Norms of certain functions of the Laplace operator on the ax+b groups
Rauan Akylzhanov, Yulia Kuznetsova, Michael Ruzhansky, Haonan Zhang
aa r X i v : . [ m a t h . F A ] J a n NORMS OF CERTAIN FUNCTIONS OF THE LAPLACEOPERATOR ON THE ax ` b GROUPS
RAUAN AKYLZHANOV, YULIA KUZNETSOVA, MICHAEL RUZHANSKY,AND HAONAN ZHANG
Abstract.
The aim of this paper is to find new estimates for thenorms of functions of the (minus) Laplace operator L on the ‘ ax ` b ’groups.The central part is devoted to spectrally localized wave propa-gators, that is, functions of the type ψ p? L q exp p it ? L q , with ψ P C p R q . We show that for t Ñ `8 , the convolution kernel k t ofthis operator satisfies } k t } — t, } k t } — , so that the upper estimates of D. M¨uller and C. Thiele (StudiaMath., 2007) are sharp.As a necessary component, we recall the Plancherel density of L and spend certain time presenting and comparing different ap-proaches to its calculation. Using its explicit form, we estimateuniform norms of several functions of the shifted Laplace-Beltramioperator ˜∆, closely related to L . The functions include in partic-ular exp p´ t ˜∆ γ q , t ą , γ ą
0, and p ˜∆ ´ z q s , with complex z, s . Introduction
Let G denote the ‘ ax ` b ’ group of dimension n ě
1, parameterizedas G “ tp x, y q P R ˆ R n u , with multiplication given by p x, y q ¨ p x , y q “ p x ` x , y ` e x y q ;in the case n “
1, it is the group of affine transformations of thereal line. In this form, the right Haar measure is just m r “ dxdy . Thisgroup is well-known to be non-unimodular, solvable, and of exponentialgrowth; the modular function is given by δ p x, y q “ e ´ nx .Let X “ BB x and Y k “ e x BB y k , 1 ď k ď n , denote left-invariantvector fields on G , given also by exp p tX q “ p t, q , exp p tY k q “ p , te k q , t P R (where p e k q is the standard basis in R n ). We consider the (minus)Laplace operator L “ ´ X ´ ř nk “ Y k . A detailed exposition of thissetting can be found in [18]. L is a positive self-adjoint operator on L p G, m r q , and for any boundedBorel function ψ on R one can define, by the spectral theorem, abounded operator ψ p L q on L p G, m r q . For ψ good enough, this is a(right) convolution operator with a kernel k ψ , for which an explicit formula is available [21, Proposition 4.1]. There has been much workto determine in which cases ψ p L q is also bounded from L p p G, m r q to L q p G, m r q for different p and q (see [9] and references therein, or morerecently [10, 16, 21]).Let us put this into a wider context. If we consider a connected Liegroup G as a Riemannian manifold, endowed with the left-invariantdistance, we obtain the Laplace–Beltrami operator ∆ on G . In theunimodular case, ∆ equals to the Laplacian L defined as above, thatis, L “ ´ ř k X k where p X k q is a basis of the Lie algebra of G . If G is non-unimodular, however, these two operators are different and canhave significantly different behaviour.By the classical H¨ormander-Mikhlin theorem, a function of the Lapla-cian m p ∆ q is a multiplier on L p p R n q if r n { s` m decreasequickly enough. One can observe a similar behaviour on any group ofpolynomial growth [1]. On the other hand, on symmetric spaces ofnon-compact type, a class which includes the ‘ax+b’ groups and im-plies exponential growth, the function m must be holomorphic in astrip around the spectrum of ∆ in order to generate an L p -multiplier, p ‰ L as defined above: on the AN groups, in particular on ‘ax+b’ groups, a finite number of deriva-tives is sufficient for an L p multiplier [14, 10, 16]. One can ask thenwhat is the class of groups carrying Laplacians of different kind, as the‘ax+b’ groups do. One could conjecture that these are solvable groupsof exponential growth; this has been disproved however by an exampleof a semidirect product of R with the p n ` q -dimensional Heisenberggroup H n [7]. This question remains open, and the criterium might bethe symmetry of the group algebra L p G q [7].Looking closer at the multipliers, one can ask how far goes theressemblance between the Laplacian L of “H¨ormander–Mikhlin type”and that of R n . In particular, in applications to PDEs oscillating func-tions of the type A t “ ψ p? L q cos p t ? L q , B t “ ψ p? L q sin p t ? L q? L are of special importance. With ψ ”
1, they give the solutions of thewave equation: the function u t “ A t f ` B t g solves the Cauchy problem B u B t “ ´ L u,u p , x q “ f p x q , B u B t p , x q “ g p x q , for a priori f, g P L p G q . One cannot of course apply the H¨ormander–Mikhlin theorem in this case. By other methods, one shows that the ORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 3 wave propagators cos p t ? ∆ q , sin p t ? ∆ q? ∆ are bounded on L p p R n q if andonly if ˇˇ p ´ ˇˇ ă n ´ [23], so that for a general p a “localizing” factor ψ should be introduced.With ψ p s q “ p ` s q ´ α , if the norms of A t , B t decrease in time, onecan set } f } H α,p “ }p ` ∆ q α f } p and get so called dispersive estimatesin terms of Sobolev norms of f and g . In R n , n ą } u t } ď Ct ´p n ´ q{ p} f } H α, ` } g } H α, q , for α big enough. More on this topic can be found in [5, 26].For the (shifted) Laplace–Beltrami operator on hyperbolic spaces, inparticular for the ax ` b groups, dispersive estimates have been obtainedby D. Tataru [27], with exponential decay in time. See also [20, 24, 4].In this context it was surprising that D. M¨uller and C. Thiele [21] didnot get any decay in time for the Laplace operator on the ax ` b group:they show that the L norms of the kernels of A t , B t are bounded by C p ` | t |q , for ψ P C p R q quickly decreasing, while the uniform normsof the same kernels are just bounded by a constant, for a compactlysupported ψ .In the present paper, we prove that these estimates are actuallysharp: if ψ is a nonzero function with certain decrease rate, then thekernels k t of both A t and B t satisfy } k t } ě C n,ψ t and } k t } ě C n,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 ψ p s q “p ` s q ´ α . This means that dispersive L ´ L estimates do not holdfor the Laplacian L on the ax ` b groups. The main part of the proofsis concentrated in Section 5.The rest of the paper is organized as follows. In Section 2, we collectnecessary notations and conventions.To explain the sequel, let us denote by ˜∆ the shifted Laplace–Beltrami operator (more details in Section 3). The two Laplaciansare linked by ˜∆ “ δ ´ { L δ { (here δ stands for multiplication by themodular function), and the kernels of ψ p ˜∆ q and ψ p L q are related as˜ k ψ “ δ ´ { k ψ , both acting by right convolution.It is known that the L -norm of ˜ k ψ is the same as of k ψ and is givenby the integral } k ψ } L p G,m r q “ ż R | ψ | ρ with a density ρ expressed via the Harish–Chandra c -function [17, 9].This appears as a building block in several multiplier estimates [9, 15,16]. In Section 3, we take time to write down explicit formula for thespectral density of the Laplacian and to relate to each other severalways to obtain it. We put forward the fact, seemingly not discussed R. AKYLZHANOV, YU. KUZNETSOVA, M. RUZHANSKY, AND H. ZHANG before, that the same density can be used also to estimate uniformnorms of ˜ k ψ .In Section 4, we show that this method makes it an easy calculationto obtain exact asymptotics of the uniform norms. We find first theexact norms of the heat kernels (of course well known), and then moregenerally of exp p´ t ˜∆ γ q , γ ą
