On the translation invariant operators in l p (Z d )

Transcription

1 On the translation invariant operators in l p Z d ) Béchir Amri and Khawla Kerfaf arxiv:9.2924v [math.ca] 9 Jan 29 Taibah University, College of Sciences, Department of Mathematics, P. O. BOX 32, Al Madinah AL Munawarah, Saudi Arabia. bechiramri69@gmail.com Université Tunis El Manar, Faculté des sciences de Tunis, Laboratoire d Analyse Mathématique et Applications, LRES, 292 El Manar I, Tunisie. kh.karfaf@gmail.com Abstract In this paper we study boundedness of translation invariant operators in the discrete space l p Z d ). In this context a Mikhlin type multiplier theorem is given, yielding boundedness for certain known operators. We also give l p l q boundedness of a discrete wave equation. Keywords. Discrete Fourier transforms, Discrete Laplacian, Calderón-Zygmund operators. 2 Mathematics Subject Classification. Primary 39A2; Secondary 47B38, 35L5. Introduction It is well known that translation invariant operator froml p intol q may be represented by convolution with a tempered distribution, or equivalently by Fourier multiplier transformation. This was originally proved in the classical article of Hörmander [4]. Through many aspects of harmonic analysis, many studies have been devoted to the topic of the L p -bounded of translation invariant operator. The most famous are the works of Calderón and Zygmund on the singular integral operators, with a large number of generalizations. In this paper we consider translation invariant operator T on Z d. The problem is essentially the multiplier problem, F Z dtf)) = mf Z df) where the function m is defined on the Torus R d /Z d. In this setting a Hörmander s type theorem for l p l q boundedness of T and an l p -theorem of Mikhlin- type are given. We apply our results to get L p -estimate for the discrete wave equation. We begin by introducing the following notations. Let T d = R d /Z d be the d- dimensional torus. Functions ont d are functions f onr d that satisfy fx+n) = fx) for all x R d and n Z d. Such functions are called -periodic in every coordinate.

2 Haar measure on T d is the restriction of d-dimensional Lebesgue measure to the set [,) d. This measure is still denoted by dx and given by fx)dx = fx)dx. T d [,) d We denote by L p T d ), p the Lebesgue space L p [,) d,dx). The inner product of the Hilbert space L 2 T d ) is given by f,g = fξ)gξ)dξ. T d The functions ψ n : ξ e 2iπξ.n, indexed by n Z d, form a complete orthonormal system of L 2 T d ) where for x = x,...,x d ) and y = y,...,y d ) in R d x.y = x y +...+x d y d and x = x.x) /2. By l p = l p Z d ), p <, we denote the usual Banach space of p-summable complex-valued function f = fn)) n Z n equipped with the norm ) p f l p = n Z d fn) p and l the space of bounded function on Z d with f l = sup n Z d fn). We note the following elementary embedding relations l q l p ; and f l q f l p, p q..) For f l 2 Z d ) its Fourier transform is given by F Z df)ξ) = n Z d fn)e i2πn.ξ, ξ R d. The Fourier transform F Z d is an isometry from l 2 Z d ) into L 2 T d ) and its inverse F Z d is given by F u)n) = uξ)e i2πξ.n dx, u L 2 T d ). Z d T d By Riesz-Thorin convexity theorem, the Fourier transform F Z d and its inverse F Z d satisfy the Hausdorff-Young inequalities F Z df) L p f lp,.2) and F u) Z d l p u Lp,.3) for, < p 2 and /p+/p =. Convolution product of two functions f and g of l 2 Z d ) is defined by f Z d gn) = g Z d fn) = k Z d fk)gn k); n Z d. If f, g l Z d ) then f Z d g l Z d ) and F Z df Z d g) = F Z df)f Z dg). Suppose f l p Z d ) and g l q Z d ) with p,q,r, + = +. Then p q r f Z d g l r f l p g l q Young s Inequality)..4) 2

