Phân tích fipronil và các chất chuyển hóa của fipronil trong trứng bằng LC MSMS

NORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR ON THE ax ` b GROUPS arXiv:2101.00584v2 [math.FA] Apr 2021 RAUAN AKYLZHANOV, YULIA KUZNETSOVA, MICHAEL RUZHANSKY, AND HAONAN ZHANG Abstract The aim of this paper is to find new estimates for the norms of functions of the (minus) Laplace operator L on the ‘ax`b’ groups The central part is devoted to spectrally ? wave propa? localized gators, that is, functions of the type ψp Lq exppit Lq, with ψ P C0 pRq We show that for t Ñ `8, the convolution kernel kt of this operator satisfies }kt }1 — t, }kt }8 — 1, so that the upper estimates of D Mă uller and C Thiele (Studia Math., 2007) are sharp As a necessary component, we recall the Plancherel density of L and spend certain time presenting and comparing different approaches to its calculation Using its explicit form, we estimate uniform norms of several functions of the shifted Laplace-Beltrami ˜ closely related to L The functions include in particoperator ∆, ˜ γ q, t ą 0, γ ą 0, and p∆ ˜ ´ zqs , with complex z, s ular expp´t∆ Introduction Let G denote the ‘ax ` b’ group of dimension n ě 1, parameterized as G “ tpx, yq P R ˆ Rn u, with multiplication given by px, yq ă px1 , y 1q px ` x1 , y ` ex y 1q; in the case n “ 1, it is the group of affine transformations of the real line In this form, the right Haar measure is just mr “ dxdy This group is well-known to be non-unimodular, solvable, and of exponential growth; the modular function is given by δpx, yq “ e´nx B B Let X “ , ď k ď n, denote left-invariant and Yk “ ex Bx Byk vector fields on G, given also by expptXq “ pt, 0q, expptYk q “ p0, tek q, t P R (where pek q is the standard basis in Rn ) We consider the (minus) ř Laplace operator L “ ´X ´ nk“1 Yk2 A detailed exposition of this setting can be found in [20] L is a positive self-adjoint operator on L2 pG, mr q, and for any bounded Borel function ψ on R one can define, by the spectral theorem, a bounded operator ψpLq on L2 pG, mr q For ψ good enough, this is a 2010 Mathematics Subject Classification 22E30; 43A15; 42B15; 35L05; 43A90 R AKYLZHANOV, YU KUZNETSOVA, M RUZHANSKY, AND H ZHANG (right) convolution operator with a kernel kψ , for which an explicit formula is available [23, Proposition 4.1] There has been much work to determine in which cases ψpLq is also bounded from Lp pG, mr q to Lq pG, mr q for different p and q (see [11] and references therein, or more recently [12, 18, 23]) Let us put this into a wider context If we consider a connected Lie group G as a Riemannian manifold, endowed with the left-invariant distance, we obtain the Laplace–Beltrami operator ∆ on G In the unimodularřcase, ∆ equals to the Laplacian L defined as above, that is, L “ ´ k Xk2 where pXk q is a basis of the Lie algebra of G If G is non-unimodular, however, these two operators are different and can have signicantly dierent behaviour By the classical Hăormander-Mikhlin theorem, a function of the Laplacian mp∆q is a multiplier on Lp pRn q if rn{2s`1 derivatives of m decrease quickly enough One can observe a similar behaviour on any group of polynomial growth [1] On the other hand, on symmetric spaces of non-compact type, a class which includes the ‘ax+b’ groups and implies exponential growth, the function m must be holomorphic in a strip around the spectrum of ∆ in order to generate an Lp -multiplier, p ‰ [10, 30, 2] It turns out, surprisingly, that the volume growth does not determine the behaviour of the distinguished Laplacian L as defined above: on the AN groups, in particular on ‘ax+b’ groups, a finite number of derivatives is sufficient for an Lp multiplier [16, 12, 18] One can ask then what is the class of groups carrying Laplacians of different kind, as the ‘ax+b’ groups One could conjecture that these are solvable groups of exponential growth; this has been disproved however by an example of a semidirect product of R with the p2n ` 1q-dimensional Heisenberg group Hn [9] This question remains open, and the criterium might be the symmetry of the group algebra L1 pGq [9] Looking closer at the multipliers, one can ask how far goes the ressemblance between the Laplacian L of HăormanderMikhlin type and that of Rn In particular, in applications to PDEs oscillating functions of the type ? ? ? ? sinpt Lq At “ ψp Lq cospt Lq, Bt “ ψp Lq ? L are of special importance With ψ ” 1, they give the solutions of the wave equation: the function ut “ At f ` Bt g solves the Cauchy problem B2u “ ´Lu, Bt2 Bu up0, xq “ f pxq, p0, xq “ gpxq, Bt for a priori f, g P L2 pGq One cannot of course apply the Hăormander Mikhlin theorem in this case By other methods, one shows that the NORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR ? ? sinpt ∆q ? wave propagators cospt ∆q, are bounded on Lp pRn q if and ∆ ˇ ˇ [25], so that for a general p a “localizing” factor only if ˇ p1 ´ 21 ˇ ă n´1 ψ should be introduced With ψpsq “ p1 ` s2 q´α , if the norms of At , Bt decrease in time, one can set }f }H α,p “ }p1 ` ∆2 qα f }p and get so called dispersive estimates in terms of Sobolev norms of f and g In Rn , n ą for example, }ut }8 ď Ct´pn´1q{2 p}f }H α,1 ` }g}H α,1 q, for α big enough More on this topic can be found in [7, 28] For the (shifted) Laplace–Beltrami operator on hyperbolic spaces, in particular for the ax`b groups, dispersive estimates have