0. Further we pass to a more technicaltask of estimating the norms of p ˜∆ ´ z q ´ s with z, s complex ( z outsideof the spectrum of ˜∆, and ℜ s ą p n ` q{ p ˜∆ ´ z q ´ s (with z outside of the positivehalf-line) is bounded if and only if ℜ s ě p n ` q{ s ‰ p n ` q{
2. Goingrather explicitly into integration with the help of special functions, weget estimates of actual uniform norms of these functions in the sameregion except for the border, and obtain asymptotic bounds for ℑ z Ñ | z | Ñ 8 .The last Section 5 contains, as mentioned above, the lower estimatesof L and uniform norms of the operators A t , B t above.2. Notations and conventions
We have chosen to work (mainly) with the right Haar measure m r “ dxdy and the right convolution: p f ˚ r g qp x q “ ż G f p xy q g p y ´ q δ p y q dm r p y q . One can alternatively opt to the left Haar measure m l “ e ´ nx dx dy , sothat m r “ δ ´ m l , and the left convolution p f ˚ l g qp x q “ ż G f p y q g p y ´ x q dm l p y q . For a function f on G , denote q f p g q “ f p g ´ q . For every p , the map τ : f ÞÑ q f is an isometry from L p p G, m r q to L p p G, m l q (or back). If R k : f ÞÑ f ˚ r k is a right convolution operator with the kernel k , then τ R k τ “ L q k is a left convolution operator with the kernel q k , and } L q k : L p p G, m l q Ñ L p p G, m l q} “ } R k : L p p G, m r q Ñ L p p G, m r q} . This means that choosing one or another convention changes nothingin norm estimates.Below we do not use the symbol ˚ r anymore and write just ˚ instead.We will use } ¨ } p and L p p G q to denote } ¨ } L p p G,m r q and L p p G, m r q ,respectively. The distance . The left-invariant Riemannian distance on G is givenby d ` p x, y q , p , q ˘ “ : R p x, y q , where R p x, y q “ arcch ´ ch x ` | y | e ´ x ¯ . (2.1) ORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 5
This implies in particular that | x | ď R and | y | “ e x p ch R ´ ch x q ď e x ` R . The Plancherel weight . Every bounded left-invariant operator on L p G, m r q belongs to the right von Neumann algebra V N R p G q , definedas the strong operator closure of the set of right translation operators.This applies, in particular, to ψ p L q with a bounded function ψ . Sim-ilarly, every right-invariant operator on L p G, m l q belongs to the leftvon Neumann algebra V N L p G, m l q .On V N L p G q , we have the Plancherel weight ϕ (see more in [22]),which can be viewed as an integral of operators. For an operator L k ofleft convolution with the kernel k P L p G, m l q X L p G, m l q , one has ϕ p L ˚ k L k q “ } k } L p G,m l q . If k “ g ˚ ˚ g for some g and is moreover continuous, then ϕ p L k q “ k p e q .Almost all the literature on Plancherel weights assumes this V N L p G q convention, but one can also consider a similar weight ϕ r on V N R p G q by setting ϕ r p R k R ˚ k q “ } k } L p G,m r q (2.2)for an operator of right convolution with the kernel k . The two algebrasare isomorphic by A ÞÑ τ Aτ , and ϕ r p A q “ ϕ p τ Aτ q . This isomorphismtransfers also, as mentioned above, R k P V N R p G q to L q k P V N L p G q ,where q k p g q “ k p g ´ q .3. Spectral measure of the Laplacian
The aim of this section is to write explicitly the Plancherel measure ofthe Laplace operator and show the relation between several apparentlydifferent approaches to its calculation.3.1.
Plancherel measure and the Harish-Chandra c -function. The Plancherel measure for the spherical transform on a connectedsemisimple Lie group is given by a so called c -function found by Harish-Chandra. Many faces of this function are described in an excellentsurvey [17], of which we will need only a few facts.Recall first that the Laplace–Beltrami operator ∆ has a spectralgap: its spectrum is r σ, `8q where σ is a constant depending on thegroup, σ “ n in our case. For this reason, it is of course more regularthan L , so it makes sense to consider the shifted operator ˜∆ “ ∆ ´ σ which has r , `8q as its spectrum. The two operators are now linkedby ˜∆ “ δ ´ { L δ { (here δ stands for multiplication by the modularfunction).Next, it is known that the L -norm of a radial function on G canbe expressed via the Harish-Chandra c -function [17, 9]. This can be R. AKYLZHANOV, YU. KUZNETSOVA, M. RUZHANSKY, AND H. ZHANG applied to the convolution kernels of ˜∆ and functions of it, as thesekernels are radial. If ˜ k f is the kernel of f p ˜∆ q , we have } ˜ k f } “ c G ż R | f p λ q| | c p λ q| ´ dλ. If k f is the right convolution kernel of f p L q , then [9] k f “ δ { ˜ k f and } k f } L p G,m r q “ } ˜ k f } L p G,m r q (this equality is verified by a direct calcula-tion, knowing that ˜ k f p x ´ q “ ˜ k f p x q ), so that the formula above is validfor k f as well. An explicit formula of the c -function is known, and forthe n -dimensional ‘ax+b’ group it is as follows [17]: for λ P R , c p λ q “ c ´ iλ Γ p iλ q Γ p p n ` ` iλ qq Γ p p n ` iλ qq . Since Γ p z q Γ p z ` q “ ? π ´ z Γ p z q [6, 1.2], this simplifies up to c p λ q “ c Γ p iλ q Γ p n ` iλ q , with c “ π ´ { n { ´ c . Denote ρ p u q “ c u ´ { | c p? u q| ´ , so that } k f } “ C ż R | f p u q| ρ p u q du (we will also write ρ n to indicate the dimension). If n “ l is even, 1 { c is a polynomial: 1 c p λ q “ c ´ l ´ ź j “ p j ` iλ q , so that ρ l p u q “ ? u l ´ ź j “ p j ` u q . If n “ l ` c p λ q “ c l ´ ź j “ p j ` ` iλ q Γ p ` iλ q Γ p iλ q , and ρ l ` p u q “ u ´ { l ´ ź j “ ´` j ` ˘ ` u ¯ | Γ p ` i ? u q| | Γ p i ? u q| , so that ρ l ` p u q “ l ´ ź j “ ` p j ` { q ` u ˘ ρ p u q . Moreover, the reflectionformula Γ p z q Γ p ´ z q “ π { sin p πz q and the conjugation identity Γ p z q “ Γ p ¯ z q imply that for real v , | Γ p ` iv q| | Γ p iv q| “ ´ πiv sin p πiv q π cos p πiv qq “ v sh p πv q ch p πv q , ORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 7 and ρ l ` p u q “ l ´ ź j “ ´` j ` ˘ ` u ¯ sh p π ? u q ch p π ? u q . These formulas imply that ρ p u q „ u p n ´ q{ as u Ñ `8 , and ρ p u q „ u { as u Ñ
0. These latter estimates have been used in application tothe norm estimates of the Laplacian and its functions in [9, 15, 16].3.2.
Uniform norms of the kernels.
