EXTENSIONS OF WEAK-TYPE MULTIPLIERS 2 P. MOHANTY AND S. MADAN 0 0 2 Abstract. In this paper we prove that if Λ ∈ Mp(RN) and has n a compact support then Λ is a weak summability kernelfor 1 0 x x such that Cp (1.1) µ{x ∈ G : |Tf(x)| > t} ≤ kfkp tp p for all f ∈ Lp(G) and t > 0. (Here Lp,∞ denotes the standard weak Lp spaces.) Asmar, Berkson and Gillespie in [3] proved that for all such operators T there exists a φ ∈ L∞(Gˆ) such that (Tf)∧ = φfˆ for all f ∈ L2 ∩ Lp(G). We will also call such φ’s to be multipliers of weak type (p,p). Let M(w)(Gˆ) denote the space of multipliers of weak type p (w) (p,p) for 1 ≤ p < ∞, and let N (φ) be the smallest constant C such p that inequality (1.1) holds. In this paper we are concerned with extensions of weak type multipliers from ZN to RN through summability kernels. For similar results on strong type multipliers , see [6], [4]. Here we identify TN with [0,1)N and for f ∈ L1(RN) we define its Fourier transform as fˆ(ξ) = f(x) e−2πiξ·xdx for ξ ∈ RN. Let us define summability ker- RRN nels for weak type multipliers as follows Definition 1.1. A bounded measurable function Λ : RN −→ C is called a weak summability kernel for M(w)(RN) if for φ ∈ M(w)(ZN) p p the function W (ξ) = φ(n)Λ(ξ − n) is defined and belongs to φ,Λ nP∈ZN M(w)(RN). p This definition is just the weak type analouge of summability kernel for strong type multipliers [4]. We first cite two important results EXTENSIONS OF WEAK-TYPE MULTIPLIERS 3 regarding the summability kernels of strong type multipliers from the work of Jodeit [6] and of Berkson, Paluszynski and Weiss [4]: Theorem 1.1. [6] Let S ∈ L1(RN) and supp S ⊆ [1, 3]N with τ = 4 4 |sˆ(n)| < ∞ , where s is the 1-periodic extension of S, then the nP∈ZN function defined by W (ξ) = φ(n)Sˆ(ξ − n) belongs to M (RN), φ,Sˆ p nP∈ZN for 1 ≤ p < ∞ with kW k ≤ C τkφk φ,Sˆ Mp(RN) p Mp(ZN) Theorem 1.2. [4] For 1 ≤ p < ∞, let Λ ∈ M (RN) and suppΛ ⊆ p [1, 3]N. For φ ∈ M (ZN) define W (ξ) = φ(n)Λ(ξ − n) on 4 4 p φ,Λ nP∈ZN RN. Then W ∈ M(RN) and kW k ≤ C kΛk kφk φ,Λ p φ,Λ Mp(RN) p Mp(RN) Mp(ZN) where C is a constant. (Further, if Λ has arbitary compact p support the same result holds except that the constant C necessarily p depends on the support of Λ, as shown in [4] Asmar, Berkson and Gillespie proved a weak type analogue of The- orem 1.1 in [2]. In this same paper they also proved that Λ defined by N Λ(ξ) = max(1 −|ξ |,0) for ξ = (ξ ,...,ξ ) is a weak type summa- j 1 N jQ=1 bility kernel. In this paper, we prove the weak type analouge of The- orem 1.2 in §2, for 1 < p < ∞. In §3 we relax the hypothesis that supp Λ ⊆ [1, 3]N. For the proof of our main result , as in [4], we will 4 4 obtain the weak type inequalities by applying the technique of transference couples due to