been obtained by D Tataru [29], with exponential decay in time See also [22, 26, 6] In this context it was surprising that D Mă uller and C Thiele [23] did not get any decay in time for the Laplace operator on the ax ` b group: they show that the L1 norms of the kernels of At , Bt are bounded by Cp1 ` |t|q, for ψ P C0 pRq quickly decreasing, while the uniform norms of the same kernels are just bounded by a constant, for a compactly supported ψ In the present paper, we prove that these estimates are actually sharp: if ψ is a nonzero function with certain decrease rate, then the kernels kt of both At and Bt satisfy }kt }1 ě Cn,ψ t and }kt }8 ě Cn,ψ for t large enough For the uniform estimate, there is no need to suppose ψ compactly supported, so that both estimates are applicable to ψpsq “ p1 ` s2 q´α This means that dispersive L1 ´ L8 estimates not hold for the Laplacian L on the ax ` b groups The main part of the proofs is concentrated in Section We thank J.-Ph Anker for indicating us the following result [5], valid in a more general situation of Damek–Ricci spaces: The convolution kernel of exppitLq is not bounded for any nonzero t P R, so that the dispersive L1 ´ L8 estimates not hold for the Schrăodinger equation associated to the distinguished Laplacian L on these spaces However, Strichartz estimates hold, in a weighted form This suggests that for the wave equation Strichartz estimates might hold as well The rest of the paper is organized as follows In Section 2, we collect necessary notations and conventions ˜ the shifted Laplace– To explain the sequel, let us denote by ∆ Beltrami operator (more details in Section 3) The two Laplacians ˜ “ δ ´1{2 Lδ 1{2 (here δ stands for multiplication by the are linked by ∆ ˜ and ψpLq are related as modular function), and the kernels of ψp∆q ´1{2 ˜ kψ “ δ kψ , both acting by right convolution It is known that the L2 -norm of k˜ψ is the same as of kψ and is given by the integral ż }kψ }2L2 pG,mr q “ R |ψ|2 ρ R AKYLZHANOV, YU KUZNETSOVA, M RUZHANSKY, AND H ZHANG with a density ρ expressed via the Harish–Chandra c-function [19, 11] This appears as a building block in several multiplier estimates [11, 17, 18] In Section 3, we take time to write down explicit formula for the spectral density of the Laplacian and to relate to each other several ways to obtain it We put forward the fact, seemingly not discussed before, that the same density can be used also to estimate uniform norms of k˜ψ In Section 4, we show that this method makes it an easy calculation to obtain exact asymptotics of the uniform norms We find first the exact norms of the heat kernels (of course well known), and then more ˜ γ q, γ ą Further we pass to a more technical generally of expp´t∆ ˜ ´ zq´s with z, s complex (z outside task of estimating the norms of p∆ ˜ and ℜs ą pn ` 1q{2) J.-P Anker[3] has shown, of the spectrum of ∆, ´s ˜ among others, that the kernel of p∆´zq (with z outside of the positive half-line) is bounded if and only if ℜs ě pn`1q{2, s ‰ pn`1q{2 Going rather explicitly into integration with the help of special functions, we get estimates of actual uniform norms of these functions in the same region except for the border, and obtain asymptotic bounds for ℑz Ñ and for |z| Ñ The last Section contains, as mentioned above, the lower estimates of L1 and uniform norms of the operators At , Bt above Notations and conventions We have chosen to work (mainly) with the right Haar measure mr “ dxdy and the right convolution: ż pf ˚r gqpxq “ f pxyqgpy ´1qδpyqdmr pyq G One can alternatively opt to the left Haar measure ml “ e´nx dx dy, so that mr “ δ ´1 ml , and the left convolution ż pf ˚l gqpxq “ f pyqgpy ´1xqdml pyq G For a function f on G, denote fqpgq “ f pg ´1q For every p, the map τ : f ÞĐ fq is an isometry from Lp pG, mr q to Lp pG, ml q (or back) If Rk : f ÞĐ f ˚r k is a right convolution operator with the kernel k, then τ Rk τ “ Lkq is a left convolution operator with the kernel q k, and }Lkq : Lp pG, ml q Ñ Lp pG, ml q} “ }Rk : Lp pG, mr q Ñ Lp pG, mr q} This means that choosing one or another convention changes nothing in norm estimates Below we not use the symbol ˚r anymore and write just ˚ instead We will use } ă }p and Lp pGq to denote } ¨ }Lp pG,mr q and Lp pG, mr q, respectively NORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR The ` distance.˘ The left-invariant Riemannian distance on G is given by d px, yq, p0, 0q “: Rpx, yq, where ´ ´x ¯ (2.1) Rpx, yq “ arcch chx ` |y| e This implies in particular that |x| ď R and |y|2 “ 2ex pchR ´ chxq ď 2ex`R The Plancherel weight Every bounded left-invariant operator on L2 pG, mr q belongs to the right von Neumann algebra V NR pGq, defined as the strong operator closure of the set of right translation operators This applies, in particular, to ψpLq with a bounded function ψ Similarly, every right-invariant operator on L2 pG, ml q belongs to the left von Neumann algebra V NL pG, ml q On V NL pGq, we have the Plancherel weight ϕ (see more in [24]), which can be viewed as an integral of operators For an operator Lk of left convolution with the kernel k P L2 pG, ml q X L1 pG, ml q, one has ϕpL˚k Lk q “ }k}2L2 pG,ml q If k “ g ˚ ˚ g for some g and is moreover continuous, then ϕpLk q “ kpeq Almost all the literature on Plancherel weights assumes this V NL pGq convention, but one can also consider a similar weight ϕr on V NR pGq by