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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:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
##### Keywords:
fundamental solution to the polyharmonic equation
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##### References:
 [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). [4] V. S. Vladimirov, Generalized Functions in Mathematical Physics [in Russian], Nauka, Moscow (1976). · Zbl 0313.32001 [5] S. L. Sobolev, Introduction to the Theory of Cubature Formulas [in Russian], Nauka (1974). [6] S. L. Sobolev, Selected Topics of the Theory of Function Spaces and Generalized Functions [in Russian], Nauka, Moscow (1989). · Zbl 0667.46025 [7] P. Antosik, J. Mikusiński, and R. Sikorski, Theory of Distributions. The Sequential Approach [Russian translation], Mir, Moscow (1976).
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