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Fractional integrals and derivatives: mapping properties

  • Humberto Rafeiro EMAIL logo und Stefan Samko EMAIL logo
Veröffentlicht/Copyright: 28. Juni 2016
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Fractional Calculus and Applied Analysis
Aus der Zeitschrift Fractional Calculus and Applied Analysis Band 19 Heft 3

Abstract

This survey is aimed at the audience of readers interested in the information on mapping properties of various forms of fractional integration operators, including multidimensional ones, in a large scale of various known function spaces.

As is well known, the fractional integrals defined in this or other forms improve in some sense the properties of the functions, at least locally, while fractional derivatives to the contrary worsen them. With the development of functional analysis this simple fact led to a number of important results on the mapping properties of fractional integrals in various function spaces.

In the one-dimensional case we consider both Riemann-Liouville and Liouville forms of fractional integrals and derivatives. In the multidimensional case we consider in particular mixed Liouville fractional integrals, Riesz fractional integrals of elliptic and hyperbolic type and hypersingular integrals. Among the function spaces considered in this survey, the reader can find Hölder spaces, Lebesgue spaces, Morrey spaces, Grand spaces and also weighted and/or variable exponent versions.

MSC 2010: Primary 26A33; Secondary 46E30
Key Words and Phrases: mapping properties; fractional integral; Riesz potential; hypersingular integrals; fractional derivatives; Lebesgue spaces; variable exponent Lebesgue spaces; Hölder spaces; variable exponent Hölder spaces; Grand spaces, Morrey spaces

Acknowledgements

H. Rafeiro was partially supported by Pontificia Universidad Javeriana under the research project “Study of mapping properties of fractional integrals and derivatives”, ID PROY: 7446.

References

[1] Adams D.R. A note on Riesz potentials Duke Math. J. 42 No 4 1975 765 77810.1215/S0012-7094-75-04265-9Suche in Google Scholar

[2] Almeida A. Hasanov J. Samko S. Maximal and potential operators in variable exponent Morrey spaces Georgian Math. J. 15 No 2 2008 195 20810.1515/GMJ.2008.195Suche in Google Scholar

[3] Bennett C. Rudnick K. On Lorentz-Zygmund spaces Diss. Math. 175 1980 67Suche in Google Scholar

[4] Berg J. Löfström J. Interpolation Spaces. An Introduction Springer Berlin 197610.1007/978-3-642-66451-9Suche in Google Scholar

[5] Cianchi A. Strong and weak type inequalities for some classical operators in Orlicz spaces J. London Math. Soc. (2) 60 No 1 1999 187 20210.1112/S0024610799007711Suche in Google Scholar

[6] Cianchi A. Edmunds D.E. On fractional integration in weighted Lorentz spaces Q. J. Math., Oxf. II. Ser. 48 No 192 1997 439 45110.1093/qjmath/48.192.439Suche in Google Scholar

[7] Cruz-Uribe D. Fiorenza A. Variable Lebesgue Spaces. Foundations and Harmonic Analysis Birkhäuser Basel 201310.1007/978-3-0348-0548-3Suche in Google Scholar

[8] Deringoz F. Guliyev V.S. Samko S. Boundedness of maximal and singular operators on generalized Orlicz-Morrey spaces In: Operator Theory: Advances and Applications Vol. 242 Birkhäuser Basel 2014 139 15810.1007/978-3-0348-0816-3_7Suche in Google Scholar

[9] Deringoz F. Guliyev V.S. Samko S. Boundedness of the maximal operator and its commutators on vanishing generalized Orlicz-Morrey spaces Ann. Acad. Sci. Fenn. Math. 40 No 2 2015 535 54910.5186/aasfm.2015.4029Suche in Google Scholar

[10] Diening L. Harjulehto P. Hästö, M. Ruu“17 užička Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics Vol. 2017 Springer-Verlag Berlin 201110.1007/978-3-642-18363-8Suche in Google Scholar

