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Einstein-Lichnerowicz type singular perturbations of critical nonlinear elliptic equations in dimension 3
Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation
1. | Tata Institute of Fundamental Research, Centre For Applicable Mathematics, Bangalore-560065, India |
2. | Department of Mathematics, Indian Institute of Science, Bangalore 560012, India |
In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with Garofalo in [
References:
[1] |
F. J. Almgren, Jr., Dirichlet's problem for multiple valued functions and the regularity of mass minimizing integral currents, Minimal Submanifolds and Geodesics, North-Holland, Amsterdam-New York, (1979), 1–6. |
[2] |
N. Aronszajn,
A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl., 36 (1957), 235-249.
|
[3] |
N. Aronszajn, A. Krzywicki and J. Szarski,
A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat., 4 (1962), 417-453.
doi: 10.1007/BF02591624. |
[4] |
V. Arya and A. Banerjee, Strong backward uniqueness for sublinear parabolic equations, NoDEA Nonlinear Differential Equations Appl., 27 (2020), 18 pp.
doi: 10.1007/s00030-020-00657-5. |
[5] |
L. Bakri,
Carleman estimates for the Schrödinger operator. Applications to quantitative uniqueness, Comm. Partial Differential Equations, 38 (2013), 69-91.
doi: 10.1080/03605302.2012.736912. |
[6] |
L. Bakri,
Quantitative uniqueness for Schrödinger operator, Indiana Univ. Math. J., 61 (2012), 1565-1580.
doi: 10.1512/iumj.2012.61.4713. |
[7] |
A. Banerjee,
Sharp vanishing order of solutions to stationary Schrödinger equations on Carnot groups of arbitrary step, J. Math. Anal. Appl., 465 (2018), 571-587.
doi: 10.1016/j.jmaa.2018.05.029. |
[8] |
A. Banerjee and N. Garofalo,
Quantitative uniqueness for elliptic equations at the boundary of $C^{1, Dini}$ domains, J. Differential Equations, 261 (2016), 6718-6757.
doi: 10.1016/j.jde.2016.09.001. |
[9] |
A. Banerjee and N. Garofalo,
Quantitative uniqueness for zero-order perturbations of generalized Baouendi-Grushin operators, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 189-207.
doi: 10.13137/2464-8728/13156. |
[10] |
A. Banerjee, N. Garofalo and R. Manna, Carleman estimates for Baouendi-Grushin operators with applications to quantitative uniqueness and strong unique continuation, Applicable Analysis, (2019), arXiv: 1903.08382.
doi: 10.1080/00036811.2020.1713314. |
[11] |
A. Banerjee and A. Mallick,
On the strong unique continuation of a degenerate elliptic operator with Hardy type potential, Ann. Mat. Pura Appl., 199 (2020), 1-21.
doi: 10.1007/s10231-019-00864-7. |
[12] |
A. Banerjee and R. Manna,
Space like strong unique continuation for sublinear parabolic equations, J. Lond. Math. Soc. (2), 102 (2020), 205-228.
doi: 10.1112/jlms.12317. |
[13] |
S. M. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. Math. France, 95 (1967), 45–87. |
[14] |
J. Bourgain and C. Kenig,
On localization in the continuous Anderson-Bernoulli model in higher dimension, Invent. Math., 161 (2005), 389-426.
doi: 10.1007/s00222-004-0435-7. |
[15] |
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. |
[16] |
L. A. Caffarelli, S. Salsa and L. Silvestre,
Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.
doi: 10.1007/s00222-007-0086-6. |
[17] |
T. Carleman, Sur un problème d'unicité pur les systemes d'équations aux dérivées partielles à deux variables indépendantes, Ark. Mat., Astr. Fys., 26 (1939), 9 pp. |
[18] |
S. Chanillo and E. Sawyer,
Unique continuation for $\Delta + \nu$ and the C. Fefferman-Phong class, Trans. Amer. Math. Soc., 318 (1990), 275-300.
doi: 10.2307/2001239. |
[19] |
H. Donnelly and C. Fefferman,
Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math., 93 (1988), 161-183.
doi: 10.1007/BF01393691. |
[20] |
H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions: Riemannian manifolds with boundary, Analysis, Et Cetera, Academic Press, Boston, MA, (1990), 251–262. |
[21] |
L. Escauriaza and S. Vessella,
Optimal three-cylinder inequalities for solutions to parabolic equations with Lipschitz leading coefficients, Contemp. Math., 333 (2003), 79-87.
