New Proof of Hörmander multiplier Theorem on compact manifolds without boundary

New Proof of Hörmander multiplier Theorem on compact manifolds without boundary Xiangjin Xu Department of athematics Johns Hopkins University Baltimore, D, 21218, USA xxu@math.jhu.edu Abstract On compact manifolds (, g) without boundary of dimension n 2, the gradient estimates for unit band spectral projection operators χ λ is proved for any second order elliptic differential operators L by maximum principle. A new proof of Hörmander ultiplier Theorem on the eigenfunction expansion of the operator L is given in this setting by using the gradient estimates and the Calderón-Zygmund argument. Keywords: gradient estimate, spectral projection operator, Hörmander ultiplier Theorem, Calderón-Zygmund decomposition athematics Subject Classification 2000: 58J40, 35P20, 35J25, 1

1 Introduction Let (, g) be a compact boundaryless manifolds (, g) of dimension n 2, and suppose that L is a second order elliptic differential operator which is positive and self-adjoint with respect to the C density dx associated to the Reimannian metric g. The purpose of this paper is to give a simple proof of sharp gradient estimates for the eigenfunctions of L and then to use these estimates to give a new proof of Hörmander multiplier theorem for the eigenfunction expansions on the compact boundaryless manifold (, g). Thus, we shall consider the eigenvalue problem (L + λ 2 )u(x) = 0, x, (1) Recall that the spectrum of L is discrete and tends to infinity. Let 0 λ 2 1 λ 2 2 λ 2 3 denote the eigenvalues, so that {λ j } is the spectrum of the first order operator P = L. Let {e j (x)} L 2 () be an associated real orthogonormal basis, define the unit band spectral projection operators, χ λ f = λ j [λ,λ+1) e j (f), where e j (f)(x) = e j (x) f(y)e j (y)dy. In [5] and [6], Sogge proved the following L estimates on χ λ, χ λ f Cλ (n 1)/2 f 2, λ 1, for a uniform constant C, which is equivalent to λ j [λ,λ+1) e j (x) 2 Cλ n 1, x, Here based on the above L estimates on χ λ f, by maximum principle, we show the following gradient estimates for χ λ f 2

Theorem 1.1 Fix a compact boundaryless Riemannian manifold (, g), there is a uniform constant C so that which is equivalent to χ λ f Cλ (n+1)/2 f 2, λ 1, λ j [λ,λ+1) e j (x) 2 Cλ n+1, x. We shall show the gradient estimates following the ideas of the interior gradient estimates for Poisson s equation in [1], where for Poisson s equation u = f, there are gradient estimates for the interior point x 0 as u(x 0 ) C d sup B u + Cd sup f, B by using maximum principle in a cube B centered at x 0 with length d. Our other main result is Hörmander ultiplier Theorem for the eigenfunction expansions on compact manifolds without boundary. Given a bounded function m(λ) L (R), one can define the multiplier operator, m(p ), by m(p )f = m(λ j )e j (f) (2) j=1 such an operator is always bounded on L 2 (). However, if one considers on any other space L p (), it is known that some smoothness assumptions on the function m(λ) are needed to ensure the boundedness of m(p ) : L p () L p (). (3) When m(λ) is C and, moreover, in the symbol class S 0, i.e., ( d dλ )α m(λ) C α (1 + λ ) α, α = 0, 1, 2, 3

