Abstract
We study linear elliptic pseudodifferential operators in the improved scale of functional Hilbert spaces on a smooth closed manifold. Elements of this scale are isotropic Hörmander-Volevich-Paneyakh spaces. We investigate the local smoothness of a solution of an elliptic equation in the improved scale. We also study elliptic pseudodifferential operators with parameter.
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References
L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 3: Pseudo-Differential Operators [Russian translation], Mir, Moscow (1987).
M. S. Agranovich, “Elliptic operators on closed manifolds,” in: VINITI Series in Contemporary Problems of Mathematics (Fundamental Trends) [in Russian], Vol. 63, VINITI, Moscow (1990), pp. 5–129.
L. Hörmander, Linear Partial Differential Operators [Russian translation], Mir, Moscow (1965).
L. P. Volevich and B. P. Paneyakh, “Some spaces of generalized functions and imbedding theorems,” Usp. Mat. Nauk, 20, No. 1, 3–74 (1965).
V. A. Mikhailets and A. A. Murach, “Elliptic operators in the improved scale of functional spaces,” Ukr. Mat. Zh., 57, No. 5, 689–696 (2005).
V. A. Mikhailets and A. A. Murach, “Improved scales of spaces and elliptic boundary-value problems. I,” Ukr. Mat. Zh., 58, No. 2, 217–235 (2006).
V. A. Mikhailets and A. A. Murach, “Improved scales of spaces and elliptic boundary-value problems. II,” Ukr. Mat. Zh., 58, No. 3, 352–370 (2006).
P. I. Lizorkin, “Spaces of generalized smoothness,” in: H. Triebel, Theory of Function Spaces [Russian translation], Mir, Moscow (1986), pp. 381–415.
D. D. Haroske and S. D. Moura, “Continuity envelopes of spaces of generalized smoothness, entropy and approximation numbers,” J. Approxim. Theory, 128, 151–174 (2004).
W. Farkas and H.-G. Leopold, “Characterization of function spaces of generalized smoothness,” Ann. Math. Pura Appl., 185, No. 1, 1–62 (2006).
G. Shlenzak, “Elliptic problems in the improved scale of spaces,” Vestn. Mosk. Univ., No. 4, 48–58 (1974).
V. A. Mikhailets and A. A. Murach, “Elliptic operator in the improved scale of spaces on a closed manifold,” Dopov. Nats. Akad. Nauk Ukr., No. 10, 27–33 (2006).
V. A. Mikhailets and A. A. Murach, “Regular elliptic boundary-value problem for a homogeneous equation in a two-sided improved scale of spaces,” Ukr. Mat. Zh., 58, No. 11, 1536–1555 (2006).
V. A. Mikhailets and A. A. Murach, “Elliptic operator with homogeneous regular boundary conditions in a two-sided improved scale of spaces,” Ukr. Mat. Visn., 3, No. 4, 547–580 (2006).
V. A. Mikhailets and A. A. Murach, “Improved scales of spaces and elliptic boundary-value problems. III,” Ukr. Mat. Zh., 59, No. 5, 679–701 (2007).
E. Seneta, Regularly Varying Functions [Russian translation], Nauka, Moscow (1985).
J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications [Russian translation], Mir, Moscow (1971).
V. A. Mikhailets and A. A. Murach, “Interpolation with functional parameter and spaces of differentiable functions,” Dopov. Nats. Akad. Nauk Ukr., No. 6, 13–18 (2006).
M. F. Atiyah and I. M. Singer, “The index of elliptic operators on compact manifolds,” Bull. Amer. Math. Soc., 69, No. 3, 422–433 (1963).
Yu. M. Berezanskii, Expansion in Eigenfunctions of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).
S. Agmon, “On the eigenfunctions and on the eigenvalues of general elliptic boundary-value problems,” Commun. Pure Appl. Math., 15, No. 2, 119–147 (1962).
M. S. Agranovich and M. I. Vishik, “Elliptic problems with parameter and parabolic problems of general form” Usp. Mat. Nauk, 19, No. 3, 53–161 (1964).
A. N. Kozhevnikov, “Spectral problems for pseudodifferential Douglis-Nirenberg elliptic systems and their applications,” Mat. Sb., 93(134), No. 1 (9), 60–88 (1973).
G. Grubb, Functional Calculus of Pseudo-Differential Boundary Problems, Birkhäuser, Boston (1996).
Ya. A. Roitberg, Elliptic Boundary Value Problems in the Spaces of Distributions, Kluwer, Dordrecht (1996).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 6, pp. 798–814, June, 2007.
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Murach, A.A. Elliptic pseudodifferential operators in the improved scale of spaces on a closed manifold. Ukr Math J 59, 874–893 (2007). https://doi.org/10.1007/s11253-007-0056-6
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DOI: https://doi.org/10.1007/s11253-007-0056-6