Abstract
In the present paper, using the theory of boundary values of analytic semigroups, we find necessary and sufficient conditions to guarantee that the operator \(i(-\Delta )^{{\alpha }/{2}}\) generates a strongly continuous semigroup in \(L^p(\mathbb {R}^n)\).
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References
Arendt, W., El-Mennaoui, O., Hieber, M.: Boundary values of holomorphic semigroups. Proc. Amer. Math. Soc. 125, 635–647 (1997)
Blumenthal, R.M., Getoor, R.K.: Some theorems on stable processes. Trans. Amer. Math. Soc. 95, 263–273 (1960)
El-Mennaoui, O., Keyantuo, V.: Trace theorems for holomorphic semigroups and the second order Cauchy problem. Proc. Amer. Math. Soc. 124, 1445–1458 (1996)
El-Mennaoui, O., Keyantuo, V., Sani, A.: Fractional integration of imaginary order in vector-valued Hölder spaces. Arch. Math. (Basel) 120, 493–506 (2023)
Grafakos, L.: Classical Fourier Analysis. Third Edition. Graduate Texts in Mathematics, vol. 249. Springer, New York (2014)
Hörmander, L.: Estimates for translation invariant operators in \(L^p\) spaces. Acta Math. 104, 93–140 (1960)
Huang, S., Wang, M., Zheng, Q., Duan, Z.: \(L^p\) estimates for fractional Schrödinger operators with Kato class potentials. J. Differential Equations 265, 4181–4212 (2018)
Kaleta, K., Sztonyk, P.: Estimates of transition densities and their derivatives for jump Lévy processes. J. Math. Anal. Appl. 431, 260–282 (2015)
Kemppainen, J., Siljander, J., Zacher, R.: Representation of solutions and large-time behavior for fully nonlocal diffusion equations. J. Differential Equations 263, 149–201 (2017)
Kilbas, A.A., Saigo, M.: \(H\)-transforms. Theory and Applications. Analytical Methods and Special Functions, 9. Chapman & Hall/CRC, Boca Raton (2004)
Komatsu, H.: Fractional powers of operators. Pac. J. Math. 19, 285–346 (1966)
Maekawa, Y., Miura, H.: On fundamental solutions for non-local parabolic equations with divergence free drift. Adv. Math. 247, 123–191 (2013)
Mainardi, F., Luchko, Y., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, 153–192 (2001)
Sin, C.: Cauchy problem for nonlocal diffusion equations modelling Lévy flights. Electron. J. Qual. Theory Differ. Equ. 2022, Paper No. 18, 22 pp. (2022)
Sin, C.: Gevrey type regularity of the Riesz–Feller operator perturbed by gradient in \(L^p(\mathbb{R} )\). Complex Anal. Oper. Theory 17, Paper No. 49, 18 pp. (2023)
Sin, C., Jo, K.: Regularity of semigroups for exponentially tempered stable processes with drift. J. Math. Anal. Appl. 526, Paper No. 127247, 28 pp. (2023)
Wang, M., Ma, Q., Duan, J.: Gevrey semigroup generated by \(-(\Lambda ^\alpha +b\cdot \nabla )\) in \(L^p(\mathbb{R} ^n)\). J. Math. Anal. Appl. 481, 123480, 17 pp. (2020)
Yosida, K.: Functional Analysis. Springer, Berlin (1980)
Zhao, S., Zheng, Q.: Uniform complex time heat kernel estimates without Gaussian bounds. Adv. Nonlinear Anal. 12, Paper No. 20230114, 24 pp. (2023)
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Sin, CS. Boundary values of analytic semigroups generated by fractional Laplacians. Arch. Math. (2024). https://doi.org/10.1007/s00013-024-02004-x
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DOI: https://doi.org/10.1007/s00013-024-02004-x