It has been known for some time that (3) holds for all 1 < p < on compact manifolds (see [10]). One assumes the following regularity assumption: Suppose that m L (R), let L 2 s(r) denote the usual Sobolev space and fix β C 0 ((1/2, 2)) satisfying β(2 j t) = 1, t > 0, for s > n/2, there is sup λ>0 λ 1+s β( /λ)m( ) 2 L = sup β( )m(λ ) 2 2 s L <. (4) 2 s λ>0 One can see this is the sharp assumption to ensure the boundedness of m(p ) on L p () for all 1 < p <, since for a special class of multiplier operators, Riesz means S δ λf = λ j λ(1 λ 2 j/λ 2 ) δ e j (f), such an assumption is sharp (see in [6]). any authors have studied this problem under different setting. Hörmander [2] first proved the boundedness of m(p ) for R n under the assumption (4), using the Calderón-Zygmund decomposition and the estimates on the integral kernel of the multiplier operator. Stein and Weiss [9] studied the boundedness of m(p ) for multiple Fourier series, which can be regarded as the case on the flat torus T n. By studying the paramatrix of the wave kernel of m(p ), Seeger and Sogge [4] and Sogge [6] proved the following Hömander multiplier Theorem for the eigenfunction expansions on compact manifolds without boundary: Theorem 1.2 Let m L (R) satisfy (4), then there are constants C p such that m(p )f L p () C p f L p (), 1 < p <. (5) Here we shall give a new proof for above theorem by using the L estimates and the gradient estimates on χ λ f, without using the paramatrix of the wave kernel of m(p ) as done in [4] and [6]. Since the complex conjugate of m satisfies the same hypotheses (4), we need only to prove Theorem 1.2 for exponents 1 < p 2. This 4

will allow us to exploit orthogonality, and since m(p ) is bounded on L 2 (), also reduce Theorem 1.2 to show that m(p ) is weak-type (1, 1) by the arcinkiewicz Interpolation Theorem. The weak-type (1,1) estimates of m(p ) will involve a splitting of m(p ) into two pieces m(p ) = m(p ) + r(p ). For the remainder r(p ), one can obtain the strong (1,1) estimates by the L estimates on χ λ f as done in [6]. For the main term m(p ), in [4] and [6], the authors used the paramatrix of the wave kernel of m(p ) to get the required estimates on the integral kernel of m(p ), and applying these estimates, proved the weak-type (1,1) estimates on m(p ). As people know, the the paramatrix construction of the wave equation does not work well for general compact manifolds with boundary unless one assume the boundary is geodesically concave. Here we use another approach to prove the weak-type (1,1) estimates on m(p ), which works for any compact manifolds with boundary (One may see it from [11] and [12] for the proof of Hörmander ultiplier Theorem of Dirichlet Laplacian and Neumann Laplacian on compact manifolds with boundary). We make a second decomposition {K λ,l (x, y)} l= for each term K λ (x, y), which comes from the dyadic decomposition of the integral kernel K(x, y) of m(p ) K(x, y) = K 2 k(x, y) + K 0 (x, y), k=1 ( ) such that K λ (x, y) = l= K λ,l (x, y) and m(p )f(x) = T 2,l(P )f(x), where T k λ,l (P )f(x) = k=0 l= K λ,l (x, y)f(y)dy. We shall give the definitions of K λ,l (x, y) in the proof of Theorem 1.2. And for each operator T λ,l (P ), we shall prove the L 1 L 2 estimates by a rescaling argument of the proof for the remainder r(p ) using the L estimates and gradient estimates on 5

χ λ f which we obtained in Theorem 1.1. With the support properties and the finite propagation speed properties, one has the relation (10) between λ and l, which is one key observation when we apply the Calderón-Zygmund decomposition to show the weak-type (1,1) estimates on m(p ). In what follows we shall use the convention that C will denote a constant that is not necessarily the same at each occurrence. Acknowledgement: The author would like to thank his advisor, Professor C.D. Sogge, brings the problems to him and a number helpful conversations on these problems and his research. The results in this paper come from part of the author s Ph.D. thesis [12] in Johns Hopkins University. 2 Gradient estimates In this section, we shall prove Theorem 1.1. by using maximum principle. Proof of Theorem 1.1. Now we fix x 0. We shall use the maximum principle in the cube centered at x 0 with length d = (λ + 1) 1 to prove the same gradient estimates for χ λ f as for Poisson s equation. Define the geodesic coordinates x = (x 1,, x n ) centered at point x 0 as following, fixed an orthogonormal basis {v i } n i=1 T x0, identity x = (x 1,, x n ) R n with the point exp( n i=1 x i v i ). In this coordinate, the elliptic operator L can be written as L = n i,j=1 2 a ij (x) + x i x j n i=1 b i (x) x i + c(x), where a ij (x), b i (x), c(x) C (), and c(x) 0 which comes from L is an elliptic 6

