Abstract:
This article gives a necessary and sufficient condition for a p-integrable function to have partial derivatives of specified orders which are pth power integrable over Rn. This condition is expressed using integrals of differences which in general converge conditionally in the Lp-norm. We also prove a Fubini theorem for these function spaces.
Bibliography: 7 titles.
\Bibitem{Liz70}
\by P.~I.~Lizorkin
\paper Description of the spaces $L_p^r(R^n)$ in terms of singular difference integrals
\jour Math. USSR-Sb.
\yr 1970
\vol 10
\issue 1
\pages 77--89
\mathnet{http://mi.mathnet.ru/eng/sm3362}
\crossref{https://doi.org/10.1070/SM1970v010n01ABEH001587}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=257727}
\zmath{https://zbmath.org/?q=an:0198.18803}
Linking options:
https://www.mathnet.ru/eng/sm3362
https://doi.org/10.1070/SM1970v010n01ABEH001587
https://www.mathnet.ru/eng/sm/v123/i1/p79
This publication is cited in the following 18 articles:
A. O. Leont'eva, “Bernstein Inequality for the Riesz Derivative of Order $0<\alpha<1$ of Entire Functions of Exponential Type in the Uniform Norm”, Math. Notes, 115:2 (2024), 205–214
A. O. Leont'eva, “Bernstein inequality for Riesz derivative of fractional order less than 1 of entire function of exponential type”, Dokl. Math., 108:3 (2023), 524–527
V. Litovchenko, “THE MAXIMUM PRINCIPLE FOR THE EQUATION OF LOCAL FLUCTUATIONS OF RIESZ GRAVITATIONAL FIELDS OF PURELY FRACTIONAL ORDER”, BMJ, 9:2 (2021), 81
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
L. N. Lyakhov, M. V. Polovinkina, “The Space of Weighted Bessel Potentials”, Proc. Steklov Inst. Math., 250 (2005), 178–182
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
A. N. Karapetyants, V. A. Nogin, “Characterization of functions in anisotropic spaces of complex order”, Russian Math. (Iz. VUZ), 42:5 (1998), 22–28
B. S. Rubin, “Popravka k state “Multiplikatornye operatory, svyazannye s zadachei Koshi dlya volnovogo uravneniya. Raznostnaya regulyarizatsiya””, Matem. sb., 181:2 (1990), 286–287
B. S. Rubin, “Multiplier operators connected with the Cauchy problem for the wave equation. Difference regularization”, Math. USSR-Sb., 68:2 (1991), 391–416
Rubin B., “A Difference Regularization of Potential Type Operations in Lp-Spaces”, Math. Nachr., 144 (1989), 119–147
A. N. Kochubei, “Parabolic pseudodifferential equations, hypersingular integrals, and Markov processes”, Math. USSR-Izv., 33:2 (1989), 233–259
Kaljabin G. Lizorkin P., “Spaces of Functions of Generalized Smoothness”, Math. Nachr., 133 (1987), 7–32
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
Pavlov P. Samko S., “Description of Spaces Lp-Alpha-(Sn-1) in Terms of Spherical Hypersingular Integrals”, 276, no. 3, 1984, 546–550
H. Dappa, W. Trebels, “On hypersingular integrals and anisotropic Bessel potential spaces”, Trans. Amer. Math. Soc., 286:1 (1984), 419
G. A. Kalyabin, “Trace spaces for generalized anisotropic Liouville classes”, Math. USSR-Izv., 12:2 (1978), 289–297
G. A. Kalyabin, “Characterization of spaces of generalized Liouville differentiation”, Math. USSR-Sb., 33:1 (1977), 37–42
S. G. Samko, “On spaces of Riesz potentials”, Math. USSR-Izv., 10:5 (1976), 1089–1117
Citation in format AMSBIB
Contact us: