August  2021, 15(4): 641-681. doi: 10.3934/ipi.2021009

Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems

1. 

Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD) FI-40014 University of Jyväskylä, Finland

2. 

Seminar for Applied Mathematics, Department of Mathematics, ETH Zurich, Rämistrasse 101, CH-8092 Zürich, Switzerland

* Corresponding author

Received  March 2020 Revised  September 2020 Published  August 2021 Early access  January 2021

Fund Project: G.C. was partially supported by the European Research Council under Horizon 2020 (ERC CoG 770924). K.M. and J.R. were partially supported by Academy of Finland (Centre of Excellence in Inverse Modelling and Imaging, grant numbers 284715 and 309963)

We prove a unique continuation property for the fractional Laplacian $ (-\Delta)^s $ when $ s \in (-n/2, \infty)\setminus \mathbb{Z} $ where $ n\geq 1 $. In addition, we study Poincaré-type inequalities for the operator $ (-\Delta)^s $ when $ s\geq 0 $. We apply the results to show that one can uniquely recover, up to a gauge, electric and magnetic potentials from the Dirichlet-to-Neumann map associated to the higher order fractional magnetic Schrödinger equation. We also study the higher order fractional Schrödinger equation with singular electric potential. In both cases, we obtain a Runge approximation property for the equation. Furthermore, we prove a uniqueness result for a partial data problem of the $ d $-plane Radon transform in low regularity. Our work extends some recent results in inverse problems for more general operators.

Citation: Giovanni Covi, Keijo Mönkkönen, Jesse Railo. Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems. Inverse Problems and Imaging, 2021, 15 (4) : 641-681. doi: 10.3934/ipi.2021009
References:
[1]

H. Abels, Pseudodifferential and Singular Integral Operators, De Gruyter Graduate Lectures, De Gruyter, Berlin, 2012, An introduction with applications.

[2]

A. Abouelaz, The $d$-plane Radon transform on the torus $\Bbb T^n$, Fract. Calc. Appl. Anal., 14 (2011), 233-246.  doi: 10.2478/s13540-011-0014-8.

[3]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, vol. 165 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010. doi: 10.1090/surv/165.

[4]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[5]

A. Behzadan and M. Holst, Multiplication in Sobolev spaces, revisited, arXiv: 1512.07379.

[6]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.

[7]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, vol. 20 of Lecture Notes of the Unione Matematica Italiana, Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3.

[8]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.

[9]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[10]

X. CaoY.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Probl. Imaging, 13 (2019), 197-210.  doi: 10.3934/ipi.2019011.

[11]

M. Cekić, Y.-H. Lin and A. Rüland, The Calderón problem for the fractional Schrödinger equation with drift, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 91, 46pp. doi: 10.1007/s00526-020-01740-6.

[12]

S. N. Chandler-WildeD. P. Hewett and A. Moiola, Sobolev spaces on non-Lipschitz subsets of $\Bbb{R}^n$ with application to boundary integral equations on fractal screens, Integral Equations Operator Theory, 87 (2017), 179-224.  doi: 10.1007/s00020-017-2342-5.

[13]

M. Courdurier, F. Noo, M. Defrise and H. Kudo, Solving the interior problem of computed tomography using a priori knowledge, Inverse Problems, 24 (2008), 065001, 27pp. doi: 10.1088/0266-5611/24/6/065001.

[14]

G. Covi, An inverse problem for the fractional Schrödinger equation in a magnetic field, Inverse Problems, 36 (2020), 045004, 24pp. doi: 10.1088/1361-6420/ab661a.

[15]

G. Covi, Inverse problems for a fractional conductivity equation, Nonlinear Anal., 193 (2020), 111418, 18pp. doi: 10.1016/j.na.2019.01.008.

[16]

G. Covi, K. Mönkkönen, J. Railo and G. Uhlmann, The higher order fractional Calderón problem for linear local operators: uniqueness, arXiv: 2008.10227.

[17]

A. D'Agnolo and M. Eastwood, Radon and Fourier transforms for $\mathcal{D}$-modules, Adv. Math., 180 (2003), 452-485.  doi: 10.1016/S0001-8708(03)00011-2.

[18]

S. DipierroO. Savin and E. Valdinoci, All functions are locally $s$-harmonic up to a small error, J. Eur. Math. Soc., 19 (2017), 957-966.  doi: 10.4171/JEMS/684.

