Weak type \((1,1)\) bounds for Schrödinger groups. (English) Zbl 1518.35262
Summary: Let \(L\) be a nonnegative self-adjoint operator acting on \({L^2}(X)\), where \(X\) is a space of homogeneous type of dimension \(n\). Suppose that the heat kernel of \(L\) satisfies a Gaussian upper bound. It is known that the operator \({(I+L)^{-s}}{e^{itL}}\) is bounded on \({L^p}(X)\) for \(s > n|1/2-1/p|\) and \(p\in (1,\infty)\) (see, e.g., [G. Carron et al., J. Evol. Equ. 2, No. 3, 299–317 (2002; Zbl 1328.35064); M. Hieber, Math. Ann. 291, No. 1, 1–16 (1991; Zbl 0724.34067);
S. Sjöstrand, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 24, 331–348 (1970; Zbl 0201.14901)]). The index \(s=n|1/2-1/p|\) was only obtained recently in
[the authors, Math. Ann. 378, No. 1-2, 667–702 (2020; Zbl 1447.42025); “Sharp endpoint estimates for Schrödinger groups on Hardy spaces”, Preprint, arXiv:19031705], and this range of \(s\) is sharp since it is precisely the range known in the case where \(L\) is the Laplace operator \(\Delta\) on \(X={\mathbb{R}^n}\) [A. Miyachi, J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 157–179 (1980; Zbl 0433.42019)]. In this paper, we establish that for \(p=1\), the operator \({(1+L)^{-n/2}}{e^{itL}}\) is of weak type \((1,1)\), that is, there is a constant \(C\), independent of \(t\) and \(f\), such that
\[
\begin{aligned}
\mu (&\{x:|{(I+L)^{-n/2}}{e^{itL}}f(x)| > \lambda \})\\
&\le C{\lambda^{-1}}{(1+|t|)^{n/2}}\| f{\|_{{L^1}(X)}}, \quad t\in \mathbb{R}
\end{aligned}
\]
(for \(\lambda > 0\) when \(\mu (X)=\infty\) and \(\lambda > \mu{(X)^{-1}}\| f{\|_{{L^1}(X)}}\) when \(\mu (X)< \infty)\). Moreover, we also show that the index \(n/2\) is sharp when \(L\) is the Laplacian on \({\mathbb{R}^n}\) by providing an example.
Our results are applicable to Schrödinger groups for large classes of operators including elliptic operators on compact manifolds, Schrödinger operators with nonnegative potentials and Laplace operators acting on Lie groups of polynomial growth or irregular nondoubling domains of Euclidean spaces.
Our results are applicable to Schrödinger groups for large classes of operators including elliptic operators on compact manifolds, Schrödinger operators with nonnegative potentials and Laplace operators acting on Lie groups of polynomial growth or irregular nondoubling domains of Euclidean spaces.
MSC:
| 35J10 | Schrödinger operator, Schrödinger equation |
| 35K08 | Heat kernel |
| 47B25 | Linear symmetric and selfadjoint operators (unbounded) |
References:
| [1] | G. Alexopoulos, Oscillating multipliers on Lie groups and Riemannian manifolds, Tohoku Math. J. (2) 46 (1994), 457-468. · Zbl 0835.22008 |
| [2] | M. Balabane and H. A. Emami-Rad, \[{L^p}\]estimates for Schrödinger evolution equation, Trans. Amer. Math. Soc. 292 (1985), 357-373. · Zbl 0588.35029 |
| [3] | S. Blunck, Generalized Gaussian estimates and Riesz means of Schrödinger groups, J. Aust. Math. Soc. 82 (2007), 149-162. · Zbl 1116.43002 |
| [4] | P. Brenner, The Cauchy problem for systems in \[{L_p} and {L_{p,\alpha }} \], Ark. Mat. 2 (1973), 75-101. · Zbl 0256.35006 |
| [5] | T. A. Bui, P. D’Ancona, and F. Nicola, \[ Sharp {L^p} \]estimates for Schrödinger groups on space of homogeneous type, Rev. Mat. Iberoam. 36 (2020), no. 2, 455-484. · Zbl 1448.35352 |
| [6] | F. Cacciafesta and P. D’Ancona, \[ Weighted {L^p} \]estimates for powers of selfadjoint operators, Adv. Math. 229 (2012), 501-530. · Zbl 1245.42010 |
| [7] | G. Carron, T. Coulhon, and E. M. Ouhabaz, Gaussian estimates and \[{L^p} \]-boundedness of Riesz means, J. Evol. Equ. 2 (2002), 299-317. · Zbl 1328.35064 |
| [8] | S. Chanillo, D. S. Kurtz, and G. Sampson, \[ Weighted weak (1,1) and weighted {L^p} \]estimates for oscillating kernels, Trans. Amer. Math. Soc. 295 (1986), 127-145. · Zbl 0594.42007 |
| [9] | P. Chen, X. T. Duong, J. Li, and L. X. Yan, \[ Sharp endpoint {L^p} \]estimates for Schrödinger groups, Math. Ann. 378 (2020), no. 1-2, 667-702. · Zbl 1447.42025 |
| [10] | P. Chen, X. T. Duong, J. Li, and L. X. Yan, Sharp endpoint estimates for Schrödinger groups on Hardy spaces, 2019, Available at arXiv:19031705. |
