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Izv. Akad. Nauk SSSR Ser. Mat., 1976, Volume 40, Issue 5, Pages 1143–1172 (Mi izv2237)  

This article is cited in 15 scientific papers (total in 15 papers)

On spaces of Riesz potentials

S. G. Samko

Abstract: In connection with problems which arise in the theory of integral equations of the first kind with a potential-type kernel we investigate the space of Riesz potentials $I^\alpha(L_p)=\{f=K^\alpha\varphi;\varphi\in L_p(R^n),1<p<n/\alpha\}$, where $K^\alpha$ is the Riesz integration operator ($\widehat{K^\alpha\varphi}(x)=|(x)|^{-\alpha}\widehat\varphi(x)$). We give a description of the space $I^\alpha(L_p)$ in terms of differences of singular integrals, establish a theorem on denseness of $C^\infty_0(R^n)$ in $I^\alpha(L_p)$, and indicate a “weight” invariant description of $I^\alpha(L_p)$.
Bibliography: 44 titles

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English version:
Mathematics of the USSR-Izvestiya, 1976, 10:5, 1089–1117

Bibliographic databases:

UDC: 517.9
MSC: Primary 46E35; Secondary 45A05
Received: 16.04.1974

Citation: S. G. Samko, “On spaces of Riesz potentials”, Izv. Akad. Nauk SSSR Ser. Mat., 40:5 (1976), 1143–1172; Math. USSR-Izv., 10:5 (1976), 1089–1117

Citation in format AMSBIB
\by S.~G.~Samko
\paper On spaces of Riesz potentials
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1976
\vol 40
\issue 5
\pages 1143--1172
\jour Math. USSR-Izv.
\yr 1976
\vol 10
\issue 5
\pages 1089--1117

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    This publication is cited in the following articles:
    1. V. A. Nogin, “The weighted spaces $L^\alpha_{p,r}(\rho_1,\rho_2)$ of differentiable functions of fractional smoothness”, Math. USSR-Sb., 59:1 (1988), 209–221  mathnet  crossref  mathscinet  zmath
    2. A. N. Kochubei, “Parabolic pseudodifferential equations, hypersingular integrals, and Markov processes”, Math. USSR-Izv., 33:2 (1989), 233–259  mathnet  crossref  mathscinet  zmath
    3. B. S. Rubin, “Multiplier operators connected with the Cauchy problem for the wave equation. Difference regularization”, Math. USSR-Sb., 68:2 (1991), 391–416  mathnet  crossref  mathscinet  zmath  isi
    4. B. S. Rubin, “Popravka k state “Multiplikatornye operatory, svyazannye s zadachei Koshi dlya volnovogo uravneniya. Raznostnaya regulyarizatsiya””, Matem. sb., 181:2 (1990), 286–287  mathnet  mathscinet  zmath
    5. Stefan G. Samko, “Inversion theorems for poterntial-type integral transforms in a and onsn-1”, Integral Transforms and Special Functions, 1:2 (1993), 145  crossref
    6. A. N. Karapetyants, V. A. Nogin, “Characterization of functions in anisotropic spaces of complex order”, Russian Math. (Iz. VUZ), 42:5 (1998), 22–28  mathnet  mathscinet  elib
    7. R.A. Cerutti, S.E. Trione, “The inversion of Marcel Riesz ultrahyperbolic causal operators”, Applied Mathematics Letters, 12:6 (1999), 25  crossref
    8. R Cerutti, “Some properties of the generalized causal and anticausal Riesz potentials”, Applied Mathematics Letters, 13:4 (2000), 129  crossref  elib
    9. A. P. Chegolin, “The classes $L_{p,r}^\alpha$ of Lizorkin–Samko type associated with complex powers of the telegraph operator”, Russian Math. (Iz. VUZ), 46:7 (2002), 56–62  mathnet  mathscinet  zmath  elib
    10. Karasev D.N., Nogin V.A., “Inversion of some potential-type operators with oscillating kernels in the elliptic and non-elliptic cases”, Integral Transforms and Special Functions, 13:6 (2002), 529–545  crossref  isi  elib
    11. Takahide Kurokawa, “Higher Riesz transforms and derivatives of the Riesz kernels”, Integral Transforms and Special Functions, 15:1 (2004), 51  crossref
    12. Sinem Sezer, Ilham A. Aliev, “A new characterization of the Riesz potential spaces with the aid of a composite wavelet transform”, Journal of Mathematical Analysis and Applications, 372:2 (2010), 549  crossref
    13. E. V. Arbuzov, “O svoistvakh integralnogo operatora Koshi s ostsilliruyuschim yadrom”, Sib. elektron. matem. izv., 10 (2013), 3–9  mathnet
    14. È. V. Arbuzov, “On the properties of a Riesz potential with oscillating kernel”, Siberian Math. J., 55:2 (2014), 201–209  mathnet  crossref  mathscinet  isi
    15. S. G. Samko, S. M. Umarkhadzhiev, “Ob opisanii prostranstva rissovykh potentsialov funktsii iz banakhovykh prostranstv s nekotorymi apriornymi svoistvami”, Vladikavk. matem. zhurn., 20:2 (2018), 95–108  mathnet  crossref
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