setting ϕr pRk Rk˚ q “ }k}2L2 pG,mr q (2.2) for an operator of right convolution with the kernel k The two algebras are isomorphic by A ÞĐ τ Aτ , and ϕr pAq “ ϕpτ Aτ q This isomorphism transfers also, as mentioned above, Rk P V NR pGq to Lkq P V NL pGq, where q kpgq “ kpg ´1 q Spectral measure of the Laplacian The aim of this section is to write explicitly the Plancherel measure of the Laplace operator and show the relation between several apparently different approaches to its calculation 3.1 Plancherel measure and the Harish-Chandra c-function The Plancherel measure for the spherical transform on a connected semisimple Lie group is given by a so called c-function found by HarishChandra Many faces of this function are described in an excellent survey [19], of which we will need only a few facts Recall first that the Laplace–Beltrami operator ∆ has a spectral gap: its spectrum is rσ, `8q where σ is a constant depending on the group, σ “ n4 in our case For this reason, it is of course more regular ˜ “∆´σ than L, so it makes sense to consider the shifted operator ∆ which has r0, `8q as its spectrum The two operators are now linked R AKYLZHANOV, YU KUZNETSOVA, M RUZHANSKY, AND H ZHANG ˜ “ δ ´1{2 Lδ 1{2 (here δ stands for multiplication by the modular by ∆ function) Next, it is known that the L2 -norm of a radial function on G can be expressed via the Harish-Chandra c-function [19, 11] This can be ˜ and functions of it, as these applied to the convolution kernels of ∆ ˜ we have kernels are radial If k˜f is the kernel of f p∆q, ż }k˜f }22 “ cG |f pλ2 q|2 |cpλq|´2 dλ R If kf is the right convolution kernel of f pLq, then [11] kf “ δ 1{2 k˜f and }kf }L2 pG,mr q “ }k˜f }L2 pG,mr q (this equality is verified by a direct calculation, knowing that k˜f px´1 q “ k˜f pxq), so that the formula above is valid for kf as well An explicit formula of the c-function is known, and for the n-dimensional ‘ax+b’ group it is as follows [19]: for λ P R, c0 2´iλ Γpiλq cpλq “ 1 Γp p n ` ` iλqqΓp 21 p 12 n ` iλqq ? Since ΓpzqΓpz ` 21 q “ π21´2z Γp2zq [8, 1.2], this simplifies up to cpλq “ c1 Γpiλq , Γp n2 ` iλq ? with c1 “ π ´1{2 2n{2´1 c0 Denote ρpuq “ c21 u´1{2 |cp uq|´2 , so that ż }kf }2 “ C |f puq|2ρpuqdu R (we will also write ρn to indicate the dimension) If n “ 2l is even, 1{c is a polynomial: l´1 ź pj ` iλq, “ c´1 cpλq j“0 so that ρ2l puq “ If n “ 2l ` is odd, we have and l´1 ? ź u pj ` uq j“1 l´1 ź Γp 12 ` iλq 1 “ c1 pj ` ` iλq , cpλq Γpiλq j“0 ρ2l`1 puq “ u so that ρ2l`1 puq “ ´1{2 l´1 ź ` j“0 l´1 ´ ź ` j“0 ¯ |Γp ` i?uq|2 ˘2 ? , `u j` |Γpi uq|2 ˘ pj ` 1{2q2 ` u ρ1 puq Moreover, the reflection formula ΓpzqΓp1 ´ zq “ π{ sinpπzq and the conjugation identity Γpzq “ NORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR Γp¯ z q imply that for real v, and |Γp 12 ` ivq|2 ´πiv sinpπivq v shpπvq “ “ , |Γpivq| π cospπivqq chpπvq l´1 ´ ¯ shpπ ?uq ź ` ˘2 ? ρ2l`1 puq “ j` `u chpπ uq j“0 These formulas imply that ρpuq „ upn´1q{2 as u Ñ `8, and ρpuq „ u as u Ñ These latter estimates have been used in application to the norm estimates of the Laplacian and its functions in [11, 17, 18] 1{2 3.2 Uniform norms of the kernels Let us consider a right convolution operator on L2 pG, mr q, Rk : f ÞĐ f ˚ k (all convolutions here are taken with respect to the right Haar measure) Its adjoint is Rk˚ “ Rk˚ , where the (right) involution is defined as f ˚ pxq “ f px´1 qδpxq The composition of a pair of operators is Rk Rh “ Rh˚k Suppose now that ď f “ |g|2 and kf , kg are the convolution kernels of f pLq, gpLq, with kg P L2 pGq We have then f pLq “ gpLq˚gpLq “ gpLqgpLq˚, so that kf “ kg ˚ kg˚ “ kg˚ ˚ kg and ż kf peq “ kg pyqkg˚py ´1 qδpyqdmr pyq żG kg pyqkg pyqδpy ´1qδpyqdmr pyq “ }kg }2L2 pG,mr q “ G By the results above, kf peq “ }kg }2L2 pG,mr q “c ż R |g| ρ “ c ż f ρ (3.1) R This formula is thus valid for 0ş ď f P L1 pR, ρq For f not necessarily positive, the formula kf peq “ c R f ρ is still valid, by linearity It is well known that the uniform norm of a positive definite function is attained at the identity Given a function h P L2 pG, mr q, the convolution h˚ ˚ h is in general not positive definite However if we multiply it by δ ´1{2 , it becomes such, since this is a coefficient of the right regular representation of G on L2 pG, mr q: ż ˚ ´1{2 ´1{2 ph ˚ hqpxqδ pxq “ δ pxq hpy ´1x´1 qδpxyqhpy ´1qδpyqdmr pyq ż G 1{2 “ δ pxq hpyx´1 qδpy ´1qhpyqdmr pyq żG “ δ 1{2 pxq hpzqδpx´1 z ´1 qhpzxqdmr pzq G ż “ hpzxqδ ´1{2 pzxqhpzqδ ´1{2 pzqdmr pzq G “ xRx phδ ´1{2 q, hδ ´1{2 yr , R AKYLZHANOV, YU KUZNETSOVA, M RUZHANSKY, AND H ZHANG where xă, ¨yr denotes the inner product induced by the right Haar mea˜ is in this case sure This means that the kernel k˜f “ δ ´1{2 kf of f p∆q positive definite In particular, its uniform norm is attained at the identity: ż ˜ ˜ }kf }8 “ kf peq “ kf peq “ c f ρ, (3.2) R ď f P L1 pR, ρq If f is real-valued, but maybe not positive, then we can decompose it f “ f` ´ f´ into positive and negative part, and get the following estimate: ż ż ˜ ˜ ˜ ˜ ˜ }kf }8 ď }kf` }8 `}kf´ }8 “ kf` peq` kf´ peq “ c pf` `f´ qρ “ c |f |ρ R R ? (3.3) Finally, for f complex-valued, we have to add a factor of in the right hand side: ż ? ż ˜ ˜ ˜ }kf }8 ď }kℜf }8 ` }kℑf }8 ď c p|ℜf | ` |ℑf |qρ ď c |f |ρ R R Thus, the Plancherel measure helps to calculate not only L , but also uniform norms 3.3 Connection with the Plancherel weight and L2 -norms of