3 2 Translation invariant operators In this section we shall be concerned with the space of bounded operators T from l p to l q for p q, which commute with translations; that is, τ n T = Tτ n for all n Z d, where τ n f)k) = fn+k). It is not difficult to see that T is a convolution operator. Indeed, let K = T {} ), where A is the characteristic function of a set A. Consider first, functions f with compact support and write f = {} Z d f = n Z d fn)τ n {} ). Since T is translation invariant operator then Tf) = T {} Z d f) = n Z d fn)tτ n {} )) = n Z d fn)τ n T {} )) = K Z d f. Now for function f l Z d ) we let f j = n j fn) {n}, j Clearly the sequence f j ) j converges to f in l r Z d ) for all r < and Tf j ) K Z d f l q = K Z d f j f) l q K l q f j f l which implies that Tf j )) j converges to K Z d f in l q Z d ). But T : l p Z d ) l q Z d ) is bounded and f j ) j converges to f in l p Z d ), thus by uniqueness of the limit we may have Tf) = K Z d f. Notice that the l p l q boundedness of T implies that K l q Z d ). We state the following Theorem 2.. If T is a bounded translation invariant operator from l p Z d ) to l q Z d ), p q, then there is exists a function K l q Z d ) such that Tf) = K Z d f, f l Z d ). Translation invariant operator can also be described as Fourier multiplier transformation T m defined by F Z dt m f)) = mf Z df) where m is a bounded measurable function m on T d. An important class of T m is given by L r, T d ) for r >, that is the space of measurable functions m such that for some constant c >, dx c sr, s >. 2.) {ξ,) d, mξ) s} In particular if m satisfies the estimate for < α < d then m L d/r, T d ). mξ) c ξ r ; ξ,) d, ξ, 2.2) 3

4 Theorem 2.2. If m L α, T d ) with α > then T m is a bounded operator from l p Z d ) into l q Z d ), provided that < p 2 q <, p q = α. This result is originally proved in [4] for translation invariant operator on R d and in [?] for translation invariant operator on Z, for completeness we extended this result to Z d. The proof of Theorem 2.2 follows closely the argument of [4]. Lemma 2.3. Let ϕ be a measurable function such that for some constant c > dx c, s >. 2.3) s {ξ,) d, ϕξ) s} Then for all < p 2 there exists a constant c p > such that ) F Z df)ξ) p ϕξ) 2 p p dξ c p f l p; f l p Z d ). 2.4),) d Proof. Put dµξ) = ϕ 2 ξ) dξ and let T be the operator defined on l Z d ) by Tf) = F Z df) ϕ Noting that Tf) is well defined µ-almost everywhere on,) d, since we have that µ{ξ,) d, ϕξ) = }) =. In fact, for s > we have µ{ξ,) d, ϕξ) s}) = ϕξ) 2 dξ {ξ,) d, ϕξ) s} = 2 tdtdξ {ξ,) d, ϕξ) s} t ϕξ) s 2 tdtdξ 2c. {ξ,) d, t ϕξ)} s dt = 2cs which implies that µ{ξ,) d, ϕξ) = }) 2cs, for all s >. Now for s > and f l Z d ) we have { µ{ξ,) d, Tf)ξ) s}) µ ξ,) d, ϕξ) f l s Hence T is of weak type.). In addition, from Plancherel Theorem }) 2c f l s. µ{ξ,) d, Tf)ξ) s}) f 2 l 2 s 2, which mean that T is of weak type 2,2). We thus obtain Lemma 2.3 by using Marcinkiewicz interpolation Theorem. 4

5 Lemma 2.4. If ϕ satisfies 2.3) and < p < r < p <, then we have F Z f)ξ)ϕξ)) /r /p ) r /r dξ) Cp f l p; f l p Z d ).,) d Proof. Put a = p p)/p r) and a it s conjugate. We note the following p a + p = r, rp ) a = 2 p, r p ) a = p. a a Then using Holder s inegality,2.5) and the Hausdorff-Young inegality.2) F Z df)ξ) r ϕξ) r/p /r ) dξ),) d ) /ra ) F Z df)ξ) p ϕξ) 2 p) dξ F Z df)ξ) p /ra dξ,) d,) d c p f l p which is the desired statement. Proof of Theorem 2.2. Assume first that p q and let ϕ = m α. Clearly from 2.) the function ϕ satisfies the condition 2.3). Hence using Lamma 2.4 with r = q and the fact that /p /q = /q /p = /α, we obtain that mξ)f Z df)ξ) q dξ,) d Now the Hausdorff-Young inegality.3 ) implies ) /q T m f) l q mff) q c p f l p. c f l p. 2.5) When q < p = p ), we can apply the similar argument to the adjoint operator T m = T m, since < q 2 p < and /q /p = /α. Hence by duality it follows that T m f) l q c p f l p. This finishes the proof of Theorem 2.2. Corollary 2.5. If m satifies 2.2) with < r < d then T m is bounded from l p Z d ) into l q Z d ), provided that < p 2 q <, p q = r d. Now observe that the inequality 2.) can be restricted only tos which implies that L α, T d ) L β, T d ) for all < β α. Thus one can state Corollary 2.6. If m L α, T d ) with α > then the operator T m is bounded from l p Z d ) into l q Z d ), provided that < p 2 q <, p q α. 5