doi: 10.1090/conm/333/05955. |
[22] |
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. |
[23] |
M. M. Fall and V. Felli,
Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst., 35 (2015), 5827-5867.
doi: 10.3934/dcds.2015.35.5827. |
[24] |
B. Franchi and E. Lanconelli, Une métrique associée à une classe d'opérateurs elliptiques dégén'er'es, Rend. Sem. Mat. Univ. Politec. Torino 1983, Special Issue, (1984), 105–114. |
[25] |
B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear non uniformly elliptic operators with measurable coefficients, Ann. Sc. Norm. Sup. Pisa, 4 (1983), 523–541. |
[26] |
B. Franchi and E. Lanconelli,
An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality, Comm. Partial Differential Equations, 9 (1984), 1237-1264.
doi: 10.1080/03605308408820362. |
[27] |
N. Garofalo,
Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension, J. Diff. Equations, 104 (1993), 117-146.
doi: 10.1006/jdeq.1993.1065. |
[28] |
N. Garofalo and E. Lanconelli,
Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier (Grenoble), 40 (1990), 313-356.
doi: 10.5802/aif.1215. |
[29] |
N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245–268.
doi: 10.1512/iumj.1986.35.35015. |
[30] |
N. Garofalo and F.-H. Lin,
Unique continuation for elliptic operators: A geometric-variational approach, Comm. Pure Appl. Math., 40 (1987), 347-366.
doi: 10.1002/cpa.3160400305. |
[31] |
N. Garofalo and X. Ros-Oton,
Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian, Rev. Mat. Iberoam., 35 (2019), 1309-1365.
doi: 10.4171/rmi/1087. |
[32] |
N. Garofalo and K. Rotz,
Properties of a frequency of Almgren type for harmonic functions in Carnot groups, Calc. Var. Partial Differential Equations, 54 (2015), 2197-2238.
doi: 10.1007/s00526-015-0862-x. |
[33] |
N. Garofalo and Z. Shen,
Carleman estimates for a subelliptic operator and unique continuation, Ann. Inst. Fourier (Grenoble), 44 (1994), 129-166.
doi: 10.5802/aif.1392. |
[34] |
N. Garofalo and D. Vassilev,
Strong unique continuation properties of generalized Baouendi-Grushin operators, Comm. Partial Differential Equations, 32 (2007), 643-663.
doi: 10.1080/03605300500532905. |
[35] |
V. V. Grushin,
A certain class of hypoelliptic operators, Mat. Sb. (N.S.), 83 (1970), 456-473.
|
[36] |
V. V. Grushin,
A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold, Mat. Sb. (N.S.), 84 (1971), 163-195.
|
[37] |
L. Hörmander,
Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations, 8 (1983), 21-64.
doi: 10.1080/03605308308820262. |
[38] |
D. Jerison,
Carleman inequalities for the Dirac and Laplace operators and unique continuation, Adv. in Math., 62 (1986), 118-134.
doi: 10.1016/0001-8708(86)90096-4. |
[39] |
D. Jerison and C. Kenig,
Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. (2), 121 (1985), 463-494.
doi: 10.2307/1971205. |
[40] |
H. Koch, A. Petrosyan and W. Shi, Higher regularity of the free boundary in the elliptic Signorini problem, Nonlinear Anal., 126 (2015), 3–44.
doi: 10.1016/j.na.2015.01.007. |
[41] |
H. Koch, A. Rüland and W. Shi,
The variable coefficient thin obstacle problem: Higher regularity, Adv. Differential Equations, 22 (2017), 793-866.
|
[42] |
H. Koch and D. Tataru,
Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients, Comm. Pure Appl. Math., 54 (2001), 339-360.
doi: 10.1002/1097-0312(200103)54:3<339::AID-CPA3>3.0.CO;2-D. |
[43] |
V. Z. Meshkov,
On the possible rate of decrease at infinity of the solutions of second-order partial differential equations, Math. USSR-Sb., 72 (1992), 343-361.
doi: 10.1070/SM1992v072n02ABEH001414. |
[44] |
K. Miller, Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients, Arch. Rational Mech. Anal., 54 (1974), 105–117.
doi: 10.1007/BF00247634. |
[45] |
A. Plis,
On non-uniqueness in Cauchy problem for an elliptic second order differential equation, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 11 (1963), 95-100.