the resolvent kernels In the case n “ or 2, the kernels kλ of the resolvent Rλ “ pL ´ λq´1 are 2-summable, as seen from the bounds above This allows to obtain the Plancherel measure in a different way Comparing the formulas (2.2) and (3.1), we notice that ż8 ` ˘ ϕr f pLq “ c f dρ, for ď f P L pR, ρq This means that the measure ρ˘ can be found ` from this equality, if we are able to determine ϕr f pLq Let A be, in general, a positive self-adjoint unbounded operator on ş8 L2 pG, mr q with the spectral decomposition A “ udEpuq For a bounded measurable function f , one defines ż8 f pAq “ f puq dEpuq by the spectral theorem for self-adjoint unbounded operators By construction, for all x, y P H “ L2 pG, mr q ż8 xf pAqx, yyr “ f puq dEx,y puq with respect to the measure Ex,y : X ÞĐ xEpXqx, yy, X Ă R Borel If A is left-invariant, that is, affiliated to the right group von Neumann algebra V NR pGq, then f pAq P V NR pGq For a vector state x,x păq xă x, xy on V NR pGq, this gives already its value on f pAq Any weight ϕr , and in particular the Plancherel weight NORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR ř is the sum of a family of normal positive functionals [15], ϕr “ α ϕr,α Every ϕr,α can řbe decomposed into a countable sum of positive vector states ϕr,α “ n ζxn,α,xn,α , which implies that for positive f ż ż8 ` ˘ ÿ ϕr f pAq “ f puq dϕr,αpEpuqq “ f puq dϕr pEpuqq α 0 Linearity allows to extend this equality to arbitrary f , real or complex One can find the spectral measure of A as the strong limit [14, XII.2, Theorem 10] żb ` ˘ lim Rλ`iε ´ Rλ´iε dλ, Era,bs “ 2πi εÑ0` a where, as usual, Rz “ pA ´ zq´1 is the resolvent of A In the case of a positive operator, we can get nontrivial values of course only for a ě We can transform ˚ Rλ`iε ´ Rλ´iε “ 2iεRλ`iε Rλ´iε “ 2iεRλ`iε Rλ`iε If we return now to the Laplace operator, then Rλ`iε “ Rkλ`iε is a convolution operator with a kernel kλ`iε By the lower semi-continuity of the right Plancherel weight ϕr , żb ϕr pEra,bs q ď lim 2iε ϕr pRkλ`iε Rk˚λ`iε qdλ 2πi εÑ0` a żb lim ε }kλ`iε }2L2 pG,mr q dλ “ π εÑ0` a Using estimates in [21] in the case n “ and [23, Lemma 3.1] for general n, one can show (this is however quite a technical task) that this limit is finite if (and only if) n ď 2, and in this case bounded by żb a ϕr pEra,bs q ď dn p1 ` |λ|qn`1 dλ, a which proves that the spectral measure is absolutely continuous with density bounded by a polynomial We see however that this bound is not sharp 3.4 Application of the explicit formula for the convolution kernel Yet another approach is to use the explicit formula for the kernel [23, Proposition 4.1]: for a function ψ P C0 pRq, the kernel kψ of ψpLq is given, for any integer l ą n2 ´ 1, by ż ? ? cl ´ n x kψ px, yq “ e ψpuqrFRpx,yq,l p uq ´ FRpx,yq,l p´ uqs du, (3.4) where Rpx, yq is given by (2.1), ż8 n l FR,l puq “ Dsh,v peiuv qpchv ´ chRql´ dv, R 10 R AKYLZHANOV, YU KUZNETSOVA, M RUZHANSKY, AND H ZHANG l Dsh,v denotes the l-th composition of Dsh,v , Dsh,v pf q “ n n p´1ql 2´1´ π ´ cl “ iπΓpl ` ´ n2 q d´ f ¯ and dv shv In particular, for x “ y “ we have (note that icl P R) ż ? ? cl ψpuqrF0,l p uq ´ F0,l p´ uqs du kψ peq “ ż8 ż8 ` ? ˘ n l “ icl ψpuq Dsh,v sinpv uq pchv ´ 1ql´ dv du 0 ş As we have seen before, for ď ψ P L1 pR, ρq this is also equal to c ψρ, so that (as L1 pR, ρq X C0 pRq is dense in L1 pR, ρq) ż ? ? ˘ ? n cl icl l ` ρpuq “ rF0,l p uq´F0,l p´ uqs “ Dsh,v sinpv uq pchv´1ql´ dv 2c c (3.5) One should note that FR,l is not bounded at 0, but the difference above is, and tends to as u Ñ This equality is not obvious, but is easy to check in the case n “ 2l “ 2: we have to change the sign as icl “ ´ 4π1 ă 0, and then obtain ? ż8 ´ ` ? ? ? ˘ d sinpv uq ¯ ´ F0,1 p uq ´ F0,1 p´ uq “ ´ dv “ u “ ρ2 puq shv dv It is clear that best bounds can be obtained for l chosen so that l ´ n{2 P t0, ´1{2u, and below we assume this choice l Denote Dl,u pvq “ Dsh,v peiuv ´ e´iuv q For l “ 0, D0,u pvq “ 2i sinpuvq is an odd function of v By induction, one verifies that every Dl,u is odd, too: if f is odd (and analytic), then f pvq{shv is even (and well d defined at 0) and Dsh,v pf q “ dv pf pvq{shvq is odd This implies, in l particular, that Dl,u p0q “ Note at the same time that Dsh,v peiuv q alone can be unbounded as v Ñ Thus, the integral below converges: ż n icl ρpu q “ (3.6) Dl,u pvqpchv ´ 1ql´ dv 2c If l “ n (n even), then ż icl d ´ Dl´1,u pvq ¯ dv ρpu q “ 2c dv shv ˇ icl icl d Dl´1,u pvq ˇ “ ´ lim “´ Dl´1,u ˇ v“0 2c vÑ0 shv 2c dv Decomposing Dl,u into its Taylor series, one can show that Dl,u p0q “ P0,l puq is a polynomial of degree 2l ` On this way, one can obtain estimates ρn puq ď Cpu1{2 ` un{2q, however already known from the calculations of the c-function NORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 15 Lower norm estimates Let ψ P C0 pRq be a function which is not identically zero on r0, `8q and twice differentiable with }ψ pjq psqsk }8 ď Cψ for ď j ď 2, ď k ď n{2 ` These conditions are in particular verified for ψpsq “ p1 ` s2 q´α{2 , α ě n ` In [23], Mă uller and Thiele ? have?proved that the L -norm of the ? convolution kernel kt of ψp Lq cospt Lq, as well ? sinpt Lq as of ψp Lq ? , is bounded by Cp1 ` |t|q We prove below that L this estimate is sharp at t Ñ `8 We thank W Hebisch for the following remark, made after the first version of the paper was completed Let f be such that the convolution