Let us consider a right convo-lution operator on L p G, m r q , R k : f ÞÑ f ˚ k (all convolutions here aretaken with respect to the right Haar measure). Its adjoint is R ˚ k “ R k ˚ ,where the (right) involution is defined as f ˚ p x q “ f p x ´ q δ p x q . Thecomposition of a pair of operators is R k R h “ R h ˚ k .Suppose now that 0 ď f “ | g | and k f , k g are the convolution kernelsof f p L q , g p L q , with k g P L p G q . We have then f p L q “ g p L q ˚ g p L q “ g p L q g p L q ˚ , so that k f “ k g ˚ k ˚ g “ k ˚ g ˚ k g and k f p e q “ ż G k g p y q k ˚ g p y ´ q δ p y q dm r p y q“ ż G k g p y q k g p y q δ p y ´ q δ p y q dm r p y q “ } k g } L p G,m r q . By the results above, k f p e q “ } k g } L p G,m r q “ c ż R | g | ρ “ c ż R f ρ. (3.1)This formula is thus valid for 0 ď f P L p R , ρ q . For f not necessarilypositive, the formula k f p e q “ c ş R f ρ is still valid, by linearity.It is well known that the uniform norm of a positive definite func-tion is attained at the identity. Given a function h P L p G, m r q , theconvolution h ˚ ˚ h is in general not positive definite. However if wemultiply it by δ ´ { , it becomes such, since this is a coefficient of theright regular representation of G on L p G, m r q : p h ˚ ˚ h qp x q δ ´ { p x q “ δ ´ { p x q ż G h p y ´ x ´ q δ p xy q h p y ´ q δ p y q dm r p y q“ δ { p x q ż G h p yx ´ q δ p y ´ q h p y q dm r p y q“ δ { p x q ż G h p z q δ p x ´ z ´ q h p zx q dm r p z q“ ż G h p zx q δ ´ { p zx q h p z q δ ´ { p z q dm r p z q“ x R x p hδ ´ { q , hδ ´ { y r , where x¨ , ¨y r denotes the inner product induced by the right Haar mea-sure. This means that the kernel ˜ k f “ δ ´ { k f of f p ˜∆ q is in this case R. AKYLZHANOV, YU. KUZNETSOVA, M. RUZHANSKY, AND H. ZHANG positive definite. In particular, its uniform norm is attained at theidentity: } ˜ k f } “ ˜ k f p e q “ k f p e q “ c ż R f ρ, (3.2)0 ď f P L p R , ρ q . If f is real-valued, but maybe not positive, then wecan decompose it f “ f ` ´ f ´ into positive and negative part, and getthe following estimate: } ˜ k f } ď } ˜ k f ` } `} ˜ k f ´ } “ ˜ k f ` p e q` ˜ k f ´ p e q “ c ż R p f ` ` f ´ q ρ “ c ż R | f | ρ. (3.3)Finally, for f complex-valued, we have to add a factor of ? } ˜ k f } ď } ˜ k ℜ f } ` } ˜ k ℑ f } ď c ż R p| ℜ f | ` | ℑ f |q ρ ď c ? ż R | f | ρ. Thus, the Plancherel measure helps to calculate not only L , but alsouniform norms.3.3. Connection with the Plancherel weight and L -norms ofthe resolvent kernels. In the case n “ k λ of theresolvent R λ “ p L ´ λ q ´ are 2-summable, as seen from the boundsabove. This allows to obtain the Plancherel measure in a different way.Comparing the formulas (2.2) and (3.1), we notice that ϕ r ` f p L q ˘ “ c ż f dρ, for 0 ď f P L p R , ρ q . This means that the measure ρ can be foundfrom this equality, if we are able to determine ϕ r ` f p L q ˘ .Let A be, in general, a positive self-adjoint unbounded operator on L p G, m r q with the spectral decomposition A “ ş udE p u q . For abounded measurable function f , one defines f p A q “ ż f p u q dE p u q by the spectral theorem for self-adjoint unbounded operators. By con-struction, for all x, y P H “ L p G, m r qx f p A q x, y y r “ ż f p u q dE x,y p u q with respect to the measure E x,y : X ÞÑ x E p X q x, y y , X Ă R Borel. If A is left-invariant, that is, affiliated to the right group von Neumannalgebra V N R p G q , then f p A q P V N R p G q .For a vector state ζ x,x p¨q “ x¨ x, x y on V N R p G q , this gives already itsvalue on f p A q . Any weight ϕ r , and in particular the Plancherel weightis the sum of a family of normal positive functionals [13], ϕ r “ ř α ϕ r,α . ORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 9
Every ϕ r,α can be decomposed into a countable sum of positive vectorstates ϕ r,α “ ř n ζ x n,α ,x n,α , which implies that for positive fϕ r ` f p A q ˘ “ ÿ α ż f p u q dϕ r,α p E p u qq “ ż f p u q dϕ r p E p u qq . Linearity allows to extend this equality to arbitrary f , real or complex.One can find the spectral measure of A as the strong limit [12, XII.2,Theorem 10] E r a,b s “ πi lim ε Ñ ` ż ba ` R λ ` iε ´ R λ ´ iε ˘ dλ, where, as usual, R z “ p A ´ z q ´ is the resolvent of A . In the case of apositive operator, we can get nontrivial values of course only for a ě R λ ` iε ´ R λ ´ iε “ iεR λ ` iε R λ ´ iε “ iεR λ ` iε R ˚ λ ` iε . If we return now to the Laplace operator, then R λ ` iε “ R k λ ` iε is aconvolution operator with a kernel k λ ` iε . By the lower semi-continuityof the right Plancherel weight ϕ r , ϕ r p E r a,b s q ď πi lim ε Ñ ` ż ba iε ϕ r p R k λ ` iε R ˚ k λ ` iε q dλ “ π lim ε Ñ ` ε ż ba } k λ ` iε } L p G,m r q dλ. Using estimates in [19] in the case n “ n , one can show (this is however quite a technical task) thatthis limit is finite if (and only if) n ď
2, and in this case bounded by ϕ r p E r a,b s q ď d n ż ba p ` a | λ |q n ` dλ, which proves that the spectral measure is absolutely continuous withdensity bounded by a polynomial. We see however that this bound isnot sharp.3.4. Application of the explicit formula for the convolutionkernel.
Yet another approach is to use the explicit formula for thekernel [21, Proposition 4.1]: for a function ψ P C p R q , the kernel k ψ of ψ p L q is given, for any integer l ą n ´
1, by k ψ p x, y q “ c l e ´ n x ż ψ p u qr F R p x,y q ,l p? u q ´ F R p x,y q ,l p´? u qs du, (3.4)where R p x, y q is given by (2.1), F R,l p u q “ ż R D l sh ,v p e iuv qp ch v ´ ch R q l ´ n dv, D l sh ,v denotes the l -th composition of D sh ,v , D sh ,v p f q “ ddv ´ f sh v ¯ and c l “ p´ q l ´ ´ n π ´ n iπ Γ p l ` ´ n q . In particular, for x “ y “ ic l P R ) k ψ p e q “ c l ż ψ p u qr F ,l p? u q ´ F ,l p´? u qs du “ ic l ż ψ p u q ż D l sh ,v ` sin p v ? u q ˘ p ch v ´ q l ´ n dv du. As we have seen before, for 0 ď ψ P L p R , ρ q this is also equal to c ş ψρ ,so that (as L p R , ρ q X C p R q is dense in L p R , ρ q ) ρ p u q “ c l c r F ,l p? u q´ F ,l p´? u qs “ ic l c ż D l sh ,v ` sin p v ? u q ˘ p ch v ´ q l ´ n dv. (3.5)One should note that F R,l is not bounded at 0, but the difference aboveis, and tends to 0 as u Ñ n “ l “
2: we have to change the sign as ic l “ ´ π ă
0, and then obtain ´ ` F , p? u q ´ F , p´? u q ˘ “ ´ ż ddv ´ sin p v ? u q sh v ¯ dv “ ? u “ ρ p u q . It is clear that best bounds can be obtained for l chosen so that l ´ n { P t , ´ { u , and below we assume this choice.Denote D l,u p v q “ D l sh ,v p e iuv ´ e ´ iuv q . For l “ D ,u p v q “ i sin p uv q is an odd function of v . By induction, one verifies that every D l,u isodd, too: if f is odd (and analytic), then f p v q{ sh v is even (and welldefined at 0) and D sh ,v p f q “ ddv p f p v q{ sh v q is odd. This implies, inparticular, that D l,u p q “
0. Note at the same time that D l sh ,v p e iuv q alone can be unbounded as v Ñ
0. Thus, the integral below converges: ρ p u q “ ic l c ż D l,u p v qp ch v ´ q l ´ n dv. (3.6)If l “ n ( n even), then ρ p u q “ ic l c ż ddv ´ D l ´ ,u p v q sh v ¯ dv “ ´ ic l c lim v Ñ D l ´ ,u p v q sh v “ ´ ic l c ddv D l ´ ,u ˇˇˇ v “ . Decomposing D l,u into its Taylor series, one can show that D l,u p q “ P ,l p u q is a polynomial of degree 2 l ` ρ n p u q ď C p u { ` u n { q , how-ever already known from the calculations of the c -function. ORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 11 Upper norm estimates
Estimates of L p – L q norms. Having at our disposal uniform and L norms of the kernel k of a convolution operator A allows one toestimate its weighted L p ´ L q norms in the following cases.Suppose that 1 ă p ă q are such that 1 { r : “ { p ´ { q ě {
2. Bythe Young’s inequality (valid for any p, q, r P r , `8s with 1 { q ` “ { r ` { p , which is equivalent to our condition 1 { r “ { p ´ { q ; recallthat we are using the right Haar measure), }p Aδ ´ { r q f } L q p G q “ }p δ ´ { r f q ˚ k } L q p G q ď } k } r } f } p ,f P L p p G q , k P L r p G q , and if r ě
2, we can estimate } Aδ ´ { r } L p Ñ L q ď } k } r ď } k } ´ r } k } r . (4.1)This can be viewed alternatively as the norm of A : L p p G, δ ´ { r m r q Ñ L q p G, m r q .4.2. The heat semigroup.
