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STRONG CONTINUITY ON HARDY SPACES 6 JACEK DZIUBAŃSKIANDBŁAŻEJ WRÓBEL 1 0 Abstract. We prove the strong continuity of spectral multiplier operators associated with 2 dilations of certain functions on the general Hardy space HL1 introduced by Hofmann, Lu, n Mitrea, Mitrea, Yan. Our results include the heat and Poisson semigroups as well as the a group of imaginary powers. J 9 1 ] 1. Introduction A F In the theory of semigroups of linear operators on Banach spaces the crucial assumption is . that of strong continuity. One often encounters a situation where the semigroup T = e tL h t − t is initially defined on L2(Ω) and L is a non-negative self-adjoint operator. In this case the a spectral theorem immediately gives the strong L2(Ω) continuity lim T f f = 0, m t 0+ t L2(Ω) for f L2(Ω). Assume additionally that T extends to a loc→ally kboun−dedksemigroup [ ∈ { t}t>0 on Lp. More precisely, we impose that for each 1 p < there exists t > 0 such that p 1 ≤ ∞ T C , t [0,t ]. Since weak and strong convergence coincide for semigroups v k tkLp(Ω)→Lp(Ω) ≤ p ∈ p 2 ofoperators(seee.g.[6,Theorem5.8]),itisstraightforwardtoseethatTt isstronglycontinuous 4 on all Lp(Ω), 1 < p < . Moreover, if we assume that T is contractive on L1(Ω), then it t t>0 0 is also strongly continu∞ous on L1(Ω). Quite often the s{em}igroup T may be also defined t t>0 5 { } on function spaces other than Lp. For instance, if T = et∆ is the classical heat semigroup 0 t . on Rd, then it also acts on the atomic Hardy spaces H1. However, even in this case it is not 1 at 0 obvious that the semigroup is strongly continuous on Ha1t. 6 In this paper we impose that T satisfies the so-called Davies-Gaffney estimates (see t t>0 1 (2.3)),andthattheunderlyingspa{ce}Ωisaspaceofhomogeneous typeinthesenseofCoifman- : v Weiss [1]. Under these assumptions, as a corollary of our main result, we prove that e tL and − Xi e t√L are strongly continuous on the Hardy space H1. This Hardy space was introduced − L r by Hofmann, Lu, Mitrea, Mitrea, Yan in [8]. Our results are quite general, as there are a many operators L satisfying (2.3), e.g. Laplace-Beltrami operators on complete Riemannian manifolds (see e.g. [7, Corollary 12.4]) or Schrödinger operators with non-negative potentials. The literature on Lp spectral multipliers for operators satisfying Davies-Gaffney estimates is vast. However, as the Lp theory is not discussed in our paper, we do not provide detailed references on this subject. Instead we kindly refer the interested reader to consult e.g. [11] and references therein. There are also results for spectral multipliers on the Hardy space H1 (or L p more generally H ), see e.g. [3], [4], [5], and [9]. L The methods we use are based on [5], in which the authors proved a Hörmander-type multiplier theorem on H1. The result for semigroups (Corollary 3.2) is a consequence of Theorem L 3.1, which treats dilations of more general multipliers than e λ. Finally, using Theorem 3.1 we − also prove the strong HL1 continuity of the group of imaginary powers {Liu}u∈R, see Corollary 3.3. 