kernel kf,n of f pLq is in L1 pGn q, where Gn is the n-dimensional ‘ax ` b’ group Then kf,n is obtained from the kernel kf,n`1 on the n ` 1-dimensional group by integration over the last coordinate [18] This trivially implies that }kf,n`1}1 ě }kf,n }1 In addition, the exact asymptotics is known in the case n “ due to Hebisch, who used the transference principle by giving an isometry with functions of the Laplacian on R3 This does not allow however to compare uniform norms in a similar way, and the case n “ needs a full consideration The remarks above could simplify our integration Lemmas 5.4 and 5.5 and a few details elsewhere; these changes seem not to be major and we decided to keep the proofs as they are The integration formulas over G below are not identical to [17] but are influenced by this article 5.1 The convolution kernel According to (3.4) ? ? (which comes from [23]), the convolution kernel kt of ψp Lq exppit Lq ż8 ż8 n x its ´n ψpsqe Dl,s pvqpchv ´ chRql´ dv sds, kt px, yq “ cl e R l where we denote as before Dl,s pvq “ Dsh,v peisv ´ e´isv q Recall that l is chosen as l “ t n2 u Note first that ż8 ? }kt }2 ď |ψp uq|2 ρpuqdu ş is bounded uniformly in t, so that the same is true for Rpx,yqď1 |kt | We can therefore assume in the sequel that R ě We start by transforming the kernel with the help of the following decomposition Proposition 5.1 l Dsh,v peiuv q “ l ÿ k“0 uk eiuv qk,l pvq, 16 R AKYLZHANOV, YU KUZNETSOVA, M RUZHANSKY, AND H ZHANG where qk,l pvq “ Pk,l pshv, chvqpshvq´2l , every Pk,l is a homogeneous polynomial of two variables of degree l, and qk,l is even for even k and odd l for k odd In particular, Dsh,u pvq Ñ 0, v Ñ `8 Proof The formula is trivially true for l “ The induction step is proved by direct differentiation: l`1 iuv Dsh,v pe q l d ÿ k iuv qk,l pvq “ u e dv k“0 shv ” q pvq q pvq q pvqchv ı k,l k,l k,l iu “ u e ` ´ shv shv sh2 v k“0 ” q pvq q pvqchv ı 0,l 0,l ´ “ eiuv shv sh2 v l ” ÿ qk,l pvq qk,l pvqchv ı ql,l pvq k iuv qk´1,l pvq i ` u e ` ´ ` iul`1 eiuv shv shv shv sh v k“1 l ÿ Set k iuv $ q0,l pvq q0,l pvqchv ’ ’ k“0 ´ ’ ’ ’ sh1 v & shv q pvq qk,l pvqchv qk,l`1 pvq :“ i qk´1,l pvq ` k,l ´ 1ďkďl ’ shv shv ’ sh2 v ’ ’ ’ %i ql,l pvq k “ l ` shv Then l`1 ÿ l`1 iuv uk eiuv qk,l`1 pvq Dsh,v pe q “ (5.1) k“0 It remains to check that there exist homogeneous polynomials of two variables Pk,l`1, ď k ď l ` which are of degree l ` such that Pk,l`1 pshv, chvq :“ pshvq2l`2 qk,l`1 pvq For this, we claim that qk,l pvq “ pshvq´2l´1 Qk,l pshv, chvq with Qk,l being a two-variable homogeneous polynomial of degree l ` In fact, qk,l pvq “pshvq´2l pPk,l,1pshv, chvqchv ´ Pk,l,2pshv, chvqshvq ´ 2lpshvq´2l´1 chvPk,l pshv, chvq, where Pk,l,1px, yq :“ Bx Pk,l px, yq and Pk,l,2px, yq :“ By Pk,l px, yq are twovariable homogeneous polynomials of degree l ´ So where pshvq2l`1 qk,l pvq “ Qk,l pshv, chvq, Qk,l px, yq :“ Pk,l,1px, yqxy ´ Pk,l,2px, yqx2 ´ 2lPk,l px, yqy (5.2) is a two-variable homogeneous polynomial of degree l ` This completes the proof of the claim NORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 17 When k “ 0, by induction and by our claim: pshvq2l`2 q0,l`1 pvq “ pshvq2l`1 q0,l pvq ´ pshvq2l chpvqq0,l pvq “ Q0,l pshv, chvq ´ chpvqP0,l pshv, chvq so we can choose P0,l`1 px, yq “ Q0,l px, yq ´ yP0,l px, yq When ď k ď l, pshvq2l`2 qk,l`1pvq “ ipshvq2l`1 qk´1,l pvq ` pshvq2l`1 qk,l pvq ´ pshvq2l qk,l pvqchv “ iPk´1,l pshv, chvqshv ` Qk,l pshv, chvq ´ Pk,lpshv, chvqchv so we can choose Pk,l`1px, yq “ ixPk´1,l px, yq ` Qk,l px, yq ´ yPk,lpx, yq When k “ l ` 1, by induction: pshvq2l`2 ql`1,l`1 pvq “ ipshvq2l`1 ql,l pvq “ iPl,l pshv, chvqshv, so we can choose Pl`1,l`1px, yq “ iPl,l px, yqx Finally, statements on parity of qk,l are easy to check Note that we not check whether Pk,l are nonzero, so with a certain abuse of language we assume zero to be a homogeneous polynomial of any degree All we need to know of qk,l in this respect is contained in the Proposition below: Proposition 5.2 There exist constants bl and ak,l , ď k ď l, such that |qk,l pvq ´ ak,l e´vl | ď bl e´2vl for every v P r0, `8q Moreover, al,l ‰ Proof By Proposition 5.1, we have pshvq2l qk,l pvq “ Pk,l pshv, chvq, for some two-variable homogeneous polynomial Pk,l of degree l It is easy to see that there exists a polynomial Al of degree 4l such that e2vl pshvq2l “ pev shvq2l “ Al pev q, and a polynomial Bk,l of degree 2l such that evl Pk,l pshv, chvq “ Pk,l pev shv, ev chvq “ Bk,l pev q Thus there exist some nonzero constant ak,l and some polynomial Ck,l of degree ď 4l ´ such that evl qk,l pvq “ evl Pk,l pshv, chvq e2vl Bk,l pev q Ck,l pev q “ “ a ` k,l pshvq2l Al pev q Al pev q Hence for some constant bl ą we have or equivalently |evl qk,l pvq ´ ak,l | ď bl e´vl , |qk,l pvq ´ ak,l e´vl | ď bl e´2vl (5.3) 18 R AKYLZHANOV, YU KUZNETSOVA, M RUZHANSKY, AND H ZHANG It remains to show that al,l ‰ The function ql,l is in fact easy to ´ i ¯l calculate: by (5.1), ql,l pvq “ For l “ 0, we have a0,0 “ For shv l ą 0, by (5.3), ak,l “ lim evl qk,l pvq, vÑ`8 so that al,l “ p2iql ‰ We can write now kt px, yq (5.4) ż8 ´ nx “ cl e “ R cl ´ nx e 2 pchv ´ chRq ż8 R ż8ÿ l l´ n n pchv ´ chRql´ rsk eisv ´ p´sqk e´isv s qk,l pvqψpsq s eits dv ds k“0 l ÿ k“0 qk,l pvqr m q k pt ` vq ´ p´1qk m q k pt ´ vqs dv where mk psq “ ψpsqsk`1Ir0,`8q and m q k is the inverse Fourier transform, which we write without additional constants While mk