As an example, let us show how to recoverknown uniform bounds for the heat kernel. Set A t “ e ´ t ˜∆ , and denote(according to tradition) by q t its convolution kernel. For n “ l ` } q t } “ c ż e ´ tu th p π ? u q l ´ ź j “ ´` j ` ˘ ` u ¯ du “ c t ż e ´ x th ´ π c xt ¯ l ´ ź j “ ´` j ` ˘ ` xt ¯ dx. When t Ñ `8 , th ´ π a xt ¯ „ π a xt (on finite intervals, which is suffi-cient) and x { t Ñ
0, so that } q t } „ t ´ { π l ´ ź j “ ` j ` ˘ ż e ´ x ? x dx “ Ct ´ { . When t Ñ
0, th ´ π a xt ¯ Ñ } q t } „ t ż e ´ x ´ xt ¯ l dx “ Ct ´ l ´ “ Ct ´p n ` q{ . For n “ l even, estimates are similar and lead to the same exponentsof t . This agrees with the known bounds for the heat kernel of Daviesand Mandouvalos [11, Theorem 3.1] (of course more general as theyconcern pointwise estimates): uniformly for 0 ď R ă 8 and t ą q t p x, y q — t ´ n ` e ´ R t ´ nR p ` R ` t q n ´ p ` R q . More generally, if A t “ e ´ t ˜∆ γ , with γ ą
0, then for its kernel ˜ k t onegets by similar calculations } ˜ k t } „ C γ t ´ γ , t Ñ `8 ,t ´ n ` γ , t Ñ . Since } ˜ k t } “ } ˜ k t } , the formula (4.1) implies also, for 1 { r “ { p ´ { q ě { } A t } L p p δ ´ { r qÑ L q À C γ t ´ rγ , t Ñ `8 ,t ´ n ` rγ , t Ñ . Rational functions.
Our next aim is to estimate the norm of f p ˜∆ q “ p ˜∆ ´ z q ´ s with z outside of r , `8q . Anker [3] has shown thatthe convolution kernels of these operators are bounded if and only if ℜ s ě n ` . Below, we estimate their actual uniform norms. We getbounds in the range ℜ s ą n ` , which is the same half-plane as in [3],except for the border.Let k z,s denote the convolution kernel of f p ˜∆ q . According to (3.3), } k z,s } ď C n ż ˇˇ p u ´ z q ´ s ˇˇ ρ p u q du. We will estimate ˇˇ p u ´ z q ´ s ˇˇ “ | u ´ z | ´ ℜ s exp p arg p z ` u q ℑ s q ď | u ´ z | ´ ℜ s exp p π | ℑ s |q , which is of course not optimal, but can make us lose at most a factor ofexp p π | ℑ s |q ; we are not studying exact dependence on s , so we acceptthis lack of precision.From the asymptotics of ρ , it is clear that the integral converges ifand only if ℜ s ą n ` , which we assume below.We consider the following cases: Case 1: ℜ z ě
0, and we are especially interested in the asymptotics ℑ z Ñ
0. Let us write z “ a ` ib with real a, b . The main term in theintegral ş | u ´ z | ´ ℜ s ρ p u q du is ż a ` p ,a ´ q | u ´ z | ´ ℜ s ρ p u q du ď C n p a ` q n ´ ż | x ´ ib | ´ ℜ s dx ď C n p a ` q n ´ ż p x ` b q ´ ℜ s { dx. Though it is possible to estimate this integral by elementary functions,we want to keep control on the dependence on s , so we are using specialfunctions below.Denote σ “ ℜ s {
2. By assumption, we have σ ą {
2. The integral iscalculated with the help of the hypergeometric function F [6, 2.1.3]: I “ ż p x ` b q ´ σ dx “ ż p t ` b q ´ σ t ´ dt ORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 13 “ | b | ´ σ F ` σ,
12 ; 32 ; ´ b ˘ . If σ ´ { R Z , this equals [6, 2.1.4 (17)] I “ | b | ´ σ ” Γ p ´ σ q Γ p q Γ p q Γ p ´ σ q | b | σ F ` σ, σ ´
12 ; σ `
12 ; ´ b ˘ ` Γ p σ ´ q Γ p q Γ p σ q | b | ı (since F ` , ´ σ ; ´ b ˘ ”
1) and at b Ñ
0, is equivalent to I Γ p q „ Γ p σ ´ q Γ p σ q | b | ´ σ “ Γ p ℜ s ´ q Γ p ℜ s q | b | ´ ℜ s . If σ ´ { “ m P Z (the smallest possible value is 1), we get bysymmetry of the first two arguments of F and by [6, 2.1.4 (20)] I “ | b | ´ σ F ` σ,
12 ; 32 ; ´ b ˘ “ | b | ´ σ F ` , σ ; 32 ; ´ b ˘ “ | b | ´ σ ´ ` b ¯ ´ σ F ` , ´ σ ; 32 ; ´ b ˘ where 3 { ´ σ “ ´ m is a nonpositive integer, so that the hypergeo-metric series is a polynomial. Leaving only the leading term (as b Ñ I „ b p q m ´ p ´ m q m ´ p q m ´ p m ´ q ! ´ ´ b ˘ m ´ “ Γ p σ ´ q Γ p q Γ p σ q | b | ´ σ . The rest of the integral is estimated as follows. If a ď
1, the followingterm is absent; if a ą
1, we have ż a ´ | u ´ z | ´ ℜ s ρ p u q du ď C n a n ` p ` b q ´ ℜ s { . The last term is ż a ` | u ´ z | ´ ℜ s ρ p u q du ď C n ż a ` a ` | ` ib | ´ ℜ s p a ` q n ´ du ` C n ż a ` ´ u ¯ ´ ℜ s u n ´ du ď C n p a ` q n ` p ` b q ´ ℜ s { ` C n ℜ s p a ` q n ` ´ ℜ s ℜ s ´ n ` . Note that for any b , p ` b q ´ ℜ s { ď | b | ´ ℜ s . We conclude that (for a ě } k a ` ib,s } À C n p a ` q n ´ Γ p ℜ s ´ q Γ p ℜ s q exp p π | ℑ s |q| b | ´ ℜ s , b Ñ For a, b fixed, we obtain also } k a ` ib,s } À C n exp p π | ℑ s | ` ℜ s ln 2 q p a ` q n ` ´ ℜ s ℜ s ´ n ` , ℜ s Ó n ` . For ℑ s ‰ k a ` ib,s is actually bounded at ℜ s “ n ` , so one wouldexpect that the last bound can be improved. Case 2: | z | Ñ 8 . Here we opt not to track dependence of theconstants on s .For ℜ z “ a ě | b | ď } k a ` ib,s } ď C n,s ” p a ` q n ` ´ ℜ s ` p a ` q n ´ | b | ´ ℜ s ı (where clearly either term can be leading depending on a and b ).For ℜ z “ a ě | b | ą