[11] Dorronsoro J.R. Mean oscillation and Besov spaces Can. Math. Bull. 28 1985 474 48010.4153/CMB-1985-058-3Suche in Google Scholar

[12] Erdélyi A. On fractional integration and its application to the theory of Hankel transforms Q. J. Math., Oxf. Ser. 11 1940 293 30310.1093/qmath/os-11.1.293Suche in Google Scholar

[13] Giaquinta M. Multiple Integrals in the Calculus of Variations and Non-linear Elliptic Systems Princeton University Press Princeton 198310.1515/9781400881628Suche in Google Scholar

[14] Gopala Rao V.R. A characterization of parabolic function spaces Amer. J. Math. 99 1977 985 99310.2307/2373995Suche in Google Scholar

[15] Greco L. Iwaniec T. Sbordone C. Inverting the p-harmonic operator Manuscripta Math. 92 1997 249 25810.1007/BF02678192Suche in Google Scholar

[16] Guliyev V.S. Function Spaces, Integral Operators and Two Weighted inequalities on Homogeneous groups. Some Applications (in Russian). Baku 1999Suche in Google Scholar

[17] Guliyev V.S. Balakishiyev A.S. Parabolic fractional integral operators with rough kernel in parabolic generalized Morrey spaces Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 38 2013 47 56Suche in Google Scholar

[18] Guliyev V.S. Balakishiyev A.S. Parabolic fractional maximal and integral operators with rough kernels in parabolic generalized Morrey spaces J. Math. Inequal. 9 No 1 2015 257 27610.7153/jmi-09-23Suche in Google Scholar

[19] Guliyev V.S. Hasanov J.J. Zeren Y. Necessary and sufficient conditions for the boundedness of the Riesz potential in modified Morrey spaces J. Math. Inequal. 5 No 4 2011 491 50610.7153/jmi-05-43Suche in Google Scholar

[20] Guliyev V.S. Muradova S.A. Parabolic fractional maximal operator in parabolic local Morrey-type spaces Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 31 No 4 2011 59 72Suche in Google Scholar

[21] Guliyev V.S. Rahimova K.R. Parabolic fractional maximal operator in modified parabolic Morrey spaces J. Funct. Spaces Appl. 2012 2012 Art. ID 543475, 20 pp10.1155/2012/543475Suche in Google Scholar

[22] Guliyev V.S. Rahimova K.R. Parabolic fractional maximal operator in parabolic generalized Morrey spaces Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 37 2012 51 6610.1155/2012/543475Suche in Google Scholar

[23] Guliyev V.S. Rahimova K.R. Parabolic fractional integral operator in modified parabolic Morrey spaces Proc. A. Razmadze Math. Inst. 163 2013 59 80Suche in Google Scholar

[24] Guliyev V.S. Samko S. Maximal, potential and singular operators in the generalized variable exponent Morrey spaces on unbounded sets J. Math. Sci. (N. Y.) 193 No 2 2013 228 24810.1007/s10958-013-1449-8Suche in Google Scholar

[25] Guliyev V.S. Serbetci A. Ekincioglu I. Necessary and sufficient conditions for the boundedness of rough B-fractional integral operators in the Lorentz spaces J. Math. Anal. Appl. 336 No 1 2007 425 43710.1016/j.jmaa.2007.02.080Suche in Google Scholar

[26] Guliyev V.S. Shukurov P.S. On the boundedness of the fractional maximal operator, Riesz potential and their commutators in generalized Morrey spaces In Operator Theory: Advances and Applications Vol. 229 Birkhäuser Basel 2014 175 19910.1007/978-3-0348-0516-2_10Suche in Google Scholar

[27] Hardy G.H. Littlewood J.E. Some properties of fractional integrals, I Math. Z. 27 No 4 1928 565 60610.1007/BF01171116Suche in Google Scholar

[28] Hardy G.H. Littlewood J.E. Some properties of fractional integrals, II Math. Z. 34 1931 403 43910.1007/BF01180596Suche in Google Scholar