p0q “ for all k, the derivatives of mk may be discontinuous at 0, so we need some attention when estimating m q k Integrating by parts, we can write, for ‰ ξ P R: ż ‰ 1“ 1 iξs m q k pξq “ mk psqe ´ mk psqeiξs ds, (5.5) ξ ξ which is bounded as |m q k pξq| ď Cl,ψ |ξ|´2 (5.6) for ď k ď l One can note also that mk P L1 pRq for every k and |m q k pξq| ď Cl,ψ We will separate (5.4) into two parts: kt px, yq “ c2l pI1 ´ I2 q with ´ nx I1 “ e ż8 R pchv ´ chRq and ´ nx I2 “ e ż8 R pchv ´ chRq l´ n l ÿ k“0 l´ n l ÿ k“0 qk,l pvqm q k pt ` vqdv qk,l pvqp´1qk`1m q k pt ´ vq dv (5.7) (5.8) Bounds for }kt }1 will be derived from estimates of the integrals of I1 and I2 over a certain set tpx, yq P G : Rpx, yq P rt ` a, t ` bsu These estimates are eventually reduced to the following Lemmas 5.4 and 5.5, preceded by a short calculation in Lemma 5.3 NORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 19 5.2 Integration lemmas Lemma 5.3 For α ą ´1 and β ą α, we have ż8 pev ´ 1qα e´βv dv “ Bpα ` 1, β ´ αq ă 8, where Bpă, ăq is the beta function Proof This is a direct computation: ż8 ż8 v α ´βv pe ´ 1q e dv “ p1 ´ e´v qα e´pβ´α´1qv e´v dv 0 ż1 “ p1 ´ xqα xβ´α´1 dx “Bpα ` 1, β ´ αq ă Lemma 5.4 For any a ă b, m ě and t ą maxp|a|, bq ż x`R e´n pt ` Rq´m dxdy ď Cn,a,b t1´m RPrt`a,t`bs Proof As the integrand depends on x and |y| only, we can pass to (n-dimensional) polar coordinates in y “ pr, Φq For x, r P R, denote R1 px, rq “ Rpx, pr, 0qq, and Fa,b “ tpx, rq P R2 : R1 px, rq P rt ` a, t ` bsu We obtain ż x`R Ia,b “ e´n pt ` Rq´m dxdy RPrt`a,t`bs ż x`R1 “ Vn e´n pt ` R1 q´m r n´1 dxdr, Fa,b where Vn is the volume of the unit ball in Rn By (2.1) and inequalities just after it, the integral can be bounded by ż x`R1 n´1 e´ t´m dxdr Ia,b ď 2 Vn F ż a,b x`t`a n´1 e´ t´m dxdr ď 2 Vn Fa,b Again by (2.1), if px, rq` P Fa,b˘, then |x| ď R1 px, rq ď t ` b, and ď r ď ? ` x`R 1˘ ď exp x`t`b This implies that exp 2 ` ˘ ż ż ż exp x`t`b x`t`a x`t`a e´ t´m dxdr ď t´m e´ drdx Fa,b |x|ďt`b ż b´a b´a ´m e dx “ 4e t´m pt ` bq “ 2t |x|ďt`b ď 8e b´a t1´m 20 R AKYLZHANOV, YU KUZNETSOVA, M RUZHANSKY, AND H ZHANG (since we suppose b ă t) Lemma 5.5 For a ă b and t ą maxp1, 2|a|, bq, there exists a constant Cn,a,b ą such that ż x`R e´n dxdy ě Cn,a,b t t`aďRďt`b Proof Denote Ga,b “ tpx, yq P G : Rpx, yq P rt ` a, t ` bsu From (2.1), we have |y|2 “ 2ex pchR ´ chxq This formula implies that for px, yq such that |x| ď t ` a and 2ex pchpt ` aq ´ chxq ď |y|2 ď 2ex pchpt ` bq ´ chxq, Rpx, yq is between t ` a and t ` b, so that Ga,b contains for each x a “thick sphere” with |y| changing according to the bounds above (under condition |x| ď t ` a) Again denoting by Vn the volume of the unit ball in Rn , we can estimate the volume of this “sphere” as follows: ˘n{2 ` ˘n{2 ‰ n x “` V “ Vn 2 en chpt ` bq ´ chx ´ chpt ` aq ´ chx ` ˘n ` ˘n chpt ` bq ´ chx ´ chpt ` aq ´ chx n x2 “ cn e ` ˘n{2 ` ˘n{2 chpt ` bq ´ chx ` chpt ` aq ´ chx ` ˘n´1 x pchpt ` bq ´ chpt ` aqq n chpt ` aq ´ chx ě cn en ` ˘n{2 chpt ` bq ` ˘` ˘n´1 cn x n ě en e´ pt`bq chpt ` bq ´ chpt ` aq chpt ` aq ´ chx Since ´2t ´ a ´ b “ ´2pt ` aq ` a ´ b ď a ´ b ă 0, we can continue as 1 chpt ` bq ´ chpt ` aq “ et peb ´ ea qp1 ´ e´2t´a´b q ě et peb ´ ea qp1 ´ ea´b q, 2 and ˘n´1 x n ` V ě cn,a,b en ep1´ qt chpt ` aq ´ chx ż The integral thus can be bounded by: ż ˘n´1 x`t`b ` x ´n x`R p1´ n qt 2 e en e´n chpt ` aq ´ chx dxdy ě cn,a,b e dx |x|ďt`a Ga,b “ c1n,a,b ep1´nqt ż t`a ` ˘n´1 chpt ` aq ´ chx dx ż t`a ” ın´1 t`a e p1 ´ ex´t´a qp1 ´ e´x´t´a q “ c1n,a,b ep1´nqt dx ż t`a ě cn,a,b p1 ´ ex´t´a qn´1 dx NORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 21 By assumption t ą |a|, so that t{4 ă t ` a If ď x ď t{4, then x ´ t ´ a ď p´t{2 ´ aq ´ t{4 ď ´t{4, so that ż ż t{4 ´n x`R 2 dxdy ě c e p1 ´ e´t{4 qn´1 dx ě c3 n,a,b a,b,n t, Ga,b where in the last inequality we use the assumption t ą 5.3 Estimates of I1 and I2 5.3.1 Estimates of I1 We are interested in R ě 1, so that in (5.7) we have v ě By Lemma 5.1, |qk,l pvq| ď Cl e´lv with some constant Cl Together with (5.6), we have ż8 l ˇ ˇ ÿ n ˇ ˇ ´ nx l´ |I1 | ď e q k pt ` vqˇdv pchv ´ chRq ˇqk,l pvqm R k“0 ż8 nx n pchv ´ chRql´ e´lv pt ` vq´2 dv R ż8 nx n ´ ´2 ď Cl,ψ e pchpv ` Rq ´ chRql´ e´lv´lR dv pt ` Rq ď Cl,ψ e´ One can transform 2pchpv ` Rq ´ chRq “ pev ´ 1qpeR ´ e´v´R q “ pev ´ 1qeR p1 ´ e´v´2R q, and estimate ´ nx ż8 n ppev ´ 1qeR ql´ e´lv´lR dv ż8 npx`Rq n ´ ´2 “ Cl,ψ e pt ` Rq pev ´ 1ql´ e´lv dv |I1 | ď Cl,ψ e pt ` Rq ´2 By Lemma 5.3, the integral converges (and depends only on n), so that |I1 | ď Cn,ψ e´ npx`Rq pt ` Rq´2 Now by Lemma 5.4, for a ă b and t ą maxp|a|, bq ż |I1 |dxdy ď Cl,ψ t´1 (5.9) (5.10) RPrt`a,t`bs 5.3.2 Simplifications of I2 We can now pass to the second term: ż8 l ÿ ´ nx l´ n 2 qk,l pvqp´1qk`1m q k pt ´ vq dv I2 “ e pchv ´ chRq R ´ nx “e ż8 k“0 pchpv ` Rq ´ chRq l´ n l ÿ k“0 qk,l pv ` Rqp´1qk`1m q k pt ´ R ´ vq dv Its analysis will pass through several stages of simplification By Lemma 5.2, |qk,l pvq ´ ak,l e´vl | ď bl e´2vl 22 R AKYLZHANOV, YU KUZNETSOVA, M RUZHANSKY, AND H ZHANG Using this, we can replace I2 by ż8 l ÿ n ´ nx ak,l e´pv`Rql p´1qk`1m q k pt ´ R ´ vq dv I3 :“ e pchpv ` Rq ´ chRql´ k“0 so that ´ nx |I2 ´ I3 | ď e ď ż8 nx c1l e´ pchpv ` Rq ´ chRq ż8 l´ n l ˇ ˇ ÿ ˇ ´pv`Rql ˇ q k pt ´ R ´ vq| dv ˇqk,l pv ` Rq ´ ak,l e ˇ|m k“0 pchpv ` Rq ´ chRq ´2pv`Rql l´ n e l ÿ k“0 |m q k pt ´ R ´ vq| dv If l “ 0, then I3 is