[19]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.  doi: 10.1137/110833294.

[20]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23 (2013), 493-540.  doi: 10.1142/S0218202512500546.

[21]

G. Eskin, Global uniqueness in the inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials, Comm. Math. Phys., 222 (2001), 503-531.  doi: 10.1007/s002200100522.

[22]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[23]

M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397.  doi: 10.1080/03605302.2013.825918.

[24]

V. Felli and A. Ferrero, Unique continuation principles for a higher order fractional Laplace equation, Nonlinearity, 33 (2020), 4133-4190.  doi: 10.1088/1361-6544/ab8691.

[25]

G. B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl., 3 (1997), 207-238.  doi: 10.1007/BF02649110.

[26]

J. Frikel and E. T. Quinto, Limited data problems for the generalized Radon transform in $\Bbb R^n$, SIAM J. Math. Anal., 48 (2016), 2301-2318.  doi: 10.1137/15M1045405.

[27]

M. A. García-Ferrero and A. Rüland, Strong unique continuation for the higher order fractional Laplacian, Math. Eng., 1 (2019), 715-774.  doi: 10.3934/mine.2019.4.715.

[28]

T. Ghosh, A. Rüland, M. Salo and G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, J. Funct. Anal., 279 (2020), 108505, 42pp. doi: 10.1016/j.jfa.2020.108505.

[29]

T. GhoshM. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, Anal. PDE, 13 (2020), 455-475.  doi: 10.2140/apde.2020.13.455.

[30]

F. O. Goncharov, An iterative inversion of weighted radon transforms along hyperplanes, Inverse Problems, 33 (2017), 124005, 20pp. doi: 10.1088/1361-6420/aa91a4.

[31]

F. O. Goncharov and R. G. Novikov, An example of non-uniqueness for Radon transforms with continuous positive rotation invariant weights, J. Geom. Anal., 28 (2018), 3807-3828.  doi: 10.1007/s12220-018-0001-y.

[32]

F. O. Goncharov and R. G. Novikov, An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions, Inverse Problems, 34 (2018), 054001, 6pp. doi: 10.1088/1361-6420/aab24d.

[33]

F. B. Gonzalez, On the range of the Radon $d$-plane transform and its dual, Trans. Amer. Math. Soc., 327 (1991), 601-619.  doi: 10.2307/2001816.

[34]

B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation I. Positive potentials, SIAM J. Math. Anal., 51 (2019), 3092-3111.  doi: 10.1137/18M1166298.

[35]

B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schödinger equation II. General potentials and stability, SIAM J. Math. Anal., 52 (2020), 402-436.  doi: 10.1137/19M1251576.

[36]

H. HeckX. Li and J.-N. Wang, Identification of viscosity in an incompressible fluid, Indiana Univ. Math. J., 56 (2007), 2489-2510.  doi: 10.1512/iumj.2007.56.3037.

[37]

S. Helgason, Integral Geometry and Radon transforms, Springer, New York, 2011. doi: 10.1007/978-1-4419-6055-9.

[38]

A. Homan and H. Zhou, Injectivity and stability for a generic class of generalized Radon transforms, J. Geom. Anal., 27 (2017), 1515-1529.  doi: 10.1007/s12220-016-9729-4.

[39]

L. Hörmander, The Analysis of Linear Partial Differential Operators. I, 2nd edition, Springer Study Edition, Springer-Verlag, Berlin, 1990, Distribution theory and Fourier analysis. doi: 10.1007/978-3-642-61497-2.

[40]

J. Horváth, Topological Vector Spaces and Distributions. Vol. I, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966.

[41]

J. Ilmavirta, On Radon transforms on tori, J. Fourier Anal. Appl., 21 (2015), 370-382.  doi: 10.1007/s00041-014-9374-x.

[42]

J. Ilmavirta and K. Mönkkönen, Unique continuation of the normal operator of the x-ray transform and applications in geophysics, Inverse Problems, 36 (2020), 045014, 23pp. doi: 10.1088/1361-6420/ab6e75.

[43]

E. Katsevich, A. Katsevich and G. Wang, Stability of the interior problem with polynomial attenuation in the region of interest, Inverse Problems, 28 (2012), 065022, 28pp. doi: 10.1088/0266-5611/28/6/065022.