|
[46] |
R. Regbaoui, Strong uniqueness for second order differential operators, J. Differential Equations, 141 (1997), 201–217.
doi: 10.1006/jdeq.1997.3327. |
[47] |
A. Rüland,
Unique Continuation for sublinear elliptic equations based on Carleman estimates, J. Differential Equations, 265 (2018), 6009-6035.
doi: 10.1016/j.jde.2018.07.025. |
[48] |
A. Rüland,
On quantitative unique continuation properties of fractional Schrödinger equations: Doubling, vanishing order and nodal domain estimates, Trans. Amer. Math. Soc., 369 (2017), 2311-2362.
doi: 10.1090/tran/6758. |
[49] |
Y. Sire, S. Terracini and G. Tortone,
On the nodal set of solutions to degenerate or singular elliptic equations with an application to $s-$ harmonic functions, J. Math. Pures Appl. (9), 143 (2020), 376-441.
doi: 10.1016/j.matpur.2020.01.010. |
[50] |
N. Soave and T. Weth, The unique continuation property of sublinear equations, SIAM J. Math. Anal., 50 (2018), 3919–3938.
doi: 10.1137/17M1144325. |
[51] |
N. Soave and S. Terracini,
The nodal set of solutions to some elliptic problems: Sublinear equations, and unstable two-phase membrane problem, Adv. Math., 334 (2018), 243-299.
doi: 10.1016/j.aim.2018.06.007. |
[52] |
C. D. Sogge,
Oscillatory integrals and spherical harmonics, Duke Math. J., 53 (1986), 43-65.
doi: 10.1215/S0012-7094-86-05303-2. |
[53] |
P. R. Stinga and J. L. Torrea,
Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.
doi: 10.1080/03605301003735680. |
[54] |
G. Tortone, The nodal set of solutions to some nonlocal sublinear problems, arXiv: 2004.04652. |
[55] |
L. Wang,
Hölder estimates for subelliptic operators, J. Funct. Anal., 199 (2003), 228-242.
doi: 10.1016/S0022-1236(03)00093-4. |
[56] |
J. Zhu,
Quantitative uniqueness for elliptic equations, Amer. J. Math., 138 (2016), 733-762.
doi: 10.1353/ajm.2016.0027. |
show all references
References:
[1] |
F. J. Almgren, Jr., Dirichlet's problem for multiple valued functions and the regularity of mass minimizing integral currents, Minimal Submanifolds and Geodesics, North-Holland, Amsterdam-New York, (1979), 1–6. |
[2] |
N. Aronszajn,
A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl., 36 (1957), 235-249.
|
[3] |
N. Aronszajn, A. Krzywicki and J. Szarski,
A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat., 4 (1962), 417-453.
doi: 10.1007/BF02591624. |
[4] |
V. Arya and A. Banerjee, Strong backward uniqueness for sublinear parabolic equations, NoDEA Nonlinear Differential Equations Appl., 27 (2020), 18 pp.
doi: 10.1007/s00030-020-00657-5. |
[5] |
L. Bakri,
Carleman estimates for the Schrödinger operator. Applications to quantitative uniqueness, Comm. Partial Differential Equations, 38 (2013), 69-91.
doi: 10.1080/03605302.2012.736912. |
[6] |
L. Bakri,
Quantitative uniqueness for Schrödinger operator, Indiana Univ. Math. J., 61 (2012), 1565-1580.
doi: 10.1512/iumj.2012.61.4713. |
[7] |
A. Banerjee,
Sharp vanishing order of solutions to stationary Schrödinger equations on Carnot groups of arbitrary step, J. Math. Anal. Appl., 465 (2018), 571-587.
doi: 10.1016/j.jmaa.2018.05.029. |
[8] |
A. Banerjee and N. Garofalo,
Quantitative uniqueness for elliptic equations at the boundary of $C^{1, Dini}$ domains, J. Differential Equations, 261 (2016), 6718-6757.
doi: 10.1016/j.jde.2016.09.001. |
[9] |
A. Banerjee and N. Garofalo,
Quantitative uniqueness for zero-order perturbations of generalized Baouendi-Grushin operators, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 189-207.