exactly equal to I2 since q0,0 ” a0,0 “ For l ą 0, we need to show that |I3 ´ I2 | is small enough By definition, n n n n n pchpv ` Rq ´ chRql´ “ 2 ´l pev ´ 1ql´ eRpl´ q p1 ´ e´v´2R ql´ Since l ´ n ď and R ě 1, n n p1 ´ e´v´2R ql´ ď p1 ´ e´2 ql´ ř q k pt ´ R ´ vq| ď Cn,ψ Next, since every mk is in L1 , we have lk“0 |m These, together with Lemma 5.3, yield ż8 n n ´ nx |I2 ´ I3 | ď Cn,ψ,l e pev ´ 1ql´ e´2pv`Rql`Rpl´ q dv ż8 npx`Rq n ´Rl ´ pev ´ 1ql´ e´2vl dv ď Cn,ψ,l e ` n n ˘ npx`Rq “ Cn,ψ,l B l ´ ` 1, l ` e´ ´Rl , 2 which has, by Lemma 5.4, the integral bounded as follows (with the same assumptions on t as in the lemma): ż ż npx`Rq ´pt`aql e´ dxdy |I2 ´ I3 |dxdy ď Cn,ψ e t`aďRďt`b t`aďRďt`b ´tl ď Cn,ψ e t (5.11) Our next step is to replace I3 by ż8 l ÿ n ´l Rpl´n{2q v l´ n ´ nx 2 ak,l e´pv`Rql p´1qk`1m q k pt ´ R ´ vq dv e pe ´ 1q I4 :“ e k“0 If l “ n2 , this is an exact equality If l ´ n2 “ ´ 12 , we have to estimate ˇ ˇ n ˇ ´l v l´ n Rpl´ n qˇ l´ n 2 2 ´ pe ´ 1q e D :“ ˇpchpv ` Rq ´ chRq ˇ ˇ ˇ n n n ˇ nˇ “ 2 ´l pev ´ 1ql´ eRpl´ q ˇ1 ´ p1 ´ e´v´2R ql´ ˇ NORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 23 Set z “ e´v´2R By the choice of R and v, we have z P p0, e´2 s In this interval, the function f pzq “ p1 ´ zq´1{2 has a derivative f pzq “ p1 ´ zq´3{2 bounded by f pe´2 q, so that ˇ ˇ ˇ ˇ ´v´2R l´ n ´ p1 ´ e q ˇ ˇ “ |f p0q ´ f pzq| ď f pe´2 q z “ f pe´2 q e´v´2R This implies that n n D ď Cn pev ´ 1ql´ eRpl´ q e´v´2R and ´ nx |I3 ´ I4 | ď Cn e ż8 n n pev ´ 1ql´ eRpl´ q e´v´2R e´pv`Rql ż8 ´ npx`Rq ´2R ď Cn,ψ,l e pev ´ 1q l´ n k“0 e´pl`1qv dv ˘ npx`Rq n n “ Cn,ψ B l ´ ` 1, ` e´ ´2R , 2 with the integral bounded by ż ż ´2pt`aq |I3 ´ I4 |dxdy ď Cn,ψ e ` l ÿ e´ |ak,l ||m q k pt ´ R ´ vq|dv npx`Rq dxdy t`aďRďt`b t`aďRďt`b ď Cn,ψ e´2t t (5.12) 5.3.3 Estimates of I4 It remains now to estimate I4 from below Recall that ż8 l ÿ n ´ npx`Rq ak,l e´vl p´1qk`1m q k pt ´ R ´ vq dv pev ´ 1ql´ I4 “ cn e k“0 Set Mpξq “ l ÿ k“0 ak,l p´1qk`1m q k pξq; this is the inverse Fourier transform of l ÿ ak,l p´1qk`1sk`1 ψpsqIr0,`8q Ψpsq “ k“0 By assumption, ψ is not identically zero on r0, `8q; it is multiplied by a nonzero polynomial since al,l ‰ by Lemma 5.2, so finally the product Ψ is not identically zero Consider the integral entering in I4 , ż8 n Ipξq “ pev ´ 1ql´ e´vl Mpξ ´ vq dv (5.13) n as the convolution of M and N pvq :“ pev ´ 1ql´ e´vl 1p0,8q pvq The q For function I is continuous, and its inverse Fourier transform is ΨN every m, we can estimate ż ż1 ż8 n m v l´ n |N pvqv |dv ď pe ´ 1q dv ` Cn e´v v m dv R 24 R AKYLZHANOV, YU KUZNETSOVA, M RUZHANSKY, AND H ZHANG ď Cn1 ` Cn ´ ¯m`1 ż n n{2 e´x xm dx ď Cn2 K m m! q pmq }8 ď Cn K m m!, so by [27, with K “ maxp1, 2{nq It follows that }N q is analytic in a strip |z| ă δ with some δ ą As it is clearly 19.9] N q is not identically zero not identically zero, we can conclude that ΨN too There exist therefore α, β P R, α ă β, A ‰ and ă ε ă |A|{2 (all these constants depend on ψ and on n) such that |Ipξq ´ A| ă ε for ξ P rα, βs It follows that for R P rt ´ β, t ´ αs |I4 | ą Cn,ψ e´ npx`Rq and by Lemma 5.5, with a “ ´β, b “ ´α, ż |I4 |dxdy ě Cn,ψ t, (5.14) (5.15) RPrt`a,t`bs once t ą maxp1, 2|a|, bq 5.4 Conclusions We can now summarize the obtained estimates in the following theorem Theorem 5.6 Let ψ P C0 pRq be a function which is not identically zero on r0, `8q and twice differentiable with }ψ pjq psqsk }8 ď Cψ for ď j ď 2, ď k ď n{2 ` Let kt be the convolution kernel of one of the following operators: ? ? Et “ ψp Lq exppit Lq, ? ? Ct “ ψp Lq cospt Lq, ? ? sinpt Lq St “ ψp Lq ? L Then there exists a constant Cn,ψ depending on n and on ψ such that for t ą sufficiently large, }kt }1 ě Cn,ψ t Proof Case 1: kt is the kernel of Et By the results of Section 5.3.3, there exist a, b P R such that (5.15) holds, for t ą maxp1, 2|a|, bq For the same a, b, t we have, with (5.10): ż ´ż 1¯ cl |I1 ´ I2 |dxdy ě Cn,ψ |I2 |dxdy ´ }kt }1 ě RPrt`a,t`bs t RPrt`a,t`bs Set γ0 “ and γl “ if l ‰ We can continue, using the results above, as ´ż (5.11) 1¯ }kt }1 ě Cn,ψ |I3 |dxdy ´ γl e´lt t ´ t RPrt`a,t`bs NORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR (5.12) ě Cn,ψ (5.15) ´ż RPrt`a,t`bs ´ ´2t ě Cn,ψ t ´ e |I4 |dxdy ´ e´2t t ´ γl e´lt t ´ ´lt t ´ γl e 1¯ t´ t 25 1¯ t For t big enough, this is clearly bounded from below by Cn,ψ t, as claimed Case 2: kt is the kernel of Ct We represent cospitsq as 21 peits `e´its q; in the integral (5.4), instead of r m q k pt ` vq ´ p´1qk m q k pt ´ vqs we then obtain ‰ 1“ m q k pt ` vq ` m q k pv ´ tq ´ p´1qk m q k pt ´ vq ´ p´1qk m q k p´t ´ vq cl pI1 We separate then kt “ ´ nx I1 “ e I2 “ e´ ż8 R nx ż8 R pchv ´ chRq ´ I2 q into l´ n l ÿ k“0 n pchv ´ chRql´ l ÿ k“0 qk,l pvqrm q k pt ` vq ´ p´1qk m q k p´t ´ vqsdv, qk,l pvqrp´1qk`1m q k pt ´ vq ` m q k pv ´ tqs dv As |m q k p´t ´ vq| ď Cn,ψ pt ` vq´2 , the estimate for I1 remains the same The passage from I2 to I4 does not change either In I4 we still have (5.13), but with Mpξq “ l ÿ k“0 “ ‰ ak,l m q k p´ξq ` p´1qk`1m q k pξq If we denote by m ˜ k the function m ˜ k psq “ mk p´sq, then M is the inverse Fourier transform of Ψpsq “ l ÿ k“0 “ ‰ ak,l m ˜ k psq ` p´1qk`1mk psq which is for s ě the same as in Case 1, so that it is not identically zero for similar reasons as above This implies that I is not everywhere zero, so that we arrive at the same conclusion: }kt }1 ě Cψ,n t, for t big enough Case 