[44]

E. Klann, E. T. Quinto and R. Ramlau, Wavelet methods for a weighted sparsity penalty for region of interest tomography, Inverse Problems, 31 (2015), 025001, 22pp. doi: 10.1088/0266-5611/31/2/025001.

[45]

V. P. Krishnan and E. T. Quinto, Microlocal Analysis in Tomography, in Handbook of Mathematical Methods in Imaging (ed. O. Scherzer), Springer, New York, 2015,847–902. doi: 10.1007/978-1-4939-0790-8_36.

[46]

N. V. Krylov, All functions are locally $s$-harmonic up to a small error, J. Funct. Anal., 277 (2019), 2728-2733.  doi: 10.1016/j.jfa.2019.02.012.

[47]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.

[48]

R.-Y. Lai and Y.-H. Lin, Global uniqueness for the fractional semilinear Schrödinger equation, Proc. Amer. Math. Soc., 147 (2019), 1189-1199.  doi: 10.1090/proc/14319.

[49]

R.-Y. Lai and Y.-H. Lin, Inverse problems for fractional semilinear elliptic equations, arXiv: 2004.00549.

[50]

R.-Y. LaiY.-H. Lin and A. Rüland, The Calderón problem for a space-time fractional parabolic equation, SIAM J. Math. Anal., 52 (2020), 2655-2688.  doi: 10.1137/19M1270288.

[51]

N. Laskin, Fractional Quantum Mechanics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018. doi: 10.1142/10541.

[52]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[53]

L. Li, A semilinear inverse problem for the fractional magnetic Laplacian, arXiv: 2005.06714.

[54]

L. Li, The Calderón problem for the fractional magnetic operator, Inverse Problems, 36 (2020), 075003, 14pp. doi: 10.1088/1361-6420/ab8445.

[55]

L. Li, Determining the magnetic potential in the fractional magnetic Calderón problem, arXiv: 2006.10150.

[56]

V. G. Maz'ya and T. O. Shaposhnikova, Theory of Sobolev Multipliers, vol. 337 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009, With applications to differential and integral operators.

[57]

S. R. McDowall, An electromagnetic inverse problem in chiral media, Trans. Amer. Math. Soc., 352 (2000), 2993-3013.  doi: 10.1090/S0002-9947-00-02518-6.

[58] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. 
[59]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 77pp. doi: 10.1016/S0370-1573(00)00070-3.

[60]

D. Mitrea, Distributions, Partial Differential Equations, and Harmonic Analysis, Universitext, Springer, New York, 2013. doi: 10.1007/978-1-4614-8208-6.

[61]

G. NakamuraZ. Q. Sun and G. Uhlmann, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann., 303 (1995), 377-388.  doi: 10.1007/BF01460996.

[62]

G. Nakamura and T. Tsuchida, Uniqueness for an inverse boundary value problem for Dirac operators, Comm. Partial Differential Equations, 25 (2000), 1327-1369.  doi: 10.1080/03605300008821551.

[63]

G. Nakamura and G. Uhlmann, Global uniqueness for an inverse boundary problem arising in elasticity, Invent. Math., 118 (1994), 457-474.  doi: 10.1007/BF01231541.

[64]

F. Natterer, The Mathematics of Computerized Tomography, vol. 32 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001, Reprint of the 1986 original. doi: 10.1137/1.9780898719284.

[65]

T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.  doi: 10.1006/jfan.1995.1012.

[66]

E. T. Quinto, Singularities of the X-ray transform and limited data tomography in ${\mathbf{R}}^2$ and ${\mathbf{R}}^3$, SIAM J. Math. Anal., 24 (1993), 1215-1225.  doi: 10.1137/0524069.

[67]

E. T. Quinto, Artifacts and visible singularities in limited data X-ray tomography, Sens Imaging, 18 (2017), 9. doi: 10.1007/s11220-017-0158-7.

[68]

J. Railo, Fourier analysis of periodic Radon transforms, J. Fourier Anal. Appl., 26 (2020), Paper No. 64, 27pp. doi: 10.1007/s00041-020-09775-1.

[69] A. G. Ramm and A. I. Katsevich, The Radon Transform and Local Tomography, CRC Press, Boca Raton, FL, 1996. 
[70]

T. Reichelt, A comparison theorem between Radon and Fourier-Laplace transforms for D-modules, Ann. Inst. Fourier (Grenoble), 65 (2015), 1577-1616.  doi: 10.5802/aif.2968.

