Abstract
The paper starts with a short survey of sufficient and/or necessary conditions for a function to be a multiplier between spaces of differentiable functions which stem from Maz’ya’s well-known isoperimetric criteria for Sobolev embeddings. Another subject touched here is traces and extensions of multipliers. It is shown, in particular, that the multipliers in the space of Bessel potentialsH l p (R n), {l} > 0, l <p<∞, are traces of multipliers in a certain class of differentiable functions in R n+S with a weighted mixed norm.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adams, D. R., On the existence of capacitary strong type estimates in R n, Ark. Mat. 14 (1976) 125–140.
Adams, D. R., Hedberg, L. I., Function spaces and potential theory, Springer, 1996.
Fefferman, C, The uncertainty principle, Bull. Amer. Math. Soc. 9 (1983) 129–206.
Havin, V. P., Maz’ya, V. G., Nonlinear potential theory, Russian Math. Surveys 27 (1972), No.6, 71–148.
Kerman, R, Saywer, E. T., The trace inequality and eigenvalue estimates for Schrödinger operators, Ann. Inst. Fourier (Grenoble), 36 (1986), 207–228
Maz’ya, V. G., On some integral inequalities for functions of several variables, Prob-lemi Mat. Anal., Leningrad Univ., 1972, No. 3, 33–69.
Maz’ya, V. G., Multipliers in Sobolev spaces, In the book: Application of function theory and functional analysis methods to problems of mathematical physics. Pjatoe Sovetsko-Čehoslovackoe Soveščcanie, 1976, 181-189. Novosibirsk, 1978.
Maz′ya, V. G., Summability, with respect to an arbitrary measure, of functions from Sobolev-Slobodeckii spaces, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 92 (1979), 192–202.
Maz′ya, V. G., Sobolev spaces, Springer-Verlag, 1985.
Maz′ya, V. G., Shaposhnikova, T. O., Theory of multipliers in spaces of differentiate functions, Monographs and Studies in Mathematics 23, Pitman, 1985.
Maz′ya, V. G., Shaposhnikova, T. O., Traces and extensions of multipliers in pairs of Sobolev spaces. To appear in the book: Complex Analysis, Operator Theory, and Related Topics: S.A.Vinogradov-In Memoriam, Birkhäuser, 1999.
Maz′ya, V. G., Shaposhnikova, T. O., On pointwise interpolation inequalities for derivatives. To appear in Mathematica Bohemica, 1999.
Maz′ya, V. G., Verbitsky, I. E. Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers, Ark. Mat. 33 (1995), No. 1, 81–115.
Shaposhnikova, T. O., Equivalent norms in spaces with fractional or functional smoothness, Sibir. Mat. J. 21 (1980) 184–196.
Shaposhnikova, T. O., Bounded solutions of elliptic equations as multipliers in spaces of differentiable functions, Zapiski Nauchn. Seminar. LOMI, 194 (1986) 165–176.
Shaposhnikova, T. O., On solvability of quasilinear elliptic equations in spaces of multipliers, Izvestia Vyssh. Uch. Zav. Math. 8 (1987) 74–81.
Shaposhnikova, T. O., Multipliers in the space of Bessel potentials as traces of multipliers in weighted classes, (In Russian) Trudy Tbilis. Mat. Inst. 88 (1988) 60–63.
Shaposhnikova, T. O., Traces of multipliers in the space of Bessel potentials, Matern. Zametki, 46 (1989) 100–109.
Shaposhnikova, T. O., Applications of multipliers in Sobolev spaces to Lp-coercivity of the Neumann problem, Dokl. Acad. Nauk USSR 305 (1989) 786–789.
Shaposhnikova, T. O., On continuity of singular integral operators in Sobolev spaces, Mathematica Scandinavica 76 (1995) 85–97.
Shaposhnikova, T. O., Sobolev multipliers in a non-smooth Lp-elliptic theory, Analysis, numerics and applications of differential and integral equations, Pitman Research Notes in Mathematical Series 379, 1998, 205–208.
Sickel, W., On pointwise multipliers for in case σp,q < s < n/p. (To appear)}.
Simon, B., Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982) 447–526.
Strichartz, R. S., Multipliers on fractional Sobolev spaces, J. Math, and Mech. 16 (1967) No. 9, 1031–1060.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Basel AG
About this paper
Cite this paper
Shaposhnikova, T. (1999). Multipliers of differentiable functions and their traces. In: Rossmann, J., Takáč, P., Wildenhain, G. (eds) The Maz’ya Anniversary Collection. Operator Theory: Advances and Applications, vol 109. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8675-8_10
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8675-8_10
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9726-6
Online ISBN: 978-3-0348-8675-8
eBook Packages: Springer Book Archive