1, the correct power of b is ´ ℜ s ; to verifythis, it is sufficient to estimate ż a ` | u ´ z | ´ ℜ s ρ p u q du ď p a ` q n ` | b | ´ ℜ s , so that } k a ` ib,s } ď C n,s p a ` q n ` ” p a ` q ´ ℜ s ` | b | ´ ℜ s ı . Finally, for ℜ z ă | z | ą
1) the estimates are straightforward: ż | u ´ z | ´ ℜ s ρ p u q du ď ż | z | | z | ´ ℜ s ρ p u q du ` ż z | u n ´ ´ ℜ s du ď C n,s | z | n ` ´ ℜ s and } k z,s } ď C n,s | z | n ` ´ ℜ s . The norm } k z,s } can be estimated by the same bounds as } k z, s } ,so that the formula (4.1) leads to the bound } A z,s } ˜ L p Ñ L q ď | z | n ` ´ ℜ s p ` r q in the case ℜ z ă | z | Ñ 8 , and this can be accordingly modified inthe other cases. 5. Lower norm estimates
Let ψ P C p R q be a function which is not identically zero on r , `8q and twice differentiable with } ψ p j q p s q s k } ď C ψ for 0 ď j ď
2, 0 ď k ď n { `
3. These conditions are in particular verified for ψ p s q “p ` s q ´ α { , α ě n `
3. In [21], M¨uller and Thiele have proved thatthe L -norm of the convolution kernel k t of ψ p? L q cos p t ? L q , as wellas of ψ p? L q sin p t ? L q? L , is bounded by C p ` | t |q . We prove below thatthis estimate is sharp at t Ñ `8 . ORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 15
The integration formulas over G below are not identical to [15] butare influenced by this article.5.1. The convolution kernel.
According to (3.4) (which comes from[21]), the convolution kernel k t of ψ p? L q exp p it ? L q k t p x, y q “ c l e ´ n x ż ψ p s q e its ż R D l,s p v qp ch v ´ ch R q l ´ n dv sds, where we denote as before D l,s p v q “ D l sh ,v p e isv ´ e ´ isv q . Recall that l ischosen as l “ t n u .Note first that } k t } ď ż | ψ p? u q| ρ p u q du is bounded uniformly in t , so that the same is true for ş R p x,y qď | k t | . Wecan therefore assume in the sequel that R ě Proposition 5.1. D l sh ,v p e iuv q “ l ÿ k “ u k e iuv q k,l p v q , where q k,l p v q “ P k,l p sh v, ch v qp sh v q ´ l , every P k,l is a homogeneous poly-nomial of two variables of degree l , and q k,l is even for even k and oddfor k odd. In particular, D l sh ,u p v q Ñ , v Ñ `8 .Proof.
The formula is trivially true for l “
0. The induction step isproved by direct differentiation: D l ` ,v p e iuv q “ ddv l ÿ k “ u k e iuv q k,l p v q sh v “ l ÿ k “ u k e iuv ” iu q k,l p v q sh v ` q k,l p v q sh v ´ q k,l p v q ch v sh v ı “ e iuv ” q ,l p v q sh v ´ q ,l p v q ch v sh v ı ` l ÿ k “ u k e iuv ” i q k ´ ,l p v q sh v ` q k,l p v q sh v ´ q k,l p v q ch v sh v ı ` iu l ` e iuv q l,l p v q sh v . Set q k,l ` p v q : “ $’’’’’&’’’’’% q ,l p v q sh v ´ q ,l p v q ch v sh v k “ i q k ´ ,l p v q sh v ` q k,l p v q sh v ´ q k,l p v q ch v sh v ď k ď li q l,l p v q sh v k “ l ` . (5.1) Then D l ` ,v p e iuv q “ l ` ÿ k “ u k e iuv q k,l ` p v q . It remains to check that there exist homogeneous polynomials of twovariables P k,l ` , ď k ď l ` l ` P k,l ` p sh v, ch v q : “ p sh v q l ` q k,l ` p v q . For this, we claim that q k,l p v q “ p sh v q ´ l ´ Q k,l p sh v, ch v q with Q k,l beinga two-variable homogeneous polynomial of degree l `
1. In fact, q k,l p v q “p sh v q ´ l p P k,l, p sh v, ch v q ch v ´ P k,l, p sh v, ch v q sh v q´ l p sh v q ´ l ´ ch vP k,l p sh v, ch v q , where P k,l, p x, y q : “ B x P k,l p x, y q and P k,l, p x, y q : “ B y P k,l p x, y q are two-variable homogeneous polynomials of degree l ´
1. So p sh v q l ` q k,l p v q “ Q k,l p sh v, ch v q , where Q k,l p x, y q : “ P k,l, p x, y q xy ´ P k,l, p x, y q x ´ lP k,l p x, y q y (5.2)is a two-variable homogeneous polynomial of degree l `
1. This com-pletes the proof of the claim.When k “
0, by induction and by our claim: p sh v q l ` q ,l ` p v q “ p sh v q l ` q ,l p v q ´ p sh v q l ch p v q q ,l p v q“ Q ,l p sh v, ch v q ´ ch p v q P ,l p sh v, ch v q . so we can choose P ,l ` p x, y q “ Q ,l p x, y q ´ yP ,l p x, y q .When 1 ď k ď l , p sh v q l ` q k,l ` p v q “ i p sh v q l ` q k ´ ,l p v q ` p sh v q l ` q k,l p v q ´ p sh v q l q k,l p v q ch v “ iP k ´ ,l p sh v, ch v q sh v ` Q k,l p sh v, ch v q ´ P k,l p sh v, ch v q ch v so we can choose P k,l ` p x, y q “ ixP k ´ ,l p x, y q ` Q k,l p x, y q ´ yP k,l p x, y q .When k “ l `
1, by induction: p sh v q l ` q l ` ,l ` p v q “ i p sh v q l ` q l,l p v q “ iP l,l p sh v, ch v q sh v, so we can choose P l ` ,l ` p x, y q “ iP l,l p x, y q x .Finally, statements on parity of q k,l are easy to check. (cid:3) Note that we do not check whether P k,l are nonzero, so with a certainabuse of language we assume zero to be a homogeneous polynomial ofany degree. All we need to know of q k,l in this respect is contained inthe Proposition below: Proposition 5.2.
There exist constants b l and a k,l , ď k ď l , suchthat | q k,l p v q ´ a k,l e ´ vl | ď b l e ´ vl for every v P r , `8q . Moreover, a l,l ‰ . ORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 17
Proof.