operator. Now define the cube Q = {x = (x 1,, x n ) R n x i < d, i = 1,, n}. Denote u(x; f) = χ λ f(x), we have u C 2 (Q), and Lu(x; f) = λ j [λ,λ+1) λ 2 je j (f) := h(x; f). From the L estimate in [5], and Cauchy-Schwarz inequality, we have h(x; f) 2 = ( λ j [λ,λ+1) λ j [λ,λ+1) (λ + 1) 4 ( (λ 2 je j (x))( λ 4 je 2 j(x) λ j [λ,λ+1) ( λ j [λ,λ+1) C(λ + 1) n+3 χ λ f 2 L 2 () e j (y)f(y)dy)) 2 e 2 j(x)) χ λ f 2 L 2 () e j (y)f(y)dy) 2 Here we estimate D n u(0; f) = x n u(0; f) only, and the same estimate holds for D i u(0; f) with i = 1,, n 1 also. Now in the half-cube Q = {x = (x 1,, x n ) R n x i < d, i = 1,, n 1, 0 < x n < d.}, Consider the function ϕ(x, x n ; f) = 1 2 [u(x, x n ; f) u(x, x n ; f)], where we write x = (x, x n ) = (x 1,, x n 1, x n ). One sees that ϕ(x, 0; f) = 0, sup Q ϕ sup Q u := A, and Lϕ sup Q h := N in Q. Now consider the function ψ(x, x n ) = A d 2 [ x 2 + αx n (nd (n 1)x n )] + βnx n (d x n ) 0 7

defined on the half-cube Q, where α 1 and β 1 will be determined below. Obviously ψ(x, x n ) 0 on x n = 0 and ψ(x, x n ) A in the remaining portion of Q. Lψ(x) = A n d [2tr(a ij(x)) (2nα 2α + 1)a 2 nn (x) + 2 b i (x)x i + b n (x)(nαd i=1 (2nα 2α + 1)x n )] + Nβ[ 2a nn (x) + b n (x)(d 2x n )] + c(x)ψ(x). Since in, tr(a ij (x)) and b i (x) are bounded uniformly, c(x) 0 and a nn (x) is positive, then for a large α, we can make 2tr(a ij (x)) (2nα 2α + 1)a nn (x) 1, Fix such a α, since d = (λ + 1) 1, for large λ, we have n 2 b i (x)x i + b n (x)(nαd (2nα 2α + 1)x n ) < 1. i=1 Then the first term is negative. For second term, let β large enough, we have β[ 2a nn (x) + b n (x)(d 2x n )] < 1. For the third term, we have c(x) 0 and ψ(x) 0 in Q. Hence we have Lψ(x) N in Q. Now we have L(ψ ± ϕ) 0 in Q and ψ ± ϕ 0 on Q, from which it follows by the maximum principle that ϕ(x, x n ; f) ψ(x, x n ) in Q. Letting x = 0 in the expressions for ψ and ϕ, then dividing by x n and letting x n tend to zero, we obtain D n u(0; f) = lim ϕ(0, x n; f) αna xn 0 d x n + βdn. Note that d = (λ + 1) 1, A C(λ + 1) (n 1)/2, and N (λ + 1) (n+3)/2, then we have the estimate D n u(0; f) C(λ + 1) (n+1)/2. 8

