FOURIER TAUBERIAN THEOREMS AND APPLICATIONS

FOURIER TAUBERIAN THEOREMS AND APPLICATIONS YU. SAFAROV Abstract. The main aim of the paper is to present a general version of the Fourier Tauberian theorem for monotone functions. This result, together with Berezin s inequality, allows us to obtain a refined version the Li Yau estimate for the counting function of the Dirichlet Laplacian which implies the Li Yau estimate itself and, at the same time, the asymptotic results obtained by the variational method. Introduction Let F be a non-decreasing function, ρ be an appropriate test function on the real line R and ρ F be the convolution. Then, under certain conditions on the derivative ρ F, one can estimate the difference F ρ F. Results of this type are called Fourier Tauberian theorems. The Fourier Tauberian theorems have been used by many authors for the study of spectral asymptotics of elliptic differential operators, with F being either the counting function or the spectral function see, for example, [L], [H1], [H], [DG], [I1], [I], [S], [SV]). The required estimate for F ρ F was obtained under the assumption that the derivative ρ F admits a sufficiently good estimate. In applications F often depends on additional parameters and we are interested in uniform estimates. Then we have to assume that the estimate for ρ F holds uniformly with respect to these parameters and to take this into account when estimating F ρ F. As a result, there have been produced a number of Fourier Tauberian theorems designed for the study of various parameter dependent problems. This has been done, in particular, for semiclassical asymptotics see, for example, [PP]). Note that all the authors used the same idea of proof which goes back to the papers [L] and [H1]. The main aim of this paper is to present a general version of Fourier Tauberian theorem which does not require any a priori estimates of or growth conditions on) ρ F and holds true even if F is not polynomially bounded. Our estimates contain only convolutions of F and test functions see Section 1). This enables one to obtain results which are uniform with respect to any parameters involved. Date: July. 1991 Mathematics Subject Classification. 6A48, 35P15. Key words and phrases. Fourier Tauberian Theorems, Laplacian, counting and spectral functions. 1

YU. SAFAROV Our proof is very different from the usual one. It leads to more general results and, at the same time, allows one to evaluate constants appearing in the estimates Section ). Therefore our Tauberian theorems can be used not only for the study of asymptotics but also for obtaining explicit estimates of the spectral and counting functions. In particular, in Section 3, applying our Tauberian theorems and Berezin s inequality, we prove a refined version of the Li Yau estimate for the counting function of the Dirichlet Laplacian in an arbitrary domain of finite volume. Our inequality implies the Li Yau estimate itself and, along with that, the results on the asymptotic behaviour of the counting function which are obtained by the variational method. Note that the proof of the Berezin inequality does not use variational techniques. This implies that, even in the non-smooth case, the classical asymptotic formulae can be proved without referring to the Whitney decompositions and Dirichlet Neumann bracketing. Throughout the paper χ +, χ denote the characteristic functions of the positive and negative semi-axes, ˆft) := π) 1/ e itτ fτ) dτ is the Fourier transform of f, and τ := 1 + τ. 1. Tauberian theorems I: basic estimates Let F be a non-decreasing function on R. For the sake of definiteness, we shall always be assuming that 1.1) F τ) = 1 [F τ + ) + F τ )], τ R. 1.1. Auxiliary functions. If ρ is a continuous function on R, let us introduce the following conditions: 1 m ) ρτ) const τ m, where m > 1 ; ) c ρ, := ρτ) dτ = 1 ; 3) ρ is even; 4) ρ ; 5) supp ˆρ [ 1, 1]. For every m the functions ρ satisfying 1 m ) 5) do exist see, for example, [H], Section 17.5, or Example 1.1 below). However, we will not be always assuming that ρ satisfies all these five conditions. Example 1.1. Let l be a positive integer and 1.) γτ) := π τ + π l s) l sin l τ + s) ds. l The function γ satisfies 3), 5), and 1.3) c γ τ l γτ) c + γ τ l with some positive constants c ± γ. Indeed, 3) and 1.3) are obvious, and 5) follows from the fact that π) 1/ τ 1 sin τ is the Fourier transform of the