[71]

M. Riesz, Intégrales de Riemann-Liouville et potentiels, Acta Sci. Math. Szeged, 9 (1938), 1-42. 

[72]

X. Ros-Oton, Nonlocal equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.  doi: 10.5565/PUBLMAT_60116_01.

[73]

A. Rüland, Unique continuation for fractional Schrödinger equations with rough potentials, Comm. Partial Differential Equations, 40 (2015), 77-114.  doi: 10.1080/03605302.2014.905594.

[74]

A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003, 21pp. doi: 10.1088/1361-6420/aaac5a.

[75]

A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, Nonlinear Anal., 193 (2020), 111529, 56pp. doi: 10.1016/j.na.2019.05.010.

[76]

A. Rüland and M. Salo, Quantitative approximation properties for the fractional heat equation, Math. Control Relat. Fields, 10 (2020), 1-26.  doi: 10.3934/mcrf.2019027.

[77]

M. Salo, Recovering first order terms from boundary measurements, J. Phys.: Conf. Ser., 73 (2007), 012020. doi: 10.1088/1742-6596/73/1/012020.

[78]

M. Salo, Calderón problem, 2008, http://users.jyu.fi/~salomi/lecturenotes/calderon_lectures.pdf, Lecture notes.

[79]

M. Salo, Fourier analysis and distribution theory, 2013, http://users.jyu.fi/~salomi/lecturenotes/FA_distributions.pdf, Lecture notes.

[80]

M. Salo, The fractional calderón problem, Journées Équations aux Dérivées Partielles, 2017, 8pp. doi: 10.5802/jedp.657.

[81]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[82]

P. Stefanov and G. Uhlmann, Microlocal Analysis and Integral Geometry (working title), 2018, http://www.math.purdue.edu/~stefanov/publications/book.pdf, Draft version.

[83] F. Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York-London, 1967. 
[84]

G. Uhlmann, Inverse problems: Seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.  doi: 10.1007/s13373-014-0051-9.

[85]

L. Xiaojun, A Note On Fractional Order Poincarés Inequalities, 2012, http://www.bcamath.org/documentos_public/archivos/publicaciones/Poicare_Academie.pdf.

[86]

J. Yang, H. Yu, M. Jiang and G. Wang, High-order total variation minimization for interior tomography, Inverse Problems, 26 (2010), 035013, 29pp. doi: 10.1088/0266-5611/26/3/035013.

[87]

R. Yang, On higher order extensions for the fractional Laplacian, arXiv: 1302.4413.

[88]

Y. Ye, H. Yu and G. Wang, Exact interior reconstruction from truncated limited-angle projection data, International Journal of Biomedical Imaging, 2008 (2008), Article ID 427989. doi: 10.1155/2008/427989.

[89]

H. Yu and G. Wang, Compressed sensing based interior tomography, Phys. Med. Biol., 54 (2009), 2791–2805, https://doi.org/10.1088%2F0031-9155%2F54%2F9%2F014.

show all references

References:
[1]

H. Abels, Pseudodifferential and Singular Integral Operators, De Gruyter Graduate Lectures, De Gruyter, Berlin, 2012, An introduction with applications.

[2]

A. Abouelaz, The $d$-plane Radon transform on the torus $\Bbb T^n$, Fract. Calc. Appl. Anal., 14 (2011), 233-246.  doi: 10.2478/s13540-011-0014-8.

[3]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, vol. 165 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010. doi: 10.1090/surv/165.

[4]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[5]

A. Behzadan and M. Holst, Multiplication in Sobolev spaces, revisited, arXiv: 1512.07379.

[6]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.

[7]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, vol. 20 of Lecture Notes of the Unione Matematica Italiana, Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3.

[8]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.

[9]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[10]

X. CaoY.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Probl. Imaging, 13 (2019), 197-210.  doi: 10.3934/ipi.2019011.

[11]

M. Cekić, Y.-H. Lin and A. Rüland, The Calderón problem for the fractional Schrödinger equation with drift, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 91, 46pp. doi: 10.1007/s00526-020-01740-6.

[12]

S. N. Chandler-WildeD. P. Hewett and A. Moiola, Sobolev spaces on non-Lipschitz subsets of $\Bbb{R}^n$ with application to boundary integral equations on fractal screens, Integral Equations Operator Theory, 87 (2017), 179-224.  doi: 10.1007/s00020-017-2342-5.

