Lusin Characterisation of Hardy Spaces Associated with Hermite Operators
Tóm tắt
Let d ∈ {3, 4, 5,...} and p ∈ (0, 1]. We consider the Hermite operator L = −Δ + |x|2 on its maximal domain in L2(ℝd). Let $$H_L^p(\mathbb{R}^d)$$ be the completion of $$\left\{ {f \in {L^2}({\mathbb{R}^d}):{\mathcal{M}_L}f \in {L^p}({\mathbb{R}^d})} \right\}$$ with respect to the quasi-norm $${\left\| \cdot \right\|_{H_L^p}} = {\left\| {{\mathcal{M}_L} \cdot } \right\|_{{L^p}}}$$, where $${\mathcal{M}_L}f( \cdot ) = {\sup _{t > 0}}\left| {{e^{ - tL}}f( \cdot )} \right|$$ for all f ∈ L2(ℝd). We characterise $$H_L^p(\mathbb{R}^d)$$ in terms of Lusin integrals associated with the Hermite operator for $$p \in \left( {\frac{d}{{d + 1}},1} \right]$$.
Tài liệu tham khảo
Adams, R. A. and Fournier, J. J. F., Sobolev Spaces, Academic Press (The Netherlands, 2003).
Bui, T., Duong, X. and Ly, F., Maximal function characterizations for new local Hardy type spaces on spaces of homogeneous type, Trans. Amer. Math. Soc., 370 (2018), 7229–7292.
Bui, H., Duong, X. and Yan, L., Calderon reproducing formulas and new Besov spaces associated with operators, Adv. Math., 229 (2012), 2449–2502.
Coifman, R., Meyer, Y. and Stein, E., Some new function spaces and their applications to harmonic analysis, J. Funct. Anal., 62 (1985), 304–335.
Grafakos, L., Modern Fourier Analysis, 2nd ed., Graduate Texts in Mathematics, vol. 250, Springer-Verlag (Berlin, etc., 2009).
Hofmann, S., Lu, G., Mitrea, D., Mitrea, M. and Yan, L., Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Mem. Amer. Math. Soc., 214 2011.
Huang, J., The boundedness of Riesz transforms for Hermite expansions on the Hardy spaces, J. Math. Anal. Appl., 385 (2012), 559–571.
Huang, J., Li, P. and Liu, Y., Poisson semigroup, area function, and the characterization of Hardy space associated to degenerate Schrodinger operators, Banach J. Math. Anal., 10 (2016), 727–749.
Kato, T., Perturbation Theory for Linear Operators, 2nd ed., Grundlehren der mathematischen Wissenschaften, vol. 132, Springer-Verlag (Berlin, etc., 1980).
Russ, E., The atomic decomposition for tent spaces on spaces of homogeneous type, in: CMA/AMSI Research Symposium “Asymptotic Geometric Analysis, Harmonic Analysis, and Related Topics”, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 42, Austral. Nat. Univ. (Canberra, 2007), pp. 125–135.
Shen, Z., Lp estimates for Schrodinger operators with certain potentials, Ann. Inst. Fourier, 45 (1995), 513–546.
Stein, E. M., Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press (Princeton, NJ, 1995).
Song, L. and Yan, L., A maximal function characterization for Hardy spaces associated to nonnegative self-adjoint operators satisfying Gaussian estimates, Adv. Math., 287 (2016), 463–484.
Thangavelu, S., Riesz transforms and the wave equation for the Hermite operator, Comm. Partial Differential Equations, 15 (1990), 1199–1215.
Thangavelu, S., Lectures on Hermite and Laguerre Expansions, Math. Notes, vol. 42, Princeton University Press (Princeton, NJ, 1993).