FOURIER TAUBERIAN THEOREMS 3 characteristic function of the interval [ 1, 1]. If ρτ) := c 1 γ,γτ) then the conditions ) 5) are fulfilled and 1 m ) holds with m = l 1. We shall always be assuming 1 m ). Let ρµ) dµ, τ >, τ ρ 1,1 τ) :=, τ =, τ ρµ) dµ, τ <, and, if 1 m ) holds with m >, ρ 1, τ) := One can easily see that τ µ ρµ) dµ, ρ 1, τ) := { τ µ τ ρλ) dλ dµ, τ, µ ρλ) dλ dµ, τ. ρ 1, τ) const τ m, ρ 1,1 τ) const τ m 1, ρ 1, τ) const τ m for all τ. Integrating by parts, we obtain 1.4) ρ 1, τ) = Denote c ρ,κ := τ µ ρ 1,1µ) dµ = ρ 1, τ) + τ ρ 1,1 τ), τ. µ κ ρµ) dµ, κ 1, m + 1). Under condition ), by Jensen s inequality, we have 1.5) c κ ρ,r c r ρ,κ, κ r. If the condition 3) is fulfilled then ρ 1, and ρ 1, are even continuous functions, ρ 1,1 is an odd function continuous outside the origin and 1.6) ρ 1,1 ±) = ± 1 c ρ,, ρ 1, ) = ρ 1, ) = 1 c ρ,1. Indeed, the first two equalities in 1.6) are obvious, and the last follows from 1.4). The condition 4) and 1.4) imply that 1.7) Let ρ 1, τ) ρ 1, τ), τ, ρ 1,k µ) ρ 1,k τ), k =, 1,, µ τ. 1.8) ρ δ τ) := δρδτ), ρ δ,k τ) := δ 1 k ρ 1,k δτ), k =, 1,, where δ is an arbitrary positive number. If 5) is fulfilled then 1.9) supp ˆρ δ, supp ˆρ δ [ δ, δ]. Indeed, these inclusions follow from 1.8) and the fact that ρ 1, is the convolution of the functions µ ρµ) and χ µ).

4 YU. SAFAROV 1.. Main estimates. If f is a piecewise continuous function on R 1, we denote f F τ) := lim R f F τ) := lim R R R R,R) fτ µ) F µ) dµ, fτ µ) df µ), whenever the limits exist. We shall deduce the estimates for F τ) from the following simple lemma. Lemma 1.. Let ρ satisfy the conditions 1 m ) 3) and ρ T,1 τ s) F s) as s ± for some T > and τ R. Then ρ T,1 F τ) is well defined if and only if ρ T F τ) is well defined, and 1.1) F τ) ρ T F τ) = ρ T,1 F τ). Proof. Integrating by parts, we obtain ρ T,1 τ µ) df µ) = ρ T,1 τ µ) df µ)+ R,R) R = R R,τ) τ,r) ρ T,1 τ µ) df µ) ρ T τ µ) F µ) dµ + ρ T,1 +) F τ ) ρ T,1 ) F τ + ) In view of 1.1), 1.6) and ), we have ρ T,1 τ + R) F R + ) + ρ T,1 τ R) F R ). ρ T,1 +) F τ ) ρ T,1 ) F τ + ) = F τ). Now the lemma is proved by passing to the limit as R. Theorem 1.3. Let ρ satisfy the conditions 1 m ) 4) with m >. Assume that ρ δ, τ s) F s) as s ± and ρ δ, F τ) < for some δ > and τ R. Then ρ T F τ) < and 1.11) F τ) ρ T F τ) c 1 ρ,1 δ 1 ρ δ, F τ) for all T δ. Proof. The identity 1.4) and 4) imply that ) d ρ1,1 τ) = ρτ) τ ρ 1,1τ) ρ 1, τ)) dτ ρ 1, τ) ρ 1, τ)), τ >. Therefore, in view of ) and 1.6), ρ 1,1 τ) ρ 1, τ) ρ 1,1+) ρ 1, ) = c 1 ρ,1, τ >. Taking into account 3), 1.8) and the second inequality 1.7), we obtain 1.1) ρ T,1 τ) ρ T,τ) c ρ,1 T ρ δ,τ) c ρ,1 δ, T δ >, τ R.