2010 Mathematics Subject Classification. 42B30, 47D06, 47A60. Key words and phrases. Hardy space, strong continuity,semigroup of linear operators. 1 2 JACEKDZIUBAŃSKIANDBŁAŻEJWRÓBEL 2. Preliminaries Let (Ω, d(x,y)) be a metric space equipped with a positive measure µ. We assume that (Ω, d, µ)is a space of homogeneous type inthe senseof Coifman-Weiss [1], that is, there exists a constant C > 0 such that (2.1) µ(B (x,2t)) Cµ(B (x,t)) for every x Ω, t > 0, d d ≤ ∈ where B (x,t) = y Ω : d(x,y) < t . The condition (2.1) implies that there exist constants d { ∈ } C > 0 and q > 0 such that 0 (2.2) µ(B (x,st)) C sqµ(B (x,t)) for every x Ω, t > 0, s > 1. d 0 d ≤ ∈ In what follows we set n to be the infimum over q in (2.2). 0 Let e tL be a semigroup of linear operators on L2(Ω, dµ) generated by L, where L is − t>0 { } − anon-negative, self-adjointoperator. Weassumeadditionally thatLisinjectiveonitsdomain. Throughout the paper we impose that T := e tL satisfies Davies-Gaffney estimates, that is, t − dist(U ,U )2 1 2 (2.3) T f ,f Cexp f f |h t 1 2i|≤ − ct k 1kL2(Ω)k 2kL2(Ω) (cid:18) (cid:19) for every f L2(Ω), suppf U , i= 1,2, U are open subsets of Ω. i i i i ∈ ⊂ Davies-Gaffneyestimatesareequivalenttothefinitespeedpropagationofthewaveequation; the reader interested in this topic is kindly referred to [2]. The finite speed propagation of the wave equation is used in the proof of [5, Lemma 4.8] (our Lemma 2.3), which is an important ingredient in the proof of our main Theorem 3.1. For f L2(Ω) we consider the square function S f associated with L defined by h ∈ 1/2 dµ(y) dt S f(x)= t2LT f(y)2 , h | t2 | V(x,t) t ZZΓ(x) ! where Γ(x)= (y,t) Ω (0, ) : d(x,y) t . { ∈ × ∞ ≤ } We define the Hardy space H1 = H1 (Ω) as the (abstract) completion of L L,Sh f L2(Ω) : S f < h L1(Ω) { ∈ k k ∞} in the norm f = S f . k kHL1 k h kL1(Ω) It was proved in Hofmann, Lu, Mitrea, Mitrea, Yan [8] that under our assumption (2.3) the space H1 admits the following atomic decomposition. L Let M 1, M N. A function a is a (1,2,M)-atom for H1 if there exist a ball B = ≥ ∈ L B (y ,r) = y Ω : d(y,y )< r and a function b (LM) such that d 0 0 { ∈ } ∈ D a = LMb; suppLkb B, k = 0,1,...,M; ⊂ (r2L)kb r2Mµ(B) 1/2, k = 0,1,...,M. L2(Ω) − k k ≤ We say that f = λ a is a (1,2,M) atomic representation (of f) if λ l1, each a is j j j { j}∞j=0 ∈ j a (1,2,M) atom, and the sum converges in L2. Then we set P H1 = f: f has an atomic (1,2,M)-representation , L,at,M (cid:26) (cid:27) STRONG CONTINUITY ON HARDY SPACES 3 with the norm given by ∞ ∞ f = inf λ : f = λ a is an atomic (1,2,M) representation . H1 j j j k k L,at,M | | (cid:26)j=0 j=0 (cid:27) X X The space H1 is defined as