[29] Hästö P.A. Local-to-global results in variable exponent spaces Math. Res. Lett. 16 No 2–3 2009 263 27810.4310/MRL.2009.v16.n2.a5Suche in Google Scholar

[30] Iwaniec T. Sbordone C. On the integrability of the Jacobian under minimal hypotheses Arch. Ration. Mech. Anal. 119 No 2 1992 129 14310.1007/BF00375119Suche in Google Scholar

[31] Karapetyants N.K. Ginzburg A.I. Fractional integrals and singular integrals in the Hölder classes of variable order Integral Transform. Spec. Funct. 2 No 2 1994 91 10610.1080/10652469408819043Suche in Google Scholar

[32] Karapetyants N.K. Ginzburg A.I. Fractional integrodifferentiation in Hölder classes of arbitrary order Georgian Math. J. 2 No 2 1995 141 15010.1007/BF02257475Suche in Google Scholar

[33] Karapetyants N.K. Rubin B.S. Riesz radial potentials on the disc and fractional integration operators Sov. Math. Dokl. 25 No 2 1982 522 525Suche in Google Scholar

[34] Karapetyants N.K. Rubin B.S. Riesz radial potentials on the disc and fractional integration operators (In Russian) DAN SSR 263 No 6 1982 1299–1302; Engl. transl. in Soviet Math. Dokl. 25 No 2 1982 522 525Suche in Google Scholar

[35] Karapetyants N.K. Rubin B.S. Local properties of fractional integrals and the BMO spaces on the segment of real axis. Rostov-on-Don Dep. in VINITI 06.02.1986, No 669-B, 43 ppSuche in Google Scholar

[36] Karapetyants N.K. Rubin B.S. On fractional integrals with limiting exponent Sov. Math. 32 1988 98 102Suche in Google Scholar

[37] Karapetiants N.K. Samko S.G. Multidimensional integral operators with homogeneous kernels Fract. Calc. Appl. Anal. 2 No 1 1999 67 96Suche in Google Scholar

[38] Karapetiants N.K. Samko S.G. Equations with Involutive Operators Birkhäuser Boston 200110.1007/978-1-4612-0183-0Suche in Google Scholar

[39] Karapetiants N.K. Shankishvili L.D. A short proof of Hardy-Littlewood-type theorem for fractional integrals in weighted Hölder spaces Fract. Calc. Appl. Anal. 2 No 2 1999 177 192Suche in Google Scholar

[40] Kipriyanov I.A. Ivanov L.A. Riesz potentials on Lorentz spaces (In Russian) Math. USSR, Sb. 58 1987 467 47510.1070/SM1987v058n02ABEH003114Suche in Google Scholar

[41] Kipriyanov I.A. Ivanov L.A. On the theory of Riesz potentials on Lorentz spaces (In Russian) Proc. Steklov Inst. Math. 180 1989 156 157Suche in Google Scholar

[42] Kiryakova V. Generalized Fractional Calculus and Applications. Pitman Res. Notes in Math. Ser. Longman and J. Wiley Harlow - N. York 1994Suche in Google Scholar

[43] Kober H. On fractional intgerals and derivatives Quart. J. Math., Ser (2) 11 1940 193 21110.1093/qmath/os-11.1.193Suche in Google Scholar

[44] Kokilashvili V.M. Weighted inequalities for maximal functions and fractional integrals in Lorentz spaces Math. Nachr. 133 1987 33 4210.1002/mana.19871330103Suche in Google Scholar

[45] Kokilashvili V. Krbec M. Weighted Inequalities in Lorentz and Orlicz Spaces. World Scientific River Edge - NJ 199110.1142/1367Suche in Google Scholar

[46] Kokilashvili V. Meskhi A. Boundedness of maximal and singular operators in Morrey spaces with variable exponent Armen. J. Math. 1 No 1 2008 18 28Suche in Google Scholar

