤ j n (N +1) n n/p+Np+1 C x (c , 0) − − p δ − , ≤ | − j | j

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p m(τj ) (x)dx x 2Q [ δ ,δ ] Z 6∈ j × − j j

np n+(Np+1)p np (Np+1)p Cδj − x (cj, 0) − − dx ≤ x (c ,0) δ | − | Z| − j |≥ j = C. This completes the example.

p p 3. The spaces hr(Ω) and hz(Ω) Let Ω be a bounded domain in Rn, with smooth boundary. Let us recall the p p deﬁnitions of the spaces hr (Ω) and hz(Ω), given in [CKS]: p p n h (Ω) = f 0(Ω) : F h (R ),F Ω =f r { ∈D ∃ ∈ | } and p p n hz(Ω) = f 0(Ω) : F h (R ),F Ω =f,F Rn Ω =0 . { ∈D ∃ ∈ | | \ } p p p Clearly hz(Ω) is a subspace of hr(Ω). It was shown in [CDS] that in fact hr(Ω) = p n n hz(Ω) for all values of p with n+j+1 0 . { } p We will construct a distribution f hr(Ω), supported in U, such that there is no distribution G hp(Rn) satisfying∈ ∈ G Ω = f | and

G Rn Ω =0. | \ p Note that the space of distributions in hr(Ω) which are supported in U does not depend on the choice of coordinates. Let us construct f as follows. For j =1,2,... , consider the cylinder n 1 (j+1) (j+1) j S = B − (0, 2− ) [2− , 2− ], j × n 1 where B − (0,r) denotes the (n 1)-dimensional ball of radius r in the variables − x0 = x1,... ,xn 1.NotethatSj is contained in the (much larger) cube − (j+1) (j+1) n 1 (j+1) (j+1) Q =[ 2− , 2− ] − [2− , 3 2− ] j − × · jn of volume Qj =2− .Set | | 1/p a =Q − χ . j | j| Sj (1+1/p) p p If λ = j− ,then λ < so by the atomic decomposition for h (see j j ∞ r [CKS], Theorem 2.7,) the distribution f 0(Ω) deﬁned by P ∈D ∞ f = λjaj j=1 X

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p is an element of hr(Ω). Furthermore, f is supported in Γ B(0, 1), where Γ is the cone ∩ n x R : x0 xn <1 . { ∈ | |≤ n} In order to extend f to a distribution F 0(R ), we define distributions τ by ∈S j 1 τ (ϕ)= a (x) ϕ(x) ∂αϕ(0)xα dx j j  − α!  Z α Np 1 | |≤X− for every ϕ (Rn). Since   ∈S 1 τ (ϕ) = a (x) ϕ(x) ∂αϕ(0)xα dx | j | j  − α!  Z α Np 1 | |≤X −   jn/p Np C 2 ϕ Np x dx ≤ k kC | | ZQj j(n/p Np n) C2 − − ϕ Np ≤ k kC = C ϕ Np , k kC n and λ < , we can define a distribution F 0(R )by j ∞ ∈S P ∞ F= λjτj. j=1 X This distribution is supported in B(0, 1) Ω and satisfies ∩ ∞ ∞ F Ω = λ τ Ω = λ a = f. | j j| j j j=1 j=1 X X n Now suppose there was a distribution G 0(R ) with G supported on Ωand ∈S p n p n GΩ=f. We want to show that G cannot be an element of h (R ). Since h (R ) is| closed under multiplication by smooth functions of compact support, we may assume G is supported in V = B(0, 2). n Consider the distribution G F 0(R ). Since both G and F are supported − ∈S on Ω, and G Ω = f = F Ω, we must have that G F is supported on ∂Ω, and in | n 1 | n − particular on B − (0, 2) = B(0, 2) ∂R+.ThusG Fis of the form of equation (1). Let N denote the order of G ∩F in the normal− direction. − Claim 1. If N

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for k Np +1.Nowtakeϕto be a smooth function of n 1 variables, supported n≤1 n 1 − in B − (0, 2), with ϕ 0, ϕ =1onB − (0, 1) and ≥ ∂α(ϕ) 1 | |≤ for α N + 1. Deﬁning φ by | |≤ p φ(x)=ϕ(x0)ψ(xn), we see that φ is supported in B(0, 3), x Np φ(x)= n Np! n 1 for x B − (0, 1) [0, 1], and ∈ × ∂α(φ) 1 | |≤ for α N +1. | |≤ p Suppose x Γ, x =0.Letx∗ =(x0,0), and deﬁne ∈ 6 n N n y x∗ φ (y)=3− − p x− φ − . x | | x | | n Then φ is supported in the ball B(x∗, 3 x ), with φ 0onR and x | | x ≥ + Np yn φx(y)= N !(3 x )n+Np p | | n 1 for y S = B − (x0, x ) [0, x ]. Furthermore ∈ x | | × | | α n α ∂ (φ ) (3 x )− −| | | x |≤ | | for α N +1.Thusφ is a normalized bump function. | |≤ p x Note that, for k

G(φxR)=F(φx) λ a(y)φ(y)dy ≥ j j x S S Sj Xj⊂x Z 1 = ( ) Np n+N λj aj y yn dy Np!(3 x ) p S S Sj | | Xj ⊂ x Z n N j(n/p N n) = C (3 x )− − p λ 2 − p− Np | | j S S Xj ⊂ x n N 1/p 1 C0 x − − p j− − ≥ Np | | j log x +2 ≥− X2 n 1/p n/p 1 − x− log . ≈|| x | |