doi: 10.13137/2464-8728/13156. |
[10] |
A. Banerjee, N. Garofalo and R. Manna, Carleman estimates for Baouendi-Grushin operators with applications to quantitative uniqueness and strong unique continuation, Applicable Analysis, (2019), arXiv: 1903.08382.
doi: 10.1080/00036811.2020.1713314. |
[11] |
A. Banerjee and A. Mallick,
On the strong unique continuation of a degenerate elliptic operator with Hardy type potential, Ann. Mat. Pura Appl., 199 (2020), 1-21.
doi: 10.1007/s10231-019-00864-7. |
[12] |
A. Banerjee and R. Manna,
Space like strong unique continuation for sublinear parabolic equations, J. Lond. Math. Soc. (2), 102 (2020), 205-228.
doi: 10.1112/jlms.12317. |
[13] |
S. M. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. Math. France, 95 (1967), 45–87. |
[14] |
J. Bourgain and C. Kenig,
On localization in the continuous Anderson-Bernoulli model in higher dimension, Invent. Math., 161 (2005), 389-426.
doi: 10.1007/s00222-004-0435-7. |
[15] |
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. |
[16] |
L. A. Caffarelli, S. Salsa and L. Silvestre,
Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.
doi: 10.1007/s00222-007-0086-6. |
[17] |
T. Carleman, Sur un problème d'unicité pur les systemes d'équations aux dérivées partielles à deux variables indépendantes, Ark. Mat., Astr. Fys., 26 (1939), 9 pp. |
[18] |
S. Chanillo and E. Sawyer,
Unique continuation for $\Delta + \nu$ and the C. Fefferman-Phong class, Trans. Amer. Math. Soc., 318 (1990), 275-300.
doi: 10.2307/2001239. |
[19] |
H. Donnelly and C. Fefferman,
Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math., 93 (1988), 161-183.
doi: 10.1007/BF01393691. |
[20] |
H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions: Riemannian manifolds with boundary, Analysis, Et Cetera, Academic Press, Boston, MA, (1990), 251–262. |
[21] |
L. Escauriaza and S. Vessella,
Optimal three-cylinder inequalities for solutions to parabolic equations with Lipschitz leading coefficients, Contemp. Math., 333 (2003), 79-87.
doi: 10.1090/conm/333/05955. |
[22] |
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. |
[23] |
M. M. Fall and V. Felli,
Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst., 35 (2015), 5827-5867.
doi: 10.3934/dcds.2015.35.5827. |
[24] |
B. Franchi and E. Lanconelli, Une métrique associée à une classe d'opérateurs elliptiques dégén'er'es, Rend. Sem. Mat. Univ. Politec. Torino 1983, Special Issue, (1984), 105–114. |
[25] |
B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear non uniformly elliptic operators with measurable coefficients, Ann. Sc. Norm. Sup. Pisa, 4 (1983), 523–541. |
[26] |
B. Franchi and E. Lanconelli,
An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality, Comm. Partial Differential Equations, 9 (1984), 1237-1264.
doi: 10.1080/03605308408820362. |
[27] |
N. Garofalo,
Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension, J. Diff. Equations, 104 (1993), 117-146.
doi: 10.1006/jdeq.1993.1065. |
[28] |
N. Garofalo and E. Lanconelli,
Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier (Grenoble), 40 (1990), 313-356.
doi: 10.5802/aif.1215. |
[29] |
N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245–268.
doi: 10.1512/iumj.1986.35.35015. |
[30] |
N. Garofalo and F.-H. Lin,
Unique continuation for elliptic operators: A geometric-variational approach, Comm. Pure Appl. Math., 40 (1987), 347-366.
doi: 10.1002/cpa.3160400305. |
[31] |
N. Garofalo and X. Ros-Oton,
Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian, Rev. Mat. Iberoam., 35 (2019), 1309-1365.
doi: 10.4171/rmi/1087. |
[32] |
N. Garofalo and K. Rotz,
Properties of a frequency of Almgren type for harmonic functions in Carnot groups, Calc. Var. Partial Differential Equations, 54 (2015), 2197-2238.
doi: 10.1007/s00526-015-0862-x. |
[33] |
N. Garofalo and Z. Shen,
Carleman estimates for a subelliptic operator and unique continuation, Ann. Inst. Fourier (Grenoble), 44 (1994), 129-166.