[47] Kokilashvili V. Meskhi A. Maximal and Calderón-Zygmund operators in grand variable exponent Lebesgue spaces Georgian Math. J. 21 No 4 2014 447 46110.1515/gmj-2014-0047Suche in Google Scholar

[48] Kokilashvili V. Meskhi A. Rafeiro H. Riesz type potential operators in generalized grand Morrey spaces Georgian Math. J. 20 No 1 2013 43 6410.1515/gmj-2013-0009Suche in Google Scholar

[49] Kokilashvili V. Meskhi A. Rafeiro H. Samko S. Integral Operators in Non-standard Function Spaces. Volume 1: Variable Exponent Lebesgue and Amalgam Spaces. Ser. Operator Theory: Advances and Applications Vol. 248 Birkhäuser Basel 201610.1007/978-3-319-21015-5_1Suche in Google Scholar

[50] Kokilashvili V. Meskhi A. Rafeiro H. Samko S. Integral Operators in Non-standard Function Spaces. Volume 2: Variable Exponent Hölder, Morrey-Campanato and Grand Spaces. Ser. Operator Theory: Advances and Applications Vol. 249 Birkhäuser Basel 201610.1007/978-3-319-21018-6Suche in Google Scholar

[51] Kufner A. John O. Fučìk S. Function Spaces Noordhoff International Publishing Leiden 1977Suche in Google Scholar

[52] Liu L. Weighted Lorentz norm inequalities for Riemann-Liouville fractional integral Northeast. Math. J. 14 No 4 1998 409 414Suche in Google Scholar

[53] Macìas R.A. Segovia C. Weighted norm inequalities for parabolic fractional integrals Studia Math. 61 No 3 1977 279 29110.4064/sm-61-3-279-291Suche in Google Scholar

[54] R.A. Macìas Segovia Weighted norm inequalities for parabolic fractional integrals Studia Math. 61 No 3 (1977) 279 29110.4064/sm-61-3-279-291Suche in Google Scholar

[55] Meskhi A. Criteria for the boundedness of potential operators in Grand Lebesgue spaces ArXiv e-prints: 1007.1185 July 2010Suche in Google Scholar

[56] Meskhi A. Integral operators in Grand Morrey spaces ArXiv e-prints: 1007.1186 July 2010Suche in Google Scholar

[57] Meskhi A. Maximal functions, potentials and singular integrals in grand Morrey spaces Complex Var. Elliptic Equ. 56 No 10–11 2011 1003 101910.1080/17476933.2010.534793Suche in Google Scholar

[58] Mikhlin S.G. Multi-dimensional Singular Integrals and Integral Equations Pergamon Press Oxford-New York-Paris 196510.1016/B978-0-08-010852-0.50011-6Suche in Google Scholar

[59] Mizuhara T. Boundedness of some classical operators on generalized Morrey spaces In: Igari S. (Ed.) Harmonic AnalysisICM 90 Satellite Proc. Springer 1991 183 18910.1007/978-4-431-68168-7_16Suche in Google Scholar

[60] Muckenhoupt B. Wheeden R.L. Weighted norm inequalities for fractional integrals Trans. Amer. Math. Soc. 192 1974 261 27410.1090/S0002-9947-1974-0340523-6Suche in Google Scholar

[61] Murdaev H.M. The estimate of the modulus of continuity of the integrals and derivatives of fractional order (Russian). Groznyi, Dep. in VINITI 14.06.1985 No 4209 16Suche in Google Scholar

[62] Nakai E. Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces Math. Nachr. 166 1994 95 10310.1002/mana.19941660108Suche in Google Scholar

[63] O’Neil R.O. Fractional integration in Orlicz spaces Trans. Amer. Math. Soc. 115 No 3 1965 300 32810.1090/S0002-9947-1965-0194881-0Suche in Google Scholar

[64] L.-E. Persson, Samko N. Weighted Hardy and potential operators in the generalized Morrey spaces J. Math. Anal. Appl. 377 No 2 2011 792 80610.1016/j.jmaa.2010.11.029Suche in Google Scholar