Berkson, Paluszynˇski, and Weiss [4]. Definition 1.2. For a locally compact group G, a transference couple is a pair (S,T) = ({S },{T }) , u ∈ G, of strongly continuous map- u u pings defined on G with values in B(X), where X is a Banach space, satisfying 4 P. MOHANTY AND S. MADAN (i) C = sup{kS k : u ∈ G} < ∞ S u (ii) C = sup{kT k : u ∈ G} < ∞ T u (iii) S T = T ∀u,v ∈ G v u vu In §4, as an application of our result, we prove a weak type analogue of an extension theorem by de Leeuw. 2. Weak-Type Inequality for Transference Couples and The Main Theorem Let Λ ∈ L∞(RN) and supp Λ ⊆ [1, 3]N. Consider the following 4 4 transference couple (S,T) used by Berkson, Paluszyn´ski, and Weiss in [4]. For u ∈ TN the family T = {T } is given by u (2.2) (T f)∧(ξ) = Λ(ξ −n)e2πiu.nfˆ(ξ), for f ∈ Lp(RN) u X n∈ZN and the family S = {S } is defined by u (2.3) (S f)∧(ξ) = b(ξ −n)e2πiu.nfˆ(ξ). for f ∈ Lp(RN), u X n∈ZN N where b(ξ) = b (ξ ) for ξ = (ξ ,......,ξ ) and for each i, b is the i i 1 N i iQ=1 continuous function defined on R as b (x) = 1 if x ∈ [1, 3], linear in i 4 4 [0, 1)∪(3,1] and 0 otherwise. It is easy to see that 4 4 ˇ (2.4) S f(x) = β (l)f(x+u−l) a.e., u u X l∈ZN ˇ where β is the inverse Fourier transform of the function β (ξ) = u u b(ξ)e2πiξ.u, given explicitily by N ˇ ˇ β (ξ) = β (ξ ), u ui i Y i=1 EXTENSIONS OF WEAK-TYPE MULTIPLIERS 5 where (2.5) 2e2πi(ξi+2ui)(cosπ(ξ −u )−cosπ(ξ −u )) if ξ 6= u βˇ (ξ ) =  π2(ξi−ui)2 2 i i i i i i ui i 3e2πi(ξi+2ui) if ξ = u . 4 i i   Then by a straightforward calculation using Eqn.(2.5) we have ˇ (2.6) |β (l)| ≤ β(l) = C < ∞, u X X l∈ZN l∈ZN N where β(l) = β (l ) and i=1 i i Q 1 if l > 1 (li−1)2 i  βi(li) = (li+11)2 if li < 1 kb k otherwise.  i 1     Inthe following theorem we shall show that the operator transferred by T (ofthetransfernce couple (S,T)defined inEqn. (2.2)andEqn. (2.3)) given by Hkf(.) = Z k(u)Tu−1f(.)du, TN where k ∈ L1(TN) and f ∈ Lp(RN), satisfies a weak (p,p) inequality. Theorem 2.1. Let(S,T) bethe transferencecouple asdefinedin Eqn.(2.2) and Eqn. (2.3). Then for 1 < p < ∞ and t > 0 C C λ{x ∈ RN : |H f(x)| > t} ≤ ( pC N(w)(k)kfk )p, k t T p p where λ denotes the Lebesgue measure of RN, C = β(l) as in l∈PZN Eqn. (2.6), C is the uniform bound for the family T = {T }, and T u C = p . p p−1 6 P. MOHANTY AND S. MADAN Proof: Assumef ∈ S(RN). Fort > 0defineE = {x : |H f(x)| > t}. t k Notice that ˇ Hkf(x) = Sv−1SvHkf(x) = βv−1(l) TN k(u)Tu−1vf(x−v−l)du| > l∈PZN R t}. Let Ft = {(v,x) ∈ TN ×RN : | βˇv−1(l) TN k(u)Tu−1vf(x − l∈PZN R l)du| > t}. Then, using translation invariant of Lebesgue measure λ(Et) = λ{x ∈ RN : |Sv−1 Z k(u)Tu−1vf(x)du| > t} TN = λ{x ∈ RN : | βˇv−1(l)Z k(u)Tu−1vf(x−l)du| > t} X TN l∈ZN = χ (v,x)dxdv F Z Z t TN RN ˇ = Z |{v : | βv−1(l)Z k(u)Tu−1vf(x−l)du| > t}|dx, RN