[13]

M. Courdurier, F. Noo, M. Defrise and H. Kudo, Solving the interior problem of computed tomography using a priori knowledge, Inverse Problems, 24 (2008), 065001, 27pp. doi: 10.1088/0266-5611/24/6/065001.

[14]

G. Covi, An inverse problem for the fractional Schrödinger equation in a magnetic field, Inverse Problems, 36 (2020), 045004, 24pp. doi: 10.1088/1361-6420/ab661a.

[15]

G. Covi, Inverse problems for a fractional conductivity equation, Nonlinear Anal., 193 (2020), 111418, 18pp. doi: 10.1016/j.na.2019.01.008.

[16]

G. Covi, K. Mönkkönen, J. Railo and G. Uhlmann, The higher order fractional Calderón problem for linear local operators: uniqueness, arXiv: 2008.10227.

[17]

A. D'Agnolo and M. Eastwood, Radon and Fourier transforms for $\mathcal{D}$-modules, Adv. Math., 180 (2003), 452-485.  doi: 10.1016/S0001-8708(03)00011-2.

[18]

S. DipierroO. Savin and E. Valdinoci, All functions are locally $s$-harmonic up to a small error, J. Eur. Math. Soc., 19 (2017), 957-966.  doi: 10.4171/JEMS/684.

[19]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.  doi: 10.1137/110833294.

[20]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23 (2013), 493-540.  doi: 10.1142/S0218202512500546.

[21]

G. Eskin, Global uniqueness in the inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials, Comm. Math. Phys., 222 (2001), 503-531.  doi: 10.1007/s002200100522.

[22]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[23]

M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397.  doi: 10.1080/03605302.2013.825918.

[24]

V. Felli and A. Ferrero, Unique continuation principles for a higher order fractional Laplace equation, Nonlinearity, 33 (2020), 4133-4190.  doi: 10.1088/1361-6544/ab8691.

[25]

G. B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl., 3 (1997), 207-238.  doi: 10.1007/BF02649110.

[26]

J. Frikel and E. T. Quinto, Limited data problems for the generalized Radon transform in $\Bbb R^n$, SIAM J. Math. Anal., 48 (2016), 2301-2318.  doi: 10.1137/15M1045405.

[27]

M. A. García-Ferrero and A. Rüland, Strong unique continuation for the higher order fractional Laplacian, Math. Eng., 1 (2019), 715-774.  doi: 10.3934/mine.2019.4.715.

[28]

T. Ghosh, A. Rüland, M. Salo and G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, J. Funct. Anal., 279 (2020), 108505, 42pp. doi: 10.1016/j.jfa.2020.108505.

[29]

T. GhoshM. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, Anal. PDE, 13 (2020), 455-475.  doi: 10.2140/apde.2020.13.455.

[30]

F. O. Goncharov, An iterative inversion of weighted radon transforms along hyperplanes, Inverse Problems, 33 (2017), 124005, 20pp. doi: 10.1088/1361-6420/aa91a4.

[31]

F. O. Goncharov and R. G. Novikov, An example of non-uniqueness for Radon transforms with continuous positive rotation invariant weights, J. Geom. Anal., 28 (2018), 3807-3828.  doi: 10.1007/s12220-018-0001-y.

[32]

F. O. Goncharov and R. G. Novikov, An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions, Inverse Problems, 34 (2018), 054001, 6pp. doi: 10.1088/1361-6420/aab24d.

[33]

F. B. Gonzalez, On the range of the Radon $d$-plane transform and its dual, Trans. Amer. Math. Soc., 327 (1991), 601-619.  doi: 10.2307/2001816.

[34]

B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation I. Positive potentials, SIAM J. Math. Anal., 51 (2019), 3092-3111.  doi: 10.1137/18M1166298.

[35]

B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schödinger equation II. General potentials and stability, SIAM J. Math. Anal., 52 (2020), 402-436.  doi: 10.1137/19M1251576.

[36]

H. HeckX. Li and J.-N. Wang, Identification of viscosity in an incompressible fluid, Indiana Univ. Math. J., 56 (2007), 2489-2510.  doi: 10.1512/iumj.2007.56.3037.