the (abstract) completion of H1 . L,at,M L,at,M Theorem 4.14 of [8] asserts that for each M > n /4 there exists a constant C > 0 such that 0 C 1 f f C f . − k kHL1 ≤ k kHL1,at,M ≤ k kHL1 In [8] the authors gave also a molecular description of H1. Fix ε > 0 and M > n /4, L 0 M N. We say that a function a˜ is a (1,2,M,ε)-molecule associated to L if there exist a ∈ function ˜b (LM) and a ball B = B (y ,r) such that d 0 ∈ D a˜ = LM˜b; (r2L)k˜b r2M2 jεµ(B(y ,2jr)) 1/2 k kL2(UjB)) ≤ − 0 − for k = 0,1,...,M, j = 0,1,2,..., where U = B, U (B)= B (y ,2jr) B (y ,2j 1r) for j 1. 0 j d 0 d 0 − \ ≥ Thedecompositionf = λ a˜ isa(1,2,M,ε) molecularrepresentation(off)if λ l1, j j j { j}∞j=0 ∈ each a˜ is a (1,2,M,ε) molecule, and the sum converges in L2. Then we define j P H1 = f L2(Ω): f has a molecular (1,2,M,ε)-representation , L,mol,M,ε ∈ (cid:26) (cid:27) with the norm given by ∞ ∞ f = inf λ : f = λ a˜ is a molecular (1,2,M,ε) representation . H1 j j j k k L,mol,M,ε | | (cid:26)j=0 j=0 (cid:27) X X The space H1 is defined as the (abstract) completion of H1 . L,mol,M,ε L,mol,M It was proved in [8, Corollary 5.3] that for each M > n /4 and ε > 0 it holds H1 = 0 L,at,M H1 , with the equivalence of the norms. Moreover, we have H1 = H1 and, conse- L,mol,M,ε L L,at,M quently, H1 = H1 = H1 , for N,M > n /4. L L,at,M L,mol,N,ε 0 The following lemma is a slight extension of the observation following the proof of [8, Corol- lary 5.3]. Lemma 2.1. Let T be an operator which is bounded on L2. Assume that there are ε > 0 and positive integers M,N > n /4 such that T maps (1,2,M) atoms uniformly to (1,2,N,ε) 0 molecules. More precisely, we impose that there is an A > 0 such that T(a) H1 k k L,mol,N,ε ≤ A a for all (1,2,M) atoms a. Then T has the unique bounded extension Text to H1 k kH1L,at,M L which satisfies Textf CA f . k kHL1 ≤ k kHL1 Proof. By density of H1 in H1 = H1 it is enough to prove that T is bounded from L,at,M L,at,M L H1 to H1 . L,at,M L,mol,N,ε Takef H1 ,sothatf = λ a ,wherea are(1,2,M)atoms, λ l1,andthesum ∈ L,at,M j j j j { j} ∈ converges in L2. We chose λ and a in a way that λ 2 f . The L2 boundedness j Pj j| j| ≤ k kH1L,at,M of T implies that Tf = λ T(a ) is a (1,2,N,ε) molecular representation of Tf. Therefore, j j j P PTf A λ 2A f , H1 j H1 k k L,mol,N,ε ≤ | |≤ k k L,at,M j X 4 JACEKDZIUBAŃSKIANDBŁAŻEJWRÓBEL and the proof is completed. (cid:3) Let E := E be the spectral measure of √L so that √L Lf = ∞λ2dE(λ)f. Z0 Then,foraboundedBorel-measurablefunctionm: [0, ) Cthespectralmultiplieroperator ∞ → m(√L) is given on L2(Ω) by m(√L)f = ∞m(λ)dE(λ)f. Z0 Using Lemma 2.1 with 2M in place of M and N = M > n /4 we deduce the following 0 enhancement of [5, Theorem 4.2]. Theorem 2.2. Assume that m is a bounded function defined on [0, ) and such that for some ∞ real number α> (n +1)/2 and any nonzero function η C (2 1,2) we have 0 ∈ c∞ − (2.4) m η,α := sup