The same estimate holds for D i u(0), i = 1,, n 1. Hence we have u(0) C(λ + 1) (n+1)/2. Since the estimate is for any x 0, Theorem 1.1 is proved. Q.E.D. For Riemannian manifolds without boundary, in [8], the authors proved that for generic metrics on any manifold one has the bounds e j L () = o(λ (n 1)/2 j ) for L 2 normalized eigenfunctions. For Laplace-Beltrami operator g, there are eigenvalues { λ 2 j}, where 0 λ 2 0 λ 2 1 are counted with multiplicity. Let {e j (x)} be an associated orthogonal basis of L 2 normalized eigenfunctions. If λ 2 is in the spectrum of g, let V λ = {u g u = λ 2 u} denote the corresponding eigenspace. We define the eigenfunction growth rate in term of L (λ, g) = and the gradient growth rate in term of L (, λ, g) = sup u L, u V λ ; u L 2 =1 sup u L. u V λ ; u L 2 =1 In [8], Sogge and Zelditch proved the following results L (λ, g) = o(λ (n 1)/2 j ) for a generic metric on any manifold without boundary. Here we have the following estimates on the gradient growth rate Theorem 2.1 L (, λ, g) = o(λ (n+1)/2 j ) for a generic metric on any manifold without boundary. And the bounds are uniform if there is a uniformly bound on the norm of tr(g ij (x)) for (, g). 9

Proof. For a compact Riemannian manifold (, g) without boundary, we can apply Theorem 1.1 to any point in. From Theorem 1.4 in [8], we have L (λ, g) = o(λ (n 1)/2 j ) for a generic metric on any compact manifold without boundary. Fix any such a metric on the manifold and a L 2 normalized eigenfunction u(x), apply Theorem 1.1 to u(x) at each point x 0, we have u(x 0 ) αna d + βdn. where α and β are constants depending on the norm of tr(g ij (x)) at only, which can been seen in the above proof of Theorem 1.1, and A = sup Q u = o(λ (n 1)/2 j ), and N = sup Q λ 2 u = (λ (n+3)/2 j ), where the cube Q = {x = (x 1,, x n ) R n x i < d, i = 1,, n}, we choose d = (λ + 1) 1 / n. Hence we have u(x 0 ) = o(λ (n+1)/2 j ) holds for all L 2 normalized eigenfunction u(x) V λ, furthermore, the bounds are uniform when those metrics of (, g) have a uniformly bound on the norm of tr(g ij (x)) from the proof. Hence we have our Theorem. Q.E.D. 3 Hörmander ultiplier Theorem In this section, we shall see how the L estimates and gradients estimates for χ λ imply Hörmander multiplier Theorem. As discussed in Introduction, we reduce 10

Theorem 1.2 to show that m(p ) is weak-type (1, 1), i.e., µ{x : m(p )f(x) > α} α 1 f L 1, (6) where µ(e) denotes the dx measure of E. Since the all eigenvalues of L are non-negative, we may assume m(t) is an even function on R. Then we have m(p )f(x) = 1 2π R ˆm(t)e itp f(x)dt = 1 π R + ˆm(t) cos(tp )f(x)dt, where P = L, and the cosine transform u(t, x) = cos(tp )f(x) is the solution of the following Cauchy problem of the wave equation: ( 2 t 2 L)u(t, x) = 0, u(0, x) = f(x), u t(0, x) = 0. We shall use the finite propagation speed of solutions of the wave equation in Part 2 of the proof to get the key observation (10). Proof of Theorem 1.2. The proof of the weak-type (1,1) estimate will involve a splitting of m(p ) into two pieces: a main piece which one need carefully study, plus a remainder which has strong (1,1) estimate by using the L estimates for the unit spectral projection operators as done in [6]. Specifically, define ρ C 0 (R) as ρ(t) = 1, for t ɛ, ρ(t) = 0, for t ɛ. (7) 2 where ɛ > 0 is a given small constant related to the manifold, which will be specified later. Write m(p ) = m(p ) + r(p ), where m(p ) = (m ˇρ)(P ) = 1 e itp ρ(t) ˆm(t)dt 2π r(p ) = (m (1 ρ)ˇ)(p ) = 1 e itp (1 ρ(t)) ˆm(t)dt 2π 11

