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2008

Criteria for Bochner's extension problem

Ruzhansky, Michael, Sugimoto, Mitsuru

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A necessary and sufficient condition for the resolution of the weak extension problem is given.

This criterion is applied to also give a criterion for the solvability of the classical Bochner's extension problem in the $L^p$-category.
The solution of the $L^p$-extension problem by Bochner giving the relation between the order of the operator, the dimension, and index $p$, for which the $L^p$-extension property holds, can be viewed as a subcritical case of the general $L^p$-extension problem.
In general, this property fails in some critical and in all supercritical cases.

Criteria for Bochner's extension problem · Around

Built on

S. Bochner, Weak solutions of linear partial differential equations, J. Math. Pures Appl
1956
Earlier work this paper cites.
L. Hörmander, Linear partial differential operators
1963
Earlier work this paper cites.
W. Littman, Polar sets and removable singularities of partial differential equations, Ark. Mat
1967
Earlier work this paper cites.
R. Harvey and J. Polking, Removable singularities of solutions of linear partial differential equations, Acta Math
1970
Earlier work this paper cites.

Similar

J. Polking, A survey of removable singularities. Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983), 261–292, Math. Sci. Res. Inst. Publ., 2, Springer, New York, 1984
1984
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N. Shimakura, Partial differential operators of elliptic type
1992
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M. Sugimoto and M. Uchida, A generalization of Bochner’s extension theorem and its application, Ark. Mat
2000
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M. Sugimoto, A weak extension theorem for inhomogeneous differential equations, Forum Math
2001
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Then

F. John, Plane waves and spherical means applied to partial differential equations
2004
Later among the works it cites.
A. V. Pokrovskii, Removable singularities of weak solutions of linear partial differential equations, Math. Notes
2005
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M. Sugimoto, A new aspect of the L p L^{p} -extension problem for inhomogeneous differential equations, Modern trends in pseudo-differential operators, 153–159, Oper. Theory Adv. Appl
2007
Later among the works it cites.

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