[65] du Plessis N. Some theorems about the Riesz fractional integral Trans. Amer. Math. Soc. 80 No 1 1955 124 13410.1090/S0002-9947-1955-0086938-3Suche in Google Scholar

[66] Rafeiro H. A note on boundedness of operators in grand grand Morrey spaces In: Operator Theory: Advances and Applications Vol. 229 Birkhäuser Basel 2008 343 356Suche in Google Scholar

[67] Rafeiro H. Samko N. Samko S. Morrey-Campanato Spaces: an Overview In: Operator Theory: Advances and Applications Vol. 228 Birkhäuser Basel 2013 293 32410.1007/978-3-0348-0537-7_15Suche in Google Scholar

[68] Rafeiro H. Samko S. Characterization of the range of one-dimensional fractional integration in the space with variable exponent In: Operator Theory: Advances and Applications Vol. 181 Birkhäuser Basel 2008 393 41610.1007/978-3-7643-8684-9_20Suche in Google Scholar

[69] Reimann H.M. Rychener T. Funktionen beschränkter mittlerer Oszillation. Lecture Notes in Mathematics Vol. 487 Springer-Verlag Berlin 197510.1007/BFb0081825Suche in Google Scholar

[70] Ross B. Samko S.G. Fractional integration operator of variable order in the Hölder spaces Hƛ(x) Internat. J. Math. Math. Sci. 18 No 4 1995 777 78810.1155/S0161171295001001Suche in Google Scholar

[71] Rubin B.S. The fractional integrals and Riesz potentials with radial density in the spaces with power weight (Russian) Izv. Akad. Nauk Armyan. SSR, Ser. Mat. 21 No 5 1986 488 503Suche in Google Scholar

[72] Sadosky C. A note on parabolic fractional and singular integrals Studia Math. 26 1965 1966) 327 33510.4064/sm-26-3-327-335Suche in Google Scholar

[73] Samko N. Weighted Hardy and singular operators in Morrey spaces J. Math. Anal. Appl. 350 No 1 2009 56 7210.1016/j.jmaa.2008.09.021Suche in Google Scholar

[74] Samko N. Weighted Hardy and potential operators in Morrey spaces J. Funct. Spaces Appl. 2012 Art. ID 678171 2110.1155/2012/678171Suche in Google Scholar

[75] Samko N. Maximal, potential and singular operators in vanishing generalized Morrey spaces J. Glob. Optim. 57 No 4 2013 1385 139910.1007/s10898-012-9997-xSuche in Google Scholar

[76] Samko N. Samko S.G. Vakulov B.G. Weighted Sobolev theorem in Lebesgue spaces with variable exponent J. Math. Anal. Appl. 335 No 1 2007 560 58310.1016/j.jmaa.2007.01.091Suche in Google Scholar

[77] Samko N. Vakulov B. Spherical potentials of complex order in weighted generalized Hölder spaces with radial oscillating weights Oper. Matrices 1 No 2 2007 283 30010.7153/oam-01-19Suche in Google Scholar

[78] Samko N. Vakulov B. On generalized spherical fractional integration operators in weighted generalized Hölder spaces on the unit sphere In: Operator Theory: Advances and Applications Vol. 181 Birkhäuser Basel 2008 429 43710.1007/978-3-7643-8684-9_22Suche in Google Scholar

[79] Samko N. Vakulov B. Spherical fractional and hypersingular integrals in generalized Hölder spaces with variable characteristic Math. Nachrichten 284 2011 355 36910.1002/mana.200810113Suche in Google Scholar

[80] Samko S.G. Fractional integration and differentiation of variable order Anal. Math. 21 No 3 1995 213 23610.1007/BF01911126Suche in Google Scholar

[81] Samko S.G. Hypersingular Integrals and their Applications. Taylor & Francis New-York 200210.1201/9781482264968Suche in Google Scholar