6 We now study L p -boundedness of the multiplier operator T m. We begin by the following: Theorem 2.7. If m is a C d+ - function on T d then T m is a bounded operator from l p Z d ) into itself for all p. Proof. We note first that the kernel of T m is given by Kn) = mξ)e 2iπξ.n dξ, n Z d.,) d Using integrations by parts we have n γ...n γ d d Kn) = γ ξ... γ d ξ d mξ) e 2iπξ.n dξ c,) d for all γ,...,γ d N with γ +...+γ d d+. It follows that Kn) c + n )...+ n j ) n d ) for all j =,...,d. By varying j from to d we deduce the following estimate Kn) c + n )...+ n j )...+ n d ) ) +/d which implies that the kernel K is in l Z d ). This yields the result. Our main result is the following Hörmander-Mihlin type multiplier theorem where we may consider T m as a Calderón-Zygmund operator. Theorem 2.8. Let m be a bounded function on the torus T d. We assume that m is C d+ -function on R d \Z d and satisfies the Mikhlin condition, α ξmξ) c ξ α, ξ /2,/2) d 2.6) for all α = α,...,α d ) N d with α = α +...+α d d+. Then T m extended to a bounded operator from l p into itself for all < p <. Proof. Taking ψ a C -function on R d such that ψξ) = for ξ /6 and ψξ) = for ξ /8. In /2,/2) d we split m into m = ψ)m+mψ = m +m 2. Since m = m near the sides ξ j = /2, then m can be extended to C d+ -function on T d and by Theorem 2.7 the operator T m is bounded on l p for all p. It is therefore enough to prove boundedness of T m2. Introduce the function K by Kx) = m 2 ξ)e i2πξ.x dξ, x R d /2,/2) d 6

7 and K its restriction to Z d. One can write T m2 f) = K Z d f, f l. Our aim is to prove that T m2 is a Calderon-Zygmund operator. We consider here Z d as a space of homogeneous type in the sense of Coifman and Weiss [?], equipped with the Euclidean metric r,s) r s and the counting measure. Precisely, we will prove that the kernel K satisfies the integral Hörmander condition: there exists constant c > such that for all s Z d, Kr s) Kr) c. 2.7) r Z d, r 2 s To begin, let φ be a C - function on R, such that suppφ) {t R; /2 t 2} and φ2 j t) =, t. So we have and we may write Notice that m 2 ξ) = j Z φ2 j ξ ) =, ξ 8. m 2 ξ)φ2 j ξ ) = m j ξ), ξ /2,/2) d. m j ξ) c 2.8) for some constant c >, since this sum contains at most three non-null terms. We now set K j x) = m j ξ)e i2πξ.x dξ, x R d. /2,/2) d By Fubini s Theorem and 2.8) the sum j K j converges and K j x) = /2,/2) d m j ξ)e i2πξ.x dξ = m 2 ξ)e i2πξ.x dξ = Kx), /2,/2) d Next we shall give estimates of the kernels K. Observe first that m 2 satisfies the condition 2.6) and from this estimate and the compactness of suppϕ) one can obtain the following α m j ξ) c 2 j α ; and α ξ i m j ξ) ) c 2 j α ) ; for i =,...,d and for α d+. It follows that x s K j x) c α m j L c 2 jd s) 2.9) α =s 7

8 and x s K j x) x i c α ξ i m j ξ)) L c 2 jd+ s). 2.) α =s Using 2.9) with s = and s = d+ we get K j x) = 2 j x K j x) + Similarly by 2.) with s = and s = d+2 K j x) x i = 2 j x K j x) x i + 2 j > x K j x) c x d. 2 j > x K j x) x i c x d. Therefore we obtain the following estimates Kx) c+ x ) d K x) x i c+ x ) d, i =,...,d. 2.) Note that K is a C -function and all of its derivatives are bounded. Now we come to the proof of 2.7). Let r,s Z d with r 2 s. By mean value Theorem we have Kr s) Kr) = Kr s) Kr) s Using 2.) and the fact that we obtain the following Now use that we have r 2 s + r ) d+ c r ts r s r 2 Kr s) Kr) max r i ) r d /2 max r i ), i d i d max i d r i ) 2d /2 s d K r ts) x i dt. i= c s + r ) d+. 2.2) +max i d r i )) d+ It is not hard to see that r Z d, max i d r i )=k ck d. 8

