On isomorphisms of Hardy spaces for certain Schrodinger¨ operators∗
Jacek Dziubański∗∗
joint works with Jacek Zienkiewicz∗ ∗∗ Instytut Matematyczny, Uniwersytet Wrocławski, Poland
Conference in Honor of Aline Bonami, Orleans, 10-13.06.2014.
Jacek Dziubański Hardy spaces Hardy spaces for semigroups
d Let L = −∆ + V be a Schr¨odinger operator on R , V ≥ 0, V 6≡ 0. −tL Kt (x, y) be the integral kernels of {Kt }t>0 = {e }t>0. The Feynman-Kac formula implies that
−d/2 2 0 ≤ Kt (x, y) ≤ (4πt) exp − |x − y| /4t = Pt (x − y).
One possible definition of the Hardy space associated with L is by means of the maximal function
MLf (x) = sup |Kt f (x)|. t>0
1 d 1 We say that an L (R )-function f belongs to HL if 1 d MLf ∈ L ( ). Then we set kf k 1 = kMLf k 1 d . R HL,max L (R )
Jacek Dziubański Hardy spaces Hardy spaces for semigroups
These spaces are generalizations of the classical Hardy spaces, which could be thought as the spaces associated with the classical 1 d 1 heat semigroups, H (R ) = H∆. The very original works in this field are due to C. Fefferman, E. Stein, G.Weiss, R. Coifman, R. Latter, also D. Burkholder, R. Gundy, M. Silverstein, then L. Carleson, R. Macias, C. Segovia, P.W. Jones, Y. Meyer, R. Rochberg, S. Semmes, A. Uchiyama, J.M. Wilson, G. Folland, M. Christ, D. Geller, D. Goldberg, ... There are many characterizations of the Hardy spaces. Among them one plays an important role, namely characterization by atoms.
Jacek Dziubański Hardy spaces Hardy spaces for semigroups
1 d A function a is an (1, ∞)-atom for the Hardy space H (R ) if there is a ball B such that supp a ⊂ B (support condition); −1 kakL∞ ≤ |B| (size condition); R a = 0 (cancellation condition). Then kakH1 ≤ C for every atom a. The atomic norm kf k 1 d is defined as Hatom(R ) X kf k 1 d = inf |λ |, Hatom(R ) j j P where the infimum is taken over all decompositions f = j λj aj . Theorem (Coifman, Latter) 1 d 1 d The spaces H (R ) and Hatom(R ) coincide and the norms kf k 1 d and kf k 1 d are equivalent. H (R ) Hatom(R )
Jacek Dziubański Hardy spaces Hardy spaces for semigroups
Question: Let (X , d, µ) be the metric space and let e−tA be a semigroup of linear operators acting on functions on X . What can be said about the Hardy space H1 if we replace the classical heat kernel in the definition of the Hardy space by e−tA:
1 n 1 −tA o HA = f ∈ L (X ): sup |e f | = kf kH1 ? t>0 L1 A Hardy spaces associated with certain operators attracted attention of many authors (J-Ph. Anker, P. Auscher, N. Ben Salem, F. Bernicot, J. Betancor, W. Czaja, X. Duong, G. Garrigós, N. Hamda, E. Harboure, S. Hofmann, T. Hyt¨onen, Z.J. Lou, G. Lu, T. Mart´inez, G. Mauceri, S. Mayboroda, A. McIntosh, S. Meda, D. Mitrea, M. Mitrea, E. Ouhabaz, M. Picardello, P. Portal, F. Ricci, E. Russ, A. Sikora, P. Sj¨ogren, X. Tolsa, J.L. Torrea, D. Yang, L. Yan, J. M. Vallarino, J. Zhao, ...
Jacek Dziubański Hardy spaces Hardy spaces for semigroups
In a project with Jacek Zienkiewicz we have been studying Hardy spaces associated with semigroup generated by Schr¨odinger d operators on R ,
−L = ∆ − V (x), V (x) ≥ 0.
1 Our task was/is: How does the space HL look like for some classes of potentials? The best way to see the space is by their atomic characterization. 1 It turns out that the space HL and its properties depend on the potential V and on the dimension d.
Jacek Dziubański Hardy spaces Hardy spaces for semigroups
If d = 1 or if V is ”large”, then the kernel Kt (x, y) of the semigroup e−tL has faster decay comparing to the classical heat kernel, (see the Faynman-Kac formula)
Z R t x − V (bs ) ds Kt (x, y)f (y) dy = E e 0 f (bt ) .
1 This reflects, that the Hardy space HL may admit local atoms, that is, some atoms may not satisfy any cancellation condition.
Our task is to study the case of ”small” potentials V .
Jacek Dziubański Hardy spaces Hardy spaces for semigroups Example and motivation for today’s talk.
d/2+ε d Assume that supp V ⊂ B(0, 1), V ∈ L (R ) and d ≥ 3. R Define w(x) = limt→∞ Kt (x, y) dy. The limit exists and defines bounded from below and above L-harmonic function w, that is, Kt w = w, such that 0 < δ ≤ w(x) ≤ 1. 1 Then in the definition of atoms for the space HL the classical cancellation condition is replaced by Z a(x)w(x) dx = 0.
In other words the mapping
1 1 d HL 3 f (x) 7→ w(x)f (x) ∈ H (R ) is an isomorphism. This result was slightly generalized with M. Preisner.
Jacek Dziubański Hardy spaces Hardy spaces for semigroups
Elias Stein asked the question: Is it possible to characterize all d nonnegative potentials V in R , d ≥ 3, such that the Hardy space 1 −tL HL for the semigroup e , L = −∆ + V , is isomorphic to the 1 d classical Hardy space H (R ) by multiplication by a function w, 0 < c ≤ w(x) ≤ C? In other words: For which V ≥ 0 there is 0 < δ ≤ w(x) ≤ C such that
1 1 d HL 3 f 7→ wf ∈ H (R ) isomorphism? Our purpose is to give a complete answer to his question.
Jacek Dziubański Hardy spaces Hardy spaces for semigroups Step 1. If such isomorphism exists,
1 1 d HL 3 f 7→ wf ∈ H (R ), then w is L-harmonic (Kt w = w). For the proof we use the following lemma. Lemma 1 d Kt f − f ∈ HL for every f ∈ Cc (R ).
Z 0 = w(x)(Kt f (x) − f (x)) dx Z Z Z = w(x) Kt (x, y)f (y) dy dx − w(x)f (x) dx Z Z Z = Kt (x, y)w(x) dx f (y) dy − w(y)f (y) dy.
Hence, Kt w(y) = w(y). Jacek Dziubański Hardy spaces Hardy spaces for semigroups
To see the lemma take f ∈ Cc and let supp f ⊂ B(0, R). Note that MLf (x) ≤ kf kL∞ , 1 d sup0
Z t+s Z d
|Ks (Kt − f )(x)| = Ku(x, y)f (y) dy du s du Z −1 −d/2 2 . ts s exp(−c|x − y| /s)|f (y)| dy |y| 1 So ML(Kt f − f ) ∈ L . Jacek Dziubański Hardy spaces Hardy spaces for semigroups Step 2. For which V there is L-harmonic w: 0 < c ≤ w(x) ≤ C? Recall that d ≥ 3. The following are equivalent: (a) There is an L-harmonic function w: 0 < c ≤ w(x) ≤ C; R (b) Kt (x, y) dy ≥ δ > 0 (c) The global Kato norm Z V (y) −1 ∞ kV kK = c sup d−2 dy = k∆ V kL is bounded; d |x − y| x∈R −d/2 −C|x−y|2/t (d) ct e ≤ Kt (x, y) (Gaussian lower bounds). Proof of (b) ⇒ (a). Assume (b): R RR R δ ≤ Kt+s (x, y)dy = Kt (x, z)Ks (z, y)dzdy ≤ Kt (x, z)dz ≤ 1. R Hence, limt→∞ Kt (x, y)dy = w(x) exists and defines an L-harmonic function. Jacek Dziubański Hardy spaces Hardy spaces for semigroups Proof of (a) ⇒ (b). Assume (a). Then R R 1 ≥ Kt (x, y) dy ≥ Kt (x, y)w(y) dy = w(x) ≥ δ > 0. Proof of (b) ⇒ (c). Assume (a) or equivalently (b). By the perturbation formula, R 1 = Pt (x − y)dy = R RR t R Kt (x, y)dy + 0 Ps (x, z)V (z)Kt−s (z, y)dz ds dy R t R ≥ 0 Ps (x, z)V (z)δ dz ds Letting t → ∞ we get 1 ≥ δ · (−∆)−1V (x) ≥ 0, which gives (c). −1 Proof of (c) ⇒ (b). Assume kV kK = k∆ V kL∞ < ∞. Assume additionally that kV kK < 1. Then R RR t R 1 = Kt (x, y)dy + 0 Ps (x, z)V (z)Kt−s (z, y)dz ds dy R RR t R ≤ Kt (x, y)dy + Ps (x, z)V (z) dz ds ≤ R 0 Kt (x, y) dy + kV kK , which implies (b). The case kV kK ≥ 1 follows from the previous one by using the Feynman-Kac formula and the H¨older inequality. We omit details here. Jacek Dziubański Hardy spaces Hardy spaces for semigroups Clearly (d) implies (b). The proof that (b) implies (d) is more complex, however it uses only the perturbation formula and the Feynman-Kac formula and an iteration argument. Remark. The equivalence (c) ⇔ (d) is due to Yu.A. Semenov, Stability of Lp-spectrum of generalized Schr¨odinger operators and equivalence of Green’s functions, IMRN 12 (1997), 573–593. Jacek Dziubański Hardy spaces Hardy spaces for semigroups Step 3. We are in a position to state our first main result: Theorem (J.D. & J.Z.) The following are equivalent (i) the operator L admits an L-harmonic function 0 < δ ≤ w(x) ≤ 1 R 2−d (ii) kV kK = supx∈Rd V (y)|x − y| dy < ∞ (iii) there is a function 0 < δ ≤ w(x) ≤ 1 such that 1 1 d HL 3 f 7→ w(x)f (x) ∈ H (R ) is an isomorphism. Jacek Dziubański Hardy spaces Hardy spaces for semigroups Recent results. Theorem (J.D. & J.Z.) Assume that kV kK < ∞, d ≥ 3. Then the operators L1/2(−∆)−1/2 and (−∆)1/2L−1/2 are bounded on L1. Moreover, 1 1/2 −1/2 1 d HL 3 f 7→ (−∆) L f ∈ H (R ) is isomorphism of the Hardy spaces with the inverse 1 d 1/2 −1/2 1 H (R ) 3 f 7→ L (−∆) f ∈ HL. Jacek Dziubański Hardy spaces Hardy spaces for semigroups Corollary (J.D. & J.Z.) 1 The Hardy space HL admits characterization by the Riesz −1/2 1 transforms Rj = ∂xj L , that is, an L function f belongs to the 1 1 Hardy space HL if and only if Rj f ∈ L . Proof. Based on: −1/2 1/2 −1/2 −1/2 (∂xj (−∆) ) (−∆) L = ∂xj L | {z } | {z } 1 1 d | {z } (classical Riesz transform) (isomorphism from HL to H (R )) (Riesz transform for L) −1/2 1/2 −1/2 −1/2 (∂xj L ) (L (−∆) ) = ∂xj (−∆) | {z } | {z } 1 d 1 | {z } (Riesz transform for L) (isomorphism from H (R ) to HL ) (classical Riesz transform) 1 Assume that Rj f ∈ L . Then, by the first part of the theorem, there is g ∈ L1 such that f = L1/2(−∆)−1/2g. 1 −1/2 1/2 −1/2 −1/2 Hence, L 3 Rj f = (∂xj L )(L (−∆) )g = ∂xj (−∆) g. 1 d Thus g ∈ H (R ). By the second part of the theorem 1/2 −1/2 1 f = L (−∆) g ∈ HL. Jacek Dziubański Hardy spaces Hardy spaces for semigroups 1 Conversely, assume that f ∈ HL. Then, by the second part of the 1/2 −1/2 1 d theorem, g = (−∆) L f ∈ H (R ). So, the Riesz transform characterization of the classical H1 space 1 −1/2 L 3 ∂xj (−∆) g −1/2 1/2 −1/2 = ∂xj (−∆) (−∆) L f −1/2 = ∂xj L f = Rj f Jacek Dziubański Hardy spaces Hardy spaces for semigroups The case of compactly supported potentials in dimension 2. Assume that V is a nonnegative nonzero compactly supported 2 2 C -function in R . Theorem (J.D. & J.Z.) There exists a regular L-harmonic weight w such that C −1 ln(2 + |x|) ≤ w(x) ≤ C ln(2 + |x|), |∇w(x)| ≤ C(1 + |x|)−1, 1 and the space HL,max admits atomic decomposition with atoms −1 R satisfying: supp a ⊂ B, kak∞ ≤ |B| , a(x)w(x) dx = 0. Jacek Dziubański Hardy spaces Hardy spaces for semigroups References: • R. Coifman, A real variable characterization of Hp, Studia Math., 51 (1974), 269–274. • Czaja, W., Zienkiewicz, J.: Atomic characterization of the Hardy 1 space HL(R) of one-dimensional Schr¨odinger operators with nonnegative potentials, Proc. Amer. Math. Soc. 136, no. 1, 89–94 (2008). • J. Dziubański and J. Zienkiewicz, Hardy spaces H1 for Schr¨odinger operators with compactly supported potentials, Ann. Mat. Pura Appl., 184 (2005), 315–326. • J. Dziubański and J. Zienkiewicz, On Hardy spaces associated with certain Schr¨odinger operators in dimension 2, Rev. Mat. Iberoamericana, 28, no. 4, 1035–1060 (2012). • J. Dziubański, J. Zienkiewicz, On Isomorphisms of Hardy Spaces Associated with Schr¨odinger Operators, J. Fourier Anal. Appl., 19, 447–456 (2013). Jacek Dziubański Hardy spaces Hardy spaces for semigroups • J. Dziubański, J. Zienkiewicz, A characterization of Hardy spaces associated with certain Schr¨oodinger operators, to appear in Potential Analysis. • S. Hofmann, G.Z. Lu, D. Mitrea, M. Mitrea, L.X.Yan, Hardy spaces associated with non-negative self-adjoint operators satisfying Davies-Gafney estimates, Memoirs Amer. Math. Soc. 214 (2011), no. 1007. • C. Fefferman and E.M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137–193. p n • R.H. Latter, A decomposition of H (R ) in terms of atoms, Studia Math. 62 (1978), no. 1, 93–101. • Yu.A. Semenov, Stability of Lp-spectrum of generalized Schr¨odinger operators and equivalence of Green’s functions, IMRN 12 (1997), 573–593. • E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993. Jacek Dziubański Hardy spaces Hardy spaces for semigroups Thank you Aline – all the best to you! Jacek Dziubański Hardy spaces