[82] Samko S. Best constant in the weighted Hardy inequality: the spatial and spherical version Fract. Calc. Appl. Anal. 8 No 1 2005 39 52Suche in Google Scholar

[83] Samko S.G. Fractional integration and differentiation of variable order: an overview Nonlinear Dynam. 71 No 4 2013 653 66210.1007/s11071-012-0485-0Suche in Google Scholar

[84] Samko S.G. Potential operators in generalized Hölder spaces on sets in quasi-metric measure spaces without the cancellation property Nonlinear Anal. 78 2013 130 14010.1016/j.na.2012.09.020Suche in Google Scholar

[85] Samko S. A note on Riesz fractional integrals in the limiting case α(x)p(x)≡n Fract. Calc. Appl. Anal. 16 No 2 2013 370 377 10.2478/s13540-013-0023-xSuche in Google Scholar

[86] Samko S. Remark to the paper of S. Samko, “A note on Riesz fractional integrals in the limiting case α(x)p(x)≡n” from FCAA Vol. 16 No 2, 2013 Fract. Calc. Appl. Anal. 17 No 1 2014 277 278 10.2478/s13540-014-0167-3Suche in Google Scholar

[87] Samko S.G. Kilbas A.A. Marichev O.I. Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach Science Publishers New-York 1993Suche in Google Scholar

[88] Samko S. Shargorodsky E. Vakulov B. Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, II J. Math, Anal. Appl. 325 No 1 2007 745 75110.1016/j.jmaa.2006.01.069Suche in Google Scholar

[89] Samko S.G. Umarkhadzhiev S.M. On Iwaniec-Sbordone spaces on sets which may have infinite measure Azerb. J. Math. 1 No 1 2011 67 84Suche in Google Scholar

[90] Samko S.G. Umarkhadzhiev S.M. On Iwaniec-Sbordone spaces on sets which may have infinite measure: addendum Azerb. J. Math. 1 No 2 2011 143 144Suche in Google Scholar

[91] Samko S. Vakulov B. Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators J. Math. Anal. Appl. 310 No 1 2005 229 24610.1016/j.jmaa.2005.02.002Suche in Google Scholar

[92] Sawyer E. A two weight weak type inequality for fractional integrals Trans. Amer. Math. Soc. 281 No 1 1984 339 34510.1090/S0002-9947-1984-0719674-6Suche in Google Scholar

[93] Sawyer E. Wheeden R.L. Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces Amer. J. Math. 114 No 4 1992 813 87410.2307/2374799Suche in Google Scholar

[94] Sobolev S.L. On a theorem of functional analysis (In Russian) Math. Sb. 4 No 3 1938 471 497Suche in Google Scholar

[95] Stein E.M. Singular integrals, harmonic functions, differentiability properties of functions of several variables In: Singular integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc. Providence - RI 1967 316 33510.1090/pspum/010/0482394Suche in Google Scholar

[96] Stein E.M. Weiss G. Fractional integrals on n-dimensional Euclidean space J. Math. and Mech. 7 No 4 1958 503 51410.1512/iumj.1958.7.57030Suche in Google Scholar

[97] Stein E.M. Zygmund A. Boundedness of translation invariant operators on Hölder spaces and Lp-spaces Ann. Math. (2) 85 1967 337 34910.2307/1970445Suche in Google Scholar

[98] Umarkhadzhiev S.M. Generalization of the notion of grand Lebesgue space Russ. Math. 58 No 4 2014 35 4310.3103/S1066369X14040057Suche in Google Scholar

[99] Vakulov B.G. Spherical operators of potential type in generalized weighted Hölder spaces on a sphere Izv. Vyssh. Uchebn. Zaved., Sev.-Kavk. Reg., Estestv. Nauki 1999 No 4 1999 5 10Suche in Google Scholar

[100] Vakulov B.G. Spherical potentials of complex order in the variable order Holder spaces Integral Transforms Spec. Funct. 16 No 5–6 2005 489 49710.1080/10652460412331310304Suche in Google Scholar