To estimate the main term and remainder, for λ = 2 j, j = 1, 2,, define m λ (τ) = β( τ )m(τ). (8) λ Part 1: Strong (1, 1) estimate on the remainder r(p )f L 1 C f L 1. We first show r(p )f L 2 C f L 1. Here we follow the first part in proof of Theorem 5.3.1 in [6]. Define r λ (P ) = (m λ (1 ρ)ˇ)(p ) = 1 2π e itp (1 ρ(t)) ˆm λ (t)dt Notice that r 0 (P ) = r(p ) j 1 r 2 j(p ) is a bounded and rapidly decreasing function of P. Hence r 0 (P ) is bounded from L 1 to any L p space. We need only to show r λ (P )f L 2 Cλ n/2 s f L 1, λ = 2 j, j = 1, 2,. Using the L estimate on χ k, see [5] or [6], we have r λ (P )f 2 L r 2 λ (P )χ k f 2 L C 2 k=1 Hence we need only to show sup k=1 τ [k,k+1] sup k=1 τ [k,k+1] r λ (τ) 2 (1 + k) n 1 Cλ n 2s Notice since m λ (τ) = 0, for τ / [λ/2, 2λ], we have r λ (τ) 2 (1 + k) n 1 f 2 L 1 m λ (τ) = O((1 + τ + λ ) N ) r λ (τ) = O((1 + τ + λ ) N ) 12

for any N when τ / [λ/4, 4λ]. Hence we need only to show that is 4λ sup k=λ/4 τ [k,k+1] r λ (τ) 2 (1 + k) n 1 Cλ n 2s 4λ sup k=λ/4 τ [k,k+1] r λ (τ) 2 Cλ 1 2s Using the fundamental theorem of calculus and the Cauchy-Schwartz inequality, we have 4λ sup k=λ/4 τ [k,k+1] r λ (τ) 2 C( r λ (τ) 2 dτ + r λ(τ) 2 dτ) R R = C( R ˆm λ (t)(1 ρ(t)) 2 dt + R t ˆm λ (t)(1 ρ(t)) 2 dt) Recall that ρ(t) = 1, for t ɛ, by a change variables shows that this is 2 dominated by λ 1 2s R t s ˆm λ (t/λ) 2 dt = λ 1 2s λβ( )m(λ ) 2 L 2 s = λ 1 2s β( )m(λ ) 2 L 2 s Cλ 1 2s Here the first equality comes from a change variables, the second equality comes from the definition of Sobolev norm of L 2 s() and the third inequality comes from our assumption (4). Hence we have the estimate for the remainder r(p )f L 2 C f L 1. And since our manifold is compact, we have r(p )f L 1 V ol() 1/2 r(p )f L 2 C f L 1. 13

Part 2: weak-type (1, 1) estimate on the main term µ{x : m(p )f(x) > α} α 1 f L 1. The weak-type (1, 1) estimate on m(p ) would follow from the integral operator m(p )f(x) = 1 2π = 1 2π = 1 2π R R ˆm(t)ρ(t)e itp f(x)dt ˆm(t)ρ(t) e itλ k e λk (x) k 1 { R e λk (y)f(y)dydt ˆm(t)ρ(t) k 1 e itλ k e λk (x)e λk (y)dt}f(y)dy with the kernel K(x, y) = ˆm(t)ρ(t) e itλ k e λk (x)e λk (y)dt R k 1 = k 1(m ˇρ)(λ k )e λk (x)e λk (y) is weak-type (1,1). Now define the dyadic decomposition K λ (x, y) = R ˆm λ (t)ρ(t) k 1 e itλ k e λk (x)e λk (y)dt We have K(x, y) = K 2 j(x, y) + K 0 (x, y) j=1 where K 0 is bounded and vanishes when dist(x, y) is larger than a fixed constant. In order to estimate K λ (x, y), we make a second dyadic decomposition as follows K λ,l (x, y) = R ˆm λ (t)β(2 l λ t )ρ(t) k 1 e itλ k e λk (x)e λk (y)dt We have K λ (x, y) = K λ,l (x, y) l= 14