3: kt is the kernel of St In (5.4), we obtain m q k pt ` vq ´ m q k pv ´ tq ´ p´1qk m q k pt ´ vq ` p´1qk m q k p´t ´ vq, but mk changes to mk psq “ sk ψpsqIr0,`8q and not with sk`1 As a consequence, we cannot integrate by parts twice as in (5.5) but only once; this changes the estimate (5.6) as to |m q k pξq| ď Cn,ψ |ξ|´1 (5.16) 26 R AKYLZHANOV, YU KUZNETSOVA, M RUZHANSKY, AND H ZHANG Let us write down the two parts of kt , up to the constant cl {4: ż8 l ÿ ´ nx l´ n 2 pchv ´ chRq I1 “ e qk,l pvqrm q k pt ` vq ` p´1qk m q k p´t ´ vqsdv, R ´ nx I2 “ e ż8 R k“0 pchv ´ chRq l´ n l ÿ k“0 qk,l pvqrp´1qk`1m q k pt ´ vq ´ m q k pv ´ tqs dv With (5.16) instead of (5.6), we obtain ż |I1 |dxdy ď Cn,ψ RPrt`a,t`bs The estimates of |I2 ´ I3 | and |I3 ´ I4 | not change since we are using only the fact mk P L1 pRq which remains true In I4 we get (5.13) with Mpξq “ l ÿ k“0 “ ‰ ak,l ´ m q k p´ξq ` p´1qk`1m q k pξq , and complete the proof as in Case It is well known that L1 -norm of a function f is also the norm of the convolution operator g ÞĐ g ˚ f on L1 pGq We get as a corollary that upper norm estimates of [23] for Ct , St are sharp: }Ct }L1 ÑL1 — t, }St }L1 ÑL1 — t, as t Ñ Our results are valid in particular for ψpsq “ p1 ` s2 q´α , α ą n{2 ` 5.5 Lower bounds for uniform norms In [23, Corollary 7.1], Mă uller and Thiele show that?for a function ψ P C0 pRq supported in ra, bs, the ? kernel kt of Ct “ ψp Lq cospt Lq has its uniform norm bounded by a constant, with no decay in t as t Ñ `8 With a simple modification, this result can be extended to functions which are not compactly supported, but decaying quickly enough From the results obtained in Sections 5.3-5.4, it follows that this estimate is actually sharp Theorem 5.7 In the assumptions and notations of Theorem 5.6, there exists a constant Cn,ψ depending on n and on ψ such that for t ą sufficiently large, }kt }8 ě Cn,ψ Proof We can consider the cases of Et and Ct together Case 1: kt is the kernel of Et or Ct By the results of Section 5.3.3, there exist a, b P R such that (5.14) holds, for t ą maxp1, 2|a|, bq It is clear then that we should consider x “ ´R, y “ Arguing similarly to Theorem 5.6, we obtain the estimate ´ 1¯ |kt p´R, 0q| ě Cn,ψ ´ e´2t ´ γl e´lt ´ , t NORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 27 which is bounded from below by Cn,ψ , as claimed Case 2: kt is the kernel of St As noted in the proof of Theorem 5.6, we get the estimate (5.16) instead of (5.6) This changes the estimate of I1 as |I1 p´R, 0q| ď Cn,ψ t for t large enough The other estimates are as in Case 1, so that we arrive at ´ 1¯ , |kt p´R, 0q| ě Cn,ψ ´ e´2t ´ γl e´lt ´ t which is bounded from below by a constant Cn,ψ ą as t Ñ `8 As pointed out in [23], this shows that no dispersive L1 ´L8 estimates hold for the wave equation Acknowledgments Yu K thanks Professor Waldemar Hebisch for valuable discussions on multipliers on Lie groups This work was started during an ICL-CNRS fellowship of the second named author at the Imperial College London Yu K is supported by the ANR-19-CE40-0002 grant of the French National Research Agency (ANR) H Z is supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 754411 R A was supported by the EPSRC grant EP/R003025 M R is supported by the EPSRC grant EP/R003025/2 and by the FWO Odysseus grant G.0H94.18N: Analysis and Partial Differential Equations References [1] G Alexopoulos, Spectral multipliers on Lie groups of polynomial growth, Proc Amer Math Soc 120 (1994), no 3, 973–979 [2] J.-Ph Anker, Lp Fourier multipliers on Riemannian symmetric spaces of the non-compact type, Ann of Math 132 (1990), 597–628 [3] J.-Ph Anker, Sharp estimates for some functions of the laplacian on noncompact symmetric spaces, Duke Math J., Vol 65, No 2, pp 257–297 [4] J.-Ph Anker, L Ji: Heat kernel and Green function estimates on noncompact symmetric spaces, Geom Funct Anal (1999), 1035–1091 [5] J.-Ph Anker, V Pierfelice, M Vallarino, Schrăodinger equations on Damek-Ricci spaces, Comm Part Diff Eq 36 (2011), 976–997 [6] J.-Ph Anker, V Pierfelice, M Vallarino, The wave equation on hyperbolic 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CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 29 Rauan Akylzhanov, School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom Email address: akylzhanov.r@gmail.com Yulia Kuznetsova, University Bourgogne Franche-Comt´ e, 16 route de Gray, 25030 Besanc ¸ on, France Email address: yulia.kuznetsova@univ-fcomte.fr Michael Ruzhansky, Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, B 9000 Ghent, Belgium, and School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom Email address: Michael.Ruzhansky@ugent.be Haonan Zhang, Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria Email address: haonan.zhang@ist.ac.at ... also to estimate uniform norms of k˜ψ In Section 4, we show that this method makes it an easy calculation to obtain exact asymptotics of the uniform norms We find first the exact norms of the heat... left-invariant operator on L2 pG, mr q belongs to the right von Neumann algebra V NR pGq, defined as the strong operator closure of the set of right translation operators This applies, in particular, to... Laplace operator and show the relation between several apparently different approaches to its calculation 3.1 Plancherel measure and the Harish-Chandra c-function The Plancherel measure for the