FOURIER TAUBERIAN THEOREMS 5 The inequality 1.1) implies that ρ and F satisfy the conditions of Lemma 1. and that ρ T F τ) <. Obviously, 1.11) follows from 1.1) and 1.1). Remark 1.4. If T = δ then the estimate 1.11) can be rewritten in the form 1.13) ρ + δ F τ) F τ) ρ δ F τ), where ρ ± δ τ) := ρ δτ) ± c 1 ρ,1 δτ ρ δ τ). Remark 1.5. The inequality 1.11) is not precise in the sense that, apart from some degenerate situations, it never turns into an equality. The crucial point in our proof is the estimate ρ T,1 c 1 ρ,1 δ 1 ρ δ, which implies that ρ T,1 F c 1 ρ,1 δ 1 ρ δ, F τ). However, the function ρ T,1 is negative on one half-line and positive on another, so ρ T,1 F may well admit much a better estimate. Using this observation, one can try to improve our results under additional conditions on the function F. Theorem 1.6. Let [a, b] be a bounded interval. Assume that the conditions of Theorem 1.3 are fulfilled for every τ [a, b] and that ρ δ, F τ) is uniformly bounded on [a, b]. Then 1.14) T 1 δ 1 fb) ρ δ, F b) b a fτ) [F τ) ρ T F τ)] dτ b T 1 δ 1 fa) ρ δ, F a) + T 1 δ 1 f τ) ρ δ, F τ) dτ for every non-negative non-decreasing function f C 1 [a, b] and all T δ. Proof. In view of 1.7) and 1.8) we have 1.15) T ρ T, τ) T 1 ρ T, τ) δ 1 ρ δ, τ), T δ >, τ R. This estimates, 1.1) and Lemma 1. imply that the functions ρ T, F τ), ρ T,1 F τ) and ρ T F τ) are uniformly bounded on [a, b]. Since ρ T, s) = ρ T,1 s) whenever s, integrating by parts with respect to τ we obtain b fτ) ρ T,1 τ µ) df µ) dτ = fa) ρ T, a µ) df µ) a fb) ρ T, b µ) df µ) + b a a f τ) Now 1.14) follows from Lemma 1. and 1.15). If f 1 then 1.14) turns into 1.16) ρ δ, F b) T δ b a ) ρ T, τ µ) df µ) dτ. [F µ) ρ T F µ)] dµ ρ δ, F a).

6 YU. SAFAROV This estimate and the obvious inequalities 1.17) ε 1 τ τ ε imply the following corollary. τ+ε F µ) dµ F τ) ε 1 F µ) dµ, ε >, Corollary 1.7. Under conditions of Theorem 1.6 1.18) 1.19) F b) ε 1 b b ε τ ρ T F µ) dµ ε 1 T 1 δ 1 ρ δ, F b), a+ε F a) ε 1 ρ T F µ) dµ + ε 1 T 1 δ 1 ρ δ, F a) for all ε, b a] and T δ. a If 4) is fulfilled then ρ T F is a non-decreasing function. Therefore 1.18) and 1.19) imply that 1.) 1.1) F b) ρ T F b ε) ε 1 T 1 δ 1 ρ δ, F b), F a) ρ T F a + ε) + ε 1 T 1 δ 1 ρ δ, F a). Remark 1.8. It is clear from the proof that Theorems 1.3 and 1.6 remain valid with some other constants independent of δ and T ) if we drop the condition 4) and replace ρ δ, τ) with an arbitrary non-negative function γ δ such that ρ T,1 τ) const δ 1 γ δ τ) and ρ T, τ) const T 1 δ 1 γ δ τ). In particular, one can take γ δ τ) = δγδτ), where γ is the function defined by 1.3) with l = m.. Tauberian theorems II: remarks and examples.1. General remarks. From now on we shall be assuming that the function F is polynomially bounded. Then the conditions of Theorems 1.3 and 1.6 are fulfilled for all τ, a, b R 1 and T δ > whenever ρ satisfies 1 m ) with a sufficiently large m. So far we have not used the condition 5), which is not needed to prove the estimates. However, this condition often appears in applications. It implies that the convolutions ρ T F and ρ T, F are determined by the restrictions of ˆF to the interval T, T ). If.1) ˆF t) = ˆF t) T,T ) T,T ) then, under condition 5), ρ T F = ρ T F and ρ δ, F = ρ δ, F for all δ T. If F τ) behaves like a linear combination of homogeneous functions for large τ then ρ δ, F is of lower order than ρ T F, so it plays the role of an error term in asymptotic formulae. It is not always possible to find a model function F satisfying.1). However, one can often construct F in such a way that the convolutions ρ T F F )τ) and ρ δ, F F )τ) admit good estimates for large τ