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j Here we have used the fact that for x Γ, if 2− (√2 1)x ,then ∈ ≤ − n n 1 (j+1) (j+1) j n 1 S =B − (0, 2− ) [2− , 2− ] S = B − (x0, x ) [0, x ]. j × ⊂ x | | × | | p n Claim 2. If N Np, then the distribution G does not belong to h (R ), since its local grand maximal≥ function satisﬁes

p 1 m(G) (x0,xn)dx0 Cxn− n 1 ≥ ZR − for all suﬃciently small xn > 0. By Lemma 1, with N N , the distribution G F satisﬁes ≥ p −

p n 1 (n+Np)p 1 m(G F ) (x0,xn)dx0 Cxn − − = Cxn− n 1 − ≥ ZR − for all suﬃciently small xn > 0. Since

p p p m(G) (x0,xn)dx0 m(G F ) (x0,xn)dx0 m(F ) (x0,xn)dx0, n 1 ≥ n 1 − − n 1 ZR − ZR − ZR − it suﬃces to prove that

p 1 m(F ) (x0,xn)dx0 = o(xn− ) n 1 ZR − as x 0. In fact, we will show that n → p 1 m(F ) (x0,xn)dx0 C(xn log(1/xn))− n 1 ≤ ZR − for suﬃciently small xn > 0. n To prove the upper bound on the maximal function of F ,takex R+, and let x ∈ ϕt be a bump function supported in a ball of radius t 1 containing x.SinceFis supported in B(0, 1), we can assume x < 3. ≤ If t x/4, write | | ≥| | ∞ x x F (ϕt )= λjτj(ϕt ). j=1 X (j+1) x Note that if j< log (t) 4, then 2− > 8t>2t+x ,soϕ =0onS and − 2 − n t j

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On the other hand,

x 1 x λj τj(ϕt ) C 1+1 ϕt Np | |≤ j /p k kC j log (t) 4 j log (t) 4 ≥− X2 − ≥− X2 − n N 1 Ct− − p ≤ j1+1/p j log (t) 4 ≥− X2 − n/p 1/p Ct− (log(1/t))− . ≤ Thus x n/p 1/p F (ϕ ) Ct− (log(1/t))− | t |≤ n/p 1/p C x − (log(1/ x ))− , ≤ | | | | since ( x /t)n/p(log x / log t)1/p remains bounded when x << t. | | | | x | | x If t< x/4, then ϕt vanishes in a neighborhood of 0, and Sj supp(ϕt ) = only if j |3|/2 log x ,so ∩ 6 ∅ ≤ − 2 | | F(ϕx) C λ a ϕx | t |≤ j j t j C log x Z ≤−X| | 1 jn/p x C 2 ϕ 1 ≤ j1+1/p k t kL j C log x ≤−X | | n/p 1/p C x − (log(1/ x ))− . ≤ | | | | In both cases we are assuming of course that x is suﬃciently small, i.e. x a<<1, so that log(1/ x ) is bounded below. Otherwise| | we just have | |≤ | | x n/p F (ϕ ) Cx− λ C . | t |≤ | | j ≤ a Now integrating in the ﬁrst n 1 variables,X we have − p m(F ) (x0,xn)dx0 = n 1 ZR − p p m(F ) (x0,xn)dx0 + m(F ) (x0,xn)dx0 x x x x a Z| 0|≤ n Z n≤| 0|≤ p + m(F ) (x0,xn)dx0 a x 3 Z ≤| 0|≤ n 1 C xn− (log(1/xn))− dx0 ≤ x x Z| 0|≤ n n 1 +C x0 − (log(1/ x0 ))− dx0 + C x x a | | | | Z n≤| 0|≤ a 1 1 dr Cx − (log(1/x ))− + C + C. ≤ n n r2( log r) Zxn − But a a 1 dr dr 1 1 − = + x − log + C 2 2 2 n xn r ( log r) xn r log r xn Z − Z a 1 dr 1 1 ( log a)− + x − (log(1/x ))− + C ≤ − r2( log r) n n Zxn −

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so a dr 1 1 Cx − (log(1/x ))− + C r2( log r) ≤ n n Zxn − for a suﬃciently small. This concludes the proof of Theorem 2.

References [CDS] D.C. Chang, G. Dafni, and E. M. Stein, Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in Rn, Trans. Amer. Math. Soc., to appear. CMP 97:15 [CKS] D.C.Chang,S.G.Krantz,andE.M.Stein,Hp Theory on a smooth domain in RN and elliptic boundary value problems, J. Funct. Anal. 114, No. 2 (1993), 286-347. MR 94j:46032 [G] D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), 27-42. MR 80h:46052 [H] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin 1983. MR 85g:35002a [JSW] A. Jonsson, P. Sjögren, and H. Wallin, Hardy and Lipschitz spaces on subsets of Rn, Studia Math. 80, No. 2 (1984), 141-166. MR 87b:46022 [M] A. Miyachi, Hp spaces over open subsets of Rn, Studia Math. 95, No. 3 (1990), 205-228. MR 91m:42022 [S] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, New Jersey, 1993. MR 95c:42002

Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2730 E-mail address: [email protected]

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