[37]

S. Helgason, Integral Geometry and Radon transforms, Springer, New York, 2011. doi: 10.1007/978-1-4419-6055-9.

[38]

A. Homan and H. Zhou, Injectivity and stability for a generic class of generalized Radon transforms, J. Geom. Anal., 27 (2017), 1515-1529.  doi: 10.1007/s12220-016-9729-4.

[39]

L. Hörmander, The Analysis of Linear Partial Differential Operators. I, 2nd edition, Springer Study Edition, Springer-Verlag, Berlin, 1990, Distribution theory and Fourier analysis. doi: 10.1007/978-3-642-61497-2.

[40]

J. Horváth, Topological Vector Spaces and Distributions. Vol. I, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966.

[41]

J. Ilmavirta, On Radon transforms on tori, J. Fourier Anal. Appl., 21 (2015), 370-382.  doi: 10.1007/s00041-014-9374-x.

[42]

J. Ilmavirta and K. Mönkkönen, Unique continuation of the normal operator of the x-ray transform and applications in geophysics, Inverse Problems, 36 (2020), 045014, 23pp. doi: 10.1088/1361-6420/ab6e75.

[43]

E. Katsevich, A. Katsevich and G. Wang, Stability of the interior problem with polynomial attenuation in the region of interest, Inverse Problems, 28 (2012), 065022, 28pp. doi: 10.1088/0266-5611/28/6/065022.

[44]

E. Klann, E. T. Quinto and R. Ramlau, Wavelet methods for a weighted sparsity penalty for region of interest tomography, Inverse Problems, 31 (2015), 025001, 22pp. doi: 10.1088/0266-5611/31/2/025001.

[45]

V. P. Krishnan and E. T. Quinto, Microlocal Analysis in Tomography, in Handbook of Mathematical Methods in Imaging (ed. O. Scherzer), Springer, New York, 2015,847–902. doi: 10.1007/978-1-4939-0790-8_36.

[46]

N. V. Krylov, All functions are locally $s$-harmonic up to a small error, J. Funct. Anal., 277 (2019), 2728-2733.  doi: 10.1016/j.jfa.2019.02.012.

[47]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.

[48]

R.-Y. Lai and Y.-H. Lin, Global uniqueness for the fractional semilinear Schrödinger equation, Proc. Amer. Math. Soc., 147 (2019), 1189-1199.  doi: 10.1090/proc/14319.

[49]

R.-Y. Lai and Y.-H. Lin, Inverse problems for fractional semilinear elliptic equations, arXiv: 2004.00549.

[50]

R.-Y. LaiY.-H. Lin and A. Rüland, The Calderón problem for a space-time fractional parabolic equation, SIAM J. Math. Anal., 52 (2020), 2655-2688.  doi: 10.1137/19M1270288.

[51]

N. Laskin, Fractional Quantum Mechanics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018. doi: 10.1142/10541.

[52]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[53]

L. Li, A semilinear inverse problem for the fractional magnetic Laplacian, arXiv: 2005.06714.

[54]

L. Li, The Calderón problem for the fractional magnetic operator, Inverse Problems, 36 (2020), 075003, 14pp. doi: 10.1088/1361-6420/ab8445.

[55]

L. Li, Determining the magnetic potential in the fractional magnetic Calderón problem, arXiv: 2006.10150.

[56]

V. G. Maz'ya and T. O. Shaposhnikova, Theory of Sobolev Multipliers, vol. 337 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009, With applications to differential and integral operators.

[57]

S. R. McDowall, An electromagnetic inverse problem in chiral media, Trans. Amer. Math. Soc., 352 (2000), 2993-3013.  doi: 10.1090/S0002-9947-00-02518-6.

[58] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. 
[59]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 77pp. doi: 10.1016/S0370-1573(00)00070-3.

[60]

D. Mitrea, Distributions, Partial Differential Equations, and Harmonic Analysis, Universitext, Springer, New York, 2013. doi: 10.1007/978-1-4614-8208-6.

[61]

G. NakamuraZ. Q. Sun and G. Uhlmann, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann., 303 (1995), 377-388.  doi: 10.1007/BF01460996.

[62]

G. Nakamura and T. Tsuchida, Uniqueness for an inverse boundary value problem for Dirac operators, Comm. Partial Differential Equations, 25 (2000), 1327-1369.  doi: 10.1080/03605300008821551.