FOURIER TAUBERIAN THEOREMS 7 roughly speaking, it happens if the Fourier transforms of F and F have similar singularities on the corresponding interval). Then the Tauberian theorems imply estimates with the error term ± ρ T F F )τ) + ρ δ, F F ) )τ). In particular, if F is the spectral or counting function of an elliptic partial differential operator with smooth coefficients then 1.11) gives a precise reminder estimate in the Weyl asymptotic formula, and the refined estimates 1.), 1.1) allow one to obtain the second asymptotic term by letting T see [SV] for details). In applications to the second order differential operators it is usually more convenient to deal with the cosine Fourier transform of F. The following elementary observation enables one to apply our results in the case where information on the sine Fourier transform of F is not available. Proposition.1. If the cosine Fourier transforms of the derivatives F and F coincide on an interval δ, δ) then the Fourier transforms of the functions F τ) F τ) and F τ) F τ) coincide on the same interval... Test functions ρ. In this subsection we consider a class of functions ρ satisfying 1 m ) 5) and estimate the constants c ρ,κ. Lemma.. Let ζ C m+1 [ 1, 1 ] be a real-valued even function such that ζ L = 1 and ζ k) ± 1) = for k =, 1,... m 1, where ζk) denotes the kth derivative. If we extend ζ to R by zero then ρ := ˆζ) satisfies 1 m ) 5) and.) c ρ,k = ζ k) L, k =, 1,..., m. Proof. The conditions 3) and 4) are obviously fulfilled; ), 5) and.) follow from the fact that ˆρ = π) 1/ ζ ζ. Finally, 1 m ) holds true because the m + 1)th derivative of the extended function ζ coincides with a linear combination of an L 1 -function and two δ-functions. The following lemma is a consequence of the uncertainty principle. Lemma.3. If ρ is defined as in Lemma. then.3) c ρ,1 π. Proof. Let Π a be the multiplication operator and ˆΠ a be the Fourier multiplier generated by the characteristic function of the interval [ a, a]. Then the Hilbert-Schmidt norm of the operator ˆΠ a1 Π a acting in L R) is equal to π 1 a 1 a. Therefore µ ˆζ τ) dτ = ˆΠ µ Π 1/ ζ L π 1 µ ζ L = π 1 µ,

8 YU. SAFAROV which implies that c ρ,1 = µ ˆζ µ) dµ = π µ µ ˆζ τ) dτ dµ ˆζ τ) dτ dµ π 1 π 1 µ) dµ = π. Remark.4. As follows from Nazarov s theorem see [Na] or [HJ]), µ ˆφ τ) dτ b 1 e b µ, φ C 1, 1 ), µ, where b 1, b > are some absolute constants. Using the estimates for b 1, b obtained in [Na], one can slightly improve the estimate.3). Example.5. Let ν m be the first eigenvalue of the operator dm dt m on the interval 1, 1) subject to Dirichlet boundary condition, and let ζ m be the corresponding real even normalized eigenfunction. Denote ν m := ν m ) 1 m. If we define ρ as in Lemma. then, in view of.) and 1.5),.4) c ρ,m = ν m m, c ρ,κ ν κ m, κ < m. The eigenvalues ν m = νm m grow very fast as m. The following lemma gives a rough estimate for ν m. Lemma.6. We have ν m m m 3 for all m. Proof. If φt) = 1 4 t) m and L is the norm in L 1, 1 ) then.5) ν m φm) L φ L = One can easily see that 4m + 1)! m!) m + 1)! m)! m+1 m)!. m m)! m) m = m 1)... m m 1) ) m m m m 1) ) m 3. Therefore.5) implies the required estimate..3. Power like singularities. Assume that F τ) const τ + 1) n with a non-negative integer n and define { {, if n is odd, n+1, if n is odd, σ n := m n := n+ 1, if n is even,, if n is even, P n + τ, µ) := τ + µ)n + τ µ) n, Pn τ, µ) := µ τ + µ)n µ τ µ) n. Clearly, P n ± are homogeneous polynomials in τ, µ) with positive coefficients, which contain only even powers of µ.

FOURIER TAUBERIAN THEOREMS 9 Lemma.7. Let ρ be a function satisfying 3), 5) and 1 m ) with m > n. If supp F, + ) and the cosine Fourier transform of F τ) coincides on the interval δ, δ) with the cosine Fourier transform of the function nτ+ n 1 then [P +.6) ρ δ F τ) n τ, δ 1 µ) σ n δ n µ n] ρµ) dµ,.7) ρ δ F τ) P n + τ, δ 1 µ) ρµ) dµ,.8) ρ δ, F τ) δ [P n τ, δ 1 µ) + σ n δ n 1 µ n+1] ρµ) dµ for all τ >. Proof. According to Proposition.1, the Fourier transform of F τ) F τ) coincides on the interval δ, δ) with the Fourier transform of sign τ τ n = 1 σ n χ τ)) τ n. Since ρ is even, this implies that ρ δ F τ) = δ 1 σ n χ τ µ)) τ µ) n ρδµ) dµ = P n + τ, δ 1 µ) ρµ) dµ σ n δ 1 µ τ) n ρµ) dµ, ρ δ, F τ) = ρ δ, F τ) = δ 3 = δ P n τ, δ 1 µ) ρµ) dµ + σ n δ δτ 1 σ n χ τ µ)) τ µ) n µ ρδµ) dµ δτ δ 1 µ τ) n µ ρµ) dµ for all τ >. Estimating δ 1 µ τ) δ 1 µ in the integrals on the right hand sides, we arrive at.6).8). The obvious inequalities τ n + σ n ν n P + n τ, ν) τ n + n ν τ + ν ) n 1, P n τ, ν) + σ n ν n+1 n ν τ + ν ) n 1 and.6).8) imply that, for all τ >,.9) ρ δ F τ) τ n n δ 1 µ τ + δ 1 µ ) n 1 ρµ) dµ,.1) ρ δ, F τ) n µ τ + δ 1 µ ) n 1 ρµ) dµ. Note that m n is the minimal positive integer which is greater than n. If ρ is defined as in Lemma. with m = m n then, by.),.11) P n ± τ, δ 1 µ) ρµ) dµ = P n ± τ, δ 1 D t )ζ, ζ ) L.

1 YU. SAFAROV Applying.6).11) and 1.11) or 1.18), 1.19), one can obtain various estimates for F τ). Example.8. Let n = 3 and ζ be an arbitrary function satisfying conditions of Lemma. with m = m n =. If the conditions of Lemma.7 are fulfilled then.6).8),.11) and 1.19), 1.) with T = δ imply that F τ) τ 3 3ετ F τ) τ 3 + 3ετ + ε τ ε3 4 + 3 δ + ε τ + ε3 4 + 3 δ τ τ ε ε τ + τ ε + ε ) ζ L 1 εδ 4 ζ L, ) ζ L + 1 εδ 4 ζ L for all ε > and τ >. Thus, F τ) lies between the first Dirichlet eigenvalues of ordinary differential operators generated by the quadratic forms on the right hand sides of the above inequalities. Corollary.9. Under conditions of Lemma.7.1).13) for all τ >. F τ) τ n π 1 ν m n n δ 1 τ + δ 1 ν mn ) n 1, F τ) τ n + π 1 ν m n + ν mn ) n δ 1 τ + δ 1 ν mn ) n 1 Proof. If we define ρ as in Lemma. with ζ = ζ m see Example.5) then.1),.13) follow from 1.11) with T = δ,.9),.1),.3) and.4). Corollary.1. Under conditions of Lemma.7.14).15) λ λ for all λ >. F µ) dµ λn+ n + n ν m n δ λ λ + δ 1 ν mn ) n 1, F µ) dµ λn+ n + + n + 1) ν m n δ λ + δ 1 ν mn ) n Proof. Since λ F µ) dµ = λ F τ) τ dτ, Theorem 1.6 with T = δ, a =, b = λ and fτ) = τ implies.16).17) λ λ F µ) dµ F µ) dµ λ λ τ ρ δ F τ) dτ δ λ ρ δ, F λ), τ ρδ F τ) + δ ρ δ, F τ) ) dτ. Let ρ be defined as in Lemma. with ζ = ζ m. Then.14) follows from.16),.9),.1) and.4). Since τ P n + τ, ν) + Pn τ, ν) = P n+1τ, + ν), the inequality.17) and.7),.8) imply that λ F λ P + µ) dµ n+1τ, δ 1 µ) + σ n δ 1 µ n+1) ρµ) dµ dτ.

