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Representation of functionals on the spaces \(L_ p^ m(E_ n)\). (English. Russian original) Zbl 0868.46022
Sib. Math. J. 38, No. 1, 140-146 (1997); translation from Sib. Mat. Zh. 38, No. 1, 166-172 (1997).
A representation formula \[ (l,f)=\sum _{{}\alpha{}=m }{m!\over\alpha_1!\dots\alpha_n!} \int_{E_n} u^{(\alpha )} (x)f^{(\alpha )}(x) dx \] is established for functionals \(l\) on the space \(L_p^m(E_n)\) of functions with weak derivatives of order \(m\) summable to the power \(p\). The derivatives \(u^{(\alpha )}\) of \(u\) in the above representation are expressed in terms of the fundamental solution to the polyharmonic equation \(\Delta^m=0\).
The result strengthens the author’s previous results [Embedding theorems and their applications to problems of mathematical physics, Collect. Sci. Works, Novosibirsk, 137-139 (1989; Zbl 0778.46020)] devoted to the case of compactly-supported functionals.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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[1] V. I. Polovinkin, ”On realization of error functionals for cubature formulas in spaces of typeLp/m,” in: Boundary Value Problems for Partial Differential Equations [in Russian], Inst. Mat. (Novosibirisk), Novosibirsk, 1988, pp. 125–136.
[2] V. I. Polovinkin, ”On realization of compactly-supported functionals inLp/m(E n ),” in: Embedding Theorems and Their Applications to Problems of Mathematical Physics [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk, 1989, pp. 137–139. · Zbl 0778.46020
[3] M. Z. Solomyak, ”On the conjugate spaces to the spacesWp/l of S. L. Sobolev,” Dokl. Akad. Nauk SSSR,143, No. 6, 1289–1292 (1962).
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[6] S. L. Sobolev, Selected Topics of the Theory of Function Spaces and Generalized Functions [in Russian], Nauka, Moscow (1989). · Zbl 0667.46025
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