[63]

G. Nakamura and G. Uhlmann, Global uniqueness for an inverse boundary problem arising in elasticity, Invent. Math., 118 (1994), 457-474.  doi: 10.1007/BF01231541.

[64]

F. Natterer, The Mathematics of Computerized Tomography, vol. 32 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001, Reprint of the 1986 original. doi: 10.1137/1.9780898719284.

[65]

T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.  doi: 10.1006/jfan.1995.1012.

[66]

E. T. Quinto, Singularities of the X-ray transform and limited data tomography in ${\mathbf{R}}^2$ and ${\mathbf{R}}^3$, SIAM J. Math. Anal., 24 (1993), 1215-1225.  doi: 10.1137/0524069.

[67]

E. T. Quinto, Artifacts and visible singularities in limited data X-ray tomography, Sens Imaging, 18 (2017), 9. doi: 10.1007/s11220-017-0158-7.

[68]

J. Railo, Fourier analysis of periodic Radon transforms, J. Fourier Anal. Appl., 26 (2020), Paper No. 64, 27pp. doi: 10.1007/s00041-020-09775-1.

[69] A. G. Ramm and A. I. Katsevich, The Radon Transform and Local Tomography, CRC Press, Boca Raton, FL, 1996. 
[70]

T. Reichelt, A comparison theorem between Radon and Fourier-Laplace transforms for D-modules, Ann. Inst. Fourier (Grenoble), 65 (2015), 1577-1616.  doi: 10.5802/aif.2968.

[71]

M. Riesz, Intégrales de Riemann-Liouville et potentiels, Acta Sci. Math. Szeged, 9 (1938), 1-42. 

[72]

X. Ros-Oton, Nonlocal equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.  doi: 10.5565/PUBLMAT_60116_01.

[73]

A. Rüland, Unique continuation for fractional Schrödinger equations with rough potentials, Comm. Partial Differential Equations, 40 (2015), 77-114.  doi: 10.1080/03605302.2014.905594.

[74]

A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003, 21pp. doi: 10.1088/1361-6420/aaac5a.

[75]

A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, Nonlinear Anal., 193 (2020), 111529, 56pp. doi: 10.1016/j.na.2019.05.010.

[76]

A. Rüland and M. Salo, Quantitative approximation properties for the fractional heat equation, Math. Control Relat. Fields, 10 (2020), 1-26.  doi: 10.3934/mcrf.2019027.

[77]

M. Salo, Recovering first order terms from boundary measurements, J. Phys.: Conf. Ser., 73 (2007), 012020. doi: 10.1088/1742-6596/73/1/012020.

[78]

M. Salo, Calderón problem, 2008, http://users.jyu.fi/~salomi/lecturenotes/calderon_lectures.pdf, Lecture notes.

[79]

M. Salo, Fourier analysis and distribution theory, 2013, http://users.jyu.fi/~salomi/lecturenotes/FA_distributions.pdf, Lecture notes.

[80]

M. Salo, The fractional calderón problem, Journées Équations aux Dérivées Partielles, 2017, 8pp. doi: 10.5802/jedp.657.

[81]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[82]

P. Stefanov and G. Uhlmann, Microlocal Analysis and Integral Geometry (working title), 2018, http://www.math.purdue.edu/~stefanov/publications/book.pdf, Draft version.

[83] F. Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York-London, 1967. 
[84]

G. Uhlmann, Inverse problems: Seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.  doi: 10.1007/s13373-014-0051-9.

[85]

L. Xiaojun, A Note On Fractional Order Poincarés Inequalities, 2012, http://www.bcamath.org/documentos_public/archivos/publicaciones/Poicare_Academie.pdf.

[86]

J. Yang, H. Yu, M. Jiang and G. Wang, High-order total variation minimization for interior tomography, Inverse Problems, 26 (2010), 035013, 29pp. doi: 10.1088/0266-5611/26/3/035013.

[87]

R. Yang, On higher order extensions for the fractional Laplacian, arXiv: 1302.4413.

[88]

Y. Ye, H. Yu and G. Wang, Exact interior reconstruction from truncated limited-angle projection data, International Journal of Biomedical Imaging, 2008 (2008), Article ID 427989. doi: 10.1155/2008/427989.

[89]

H. Yu and G. Wang, Compressed sensing based interior tomography, Phys. Med. Biol., 54 (2009), 2791–2805, https://doi.org/10.1088%2F0031-9155%2F54%2F9%2F014.

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