Define T λ,l (P )f(x) = K λ,l (x, y)f(y)dy. From above two dyadic decompositions, we have m(p )f(x) = T 2,l(P )f(x). (9) k k=0 l= Note that, because of the support properties of ρ(t), K λ,l (x, y) vanishes if l is larger than a fixed multiple of logλ. Now we exploit the fact that the finite propagation speed of the wave equation mentioned before implies that the kernels of the operators T λ,l, K λ,l must satisfy K λ,l (x, y) = 0, if dist(x, y) C(2 l λ 1 ), since cos(tp ) will have a kernel that vanishes on this set when t belongs to the support of the integral defining K λ,l (x, y). Hence in each of the second sum of (9), there are uniform constants c, C > 0 such that cλdist(x, y) 2 l Cλ (10) must be satisfied for each λ = 2 k, we will use this key observation later. Now for T λ,l (P )s, we have the following estimates: (a). (b). T λ,l (P )f L 2 () C(2 l ) s λ n/2 f L 1 () T λ,l (P )g L 2 () C(2 l ) s 0 λ n/2 [λ max y,y 0 Ω dist(y, y 0)] g L 1 (Ω) where Ω = support(g), Ω g(y)dy = 0 and n/2 < s 0 < min{s, n/2 + 1}. We first show estimate (a). Notice that β(2 l λ t )ρ(t) = 0 when t 2 l 1 λ 1, we can use the same idea to prove estimate (a) as we prove the estimate on the 15

remainder r(p ) in Part 1, where 1 ρ(t) = 0, for t ɛ. Using orthogonality of 2 χ k for k N, and the L estimates on χ k in [5], we have T λ,l (P )f 2 L T 2 λ,l (P )χ k f 2 L 2 Hence we need only to show sup k=1 τ [k,k+1] k=1 C sup k=1 τ [k,k+1] T λ,l (τ) 2 (1 + k) n 1 f 2 L 1 T λ,l (τ) 2 (1 + k) n 1 C(2 l ) 2s λ n Notice since m λ (τ) = 0, for τ / [λ/2, 2λ], we have T λ,l (τ) = O((1 + τ + λ ) N ) for any N when τ / [λ/4, 4λ]. Then we have sup k / [λ/4,4λ] τ [k,k+1] T λ,l (τ) 2 (1 + k) n 1 C C k / [λ/4,4λ] x>1,x/ [λ/4,4λ] C(1 + λ) n 2N (1 + k + λ) 2N (1 + k) n 1 x n 1 dx (x + λ) 2N Since 2 l Cλ from our observation (10) above, we need only to show that is 4λ sup k=λ/4 τ [k,k+1] 4λ k=λ/4 τ [k,k+1] T λ,l (τ) 2 (1 + k) n 1 C(2 l ) 2s λ n sup T λ,l (τ) 2 C(2 l ) 2s λ 16

Using the same argument as in Part 1, Notice that β(2 l λ t )ρ(t) = 0 when t 2 l 1 λ 1, we have the estimate (a), T λ,l (P )f L 2 () C(2 l ) s λ n/2 f L 1 () Next we prove the estimate (b). We will use the orthogonality of {e j } j N, e λk (x)e λj (x)dx = δ kj, and the gradient estimates on χ k for all k N as in Theorem 1.1. Given function g L 1 () with Ω = support(g) and g(y)dy = 0, fix a point y 0 Ω, we have = = = T λ,l (P )g 2 L 2 K λ,l (x, y)g(y)dy 2 dx Ω [K λ,l (x, y) K λ,l (x, y 0 )]g(y)dy 2 dx Ω (here use the cancellation of g) T λ,l (λ k )e λk (x)[e λk (y) e λk (y 0 )]g(y)dy 2 dx = k 1 Ω k 1 Ω λ j [k,k+1) {T λ,l (λ j )e λj (x)[e λj (y) e λj (y 0 )]}g(y)dy 2 dx (here use the orthogonality) max {T λ,l (λ j )e λj (x)[e λj (y) e λj (y 0 )]} 2 dx k 1 y Ω λ j [k,k+1) = g 2 L {T 1 λ,l (λ j )e λj (x)[e λj (y 1 ) e λj (y 0 )]} 2 dx k 1 λ j [k,k+1) (where the maximum achieves at y 1 ) = g 2 L ( 1 y T λ,l (λ j )e λj (x)e λj (ȳ), y 1 y 0 ) 2 dx = g 2 L 1 k 1 k 1 { λ j [k,k+1) λ j [k,k+1) T λ,l (λ j )e λj (x)( e λj (ȳ), y 1 y 0 )} 2 dx 17 Ω g(y) dy 2