X TN l∈ZN where |E| denotes the measure of the subset E ⊆ TN. Thus λ(Et) ≤ Z |{v : β(l)|Z k(u)Tu−1vf(x−l)|du > t}|dx RN X TN l∈ZN = |{v : β(l)|k∗F(.,x−l)(v)| > t}|dx, where F(v,x) = T f(x) a.e.. Z v RN X l∈ZN We know that sup tλf(t)p1 = kfkLp,∞ for f ∈ Lp,∞. Also, since p > 1, t>0 k k is equivalent to a norm k k∗ ([8]), using traingle inequality for p,∞ p,∞ norms we have 1 λ(E ) ≤ k β(l)|k ∗F(.,x−l)kp dx t Z tp Lp,∞(TN) RN X l∈ZN 1 p ≤ Cp ( β(l)kk ∗F(.,x−l)k∗ )pdx, where C = p Z tp Lp,∞(TN) p p−1 RN X l∈ZN 1 ≤ C ( β(l)N(w)(k)kF(.,x−l)k )pdx, pZ tp p Lp(TN) RN X l∈ZN EXTENSIONS OF WEAK-TYPE MULTIPLIERS 7 (w) whereN (k) istheweak-type normoftheconvolution operatorf 7−→ p k ∗f for f ∈ Lp(TN). Thus, 1 λ(Et) ≤ Cpptp β(l)Np(w)(k)(Z Z |Tvf(x−l)|pdxdv)p1)p X RN TN l∈ZN 1 = Cpptp( β(l)Np(w)(k)(Z Z |Tvf(x−l)|pdxdv)p1)p X TN RN l∈ZN CC C ≤ ( p TN(w)(k)kfk )p. tp p p Hence, H f satisfies a weak (p,p) inequality. k In order to prove the weak-type analogue of Theorem 1.2 we need the following Lemma proved by Asmar, Berkson, and Gillespie in [1]. Lemma 2.1. [1]Supposethat1 ≤ p < ∞, {φ } ⊆ M(w)(Gˆ); sup{|φ (γ)| : j p j j ∈ N,γ ∈ Gˆ} < ∞ and suppose φ converges pointwise a.e. on Gˆ j to a function φ . If liminfN(w)(φ ) < ∞ then φ ∈ M(w)(Gˆ) and p j p j (w) (w) N (φ) ≤ liminfN (φ ). p p j j In the following theorem, we use the family of operators {T } u defined in ( 2.2) with Λ ∈ M (RN) and p suppΛ ⊆ [1, 3]N. In ths case, 4 4 by [4] we have C ≤ c kΛk ), T p Mp(RN where c is a constant. p Theorem 2.2. Suppose 1 < p < ∞ and Λ ∈ M (RN) is supported in p the set [1, 3]N. For φ ∈ M(w)(ZN) define p 4 4 W (ξ) = φ(n)Λ(ξ −n) on RN. φ,Λ X n∈ZN Then W ∈ M(w)(RN) and N(w)(W ) ≤ CN(w)(φ)kΛk . φ,Λ p p φ,Λ p Mp(RN) Proof: Using Lemma 2.1 we first show that it is enough to prove the theorem for φ ∈ M(w)(ZN) having finite support. Suppose the theorem p 8 P. MOHANTY AND S. MADAN is true for finitely supported φ. Then for arbitrary φ ∈ M(w)(ZN), p ˆ define φ = k φ, where k is the j-th Féjer kernel. Then for each j , j j j φ ’shavefinitesupport and(T f)∧(n) = φ (n)fˆ(n) = (T (k ∗f))∧(n). j φj j φ j So φ ∈ M(w)(ZN) for each j and N(w)(φ ) ≤ N(w)(φ). Define j p p j p W (ξ) = φ (n)Λ(ξ −n). Now liminfW (ξ) = W (ξ). Also, φj,Λ n∈PZN j j φj,Λ φ,Λ by our assumption N(w)(W ) ≤ CN(w)(φ )kΛk p φj,Λ p j Mp(RN) ≤ CN(w)(φ)kΛk p Mp(RN) and |W | ≤ 2kΛk kφ k ≤ 2kΛk kφk . Thus by Lemma 2.1, φj,Λ ∞ j ∞ ∞ ∞ applied to W ’s ,we conclude that W ∈ M(w)(RN). Hence it is φj,Λ φ,Λ p enough to assume that φ ∈ M(w)(ZN) has finite support. p Nowletφ ∈ M(w)(ZN)befinitelysupported. Definek(u) = φ(n)e−2πiu.n p n∈PZN then k ∈ L1(TN) and kˆ(n) = φ(n). For this particular k and the transference couple (S,T) defined above. We have ∧ ∧ (H f) (ξ) = (T f) (ξ). k Wφ,Λ ThusT f = H