By Proposition 5.1, we have p sh v q l q k,l p v q “ P k,l p sh v, ch v q , for some two-variable homogeneous polynomial P k,l of degree l . It iseasy to see that there exists a polynomial A l of degree 4 l such that e vl p sh v q l “ p e v sh v q l “ A l p e v q , and a polynomial B k,l of degree 2 l such that e vl P k,l p sh v, ch v q “ P k,l p e v sh v, e v ch v q “ B k,l p e v q . Thus there exist some nonzero constant a k,l and some polynomial C k,l of degree ď l ´ e vl q k,l p v q “ e vl P k,l p sh v, ch v qp sh v q l “ e vl B k,l p e v q A l p e v q “ a k,l ` C k,l p e v q A l p e v q . Hence for some constant b l ą | e vl q k,l p v q ´ a k,l | ď b l e ´ vl , (5.3)or equivalently | q k,l p v q ´ a k,l e ´ vl | ď b l e ´ vl . It remains to show that a l,l ‰
0. The function q l,l is in fact easy tocalculate: by (5.1), q l,l p v q “ ´ i sh v ¯ l . For l “
0, we have a , “
1. For l ą
0, by (5.3), a k,l “ lim v Ñ`8 e vl q k,l p v q , so that a l,l “ p i q l ‰ (cid:3) We can write now k t p x, y q (5.4) “ c l e ´ nx ż R p ch v ´ ch R q l ´ n ż l ÿ k “ r s k e isv ´ p´ s q k e ´ isv s q k,l p v q ψ p s q s e its dv ds “ c l e ´ nx ż R p ch v ´ ch R q l ´ n l ÿ k “ q k,l p v qr q m k p t ` v q ´ p´ q k q m k p t ´ v qs dv where m k p s q “ ψ p s q s k ` I r , `8q and q m k is the inverse Fourier transform,which we write without additional constants.While m k p q “ k , the derivatives of m k may be discontinuousat 0, so we need some attention when estimating q m k . Integrating byparts, we can write, for 0 ‰ ξ P R : q m k p ξ q “ ξ “ m k p s q e iξs ‰ ´ ξ ż m k p s q e iξs ds, (5.5)which is bounded as | q m k p ξ q| ď C l,ψ | ξ | ´ (5.6) for 0 ď k ď l . One can note also that m k P L p R q for every k and | q m k p ξ q| ď C l,ψ .We will separate (5.4) into two parts: k t p x, y q “ c l p I ´ I q with I “ e ´ nx ż R p ch v ´ ch R q l ´ n l ÿ k “ q k,l p v q q m k p t ` v q dv (5.7)and I “ e ´ nx ż R p ch v ´ ch R q l ´ n l ÿ k “ q k,l p v qp´ q k ` q m k p t ´ v q dv. (5.8)Bounds for } k t } will be derived from estimates of the integrals of I and I over a certain set tp x, y q P G : R p x, y q P r t ` a, t ` b su . Theseestimates are eventually reduced to the following Lemmas 5.4 and 5.5,preceded by a short calculation in Lemma 5.3.5.2. Integration lemmas.Lemma 5.3.
For α ą ´ and β ą α , we have ż p e v ´ q α e ´ βv dv “ B p α ` , β ´ α q ă 8 , where B p¨ , ¨q is the beta function.Proof. This is a direct computation: ż p e v ´ q α e ´ βv dv “ ż p ´ e ´ v q α e ´p β ´ α ´ q v e ´ v dv “ ż p ´ x q α x β ´ α ´ dx “ B p α ` , β ´ α q ă 8 . (cid:3) Lemma 5.4.
For any a ă b , m ě and t ą max p| a | , b q ż R Pr t ` a,t ` b s e ´ n x ` R p t ` R q ´ m dxdy ď C n,a,b t ´ m . Proof.
As the integrand depends on x and | y | only, we can pass to( n -dimensional) polar coordinates in y “ p r, Φ q . For x, r P R , denote R p x, r q “ R p x, p r, qq , and F a,b “ tp x, r q P R : R p x, r q P r t ` a, t ` b su .We obtain I a,b “ ż R Pr t ` a,t ` b s e ´ n x ` R p t ` R q ´ m dxdy “ V n ż F a,b e ´ n x ` R p t ` R q ´ m r n ´ dxdr, ORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 19 where V n is the volume of the unit ball in R n . By (2.1) and inequalitiesjust after it, the integral can be bounded by I a,b ď n ´ V n ż F a,b e ´ x ` R t ´ m dxdr ď n ´ V n ż F a,b e ´ x ` t ` a t ´ m dxdr. Again by (2.1), if p x, r q P F a,b , then | x | ď R p x, r q ď t ` b , and 0 ď r ď? ` x ` R ˘ ď ` x ` t ` b ˘ . This implies that ż F a,b e ´ x ` t ` a t ´ m dxdr ď t ´ m ż | x |ď t ` b e ´ x ` t ` a ż ` x ` t ` b ˘ drdx “ t ´ m ż | x |ď t ` b e b ´ a dx “ e b ´ a t ´ m p t ` b qď e b ´ a t ´ m (since we suppose b ă t ). (cid:3) Lemma 5.5.
For a ă b and t ą max p , | a | , b q , there exists a constant C n,a,b ą such that ż t ` a ď R ď t ` b e ´ n x ` R dxdy ě C n,a,b t. Proof.
Denote G a,b “ tp x, y q P G : R p x, y q P r t ` a, t ` b su . From (2.1),we have | y | “ e x p ch R ´ ch x q . This formula implies that for p x, y q such that | x | ď t ` a and2 e x p ch p t ` a q ´ ch x q ď | y | ď e x p ch p t ` b q ´ ch x q ,R p x, y q is between t ` a and t ` b , so that G a,b contains for each x a“thick sphere” with | y | changing according to the bounds above (undercondition | x | ď t ` a ). Again denoting by V n the volume of the unitball in R n , we can estimate the volume of this “sphere” as follows: V “ V n n e n x “` ch p t ` b q ´ ch x ˘ n { ´ ` ch p t ` a q ´ ch x ˘ n { ‰ “ c n e n x ` ch p t ` b q ´ ch x ˘ n ´ ` ch p t ` a q ´ ch x ˘ n ` ch p t ` b q ´ ch x ˘ n { ` ` ch p t ` a q ´ ch x ˘ n { ě c n e n x p ch p t ` b q ´ ch p t ` a qq n ` ch p t ` a q ´ ch x ˘ n ´ ` ch p t ` b q ˘ n { ě c n e n x e ´ n p t ` b q ` ch p t ` b q ´ ch p t ` a q ˘` ch p t ` a q ´ ch x ˘ n ´ . Since ´ t ´ a ´ b “ ´ p t ` a q ` a ´ b ď a ´ b ă
0, we can continue asch p t ` b q ´ ch p t ` a q “ e t p e b ´ e a qp ´ e ´ t ´ a ´ b q ě e t p e b ´ e a qp ´ e a ´ b q , and V ě c n,a,b e n x e p ´ n q t ` ch p t ` a q ´ ch x ˘ n ´ . The integral thus can be bounded by: ż G a,b e ´ n x ` R dxdy ě c n,a,b e p ´ n q t ż | x |ď t ` a e n x e ´ n x ` t ` b ` ch p t ` a q ´ ch x ˘ n ´ dx “ c n,a,b e p ´ n q t ż t ` a ` ch p t ` a q ´ ch x ˘ n ´ dx “ c n,a,b e p ´ n q t ż t ` a ” e t ` a p ´ e x ´ t ´ a qp ´ e ´ x ´ t ´ a q ı n ´ dx ě c n,a,b ż t ` a p ´ e x ´ t ´ a q n ´ dx. By assumption t ą | a | , so that t { ă t ` a . If 0 ď x ď t {