η( )m(t ) W2,α(R) < , k k k · · k ∞ t>0 where F Wp,α(R) = (I d2/dx2)α/2F Lp(R). Then the operator m(√L) extends uniquely to k k k − k a bounded operator on H1. Moreover, there exists a constant C > 0 such that L m(√L)f C m f , f H1. k kHL1 ≤ k kη,αk kHL1 ∈ L For the convenience of the reader we also restate Lemma 4.8 of [5]. Lemma 2.3. Let γ > 1/2, β > 0. Then there exists a constant C > 0 such that for every even function F W2,γ+β/2(R) and every g L2(Ω), suppg B (y ,r), we have d 0 ∈ ∈ ⊂ β d(x,y ) F(2 j√L)g(x)2 0 dµ(x) C(r2j) β F 2 g 2 | − | r ≤ − k kW2,γ+β/2k kL2(Ω) Zd(x,y0)>2r (cid:18) (cid:19) for j Z. ∈ Summarizing this section, we may use whichever of the spaces H1 or H1 , M > L,at,M L,mol,M n /4, that is convenient. 0 3. The results We are going to study strong H1 convergence of operators of the form m(tL) as t 0. L → Observe that for the strong L2 convergence it is enough to assume that m is bounded and continuous at 0. Our first main result is the following theorem. Theorem 3.1. Take κ an integer larger than (n +1)/2. Let m: [0, ) C be a continuous 0 ∞ → function which is Cκ on (0, ). Assume that m satisfies the Mikhlin condition of order κ, i.e. ∞ (3.1) sup sup λjm(j)(λ) < , | | ∞ 0 j κλ>0 ≤ ≤ and, additionally (3.2) lim λjm(j)(λ) = 0, j = 1,...,κ. λ 0+ → Then, we have the following strong H1 convergence, L (3.3) lim m(t√L)f = m(0)f, for every f H1. t 0+ ∈ L → STRONG CONTINUITY ON HARDY SPACES 5 Remark. Straightforward modifications in the proof we present below give a slightly stronger version of the theorem, with the assumption (3.1) replaced by (2.4) for some real number α larger than (n +1)/2. 0 Before proceeding to the proof let us note the following important corollary. Corollary 3.2. Both the heat semigroup e tL and the Poisson semigroup e t√L are strongly − − continuous on H1. L Proof of Theorem 3.1. Let M be an integer such that 2M κ. Then M > n /4. From 0 ≥ Theorem 2.2 and the dilation invariance of (2.4) it follows that m(tL) is well-defined and bounded on H1, uniformly in t >0. Therefore it is enough to prove (3.3) for f H1 . L ∈ L,at,2M We claim that we can further reduce the proof to demonstrating that (3.4) lim m(t√L)a m(0)a = 0, for a being a (1,2,2M)-atom. t 0+k − kHL1 → Indeed, if (3.4) is true, and f = λ a (where λ l1 and the sum defining f converges j j j { j} ∈ also in L2) then we obtain P ∞ ∞ [m(t√L) m(0)](f) = λ [m(t√L) m(0)](a ) λ [m(t√L) m(0)](a ) . k − kHL1 j − j HL1 ≤ | j|k − j kHL1 j=0 j=0 (cid:13)X (cid:13) X (cid:13) (cid:13) Now, from Theorem 2.2 it follows that [m(t√L) m(0)](a ) is uniformly bounded in t. k − j kHL1 Therefore, thanks to (3.4) we obtain lim [m(t√L) m(0)](f) = 0, as desired. t→0+k − kHL1 To prove (3.4) we will show that there is an ε > 0 such that for every a being a (1,2,2M)- atom the function (m(t√L) m(0))a is a multiple of a (1,2,M,ε) molecule and the multiple − constant tends to 0 as