FOURIER TAUBERIAN THEOREMS 11 Estimating λ [ P + n+1 τ, ν) + σ n ν n+1] dτ λn+ n + + n + 1 ν λ + ν ) n with ν = δ 1 µ and applying.4), we obtain.15). 3. Applications to the Laplace operator Let Ω R n be an open domain and dx) be the distance from x Ω to the boundary Ω. 3.1. Estimates of the spectral function. Consider the Laplacian B in Ω subject to a self-adjoint boundary condition Bx, D x )u Ω =, where B is a differential operator. Assume that the operator B is non-negative and denote by Πλ) its spectral projection corresponding to the interval [, λ). Let ex, y; λ) be the integral kernel of the operator Πλ )+Πλ+) the so-called spectral function). The Sobolev embedding theorem implies that ex, y; λ) is a smooth function on Ω Ω for each fixed λ and that ex, x; λ) is a nondecreasing polynomially bounded function of λ for each fixed x Ω. Let be the Laplacian on R n, and e x, y; λ), ẽ x, y; λ), ẽx, y; λ) be the spectral functions of the operators,, B respectively. Then where χ + τ) ex, x; τ ) = ẽx, x; τ), χ + τ) e x, x; τ ) = ẽ x, x; τ) = C n τ n +, 3.1) C n := π) n meas {ξ R n : ξ < 1}. By the spectral theorem, the cosine Fourier transform of d ẽx, y; τ) coincides with the fundamental solution ux, y; t) of the wave equation in dτ Ω, u tt = u, Bu Ω =, u t= = δx y), u t t= =. Due to the finite speed of propagation, ux, x; t) is equal to u x, x; t) whenever t dx), dx)), where u x, y; t) is the fundamental solution of the wave equation in R n. Thus, the cosine Fourier transforms of the derivatives d dτ ẽx, x; τ) and d dτ ẽx, x; τ) coincide on the time interval dx), dx)). Ap- ẽx, x; τ) we obtain the following corollary. plying.1).15) to F τ) = C 1 n Corollary 3.1. For every x Ω and all λ > we have 3.) 3.3) ex, x; λ) C n λ n/ n C n π 1 ν m n dx) λ 1/ + ν m n dx) ex, x; λ) C n λ n/ + n C n π 1 ν m n + ν mn ) dx) ) n 1, λ 1/ + ν m n dx) ) n 1,