[101] Vakulov B.G. Spherical convolution operators in Hölder spaces of variable order (in Russian) Matem. Zametki 80 No 5 2006 683–695; Transl In: Math. Notes 80 No 5 2006 645 657Suche in Google Scholar

[102] Vakulov B.G. Karapetyants N.K. Potential-type operators on a sphere with singularities at the poles Dokl. Math. 68 No 2 2003 173 176Suche in Google Scholar

[103] Vakulov B.G. Karapetiants N.K. Shankishvili L.D. Spherical hypersingular operators of imaginary order and their multipliers Fract. Calc. Appl. Anal. 4 No 1 2001 101 112Suche in Google Scholar

[104] Vakulov B.G. Karapetiants N.K. Shankishvili L.D. Spherical potentials of complex order in generalized Hölder spaces (In Russian) Izv. Nats. Akad. Nauk Armenii Mat. 36 No 2 2002 54–78; English transl In: J. Contemp. Math. Anal. 36 No 2 2002 5 29Suche in Google Scholar

[105] Vakulov B.G. Karapetiants N.K. Shankishvili L.D. Spherical convolution operators with a power-logarithmic kernel in generalized Hölder spaces (In Russian) Izv. Vyssh. Uchebn. Zaved. Mat. No 2 2003 3–14; English transl In: Russian Math. (Iz. VUZ) 47 No 2 2003 1 12Suche in Google Scholar

[106] Yuan W. Sickel W. Yang D. Morrey and Campanato Meet Besov, Lizorkin and Triebel. Springer Berlin 201010.1007/978-3-642-14606-0Suche in Google Scholar

Received: 2015-11-15
Published Online: 2016-6-28
Published in Print: 2016-6-1

© 2016 Diogenes Co., Sofia

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA-volume 19-3-2016)
  4. Survey Paper
  5. Fractional integrals and derivatives: mapping properties
  6. Research Paper
  7. Riesz fractional integrals in grand lebesgue spaces on ℝn
  8. Survey Paper
  9. United lattice fractional integro-differentiation
  10. Research Paper
  11. Integral equations of fractional order in Lebesgue spaces
  12. Research Paper
  13. General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems
  14. Research Paper
  15. Multilinear integral operators in weighted grand Lebesgue spaces
  16. Research Paper
  17. Fractional integration operator on some radial rays and intertwining for the Dunkl operator
  18. Research Paper
  19. Pseudo almost automorphy of semilinear fractional differential equations in Banach Spaces
  20. Research Paper
  21. Existence and uniqueness of global solutions of caputo-type fractional differential equations
  22. Research Paper
  23. Perfect nonlinear observers of fractional descriptor continuous-time nonlinear systems
https://doi.org/10.1515/fca-2016-0032
Schlagwörter für diesen Artikel
mapping properties; fractional integral; Riesz potential; hypersingular integrals; fractional derivatives; Lebesgue spaces; variable exponent Lebesgue spaces; Hölder spaces; variable exponent Hölder spaces; Grand spaces, Morrey spaces

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA-volume 19-3-2016)
  4. Survey Paper
  5. Fractional integrals and derivatives: mapping properties
  6. Research Paper
  7. Riesz fractional integrals in grand lebesgue spaces on ℝn
  8. Survey Paper
  9. United lattice fractional integro-differentiation
  10. Research Paper
  11. Integral equations of fractional order in Lebesgue spaces
  12. Research Paper
  13. General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems
  14. Research Paper
  15. Multilinear integral operators in weighted grand Lebesgue spaces
  16. Research Paper
  17. Fractional integration operator on some radial rays and intertwining for the Dunkl operator
  18. Research Paper
  19. Pseudo almost automorphy of semilinear fractional differential equations in Banach Spaces
  20. Research Paper
  21. Existence and uniqueness of global solutions of caputo-type fractional differential equations
  22. Research Paper
  23. Perfect nonlinear observers of fractional descriptor continuous-time nonlinear systems
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