| [11] | R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math., 242, Springer, Berlin, 1971. · Zbl 0224.43006 |
| [12] | P. D’Ancona and F. Nicola, \[ Sharp {L^p} \]estimates for Schrödinger groups, Rev. Mat. Iberoam. 32 (2016), 1019-1038. · Zbl 1353.42009 |
| [13] | E. B. Davies, Heat kernels and spectral theory, Cambridge University Press, Cambridge, 1989. · Zbl 0699.35006 · doi:10.1017/CBO9780511566158 |
| [14] | E. B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995), 141-169. · Zbl 0839.35034 |
| [15] | X. T. Duong and A. McIntosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoam. 15 (1999), no. 2, 233-265. · Zbl 0980.42007 |
| [16] | X. T. Duong and E. M. Ouhabaz, Gaussian upper bounds for heat kernels of a class of nondivergence operators, International conference on harmonic analysis and related topics, Sydney, 2002, Proc. Centre Math. Appl. Austral. Nat. Univ., 41, pp. 35-45, Austral. Nat. Univ., Canberra, 2003. · Zbl 1115.35314 |
| [17] | X. T. Duong, E. M. Ouhabaz, and A. Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal. 196 (2002), 443-485. · Zbl 1029.43006 |
| [18] | X. T. Duong and D. W. Robinson, Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal. 142 (1996), no. 1, 89-128. · Zbl 0932.47013 |
| [19] | O. El-Mennaoui and V. Keyantuo, On the Schrödinger equation in \[{L^p} spaces \], Math. Ann. 304 (1996), 293-302. · Zbl 0842.35024 |
| [20] | C. Fefferman, Inequality for strongly singular convolution operators, Acta Math. 124 (1970), 9-36. · Zbl 0188.42601 |
| [21] | C. Fefferman and E. M. Stein, \[{H^p}\]spaces of several variables, Acta Math. 129 (1972), 137-195. · Zbl 0257.46078 |
| [22] | M. Hieber, Integrated semigroups and differential operators on \[{L^p} spaces \], Math. Ann. 291 (1991), 1-16. · Zbl 0724.34067 |
| [23] | L. Hörmander, Estimates for translation invariant operators in \[{L^p} spaces \], Acta Math. 104 (1960), 93-140. · Zbl 0093.11402 |
| [24] | A. Jensen and S. Nakamura, Mapping properties of functions of Schrödinger operators between \[{L^p} \]-spaces and Besov spaces, Spectral and scattering theory and applications, Adv. Stud. Pure Math., 23, pp. 187-209, Math. Soc. Japan, Tokyo, 1994. · Zbl 0815.47012 |
| [25] | A. Jensen and S. Nakamura, \[{L^p}\]-Mapping properties of functions of Schrödinger operators and their applications to scattering theory, J. Math. Soc. Japan 47 (1995), 253-273. · Zbl 0841.35096 |
| [26] | W. B. Jurkat and G. Sampson, The complete solution to the \[({L^p},{L^q})\]mapping problem for a class of oscillating kernels, Indiana Univ. Math. J. 30 (1981), 403-413. · Zbl 0507.47013 |
| [27] | E. Lanconelli, \[ Valutazioni in {L^p}({\mathbb{R}^n})\]della soluzione del problema di Cauchy per l’equazione di Schrödinger, Boll. Un. Mat. Ital. (4) 1 (1968), 591-607. · Zbl 0167.10401 |
| [28] | N. Lohoué, Estimations des sommes de Riesz d’opérateurs de Schrödinger sur les variétés riemanniennes et les groupes de Lie, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 13-18. · Zbl 0760.58040 |
| [29] | A. McIntosh, Operators which have an \[{H_{\infty }} functional calculus \], Miniconference on operator theory and partial differential equations, North Ryde, 1986, Proc. Centre Math. Anal. Austral. Nat. Univ., 14, pp. 210-231, Australian National University, Canberra, 1986. · Zbl 0623.00012 |
| [30] | A. Miyachi, On some Fourier multipliers for \[{H^p}({R^n})\], J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 27 (1980), 157-179. · Zbl 0433.42019 |
| [31] | E. M. Ouhabaz, Analysis of heat equations on domains, London Math. Soc. Monogr. Ser., 31, Princeton University Press, Princeton, 2005. · Zbl 1082.35003 |
| [32] | B. Simon, Schrödinger semigroup, Bull. Amer. Math. Soc. 7 (1982), 447-526. · Zbl 0524.35002 |
| [33] | S. Sjöstrand, On the Riesz means of the solutions of the Schrödinger equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3) 24 (1970), 331-348. · Zbl 0201.14901 |
| [34] | E. M. Stein, Singular integral and differentiability properties of functions, 30, Princeton Univ. Press, 1970. · Zbl 0207.13501 |
| [35] | N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups, Cambridge University Press, Cambridge, 1992. · Zbl 0813.22003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