4, then x ´ t ´ a ď p´ t { ´ a q ´ t { ď ´ t {
4, so that ż G a,b e ´ n x ` R dxdy ě c n,a,b ż t { p ´ e ´ t { q n ´ dx ě c a,b,n t, where in the last inequality we use the assumption t ą (cid:3) Estimates of I and I . Estimates of I . We are interested in R ě
1, so that in (5.7) wehave v ě
1. By Lemma 5.1, | q k,l p v q| ď C l e ´ lv with some constant C l .Together with (5.6), we have | I | ď e ´ nx ż R p ch v ´ ch R q l ´ n l ÿ k “ ˇˇˇ q k,l p v q q m k p t ` v q ˇˇˇ dv ď C l,ψ e ´ nx ż R p ch v ´ ch R q l ´ n e ´ lv p t ` v q ´ dv ď C l,ψ e ´ nx p t ` R q ´ ż p ch p v ` R q ´ ch R q l ´ n e ´ lv ´ lR dv. One can transform2 p ch p v ` R q ´ ch R q “ p e v ´ qp e R ´ e ´ v ´ R q “ p e v ´ q e R p ´ e ´ v ´ R q , and estimate | I | ď C l,ψ e ´ nx p t ` R q ´ ż pp e v ´ q e R q l ´ n e ´ lv ´ lR dv “ C l,ψ e ´ n p x ` R q p t ` R q ´ ż p e v ´ q l ´ n e ´ lv dv. By Lemma 5.3, the integral converges (and depends only on n ), so that | I | ď C n,ψ e ´ n p x ` R q p t ` R q ´ . (5.9) ORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 21
Now by Lemma 5.4, for a ă b and t ą max p| a | , b q ż R Pr t ` a,t ` b s | I | dxdy ď C l,ψ t ´ . (5.10)5.3.2. Simplifications of I . We can now pass to the second term: I “ e ´ nx ż R p ch v ´ ch R q l ´ n l ÿ k “ q k,l p v qp´ q k ` q m k p t ´ v q dv “ e ´ nx ż p ch p v ` R q ´ ch R q l ´ n l ÿ k “ q k,l p v ` R qp´ q k ` q m k p t ´ R ´ v q dv. Its analysis will pass through several stages of simplification. By Lemma5.2, | q k,l p v q ´ a k,l e ´ vl | ď b l e ´ vl . Using this, we can replace I by I : “ e ´ nx ż p ch p v ` R q ´ ch R q l ´ n l ÿ k “ a k,l e ´p v ` R q l p´ q k ` q m k p t ´ R ´ v q dv so that | I ´ I | ď e ´ nx ż p ch p v ` R q ´ ch R q l ´ n l ÿ k “ ˇˇˇ q k,l p v ` R q ´ a k,l e ´p v ` R q l ˇˇˇ | q m k p t ´ R ´ v q| dv ď c l e ´ nx ż p ch p v ` R q ´ ch R q l ´ n e ´ p v ` R q l l ÿ k “ | q m k p t ´ R ´ v q| dv. If l “
0, then I is exactly equal to I since q , ” a , “
1. For l ą | I ´ I | is small enough. By definition, p ch p v ` R q ´ ch R q l ´ n “ n ´ l p e v ´ q l ´ n e R p l ´ n q p ´ e ´ v ´ R q l ´ n . Since l ´ n ď R ě p ´ e ´ v ´ R q l ´ n ď p ´ e ´ q l ´ n . Next, since every m k is in L , we have ř lk “ | q m k p t ´ R ´ v q| ď C n,ψ .These, together with Lemma 5.3, yield | I ´ I | ď C n,ψ,l e ´ nx ż p e v ´ q l ´ n e ´ p v ` R q l ` R p l ´ n q dv ď C n,ψ,l e ´ n p x ` R q ´ Rl ż p e v ´ q l ´ n e ´ vl dv “ C n,ψ,l B ` l ´ n ` , l ` n ˘ e ´ n p x ` R q ´ Rl , which has, by Lemma 5.4, the integral bounded as follows (with thesame assumptions on t as in the lemma): ż t ` a ď R ď t ` b | I ´ I | dxdy ď C n,ψ e ´p t ` a q l ż t ` a ď R ď t ` b e ´ n p x ` R q dxdy ď C n,ψ e ´ tl t. (5.11)Our next step is to replace I by I : “ e ´ nx ż n ´ l e R p l ´ n { q p e v ´ q l ´ n l ÿ k “ a k,l e ´p v ` R q l p´ q k ` q m k p t ´ R ´ v q dv. If l “ n , this is an exact equality. If l ´ n “ ´ , we have to estimate D : “ ˇˇˇ p ch p v ` R q ´ ch R q l ´ n ´ n ´ l p e v ´ q l ´ n e R p l ´ n q ˇˇˇ “ n ´ l p e v ´ q l ´ n e R p l ´ n q ˇˇˇ ´ p ´ e ´ v ´ R q l ´ n ˇˇˇ . Set z “ e ´ v ´ R . By the choice of R and v , we have z P p , e ´ s . Inthis interval, the function f p z q “ p ´ z q ´ { has a derivative f p z q “ p ´ z q ´ { bounded by f p e ´ q , so that ˇˇˇ ´ p ´ e ´ v ´ R q l ´ n ˇˇˇ “ | f p q ´ f p z q| ď f p e ´ q z “ f p e ´ q e ´ v ´ R . This implies that D ď C n p e v ´ q l ´ n e R p l ´ n q e ´ v ´ R and | I ´ I | ď C n e ´ nx ż p e v ´ q l ´ n e R p l ´ n q e ´ v ´ R e ´p v ` R q l l ÿ k “ | a k,l || q m k p t ´ R ´ v q| dv ď C n,ψ,l e ´ n p x ` R q ´ R ż p e v ´ q l ´ n e ´p l ` q v dv “ C n,ψ B ` l ´ n ` , n ` ˘ e ´ n p x ` R q ´ R , with the integral bounded by ż t ` a ď R ď t ` b | I ´ I | dxdy ď C n,ψ e ´ p t ` a q ż t ` a ď R ď t ` b e ´ n p x ` R q dxdy ď C n,ψ e ´ t t. (5.12)5.3.3. Estimates of I . It remains now to estimate I from below. Re-call that I “ c n e ´ n p x ` R q ż p e v ´ q l ´ n l ÿ k “ a k,l e ´ vl p´ q k ` q m k p t ´ R ´ v q dv. Set M p ξ q “ l ÿ k “ a k,l p´ q k ` q m k p ξ q ;this is the inverse Fourier transform ofΨ p s q “ l ÿ k “ a k,l p´ q k ` s k ` ψ p s q I r , `8q . ORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 23
By assumption, ψ is not identically zero on r , `8q ; it is multipliedby a nonzero polynomial since a l,l ‰ I , I p ξ q “ ż p e v ´ q l ´ n e ´ vl M p ξ ´ v q dv (5.13)as the convolution of M and N p v q : “ p e v ´ q l ´ n e ´ vl p , p v q . Thefunction I is continuous, and its inverse Fourier transform is Ψ q N . Forevery m , we can estimate ż R | N p v q v m | dv ď ż p e v ´ q l ´ n dv ` ż C n e ´ v n v m dv ď C n ` C n ´ n ¯ m ` ż n { e ´ x x m dx ď C n K m m !with K “ max p , { n q . It follows that } q N p m q } ď C n K m m !, so by [25,19.9] q N is analytic in a strip | z | ă δ with some δ ą
0. As it is clearlynot identically zero, we can conclude that Ψ q N is not identically zerotoo.There exist therefore α, β P R , α ă β , A ‰ ă ε ă | A |{ ψ and on n ) such that | I p ξ q ´ A | ă ε for ξ P r α, β s . It follows that for R P r t ´ β, t ´ α s| I | ą C n,ψ e ´ n p x ` R q (5.14)and by Lemma 5.5, with a “ ´ β , b “ ´ α , ż R Pr t ` a,t ` b s | I | dxdy ě C n,ψ t, (5.15)once t ą max p , | a | , b q .5.4. Conclusions.
