# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Weighted Sobolev theorem in Lebesgue spaces with variable exponent. (English) Zbl 1142.46018
This paper deals with Sobolev inequalities for Riesz potentials in variable exponent spaces with weights. The weights are radial and it is assumed that their growth is constrained by two polynomials of appropriate exponents. The variable exponent is $\log$-Hölder continuous, and the index $\alpha$ of the Riesz potential $I_\alpha$ is allowed to vary, as well.

##### MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text:
##### References:
 [1] Bari, N. K.; Stechkin, S. B.: Best approximations and differential properties of two conjugate functions. Proc. Moscow math. Soc. 5, 483-522 (1956) [2] C. Capone, D. Cruz-Uribe, A. Fiorenza, The fractional maximal operator on variable Lp spaces, preprint of Istituto per le Applicazioni del Calcolo ”Mauro Picone” -- Sezione di Napoli, RT 281/04, 2004, pp. 1 -- 22 [3] Cruz-Uribe, D.; Fiorenza, A.; Martell, J. M.; Perez, C.: The boundedness of classical operators on variable lp spaces. Ann. acad. Sci. fenn., 239-264 (2006) · Zbl 1100.42012 [4] Cruz-Uribe, D.; Fiorenza, A.; Neugebauer, C. J.: The maximal function on variable lp-spaces. Ann. acad. Sci. fenn. Math. diss. 28, 223-238 (2003) · Zbl 1037.42023 [5] Diening, L.: Maximal function on generalized Lebesgue spaces $Lp(\cdot )$. Math. inequal. Appl. 7, No. 2, 245-253 (2004) · Zbl 1071.42014 [6] L. Diening, P. Hästö, A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, in: Function Spaces, Differential Operators and Nonlinear Analysis, Proceedings of the Conference Held in Milovy, Bohemian -- Moravian Uplands, May 28 -- June 2, 2004, Math. Inst. Acad. Sci. Czech Republic, Praha [7] Karapetiants, N. K.; Samko, N. G.: Weighted theorems on fractional integrals in the generalized hölder spaces H0$\omega (\rho )$ via the indices m$\omega$ and M$\omega$. Fract. calc. Appl. anal. 7, No. 4 (2004) [8] V. Kokilashvili, On a progress in the theory of integral operators in weighted Banach function spaces, in: Function Spaces, Differential Operators and Nonlinear Analysis, Proceedings of the Conference Held in Milovy, Bohemian -- Moravian Uplands, May 28 -- June 2, 2004, Math. Inst. Acad. Sci. Czech Republic, Praha [9] Kokilashvili, V.; Samko, N.; Samko, S.: The maximal operator in variable spaces $Lp(\cdot )(\Omega ,\rho )$. Georgian math. J. 13, No. 1, 109-125 (2006) · Zbl 1097.42014 [10] Kokilashvili, V.; Samko, S.: On Sobolev theorem for the Riesz type potentials in Lebesgue spaces with variable exponent. Z. anal. Anwendungen 22, No. 4, 899-910 (2003) · Zbl 1040.42013 [11] Kokilashvili, V.; Samko, S.: Maximal and fractional operators in weighted $Lp(x)$ spaces. Rev. mat. Iberoamericana 20, No. 2, 495-517 (2004) [12] Kováčik, O.; Rákosník, J.: On spaces $Lp(x)$ and wk,$p(x)$. Czechoslovak math. J. 41, No. 116, 592-618 (1991) · Zbl 0784.46029 [13] Maligranda, Lech: Indices and interpolation. Dissertationes math. (Rozprawy mat.) 234, 49 (1985) · Zbl 0566.46038 [14] Mikhlin, S. G.: Multi-dimensional singular integrals and integral equations. (1962) [15] Muckenhoupt, B.; Wheeden, R. L.: Weighted norm inequalities for fractional integrals. Trans. amer. Math. soc. 192, 261-274 (1974) · Zbl 0289.26010 [16] Nekvinda, A.: Hardy -- Littlewood maximal operator on $Lp(x)$(Rn). Math. inequal. Appl. 7, No. 2, 255-265 (2004) · Zbl 1059.42016 [17] Ružička, M.: Electrorheological fluids: modeling and mathematical theory. Lecture notes in math. 1748 (2000) [18] Samko, N. G.: Singular integral operators in weighted spaces with generalized hölder condition. Proc. A. Razmadze math. Inst. 120, 107-134 (1999) · Zbl 0946.45006 [19] Samko, N. G.: On non-equilibrated almost monotonic functions of the Zygmund -- bary -- stechkin class. Real anal. Exchange 30, No. 2 (2005) · Zbl 1106.26012 [20] Samko, S. G.: On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators. Integral transforms spec. Funct. 16, No. 5 -- 6, 461-482 (2005) · Zbl 1069.47056 [21] Samko, S. G.: Convolution and potential type operators in $Lp(x)$. Integral transforms spec. Funct. 7, No. 3 -- 4, 261-284 (1998) · Zbl 1023.31009 [22] Samko, S. G.: Convolution type operators in $Lp(x)$. Integral transforms spec. Funct. 7, No. 1 -- 2, 123-144 (1998) · Zbl 0934.46032 [23] Samko, S. G.: Hardy -- Littlewood -- Stein -- Weiss inequality in the Lebesgue spaces with variable exponent. Fract. calc. Appl. anal. 6, No. 4, 421-440 (2003) · Zbl 1093.46015 [24] Samko, S. G.; Shargorodsky, E.; Vakulov, B.: Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, II. J. math. Anal. appl. 325, No. 1, 745-751 (2007) · Zbl 1107.47016 [25] Samko, S. G.; Vakulov, B. G.: Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators. J. math. Anal. appl. 310, 229-246 (2005) · Zbl 1079.47053 [26] Stein, E. M.; Weiss, G.: Fractional integrals on n-dimensional Euclidean space. J. math. Mech. 7, No. 4, 503-514 (1958) · Zbl 0082.27201