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A blowup alternative result for fractional nonautonomous evolution equation of Volterra type
The Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators with certain potentials
1. | Department of Mathematics Education, Incheon National University, Incheon 22012, Republic of Korea |
2. | Department of Mathematics Education, Korea University, Seoul 02841, Republic of Korea |
$\begin{equation*}\bigl(L_K+V\bigr)\mathfrak{e}_V = \delta _0\,\,\text{ in $\mathbb{R}^n$ }\end{equation*}$ |
References:
[1] |
C. Bucur,
Some observations on the Green function for the ball in the fractional Laplace framework, Comm. Pure and Appl. Anal., 15 (2016), 657-699.
|
[2] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. P.D.E., 32 (2007), 1245-1260.
|
[3] |
W. Choi and Y.-C. Kim,
Lp-mapping properties for nonlocal Schrödinger operators with certain potential, preprint, arXiv: math/0605406. |
[4] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
|
[5] |
L. Ehrenpreis,
Solution of some problems of division. Ⅰ. Division by a polynomial of derivation, Amer. J. Math., 76 (1954), 883-903.
|
[6] |
L. Ehrenpreis,
Solution of some problems of division. Ⅰ. Division by a polynomial of derivation, Amer. J. Math., 77 (1955), 286-292.
|
[7] |
M. Fall and T. Weth,
Liouville theorems for a general class of nonlocal operators, Potential. Anal., 45 (2016), 187-200.
|
[8] |
Q. Han and F. Lin,
Elliptic Partial Differential Equations Courant Lecture Notes in Mathematics, American Mathematical Society, 1997. |
[9] |
T. Kuusi, G. Mingione and Y. Sire,
Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368.
|
[10] |
N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York, 1972.
![]() |
[11] |
B. Malgrange,
Existence et approximation des solutions des équations aux d'erivées partielles et des équations de convolution, Ann. Inst. Fourier, 6 (1955/56), 271-355.
|
[12] |
M. Reed and B. Simon, Functional Analysis I,Methods of Modern Mathematical Physics, Academic Press, 1970.
![]() |
[13] |
W. Rudin,
Functional Analysis 2nd
edition, International Series in Pure and Applied Mathematics. McGraw-Hill, 1991. |
[14] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
|
[15] |
E. M. Stein, Singular Integrals and Differentiability, Princeton Univ. Press, 1970.
![]() |
show all references
References:
[1] |
C. Bucur,
Some observations on the Green function for the ball in the fractional Laplace framework, Comm. Pure and Appl. Anal., 15 (2016), 657-699.
|
[2] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. P.D.E., 32 (2007), 1245-1260.
|
[3] |
W. Choi and Y.-C. Kim,
Lp-mapping properties for nonlocal Schrödinger operators with certain potential, preprint, arXiv: math/0605406. |
[4] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
|
[5] |
L. Ehrenpreis,
Solution of some problems of division. Ⅰ. Division by a polynomial of derivation, Amer. J. Math., 76 (1954), 883-903.
|
[6] |
L. Ehrenpreis,
Solution of some problems of division. Ⅰ. Division by a polynomial of derivation, Amer. J. Math., 77 (1955), 286-292.
|
[7] |
M. Fall and T. Weth,
Liouville theorems for a general class of nonlocal operators, Potential. Anal., 45 (2016), 187-200.
|
[8] |
Q. Han and F. Lin,
Elliptic Partial Differential Equations Courant Lecture Notes in Mathematics, American Mathematical Society, 1997. |
[9] |
T. Kuusi, G. Mingione and Y. Sire,
Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368.
|
[10] |
N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York, 1972.
![]() |
[11] |
B. Malgrange,
Existence et approximation des solutions des équations aux d'erivées partielles et des équations de convolution, Ann. Inst. Fourier, 6 (1955/56), 271-355.
|
[12] |
M. Reed and B. Simon, Functional Analysis I,Methods of Modern Mathematical Physics, Academic Press, 1970.
![]() |
[13] |
W. Rudin,
Functional Analysis 2nd
edition, International Series in Pure and Applied Mathematics. McGraw-Hill, 1991. |
[14] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
|
[15] |
E. M. Stein, Singular Integrals and Differentiability, Princeton Univ. Press, 1970.
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