9 and from which r 2d /2 s + r ) d+ c k 2d /2 s k c 2 s. 2.3) Combine 2.3) with 2.2), yield that Kr s) Kr) c r 2 s which proves 2.7). The proof of Theorem 2.8 follows. In the next we shall be concerned with Mikhlin type multiplier on Z. Thus one can read Theorem 2.8 as follows Theorem 2.9. If m is a bounded C 2 function on,) such that ξ ξ)) k m k) ξ) c; k 2 2.4) then T m is a bounded operator from l p Z) into l p Z) for < p <. Clearly condition 2.4) is exactly 2.6) when extending m to a periodic function. Now we replace the interval,) by a bounded interval a,b). For f l 2 Z) we define its Fourier transform by and its inverse F a,b Z f)ξ) = fn)e i2πnξ a b a) = FZ f) n Z F a,b Z ) f)n) = b a b a ) ξ a b a fξ)e i2πnξ a b a) dξ, n Z. For a bounded function m on a,b) we define on l 2 Z) the operator Tm a,b by Tm a,b f)n) = b a = = b a mξ)f a,b Z f)ξ)ei2πnξ a b a) dξ ma+b a)ξ)f a,b Z f)a+b a)ξ)ei2πn dξ ma+b a)ξ)f Z f)ξ)e i2πn dξ, for n Z. According to Theorem 2.9 we have the following Corollary 2.. If m is a bounded C 2 -function on a bounded interval a,b), such that for some constant c > ξ a) k b ξ) k m k) ξ) c, k 2 then T a,b m is a bounded operator from lp Z) into l p Z) for < p <. 9

10 As a typical example we have m = χ a,b) the characteristic function of a,b). Theorem 2.9 can be generalized as follows. Theorem 2.. Let a j ) j s be a subdivision of [,]. If m is a bounded C 2 -function on,) such that, for some constant c >, ) ξ a j k m k) ξ) c; k 2. j s then T m is a bounded operator from l p Z) into l p Z) for < p <. Proof. Assume first that s = 2 and let = a < a < a 2 =. Let ε > and ϕ a be C function on R such that ]a 2ε,a +2ε[ ],[ and ϕξ) = for ξ ]a ε,a +ε[ and ϕξ) = for all ξ / ]a 2ε,a +2ε[. Put and mξ) = mξ)ϕξ)+mξ) ϕξ)) = m ξ)+m 2 ξ), ξ,). T m = T m +T m2 Clearly the boundedness of T m is a consequence of Theorem 2.9. However, if we consider m 2 the - periodic function such that m 2 ξ) = m 2 ξ) for ξ [,], then one can write T m2 f)n) = = = a + a = e i2πna m 2 ξ)f Z f)ξ)e i2πnξ dξ m 2 ξ)f Z f)ξ)e i2πnξ dξ m 2 ξ a )F Z f)ξ +a )e i2πnξ a ) dξ = e i2πna T a,a + m 2 m 2 ξ +a )F Z f)ξ)e i2πnξ dξ f)n) where f is the function given by fn) = e i2πna fn), n Z. Now obseve that m 2 satisfies the hypothesis of Corollary 2. on a,a +), then the boundedness of T m2 follows. Now for s 3 we proceed as follows: choose ε > such that the intervals ]a j 2ε,a j +2ε[ are disjoint for all j s and C functions ϕ j with ϕ j ξ) = for ξ ]a j ε,a j +ε[ and ϕξ) = for ξ / ]a j 2ε,a j +2ε[. Put ϕ = s s ϕ j and write m = mϕ+m ϕ) = mϕ+ s ϕ j ) s s m = m + m j