= g 2 L 1 k 1 λ j [k,k+1) g 2 L max T λ,l(τ) 2 { 1 τ [k,k+1) k 1 (here use the orthogonality) T λ,l (λ j )( e λj (ȳ), y 1 y 0 ) 2 λ j [k,k+1) g 2 L1[ max dist(y, y 0)] 2 max T λ,l(τ) 2 { y,y 0 Ω τ [k,k+1) k 1 e λj (ȳ) 2 dist(y 1, y 0 ) 2 } λ j [k,k+1) C g 2 L1[ max dist(y, y 0)] 2 max T λ,l(τ) 2 (1 + k) n+1 y,y 0 Ω τ [k,k+1) k 1 e λj (ȳ) 2 } Now using the same computation as to the estimate (a), for some constant s 0 satisfying n/2 < s 0 < min{s, n/2 + 1}, we have max T λ,l(τ) 2 (1 + k) n+1 C(2 l ) 2s 0 λ n+2. τ [k,k+1) k 1 Combine above two estimates, we proved the estimate (b), T λ,l (P )g L 2 () C(2 l ) s 0 λ n/2 [λ max y,y 0 Ω dist(y, y 0)] g L 1 (Ω) Now we use the estimates (a) and (b) to show m(p )f(x) = K(x, y)f(y)dy is weak-type (1,1). We let f(x) = g(x) + k=1 b k (x) := g(x) + b(x) be the Calderón- Zygmund decomposition of f L 1 () at the level α using the same idea as Lemma 0.2.7 in [6]. Let Q k supp(b k ) be the cube associated to b k on, and we have g L 1 + b k L 1 3 f L 1 k=1 g(x) 2 n α almost everywhere, and for certain non-overlapping cubes Q k, b k (x) = 0 for x / Q k and µ Q k α 1 f L 1. k=1 b k (x)dx = 0 18

Now we show the weak-type (1, 1) estimate for m(p ). Since {x : m(p )f(x) > α} {x : m(p )g(x) > α/2} {x : m(p )b(x) > α/2} Notice g 2 dx 2 n α g dx. Hence we use the L 2 boundedness of m(p ) and Tchebyshev s inequality to get µ{x : m(p )g(x) > α/2} Cα 2 g 2 L 2 C α 1 f L 1. Let Q k be the cube with the same center as Q k but twice the side-length. After possibly making a translation, we may assume that Q k = {x : max x j R}. Let O = Q k, we have µ O 2 n α 1 f L 1, and µ{x / O : m(p )b(x) > α/2} 2α 1 x/ O m(p )b(x) dx Hence we need only to show 2α 1 k=1 x/ Q k m(p )b k (x) dx m(p )b k (x) dx = x/ Q k x/ Q k C K(x, y)b k (y)dy dx Q k b k dx. From the double dyadic decomposition (9), we show two estimates of T λ,l (P )b k (x) on set {x : x / O }, (I) T λ,l (P )b k L 1 (x/ O ) C(2 l ) n/2 s b k L 1 (Q k ) (II) T λ,l (P )b k L 1 (x/ O ) C(2 l ) n/2 s 0 [λ max y,y 0 Q k dist(y, y 0 )] b k L 1 (Q k ) 19