f. HencefromTheorem2.1andsinceC ≤ c kΛk , Wφ,Λ k T p Mp(RN we have C λ{x ∈ RN : |T f(x)| > t} ≤ ( N(w)(φ)kΛk kfk )p. Wφ,Λ t p Mp(RN) p 3. Lattice Preserving Linear Transformations and Multipliers We shall now relax the hypothesis that supp Λ ⊆ [1, 3]N to allow 4 4 Λ to have arbitrary compact support. In fact this can be done by a partitionofidentity argument asin[4]. Herewe giveadifferent method by proving Lemma 3.2 below. Particular cases of this lemma occur in [6] and in [2]. Suppose supp Λ ⊆ [−M,M]N; define Λ (ξ) = Λ (4Mξ), M 1 EXTENSIONS OF WEAK-TYPE MULTIPLIERS 9 where Λ (ξ) = Λ(ξ − 1). So supp Λ ⊆ [1, 3]N. Thus if we define a 1 2 M 4 4 non-singular transformation A : RN −→ RN such that Ax = 4Mx then Λ = Λ ◦ A. In order to replace the support condition we need to M 1 prove Λ ◦ A−1 is a summability kernel. In the work of Jodeit and M of Asmar, Berkson and Gillespie they assume A in Lemma 3.2 to be multiplication by 2. We have combined some of the results proved by Gröchenig and Madych [5] in the following lemma which will help us to prove Lemma 3.2. In the proof of Theorem 3.1, we only use the case of a diagonal linear transform, but the more general results proved below are of some interest in their own right. Lemma 3.1. [5] Let A : RN −→ RN be a non-singular linear transformation which preserves the lattice ZN (i.e. A(ZN) ⊆ ZN). Then the following are true. (i) The number of distinct coset representatives of ZN/AZN is equal to q = |detA|. (ii) If Q = [0,1)N and k ,.....,k are the distinct coset representatives 0 1 q of ZN/AZN then the sets A−1(Q +k ) are mutually disjoint. 0 i (iii) Let Q = ∪q A−1(Q +k ), then λ(Q) = 1 and ∪ (Q+k) ≃ RN. i=1 0 i k∈ZN (iv) AQ ≃ ∪q (Q +k ). i=1 0 i Where E ≃ F if λ(F △E) = 0. The above result is essentially contained in [5]. Lemma 3.2. Let A be as in Lemma 3.1. Denote At = B, where At is the transpose of A. For φ ∈ l (ZN) define ∞ ψ(n) = φ(Bn) and 10 P. MOHANTY AND S. MADAN φ(B−1n) n ∈ BZN  η(n) =   0 otherwise.   (i) If φ ∈ M (ZN) then ψ,η ∈ M (ZN) with multiplier norms not p p exceeding the multiplier norm of φ. (ii) If φ ∈ M(w)(ZN) then ψ,η ∈ M(w)(ZN) with weak multiplier norms p p not exceeding the weak multiplier norm of φ. Proof: (i)Forf ∈ Lp(Q ),weletf againdenotetheperiodicextension 0 to RN. Define Sf(x) = f(Ax) , then Sf is also periodic and |Sf(x)|pdx = |Sf(x)|p χ (x−j)dx Z Z Q Q0 Q0 Xj = |Sf(x)|pχ (x)dx Z Q Xj Q0+j = |Sf(x)|pdx Z Q 1 = |f(x)|pdx |detA| Z AQ q 1 = |f(x)|pdx ((iv) of Lemma 3.1) q Z Xi=1 Q0+ki = |f(x)|pdx. Z Q0 Thus S is an isometry, i.e., k Sf k =k f k . Further, from Lp(Q0) Lp(Q0) the orthogonality relations of the characters (Lemma 1, [7]) we have fˆ(B−1n) if n ∈ BZN ∧  (Sf) (n) =   0 otherwise.  

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