Abstract
Let α > 0. We consider the linear span \(\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right)\) of scalar Riesz's kernels \(\left\{ {\tfrac{1}{{\left| {x - a} \right|^\alpha }}} \right\}_{a \in \mathbb{R}^n }\) and the linear span \(\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right)\) of vector Riesz's kernels \(\left\{ {\tfrac{1}{{\left| {x - a} \right|^{\alpha + 1} }}\left( {x - a} \right)} \right\}_{a \in \mathbb{R}^n }\). We study the following problems. (1) When is the intersection \(\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n } \right)\) dense in Lp(ℝn)? (2) When is the intersection \(\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n ,\mathbb{R}^n } \right)\) dense in Lp(ℝn, ℝn)? Bibliography: 15 titles.
Similar content being viewed by others
REFERENCES
A. B. Aleksandrov, “Approximation by rational functions and an analog of M. Riesz's theorem on conjugate functions for the space L p with p ∈ (0, 1),” Mat. Sb., 107(149), 3–19 (1978).
A. B. Aleksandrov and P. P. Kargaev, “Hardy classes of functions harmonic in the half-space,” Algebra Analiz, 5, 1–73 (1993).
A. P. Calderon and A. Zygmund, “On higher gradients of harmonic functions,” Studia Math., 24, 211–226 (1964).
C. Fefferman and E. M. Stein, “H p spaces of several variables,” Acta Math., 129, 137–193 (1972).
J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam (1985).
M. G. Goluzina, “On a uniqueness theorem,” Vestn. Leningr. Univ. Math., Ser. 1, No. 1, 109–111 (1987).
V. P. Havin, “The uncertainty principle for one-dimensional M. Riesz potentials,” Dokl. Akad. Nauk SSSR, 264, 559–563 (1982).
V. P. Havin and B. Joricke, The Uncertainty Principle in Harmonic Analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 28, Springer-Verlag, Berlin (1994).
P. P. Kargaev, “On positive solutions of the Darboux equation Δu = (α − 1)y −1∂u/∂y, α > 0, in the half-space y > 0,” Algebra Analiz, 8, 125–150 (1996).
E. M. Landis, Second-Order Equations of Elliptic and Parabolic Type [in Russian], Nauka, Moscow (1971).
D. M. Oberlin, “Translation-invariant operators on L p (G), 0 < p < 1. II,” Canad. J. Math., 29, 626–630 (1977).
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications [in Russian], Nauka Tekhnika, Minsk (1987).
A. I. Sergeev, “Uniqueness and approximation theorems for a one-dimensional M. Riesz potential,” Vestn. St. Petersb. Univ. Math., Ser. 1, No. 3 45–53 (1994).
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey (1970).
P. Turpin and L. Waelbroeck, “Integration et fonctions holomorphes dans les espaces localement pseudoconvexes,” C. R. Acad. Sci. Paris, Ser. A–B, 267, 160–162 (1968).
Author information
Authors and Affiliations
Additional information
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 315, 2004, pp. 5–38.
Rights and permissions
About this article
Cite this article
Aleksandrov, A.B. Approximation by M. Riesz Kernels in Lp for p<1. J Math Sci 134, 2239–2257 (2006). https://doi.org/10.1007/s10958-006-0098-6
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10958-006-0098-6