1 YU. SAFAROV 3.4) 3.5) λ λ ex, x; µ) dµ C n λ n/+1 n + ex, x; µ) dµ C n λ n/+1 n + n C n ν m n λ 1/ dx)) + n + 1) C n ν m n dx)) ) n 1 λ 1/ + ν m n dx) λ 1/ + ν ) n m n. dx) 3.. Estimates of the counting function of the Dirichlet Laplacian. In this subsection we shall be assuming that Ω <, where denotes the n-dimensional Lebesgue measure. Consider the positive operator D, where D is the Dirichlet Laplacian in Ω. Let Nλ) be the number of its eigenvalues lying below λ. The following theorem is due to F. Berezin [B]. Theorem 3.. For all λ we have 3.6) λ Nµ) dµ n + C n Ω λ n/+1. This results was reproduced in [La]. A. Laptev also noticed that the famous Li Yau estimate 3.7) Nλ) 1 + /n) n/ C n Ω λ n/, λ, see [LY]) is a one line consequence of 3.6). Indeed, 3.7) can be proved by estimating λ+θλ 3.8) Nλ) θλ) 1 Nµ) dµ and optimizing the choice of θ >. 1 + θ)n/+1 n + ) θ C n Ω λ n/ Remark 3.3. In [B] F. Berezin proved an analogue of 3.6) for general operators with constant coefficients subject to Dirichlet boundary condition. In the same way as above, applying the first inequality 3.8) and Berezin s estimates, one can easily obtain upper bounds for the corresponding counting functions see [La]). According to the Weyl asymptotic formula 3.9) Nλ) = C n Ω λ n/ + oλ n/ ), λ +, in the general case 3.9) was proved in [BS]). The coefficient in the right hand side of 3.7) contains an extra factor 1 + /n) n/. G. Pólya conjectured [P] that 3.7) holds without this factor. However, this remains an open problem. Given a positive ε, denote If Ω b ε := {x Ω : dx) ε}, Ω i ε := {x Ω : dx) > ε}. 3.1) Ω b ε const ε r, r, 1],

FOURIER TAUBERIAN THEOREMS 13 then, using the variational method [CH], one can prove that { 3.11) Nλ) C n Ω λ n/ const λ n 1)/ ln λ, r = 1, const λ n r)/, r < 1. It is well known that in the smooth case Nλ) C n Ω λ n/ = Oλ n 1)/ ) see, for example, [I1] or [SV]), but it is not clear whether this estimate remains valid for an arbitrary domain satisfying 3.1) with r = 1. There is a number of papers devoted to estimates of the remainder term in the Weyl formula. In [BL] the authors, applying the variational technique, obtain explicit estimates for the constants in 3.11). In order to prove the estimate of Nλ) from above, they imposed an additional condition on the outer neighbourhood of the boundary Ω, but this condition can probably be removed [Ne]. In [Kr] the author estimated the remainder term with the use of a different technique similar to that in [LY]); his results seem to be less precise than those obtained in [BL]. Let Nε b λ) := ex, x; λ) dx, Nελ) i := ex, x; λ) dx. Ω b ε Then Nλ) = N b ε λ) + N i ελ) for every ε >. Corollary 3.4. For all λ > and ε > we have 3.1) Nελ) i C n Ω i ε λ n/ C n,1 λ 1/ + ε 1 ν mn ) n 1 3.13) Nελ) i C n Ω i ε λ n/ + C n, λ 1/ + ε 1 ν mn ) n 1 Ω i ε Ω i ε Ω i ε dx dx), dx dx), 3.14) Nε b λ) C n,3 Ω b ε λ n/ + C n,4 λ 1/ λ 1/ + ε 1 ν mn ) n 1 where C n,1 = n C n π 1 ν m n, C n, = n C n π 1 ν m n + ν mn ), C n,3 = 1 + /n) n/ C n, C n,4 = 1 + /n) n/ n C n ν m n. Ω i ε dx dx)), Proof. The inequalities 3.1), 3.13) are proved by straightforward integration of 3.), 3.3). Theorem 3. and 3.4) imply that 3.15) λ N b ε µ) dµ = λ Nλ) dµ λ N i εµ) dµ n + C n Ω b ε λ n/+1 + n C n ν m n λ 1/ λ 1/ + ε 1 ν mn ) n 1 Ω i ε dx dx)). Now, applying the first inequality 3.8) with θ = /n, we arrive at 3.14).

