Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus.

Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus. Xuan Thinh Duong (Macquarie University, Australia) Joint work with Ji Li, Zhongshan University Harmonic Analysis and PDEs - Nagoya University, Japan September 2009

1. Introduction and background The theory of Hardy spaces in R n was initiated by E. Stein and G. Weiss (1960). It was developed into a fruitful theory in the last few decades by many authors such as C. Fefferman, R. Coifman, E. Stein, G. Weiss, D-C Chang, S. Krantz, A. Miyachi, A. Uchiyama and many others. An L 1 function f on R n is in the Hardy space H 1 (R n ) if the area integral function of the Poisson integral e t f satisfies Z Z S(f )(x) = 0 y x <t t e t f (y) 2 t 1 n dy dt 1/2 L 1 (R n ). (1) There are a number of equivalent characterizations of functions in the H 1 (R n ) space, including the atomic decomposition. One main application of the theory of Hardy spaces is the following interpolation theorem: Assume that: (i) T is a bounded linear operator on L 2 (R n ), (ii) T maps the Hardy space H 1 (R n ) into L 1 (R n ). Then we can conclude that T is bounded on L p (R n ) for all 1 < p < 2. Remark: Assume (i), then a sufficient condition for (ii) holds is that T has an associated kernel which satisfies the Hörmander condition.

Hardy spaces on domains for 0 < p 1: The theory of Hardy spaces H 1 was extended to H p, 0 < p 1 and to domains of R n as well as spaces of homogeneous type. Consider two subspaces of h p (R n ) specific to Ω when Ω is a bounded Lipschitz domain in R n. Let D(Ω) denote the space of C functions with compact support in Ω, and let D (Ω) denotes its dual, the space of distributions on Ω. The space h p r (Ω) consists of elements of D (Ω) which are the restrictions to Ω of elements of h p (R n ), i.e., h p r (Ω) = f S (R n ) : there exists F h p (R n ) such that F Ω = f = h p (R n )\ F h p (R n ) : F = 0 on Ω with [f ] h p r (Ω) = inf f h p (R n ) : f [f ]. (2) A distribution f on Ω is said to be in h p z (Ω) if the zero extension of f to R n belongs to h p (R n ), i.e., h p z (Ω) = h p (R n ) f h p (R n ) : f = 0 on ( Ω) c / f h p (R n ) : f = 0 on Ω. We can identify h p z (Ω) with a set of distributions in D (Ω), equipped with the quotient norm and it is clearly a subspace of h p r (Ω). We can also obtain atomic decomposition characterizations for these spaces.

2. Hardy spaces associated with operators To estimate singular integrals with non-smooth kernels, a theory of Hardy spaces associated to operators was introduced and developed by many authors. The following are some results which are closely related to this talk. Assume that L has a bounded holomorphic functional calculus on L 2 (R n ) and the kernel of the heat semigroup e tl has a pointwise Poisson upper bound: (i) P. Auscher, X.T. Duong and A. M c Intosh introduced the Hardy space H 1 L(R n ) associated to an operator L (about 10 years ago). (ii) X.T. Duong and L.X. Yan introduced the space BMO L (X) adapted to L (where X is doubling or subset of doubling space) and established the duality of H 1 L(R n ) and BMO L (R n ) where L denotes the adjoint operator of L. (iii) L.X. Yan studied the Hardy space H p L (Rn ) associated to an operator L for all 0 < p < 1. Recently: (iv) P. Auscher, A. M c Intosh and E. Russ studied Hardy spaces H p L, p 1, adapted to the Hodge Laplacian on Riemann manifolds with doubling measure. (v) S. Hofmann and S. Mayboroda studied BMO spaces associated with divergence form elliptic operators on R n with complex coefficients. (vi) S. Hofmann, G. Lu, D. Mitrea, M. Mitrea and L.X. Yan developed a theory of H 1 and BMO spaces adapted to a non-negative, self-adjoint operator L whose heat kernel satisfies the Davies-Gaffney bounds, in the setting of a space of homogeneous type X.

Assumptions: Let (X, d, µ) be a space of homogeneous type, i.e. X has doubling (volume) property. Let L be a linear operator of type ω on L 2 (X ) with ω < π/2, hence L generates a holomorphic semigroup e zl, 0 Arg(z) < π/2 ω. In this talk, we introduce and study Hardy spaces H p L (X ), 0 0 satisfies the Davies-Gaffney estimate. That is, there exist constants C, c > 0 such that for any open subsets U 1, U 2 X, for every f i L 2 (X ) with supp f i U i, i = 1, 2, e tl f 1, f 2 C exp where dist(u 1, U 2) := inf x U1,y U 2 d(x, y). dist(u1, U2)2 f 1 c t L 2 (X ) f 2 L 2 (X ), t > 0, (H2) The operator L has a bounded holomorphic functional calculus on L 2 (X ). That is, there exists c ν,2 > 0 such that for b H (S 0 ν), b(l) is a bounded operator on L 2 (X ) and for any g L 2 (X ). b(l)g 2 c ν,2 b g 2

Lipschitz spaces: The Hardy spaces H p (X ) when p < 1 are not spaces of functions defined on X. In general, elements of H p are distributions. Here, we introduce an appropriate space of linear functionals to define these Hardy spaces. We first introduce our adapted Lipschitz spaces Λ α,s L (Ω), α > 0 and s N, associated to an operator L on space X as follows. Let φ = L M ν be a function in L 2 (X ), where ν D(L M ). For ɛ > 0 and M N, we introduce the norm φ M p,2,m,ɛ 0 (L) h := sup 2 jɛ V (x 0, 2 j ) p 1 1 2 j 0 MX L k ν i L, 2 (U j (B 0 )) where B 0 is the ball centered at some x 0 X with radius 1, and we set n o M p,2,m,ɛ 0 (L) := φ = L M ν L 2 (X ) : φ p,2,m,ɛ < M. 0 (L) p,2,m,ɛ Let `M (L) 0 be the dual of M p,2,m,ɛ 0 (L). Let A t denote either (I + t 2 L) 1 or e t2l. p,2,m,ɛ We can check that if f `M (L) 0, then the distribution (I A t) M f belongs to L 2 loc(x ) (by using the Davies-Gaffney estimates). k=0

In order to define our adapted Lipschitz spaces Λ α,s L (Ω), we need to introduce one more space. For any 0 0 `Mp,2,M,ɛ 0 (L). (3) Definition Let L be an operator satisfying assumptions (H1) and (H2). If α 0 and an integer s ˆ nα, 2 then an element l EM,(α+1) 1 is said to belong to Λ α,s L (Ω) if» Z 1 1/2 (I (I + rbl) 2 1 ) s l(x) 2dµ(x) C, (4) µ(b) 1+2α B where B is any ball in X and C depends only on l. Let R α,s (l) be the infimum of all C for which (4) holds. The Lipschitz norm of l in Λ α,s L (Ω) is given by l α,s Λ (Ω) := R α,s (l). L One writes BMO L (X ) in place of Λ 0,1 L (X ), the adapted space of functions with bounded mean oscillations on X.

Remark: In this case the mapping l l α,s Λ (Ω) is a norm. Hence Λ α,s L L (Ω) is a normed space. We can check that this definition is independent of the choice of M > [ n(2 p) ] (up to taking out elements in the null space of the operator 4p L M, as these are annihilated by (I (I + rbl) 2 1 ) s ). Compared to the classical definition, the resolvent (I + rbl) 2 1 plays the role of averaging over the ball, and the power M > [ n(2 p) ] provides the necessary L-cancellation. 4p

Hardy spaces via molecules We now introduce the notion of a (p, 2, M, ɛ)-molecule associated to operators on a space (X, d, µ). Assume that M N and M > h n(2 p) i. (5) 4p Let us denote by D(T ) the domain of an unbounded operator T, and by T k the k-fold composition of T with itself, in the sense of unbounded operators. Assume that L satisfies assumptions (H1) and (H2). Definition For 0 < p 1, a function m(x) L 2 (X ) is called a (p, 2, M, ɛ)-molecule associated to L if there exist a function b D(L M ) and a ball B such that (i) m = L M b; (ii) For every k = 0, 1,..., M and j = 0, 1, 2,..., the estimates (r 2 BL) k b L 2 (U j (B)) r 2M B 2 jɛ V (2 j B) 1 2 1 p, (6) hold where the annuli U j (B) are defined by U 0(B) := B, and U j (B) := 2 j B\2 j 1 B for j = 1, 2,.... (7) where λb is the ball with the same center as B and with radius r λb = λr B.

Given p (0, 1], M > h i n(2 p) and ɛ > 0, we say that f = P 4p j λ jm j is a molecular (p, 2, M, ɛ)-representation of f if ` P j=0 λ j p 1/p <, each m j is a (p, 2, M, ɛ)-molecule, and the sum converges in L 2 (X ). Set H 1 L,mol,M(X ) = {f L 1 (X ) : f has a (1, 2, M, ɛ) representation} with the norm given by j X X ff f H 1 L,mol,M (X ) = inf λ j : f = λ j m j is a (1, 2, M, ɛ) representation. For 0 < p < 1, set H p L,mol,M j=0 j=0 α,m (X ) = {f `ΛL (X ) : f has a (p, 2, M, ɛ) representation}, where α = 1/p 1,, with the (quasi) metric given by j` X f p H L,mol,M (X ) = inf λ j p 1/p : f = j=0 X j=0 ff λ j m j is a (p, 2, M, ɛ) representation.

Definition For 0 < p 1, the space H p L,mol,M (X ) is defined to be the completion of H p L,mol,M (X ) with respect to the metric f H p L,mol,M (X ) above. Remark: In this case the mapping f f p H L,mol,M (X ), 0 < p < 1, is not a norm; but d(f, g) = f g p H L,mol,M (X ) is a quasi-metric. For p = 1, the mapping f f H 1 L,mol,M (X ) is a norm. We can check that H p L,mol,M (X ) is complete. In particular, HL,mol,M(X 1 ) is a Banach space. It also follows from the above definitions that H p L,mol,M 2 (X ) H p L,mol,M 1 (X ) (8) whenever 0 < p 1 and two integers M i N, i = 1, 2, with [ n(2 p) ] < M 1 M 2 <. 4p

Hardy spaces via square functions Suppose that the operator L satisfies (H1) and (H2). Set H 2 (X ) = R(L) = {Lu L 2 (X ) : u L 2 (X )} and denote by N (L) the nullspace of L. Note that L 2 (X ) = R(L) N (L) = H 2 (X ) N (L). For any x X and α > 0, let Γ α (x) = {(t, y) X (0, ) : d(x, y) < αt} and write Γ(x) in place of Γ 1 (x). For f L 2 (X ), consider the following quadratic operator associated with e tl : Z S h (f )(x) = t 2 Le t2l f (y) 2 dµ(y) Γ(x) V (x, t) The space H p L,S h (X ), 0 < p 1, is defined as the completion of in the metric {f H 2 (X ) : S h (f ) L p (X ) < } f H p L,S h (X ) = S h (f ) L p (X ). «1 dt 2, x X. (9) t Note that for p = 1, the mapping f f H 1 L,Sh (X ) is a norm. For 0 < p < 1, d(f, g) = f g H p L,S h (X ) is a quasi-metric.

3. Characterizations of Hardy spaces H p L (X ) We can show that the Hardy spaces H p L in terms of the Littlewood-Paley square function, and in terms of molecules are all equivalent, assuming that the parameter M of the molecules satisfies M > ˆ n(2 p). 4p Theorem Suppose that M > ˆ n(2 p). 4p Then Moreover, H p L,mol,M (X ) = Hp L,S h (X ). f H p L,mol,M (X ) f H p L,S h (X ), where the implicit constants depend only on p, M, n and on the constants in the Davies-Gaffney estimates and doubling property. Consequently, one may write H p L,mol (X ) in place of Hp L,mol,M (X ) when M > ˆ n(2 p), 4p as these spaces are all equivalent. Hence, we define Definition Let L be an operator satisfying (H1) and (H2) and 0 ˆ n(2 p). 4p H p L (X ) = Hp L,S h (X ) = H p L,mol (X ) = Hp L,mol,M (X ),

4. Duality of Hardy and Lipschitz spaces Theorem Suppose that the operator L satisfies assumptions (H1) and (H2). Let 0 ˆ n(2 p) 4p and any f Λ α,m L (X ) with α = 1/p 1, the linear functional given by l(g) =< f, g >, (10) initially defined on the dense subspace of M p,2,m,ɛ 0 (L), consisting of finite linear combinations of (p, 2, M, ɛ)-molecules, ɛ > 0, and where the pairing is that between M p,2,m,ɛ 0 (L) and its dual, has a unique bounded extension to (X ) with H p L,mol,M l p (H (X C f )) L Λ α,m, for some C independent of f. L (X ) Theorem Suppose that the operator L satisfies assumptions (H1) and (H2). Let 0 ˆ n(2 p), 4p every bounded linear functional l on the H p L (X ) space can be realized as (10), i.e., there exists f Λα,M L (X ) with α = 1/p 1, such that (10) holds and f α,m C l Λ L (X ) (H p (X. )) L

5. An interpolation theorem between Hardy spaces Our aim is to prove a Marcinkiewicz-type interpolation theorem. We first recall the concept of weak-type operators. If T is defined on H p L for some p > 0, we say that it is of weak-type (H p L, p) provided that µ`{x X : T (f )(x) > λ} Cλ p f p H p L (X ) for all f H p L (X ). The best constant C will be referred to as the weak-type norm of T. We can show the following result. Theorem Let L be an operator satisfying assumptions (H1) and (H2). Suppose 0 < p 1 < p 2 <, and let T be a sublinear operator from H p 1 L (X ) + Hp 2 into measurable functions on X, which is of weak-type (H p 1 L, p1) and (Hp 2 with weak-type norms C 1 and C 2, respectively. If p 1 < p < p 2, then T is bounded from H p L (X ) into Lp (X ) and where C depends only on C 1, C 2, p 1, p 2 and p. L (X ) L, p2) Tf L p (X ) C f H p L (X ), (11)

6. Application: Holomorphic functional calculi of generators of semigroups We have the following result. Theorem Suppose that L is of type ω on L 2 (X ) with 0 ω < π/2 and satisfies (H1) and (H2). Let ω < ν < π. Then, for any f H (Sν), 0 f (L) is bounded on H p L (X ) with 0 1, can be different from the Lebesgue space L p. If we add the extra assumption that the heat semigroup has an upper bound of Poisson type, then we can show that the Hardy spaces H p L, p > 1, coincide with the Lebesgue spaces Lp and our result gives boundedness of holomorphic functional calculus on L p spaces for 1 < p <.