doi: 10.5802/aif.1392. |
[34] |
N. Garofalo and D. Vassilev,
Strong unique continuation properties of generalized Baouendi-Grushin operators, Comm. Partial Differential Equations, 32 (2007), 643-663.
doi: 10.1080/03605300500532905. |
[35] |
V. V. Grushin,
A certain class of hypoelliptic operators, Mat. Sb. (N.S.), 83 (1970), 456-473.
|
[36] |
V. V. Grushin,
A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold, Mat. Sb. (N.S.), 84 (1971), 163-195.
|
[37] |
L. Hörmander,
Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations, 8 (1983), 21-64.
doi: 10.1080/03605308308820262. |
[38] |
D. Jerison,
Carleman inequalities for the Dirac and Laplace operators and unique continuation, Adv. in Math., 62 (1986), 118-134.
doi: 10.1016/0001-8708(86)90096-4. |
[39] |
D. Jerison and C. Kenig,
Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. (2), 121 (1985), 463-494.
doi: 10.2307/1971205. |
[40] |
H. Koch, A. Petrosyan and W. Shi, Higher regularity of the free boundary in the elliptic Signorini problem, Nonlinear Anal., 126 (2015), 3–44.
doi: 10.1016/j.na.2015.01.007. |
[41] |
H. Koch, A. Rüland and W. Shi,
The variable coefficient thin obstacle problem: Higher regularity, Adv. Differential Equations, 22 (2017), 793-866.
|
[42] |
H. Koch and D. Tataru,
Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients, Comm. Pure Appl. Math., 54 (2001), 339-360.
doi: 10.1002/1097-0312(200103)54:3<339::AID-CPA3>3.0.CO;2-D. |
[43] |
V. Z. Meshkov,
On the possible rate of decrease at infinity of the solutions of second-order partial differential equations, Math. USSR-Sb., 72 (1992), 343-361.
doi: 10.1070/SM1992v072n02ABEH001414. |
[44] |
K. Miller, Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients, Arch. Rational Mech. Anal., 54 (1974), 105–117.
doi: 10.1007/BF00247634. |
[45] |
A. Plis,
On non-uniqueness in Cauchy problem for an elliptic second order differential equation, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 11 (1963), 95-100.
|
[46] |
R. Regbaoui, Strong uniqueness for second order differential operators, J. Differential Equations, 141 (1997), 201–217.
doi: 10.1006/jdeq.1997.3327. |
[47] |
A. Rüland,
Unique Continuation for sublinear elliptic equations based on Carleman estimates, J. Differential Equations, 265 (2018), 6009-6035.
doi: 10.1016/j.jde.2018.07.025. |
[48] |
A. Rüland,
On quantitative unique continuation properties of fractional Schrödinger equations: Doubling, vanishing order and nodal domain estimates, Trans. Amer. Math. Soc., 369 (2017), 2311-2362.
doi: 10.1090/tran/6758. |
[49] |
Y. Sire, S. Terracini and G. Tortone,
On the nodal set of solutions to degenerate or singular elliptic equations with an application to $s-$ harmonic functions, J. Math. Pures Appl. (9), 143 (2020), 376-441.
doi: 10.1016/j.matpur.2020.01.010. |
[50] |
N. Soave and T. Weth, The unique continuation property of sublinear equations, SIAM J. Math. Anal., 50 (2018), 3919–3938.
doi: 10.1137/17M1144325. |
[51] |
N. Soave and S. Terracini,
The nodal set of solutions to some elliptic problems: Sublinear equations, and unstable two-phase membrane problem, Adv. Math., 334 (2018), 243-299.
doi: 10.1016/j.aim.2018.06.007. |
[52] |
C. D. Sogge,
Oscillatory integrals and spherical harmonics, Duke Math. J., 53 (1986), 43-65.
doi: 10.1215/S0012-7094-86-05303-2. |
[53] |
P. R. Stinga and J. L. Torrea,
Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.
doi: 10.1080/03605301003735680. |
[54] |
G. Tortone, The nodal set of solutions to some nonlocal sublinear problems, arXiv: 2004.04652. |
[55] |
L. Wang,
Hölder estimates for subelliptic operators, J. Funct. Anal., 199 (2003), 228-242.
doi: 10.1016/S0022-1236(03)00093-4. |
[56] |
J. Zhu,
Quantitative uniqueness for elliptic equations, Amer. J. Math., 138 (2016), 733-762.
doi: 10.1353/ajm.2016.0027. |
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