14 YU. SAFAROV Adding up the inequalities 3.13) and 3.14) we obtain 3.16) Nλ) C n Ω i ε λ n/ + C n,3 Ω b ε λ n/ + λ 1/ + ε 1 ν mn ) n 1 C n, dx) + C n,4 λ 1/ Since 3.17) Ω i ε dx dx)) j = ε Ω i ε s j d Ω b s ) = j dx)) dx, ε >. ε s j 1 Ω b s ds ε j Ω b ε, 3.1) and the inequalities 3.1), 3.16) with ε = λ 1/ imply 3.11). By 3.1) and 3.16) we have C n Ω b ε Ω i C n,1 ε 1 + ν ) n 1 m n ελ 1/ ελ 1/ 3.18) λ n/ Nλ) C n Ω C n,3 C n ) Ω b ε + Ω i Cn, ε ελ + C ) n,4 1 + ν ) n 1 m n 1/ ε λ ελ 1/ for all ε >. If ε then the second inequality 3.18) turns into 3.7). Since Ω b ε as ε, 3.18) implies 3.9). Moreover, taking ε = λ κ with an arbitrary κ, 1 ), we obtain the Weyl formula with a remainder estimate λ n/ Nλ) C n Ω = O Ω b λ κ + λκ 1/ ), λ +. Remark 3.5. If the condition 3.1) is fulfilled then integrating 3.4) over Ω i λ 1/, applying 3.17) and taking into account 3.6), we see that λ 3.19) λ 1 Nµ) dµ = for all r, 1]. n + C n Ω λ n/ + Oλ n r)/ ) [B] [BL] [BS] [CH] [DG] References F.A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR 37 197), 1134 1167 Russian), English transl. in Math. USSR Izv. 6 197), 1117 1151. M. van den Berg and M. Lianantonakis, Asymptotics for the spectrum of the Dirichlet Laplacian on horn-shaped regions and ζ functions on cross-sections. To appear in J. Funct. Anal. M.S. Birman and M.Z. Solomyak, The principal term of spectral asymptotics for non-smooth elliptic problems, Funktsional. Anal. i Prilozhen. 4:4 197), 1 13 Russian), English transl. in Functional Anal. Appl. 4 1971). R. Courant and D. Hilbert, Methods of Mathematical Physics, Wiley, New York, 1989. J.J. Duistermaat and V.W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 5 1975), 39 79.

[I1] [I] FOURIER TAUBERIAN THEOREMS 15 V. Ya. Ivrii, On the second term of the spectral asymptotics for the Laplace-Beltrami operator on manifolds with boundary, Funktsional. Anal. i Prilozhen. 14:, 198), 5 34 Russian), English transl. in Functional Anal. Appl. 14 198)., The asymptotic Weyl formula for the Laplace-Beltrami operator in Riemannian polyhedra and domains with conical singularities of the boundary, Dokl. Akad. Nauk SSSR 88 1986), 35 38 Russian). [H1] L. Hörmander, The spectral function of an elliptic operator, Acta Math. 11 1968), 193 18. [H], Theory of Linear Partial Differential Operators I-IV, Springer-Verlag, 1985. [HJ] V. Havin and B. Jöricke, The uncertainty principle in harmonic analysis, Springer- Verlag, 1994. [Kr] P. Kröger Estimates for sums of eigenvalues of the Laplacian, J. Funct. Anal. 16 [L] [LY] [La] 1994), 17 7. B. M. Levitan, On the asymptotic behavior of the spectral function of a self-adjoint differential second order equation, Izv. Akad. Nauk SSSR Ser. Mat. 16 195), 35 35 Russian). P. Li and S.T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 1983), 39 318. A. Laptev, Dirichlet and Neumann eigenvalue problems on domains in Euclidean space, J. Funct. Anal. 151 1997), 531 545. [Na] F. L. Nazarov, Local estimates for exponential polynomials, Algebra i Analiz, 5:4 1993), 3 66 Russian). [Ne] Yu. Netrusov, private communication, 1999. [P] G. Pólya, On the eigenvalues of vibrating membranes, Proc. London Math. Soc. 11 1961), 419 433. [PP] [S] [SV] V. Petkov and G. Popov, Semi-classical trace formula and clustering of eigenvalues for Schrödinger operators, Ann. Inst. Henri Poincaré 68 1998), 17 83. Yu. Safarov, Asymptotics of the spectral function of a positive elliptic operator without non-trapping condition, Funktsional. Anal. i Prilozhen. :3 1988), 53 65 Russian), English transl. in Functional Anal. Appl. 1988). Yu. Safarov and D. Vassiliev, The asymptotic distribution of eigenvalues of partial differential operators, American Mathematical Society, 1996. Department of Mathematics, King s College, Strand, London WCR LS, UK E-mail address: ysafarovmth.kcl.ac.uk