Application: boundedness of Riesz transforms Let L be a linear operator of type ω on L 2 (X ) with 0 ω < π/2 and satisfy (H1) and (H2). Assume that D is a densely defined linear operator on L 2 (X ) which possesses the following properties: (R1) DL 1 2 is bounded on L 2 (X ), (R2) the family of operators { tde tl } t>0 obtained by the action of D on e tl, satisfies the Davies-Gaffney estimate. We have the following theorem. Theorem Under the above assumptions (R1) and (R2), the operator DL 1 2 is bounded from H p L (X ) to Lp (X ) with 0 < p 1. Namely, for any f H p L (X ), DL 1 2 (f ) L p (X ) C f H p L (X ). (13) By interpolation, we obtain that DL 1 2 (f ) L p (X ) C f H p L (X ), for all 0 < p < 2. Remark: An example is D = on Riemann manifolds or R n.

Reference: Classical theory of Hardy spaces and related spaces: [1] E.M. Stein and G. Weiss, On the theory of harmonic functions of several variables I, The theory of H p spaces, Acta Math., 103(1960), 25-62. [2] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14(1961), 415 426. [3] C. Fefferman and E.M. Stein, H p spaces of several variables, Acta Math., 129(1972), 137 195. [4] R. Coifman, A real variable characterization of H p, Studia Math., 51(1974), 269-274. [5] R.H. Latter, A decomposition of H p (R n ) in terms of atoms, Studia Math., 62(1977), 92-102. [6] M.W. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces, Astérisque, 77(1980), 68-149. [7] A. Uchiyama and J.M. Wilson, Approximate identities and H 1 (R), Proc. Amer. Math. Soc., 88 (1983), 53-58. [8] A. Miyachi, H p space over open subsets of R n, Studia Math., 95 (1990), 205-228. [9] D-C. Chang, S.G. Krantz and E.M. Stein, H p theory on a smooth domain in R n and elliptic boundary value problems, J. Funct. Anal., 114(1993), 286-347.

Reference: Recent works on Hardy spaces [10] P. Auscher, X.T. Duong and A. M c Intosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, unpublished manuscript. [11] P. Auscher, A. M c Intosh and E. Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal., 18(2008), 192-248. [12] P. Auscher and E. Russ, Hardy spaces and divergence operator on strongly Lipschitz domain of R n, J. Funct. Anal., 201(2003), 148-184. [13] D.G. Deng, X.T. Duong, A. Sikora and L.X. Yan, Comparison of the classical BMO with the BMO spaces associated with operators and applications, Rev. Mat. Iberoamericana, 24(2008), no.1, 267-296. [14] X.T. Duong and L.X. Yan, New function spaces of BMO type, the John-Nirenberg inequality, interpolation and applications, Comm. Pure Appl. Math., 58(2005), 1375-1420. [15] X.T. Duong and L.X. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc., 18(2005), 943-973. [16] L.X. Yan, Classes of Hardy spaces associated with operators, duality theorem and applications, Trans. Amer. Math. Soc., 360(2008), 4383-4408. [17] S. Hofmann, G.Z. Lu, D. Mitrea, M. Mitrea and L.X. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, submitted 2008. [18] S. Hofmann and S. Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, to appear in Math. Ann.

Reference: Related topics [19] M.P. Gaffney, The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math., 12(1959), 1-11. [20] R.R. Coifman, G. Weiss, Analyse Harmonique Non-commutative sur Certains Espaces Homogeneous, Lecture Notes in Math. 242, Springer-Verlag, Berlin, 1971. [21] R.R. Coifman, Y. Meyer and E.M. Stein, Some new functions and their applications to harmonic analysis, J. Funct. Analysis, 62(1985), 304-335. [22] E. Russ, The atomic decomposition for tent spaces on spaces of homogeneous type, Asymptotic Geometric Analysis, Harmonic Analysis, and Related Topics, 125-135, Proceedings of the Centre for Mathematical Analysis, Australian National University, 42, Australian National University, Canberra, 2007. [23] A. M c Intosh, Operators which have an H -calculus, Miniconference on operator theory and partial differential equations, Proc. Centre Math. Analysis, ANU, Canberra, 14(1986), 210-231. [24] E.M. Stein, Harmonic Analysis: Real variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, 1993.