t 0. Note that the rate of convergence may well depend on a for → our purposes. There is no loss of generality if we assume that the associated ball B has radius 1, that is B = B(y ,1) for certain y Ω. This means that a = L2Mb where b (L2M), 0 0 ∈ ∈ D suppLkb B,and Lkb 1fork = 0,1,...,2M. Then, denoting˜b = [m(t√L) m(0)]LMb, L2 ⊂ k k ≤ − we have [m(t√L) m(0)]a = LM˜b. Our task is to study the behavior of L2-norms of Lk˜b = − [m(t√L) m(0)]Lk+Mb, k = 0,1,...,M, on the sets U (B). j − Let ψ C (1,2) be such that ψ(2 ℓλ) = 1 for λ > 0. For ℓ Z and λ R set ∈ c∞ 2 ℓ Z − 0 ∈ ∈ Ψℓ0(λ) = 1− ∞ℓ=ℓ0ψ(2−ℓ|λ|). We sPpli∈t P ∞ m(t λ ) m(0) = Ψ (λ)(m(t λ ) m(0))+ ψ(2 ℓ λ )(m(t λ ) m(0)) | | − ℓ0 | | − − | | | | − ℓX=ℓ0 and for λ R put ∈ m (λ) = ψ(2 ℓ λ )(m(t λ ) m(0)), m˜ (λ) =m (2ℓλ) = ψ(λ )(m(t2ℓ λ ) m(0)). ℓ,t − ℓ,t ℓ,t | | | | − | | | | − Fix ε > 0 and γ > 1/2 such that γ +ε+n /2 = α. Set β = n +2ε, so that γ +β/2 = α. 0 0 Recall that suppLk+Mb B and m (λ) = m ( λ). Applying Lemma 2.3 we have ℓ,t ℓ,t ⊂ − m (√L)Lk+Mb(x)2d(x,y )βdµ(x) C2 ℓβ m˜ Lk+Mb 2 , | ℓ,t | 0 ≤ − k ℓ,tkW2,αk kL2 Zd(x,y0)>2 hence, using (3.1) we arrive at m (√L)Lk+Mb(x)2dµ(x) C 2 ℓβ2 jβ Lk+Mb 2 . | ℓ,t | ≤ α − − k kL2 ZUj(B) 6 JACEKDZIUBAŃSKIANDBŁAŻEJWRÓBEL Therefore 1/2 (3.5) mℓ,t(√L)Lk+Mb|2dµ ≤ Cα1/22−jβ/22−ℓ0β/2kLk+MbkL2. (cid:16)ZUj(B)(cid:12)ℓX>ℓ0 (cid:17) (cid:12) Note that the estim(cid:12)ate above does not depend on t > 0. For the rest of the proof we fix ℓ 0 large enough. Denote n (λ) = Ψ (λ)(m(t λ ) m(0))λ2M, λ R. Clearly, n (λ) = n ( λ). Using ℓ0,t ℓ0 | | − ∈ ℓ0,t ℓ0,t − Lemma 2.3 we get Lkn (√L)b(x)2d(x,y )βdµ(x) C n 2 Lkb 2 = C n 2 Lkb 2 , | ℓ0,t | 0 ≤ k ℓ0,tkW2,γ+β/2k kL2 k ℓ0,tkW2,αk kL2 Zd(x,y0)>2 and, consequently, (3.6) Lkn (√L)b(x)2dµ(x) C2 βj n 2 Lkb 2 . | ℓ0,t | ≤ − k ℓ0,tkW2,αk kL2 ZUj(B) We claim that n (λ) = Ψ (λ)(m(t λ ) m(0))λ2M satisfies lim n = 0. Indeed ℓ0,t ℓ0 | | − t→0+k ℓ0,tkW2,α n n n + (n )(κ) . C (m(t λ ) m(0))λ2M , k ℓ0,tkW2,α ≤ k ℓ0,tkW2,κ ≈ k ℓ0,tkL2 k ℓ0,t kL2 l0k | | − kCκ[0,2l0+1] and, because of (3.2), the quantity on the right hand side of the above inequality approaches 0 as t 0+. Summarizing (3.5) and (3.6) we have proved that, for k = 0,...,M, it holds → Lk+M[m(t√L) m(0)]b(x)2dµ(x) C2 jβ(2 ℓ0β Lk+Mb 2 + n 2 Lkb 2 ) | − | ≤ − − k kL2 k ℓ0,tkW2,αk kL2 ZUj(B) C2 jβ(2 ℓ0β + n 2 )µ(B(y ,1)) 1 ≤ − − k ℓ0,tkW2,α 0 − µ(B(y ,2j)) C2 jβ(2 ℓ0β + n 2 ) 0 µ(B(y ,2j)) 1. ≤ − − k ℓ0,tkW2,α µ(B(y ,1)) 0 − 0 Using (2.2) with q = n +ε we obtain 0 Lk+M[m(t√L) m(0)]b(x)2dµ(x) C2 jβ(2 ℓ0β + n 2 )2jqµ(B(y ,2j)) 1, | − | ≤ − − k ℓ0,tkW2,α 0 − ZUj(B) which is enough for our purpose, since β q = ε, γ +β/2 = α, and lim n 2 = 0. − t→0+k ℓ0,tkW2,α To estimate Lk+M(m(t√L) m(0))b on 2B, we note that by the spectral theorem, − [m(t√L) m(0)]Lk+Mb 2 [m(t√L) m(0)]Lk+Mb 2 k − kL2(2B) ≤ k − kL2(Ω) = ∞ m(tλ) m(0)2dE (λ) 0 as t 0 | − | Lk+Mb,Lk+Mb → → Z0 thanks to the Lebesgue