11 and s T m = T m + T mj. Therefore from the above argument all the operators T mj are bounded on l p Z), < p <. 3 Applications 3. Discrete Riesz Transforms For a complex-valued function f on Z d its discrete Laplacian is given by d f)n) = d d j j fn) = j jfn), n Z d where j fn) = fn+e j ) fn) and j fn) = fn) fn e j). We have d f)n) = d ) fn+e j ) 2dfn)+fn e j ). The discrete Laplacian d is a bounded self-adjoint operator on l 2 Z d ) and one has F Z d d f))ξ) = d d ) e i2πξ j 2 F Z df)ξ) = 4 sin 2 πξ j ) F Z df)ξ). The discrete Riesz transforms R j, j =,...,d, associated with d are defined on l 2 Z d ) as the multiplier operators F Z dr j f))ξ) = e iπξ j sinπξ j ) d ) /2 F Z df)ξ), 2 k= sin2 πξ k ) its can be interpret as R j f) = j /2 d. Let us set ψ j ξ) = 2 d e iπξ j sinπξ j ) k= sin2 πξ k ) ) /2 and prove that ψ j satisfies the Mikhlin condition 2.6). This can be seen by using the fact that γ N d, γ ψ j is a linear combination of the following functions e iπξ j d sin α k πξ k ) k= d d cos β k πξ k ) sin 2 πξ k ) k= i=k ) d i= α k+β k )/2

12 where d k= α k γ + and d k= β k γ. Hence using the fact that for k =,...,d, ξ k /2,/2) and 2 ξ k sinπξ k π ξ k we obtain d sin α k πξ i ) k= d d cos β k πξ k ) k= k= sin 2 πξ k )) dk=αk+βk)/2 d ξ k= β k ξ γ. Therefore we can apply Theorem 2.8 to assert that R j is bounded on l p Z d ) for < p <. 3.2 Imaginary powers of the discrete Laplace operator d Theorem 2.8 also applies to imaginary powers of the discrete Laplacian: d ) it for t R, it is the multiplier operator with multiplier 4 ) it d sin2 πξ j ) 3.3 Strichartz type estimates for discrete wave equation We define the d-dimensional discrete wave equation by d un,t) = 2 tun,t), 3.) un,) = fn), t un,t) = gn), n,t) Z d R where f and g are a given suitable functions on Z d. Considered as a discrete counterpart of the continuous wave equation, many authors have been interested in studying this equation see, for example, [6, 5, 7] and the references therein. Putting φξ) = 2 d sin 2 πξ j ) and applying the discrete Fourier transform, considering t as a parameter, we deduce that the solution of 3.) can be written formally) in the form ) un,t) = F costφ.))f Z d Z df) n)+f sintφ.)) Z d φ.) We will prove the following version of the Strichartz estimates. u.,t) l q ct) g l p + ) F Z df) n). ) d j f l p. 3.2) for all < p 2 q <. Let us obseve first that ξ sintφξ))/φξ) is a C - function on T d and then in view of Theorem 2.7 we have sintφ.)) F F Z d Z dg)) l ct) g l p φ.) q 2

13 whenever < p q <. To prove 3.2) it suffices to show that We write As F Z d costφ.))f Z df)) l q ct) costφξ))f Z df)ξ) = costφξ) φξ) it follows from Theorem 2.2 that d j f l p. d F Z dr j j f))ξ). cosφξ))) φξ) c ξ F Z d costφ.))f Z df)) l q and by using l p -boundedness of R j, which conclude the proof of 3.2). References ct) F Z d costφ.))f Z df)) l q ct) d R j j f) l p d j f l p [] O. Ciaurri, T. Gillespie, L. Roncal, J.L. Torrea, J. L. Varona, Harmonic analysis associated with a discrete Laplacian, Journal d Analyse Mathématique, 32 27), 9-3. [2] R. R. Coifman and G. Weiss, Analyse harmonique non-commutatives sur certains espaces homogènes, Lecture Notes in Math., vol.242, Springer-Verlag, Berlin and New York, 97. [3] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc ), [4] L. Hörmander, Estimates for translation invariant operators in L p spaces, Acta Math. 496), [5] I. Egorova, E. Kopylova, and G. Teschl, Dispersion estimates for onedimensional discrete Schrödinger and wave equations, J. Spectr. Theory. [6] E. Kopylova On dispersion decay for discrete wave equations,communications in Mathematical Analysis 72), [7] A. Slavik, Discrete-Space Systems of Partial Dynamic Equations and Discrete- Space Wave Equation, Qualitative Theory of Dynamical Systems 62),