We can now summarize the obtained estimates inthe following theorem.
Theorem 5.6.
Let ψ P C p R q be a function which is not identicallyzero on r , `8q and twice differentiable with } ψ p j q p s q s k } ď C ψ for ď j ď , ď k ď n { ` . Let k t be the convolution kernel of one ofthe following operators: E t “ ψ p? L q exp p it ? L q ,C t “ ψ p? L q cos p t ? L q ,S t “ ψ p? L q sin p t ? L q? L . Then there exists a constant C n,ψ depending on n and on ψ such thatfor t ą sufficiently large, } k t } ě C n,ψ t. Proof.
Case 1: k t is the kernel of E t . By the results of Section 5.3.3,there exist a, b P R such that (5.15) holds, for t ą max p , | a | , b q . Forthe same a, b, t we have, with (5.10): } k t } ě c l ż R Pr t ` a,t ` b s | I ´ I | dxdy ě C n,ψ ´ ż R Pr t ` a,t ` b s | I | dxdy ´ t ¯ . Set γ “ γ l “ l ‰
0. We can continue, using the resultsabove, as } k t } ě C n,ψ ´ ż R Pr t ` a,t ` b s | I | dxdy ´ γ l e ´ lt t ´ t ¯ (5.12) ě C n,ψ ´ ż R Pr t ` a,t ` b s | I | dxdy ´ e ´ t t ´ γ l e ´ lt t ´ t ¯ (5.15) ě C n,ψ ´ t ´ e ´ t t ´ γ l e ´ lt t ´ t ¯ . For t big enough, this is clearly bounded from below by C n,ψ t , asclaimed. Case 2: k t is the kernel of C t . We represent cos p its q as p e its ` e ´ its q ;in the integral (5.4), instead of r q m k p t ` v q ´ p´ q k q m k p t ´ v qs we thenobtain12 “ q m k p t ` v q ` q m k p v ´ t q ´ p´ q k q m k p t ´ v q ´ p´ q k q m k p´ t ´ v q ‰ . We separate then k t “ c l p I ´ I q into I “ e ´ nx ż R p ch v ´ ch R q l ´ n l ÿ k “ q k,l p v qr q m k p t ` v q ´ p´ q k q m k p´ t ´ v qs dv,I “ e ´ nx ż R p ch v ´ ch R q l ´ n l ÿ k “ q k,l p v qrp´ q k ` q m k p t ´ v q ` q m k p v ´ t qs dv. As | q m k p´ t ´ v q| ď C n,ψ p t ` v q ´ , the estimate for I remains the same.The passage from I to I does not change either. In I we still have(5.13), but with M p ξ q “ l ÿ k “ a k,l “ q m k p´ ξ q ` p´ q k ` q m k p ξ q ‰ . If we denote by ˜ m k the function ˜ m k p s q “ m k p´ s q , then M is the inverseFourier transform ofΨ p s q “ l ÿ k “ a k,l “ ˜ m k p s q ` p´ q k ` m k p s q ‰ which is for s ě I is not everywherezero, so that we arrive at the same conclusion: } k t } ě C ψ,n t , for t bigenough. ORMS OF CERTAIN FUNCTIONS OF THE LAPLACE OPERATOR 25
Case 3: k t is the kernel of S t . In (5.4), we obtain q m k p t ` v q ´ q m k p v ´ t q ´ p´ q k q m k p t ´ v q ` p´ q k q m k p´ t ´ v q , but m k changes to m k p s q “ s k ψ p s q I r , `8q and not with s k ` . As aconsequence, we cannot integrate by parts twice as in (5.5) but onlyonce; this changes the estimate (5.6) as to | q m k p ξ q| ď C n,ψ | ξ | ´ . (5.16)Let us write down the two parts of k t , up to the constant c l { I “ e ´ nx ż R p ch v ´ ch R q l ´ n l ÿ k “ q k,l p v qr q m k p t ` v q ` p´ q k q m k p´ t ´ v qs dv,I “ e ´ nx ż R p ch v ´ ch R q l ´ n l ÿ k “ q k,l p v qrp´ q k ` q m k p t ´ v q ´ q m k p v ´ t qs dv. With (5.16) instead of (5.6), we obtain ż R Pr t ` a,t ` b s | I | dxdy ď C n,ψ . The estimates of | I ´ I | and | I ´ I | do not change since we are usingonly the fact m k P L p R q which remains true. In I we get (5.13) with M p ξ q “ l ÿ k “ a k,l “ ´ q m k p´ ξ q ` p´ q k ` q m k p ξ q ‰ , and complete the proof as in Case 2. (cid:3) It is well known that L -norm of a function f is also the norm of theconvolution operator g ÞÑ g ˚ f on L p G q . We get as a corollary thatupper norm estimates of [21] for C t , S t are sharp: } C t } L Ñ L — t, } S t } L Ñ L — t, as t Ñ 8 . Our results are valid in particular for ψ p s q “ p ` s q ´ α , α ą n { ` Lower bounds for uniform norms.
In [21, Corollary 7.1], M¨ullerand Thiele show that for a function ψ P C p R q supported in r a, b s , thekernel k t of C t “ ψ p? L q cos p t ? L q has its uniform norm bounded bya constant, with no decay in t as t Ñ `8 . With a simple modifica-tion, this result can be extended to functions which are not compactlysupported, but decaying quickly enough.From the results obtained in Sections 5.3-5.4, it follows that thisestimate is actually sharp.
Theorem 5.7.
In the assumptions and notations of Theorem 5.6, thereexists a constant C n,ψ depending on n and on ψ such that for t ą sufficiently large, } k t } ě C n,ψ . Proof.
We can consider the cases of E t and C t together. Case 1: k t is the kernel of E t or C t . By the results of Section 5.3.3,there exist a, b P R such that (5.14) holds, for t ą max p , | a | , b q . It isclear then that we should consider x “ ´ R , y “
0. Arguing similarlyto Theorem 5.6, we obtain the estimate | k t p´ R, q| ě C n,ψ ´ ´ e ´ t ´ γ l e ´ lt ´ t ¯ , which is bounded from below by C n,ψ , as claimed. Case 2: k t is the kernel of S t . As noted in the proof of Theorem 5.6,we get the estimate (5.16) instead of (5.6). This changes the estimateof I as | I p´ R, q| ď C n,ψ t for t large enough. The other estimates are as in Case 1, so that wearrive at | k t p´ R, q| ě C n,ψ ´ ´ e ´ t ´ γ l e ´ lt ´ t ¯ , which is bounded from below by a constant C n,ψ ą t Ñ `8 . (cid:3) As pointed out in [21], this shows that no dispersive L ´ L estimateshold for the wave equation. Acknowledgments.
This work was started during an ICL-CNRS fel-lowship of the second named author at the Imperial College London.Yu. K. is partially supported by the ANR-19-CE40-0002 grant of theFrench National Research Agency (ANR). H. Z. is supported by theEuropean Union’s Horizon 2020 research and innovation programmeunder the Marie Sk lodowska-Curie grant agreement No. 754411. R. A.was supported by the EPSRC grant EP/R003025. M. R. is supportedby the EPSRC grant EP/R003025/2 and by the FWO Odysseus 1 grantG.0H94.18N: Analysis and Partial Differential Equations.
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Email address : [email protected] Yulia Kuznetsova, University Bourgogne Franche-Comt´e, 16 routede Gray, 25030 Besanc¸on, France
Email address : [email protected] Michael Ruzhansky, Department of Mathematics: Analysis, Logicand Discrete Mathematics, Ghent University, Krijgslaan 281, Build-ing 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 : [email protected]
Haonan Zhang, Institute of Science and Technology Austria (ISTAustria), Am Campus 1, 3400 Klosterneuburg, Austria
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