dominated convergence theorem and the continuity of m at 0. (cid:3) We finish the paper with showing the strong convergence of the group of imaginary powers. This is achieved by using Theorems 2.2 and 3.1. Corollary 3.3. Let f H1. Then lim Liuf = f, the limit being in H1. ∈ L u→0 L Proof. Let φ(λ) be a smooth function on [0, ) which is equal to 1 on [0,2] and vanishes for ∞ λ > 4. Theorem 2.2 implies (3.7) sup Liu < . u 1k kHL1→HL1 ∞ | |≤ STRONG CONTINUITY ON HARDY SPACES 7 Moreover, from Theorem 3.1 it follows that lim φ(sL)f = f, for f H1 (the limit being in H1). Hence, a density argument together wits→h0(+3.7) show that it is e∈nouLgh to justify that L for each fixed s > 0, we have (3.8) lim(Liu 1)φ(sL)f = 0, f H1, u 0 − ∈ L → the limit being understood in H1. Let M be an integer larger than (n +3)/2. As the linear L 0 span of atoms is dense in H1, in view of (3.7) it suffices to verify (3.8) for f being a fixed L (1,2,2M) atom. Then f = L2Mb. Moreover, a = LMb is a multiple of a (1,2,M) atom, with a multiple constant that depends on f.Letm (λ) = λ2M(λ2iu 1)φ(sλ2) andletη beanon-zero u − smooth function supported in [1/2,2]. A short computation shows that ulim0st>up0 kη(·)mu(t·)kW2,M−1 = 0. → We also have (Liu 1)φ(sL)f = m (√L)(a) with a being a (1,2,M)-atom. Since M 1 > u − − (n +1)/2, using Theorem 2.2 we finish the proof of Corollary 3.3. (cid:3) 0 Remark. Corollary 3.3 seems crucial in extending various results in harmonic analysis based on the group of imaginary powers from the Lp to the H1 setting. For potential applications L see e.g. [10] or [12, Remark 3]. Acknowledgments. The research of the first named author was supported by Polish funds for sciences, National Science Centre (NCN),Poland, Research Project DEC-2012/05/B/ST1/ 00672. The research of the second named author was supported by Polish funds for sciences, National Science Centre (NCN), Poland, Research Project DEC-2014/15/D/ST1/00405, by Foundation for Polish Science - START scholarship, and by the Italian PRIN 2011 project Real and complex manifolds: geometry, topology and harmonic analysis. References [1] Ronald R. 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Soc. 110 (1990), no. 3, 639–647. MR 1028046 (91f:42010) 8 JACEKDZIUBAŃSKIANDBŁAŻEJWRÓBEL [11] Adam Sikora, Lixin Yan, and Xiaohua Yao, Sharp spectral multipliers for operators satisfying generalized Gaussian estimates, J. Funct.Anal. 266 (2014), no. 1, 368–409. MR 3121735 [12] Błażej Wróbel, On the consequences of a Mikhlin-Hörmander functional calculus: maximal and square function estimates, submitted (2015), arXiv:1507.08114. Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland E-mail address: [email protected] Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, via R. Cozzi 53 I-20125, Milano, Italy, & Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland E-mail address: [email protected]