From our observation (10), as did in [7], it suffices to show that for all geodesic balls B Rλ,l of radius R λ,l = 2 l λ 1, one has the bounds (I) T λ,l (P )b k L 1 ({x/ O } B Rλ,l ) C(2 l ) n/2 s b k L 1 (Q k ) (II) T λ,l (P )b k L 1 ({x/ O } B Rλ,l ) C(2 l ) n/2 s 0 [λ max y,y 0 Q k dist(y, y 0 )] b k L 1 (Q k ) To show (I), using the estimate (a), and Hölder inequality, we get T λ,l (P )b k L 1 ({x/ O } B Rλ,l ) V ol(b Rλ,l ) 1/2 T λ,l (P )b k L 2 C(2 l λ 1 ) n/2 (2 l ) s λ n/2 b k L 1 = C(2 l ) n/2 s b k L 1 To show (II), using the cancellation property Q k b k (y)dy = 0, the estimate (b), and Hölder inequality, we have T λ,l (P )b k L 1 ({x/ O } B Rλ,l ) V ol(b Rλ,l ) 1/2 T λ,l (P )b k L 2 C( 2l λ )n/2 (2 l ) s 0 λ n/2 [λ max y,y 0 Q k dist(y, y 0 )] b k L 1 (Q k ) = C(2 l ) n/2 s 0 [λ max y,y 0 Q k dist(y, y 0 )] b k L 1 (Q k ) From our observation (10), and estimates (I), we have l= T λ,l (P )b k L 1 (x/ O ) C 2 l cλdist(x,y) (2 l ) n/2 s b k L 1 (Q k ) C s (λdist(x, y)) n/2 s b k L 1 (Q k ) C s (λr) n/2 s b k L 1 (Q k ), and from max y,y0 Q k dist(y, y 0 ) CR, estimate (II), and n/2 < s 0 < min{s, n/2 + 1}, we have l= T λ,l (P )b k L 1 (x/ O ) C 2 l cλdist(x,y) 20 (2 l ) n/2 s 0 [λ max dist(y, y 0 )] b k L y,y 0 Q 1 (Q k ) k

C s0 (λdist(x, y)) n/2 s 0 [λ max y,y 0 Q k dist(y, y 0 )] b k L 1 (Q k ) C s0 (λr) n/2+1 s 0 b k L 1 (Q k ). Therefore, we combine the above two estimate we conclude that x/ Q k m(p )b k (x) dx j=0 l= C s 2 j R>1 C s b k L 1 T 2 j,l(p )b k L 1 (x/ O ) (λr) n/2 s b k L 1 + C s0 2 j R1 Hence we have the weak-type (1, 1) estimate on the main term µ{x : m(p )f(x) > α} α 1 f L 1. (λr) n/2+1 s 0 b k L 1 Combine Case 1 and Case 2, we have the weak-type estimate of m(p ) and we finish the proof of Theorem 1.2. Q.E.D. References [1] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, 2001 [2] L. Hörmander, The spectral function of an elliptic operator, Acta ath. 88 (1968), 341-370. [3] L. Hörmander, The analysis of linear partial differential operators III, Springer- Verlag, 1985. [4] A. Seeger and C. D. Sogge, On the boundedness of functions of pseudodifferential operators on compact manifolds. Duke ath. J. 59 (1989), 709-736. 21

[5] C. D. Sogge, Concerning the L p norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), no. 1, 123-134. [6] C. D. Sogge, Fourier integrals in classical analysis. Cambridge Tracts in athematics, 105. Cambridge University Press, Cambridge, 1993. [7] C. D. Sogge, Eigenfunction and Bochner Riesz estimates on manifolds with boundary. athematical Research Letter 9, 205-216 (2002). [8] C. D. Sogge, S. Zelditch, Riemannian manifolds with maximal eigenfunction growth. Duke ath. J. Vol. 114, No. 3, 387-437 (2002). [9] E.. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces. Princeton Uniersity Press, Princeton, 1971. [10]. Taylor, Pseudo-differential Operators. Princeton Univ. Press, Princeton N.J., 1981. [11] Xiangjin Xu, Gradient estimates for eigenfunctions of compact manifolds with boundary and the Hörmander ultiplier Theorem. (preprint) [12] Xiangjin Xu, Eigenfunction Estimates on Compact anifolds with Boundary and Hörmander ultiplier Theorem. PhD Thesis, Johns Hopkins University, ay, 2004. 22