Results 1  10
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68
A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids
, 2006
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p(x)harmonic functions with unbounded exponent in a subdomain, Ann
 Inst. H. Poincaré Anal. Non Linéaire
"... Abstract. We study the Dirichlet problem − div(∇u  p(x)−2 ∇u) = 0 in Ω, with u = f on ∂Ω and p(x) = ∞ in D, a subdomain of the reference domain Ω. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as n → ∞ of the solutions un to the corresponding ..."
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Cited by 15 (7 self)
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Abstract. We study the Dirichlet problem − div(∇u  p(x)−2 ∇u) = 0 in Ω, with u = f on ∂Ω and p(x) = ∞ in D, a subdomain of the reference domain Ω. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as n → ∞ of the solutions un to the corresponding problem when pn(x) = p(x) ∧ n, in particular, with pn = n in D. Under suitable assumptions on the data, we find that such a limit exists and that it can be characterized as the unique solution of a variational minimization problem which is, in addition, ∞harmonic within D. Moreover, we examine this limit in the viscosity sense and find the boundary value problem it satisfies in the whole of Ω. 1.
Maximal and potential operators in variable exponent Morrey spaces
 Georgian Math. J
"... Abstract. We prove the boundedness of the Hardy–Littlewood maximal operator on variable Morrey spaces Lp(·),λ(·)(Ω) over a bounded open set Ω ⊂ Rn and a Sobolev type Lp(·),λ(·) → Lq(·),λ(·)theorem for potential operators Iα(·), also of variable order. In the case of constant α, the limiting case i ..."
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Cited by 11 (3 self)
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Abstract. We prove the boundedness of the Hardy–Littlewood maximal operator on variable Morrey spaces Lp(·),λ(·)(Ω) over a bounded open set Ω ⊂ Rn and a Sobolev type Lp(·),λ(·) → Lq(·),λ(·)theorem for potential operators Iα(·), also of variable order. In the case of constant α, the limiting case is also studied when the potential operator Iα acts into BMO space.
ON SOME QUESTIONS RELATED TO THE MAXIMAL OPERATOR ON VARIABLE L p SPACES
, 2010
"... Abstract. Let P(Rn) be the class of all exponents p for which the HardyLittlewood maximal operator M is bounded on Lp(·) (Rn). A recent result by T. Kopaliani provides a characterization of P in terms of the Muckenhoupttype condition A under some restrictions on the behavior of p at infinity. We gi ..."
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Cited by 9 (0 self)
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Abstract. Let P(Rn) be the class of all exponents p for which the HardyLittlewood maximal operator M is bounded on Lp(·) (Rn). A recent result by T. Kopaliani provides a characterization of P in terms of the Muckenhoupttype condition A under some restrictions on the behavior of p at infinity. We give a different proof of a slightly extended version of this result. Then we characterize a weak type ( p(·),p(·) ) property of M in terms of A for radially decreasing p. Finally, we construct an example showing that p ∈P(Rn)does not imply p(·) − α ∈P(Rn) for all α<p− − 1. Similarly, p ∈P(Rn)doesnot imply αp(·) ∈P(Rn) for all α>1/p−. 1.
Multiplicity of solutions for a class of quasilinear problems in exterior domains with Neumann conditions
, 2003
"... We study the existence andmultiplicity of solutions for a class of quasilinear elliptic problem in exterior domain with Neumann boundary conditions. ..."
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Cited by 7 (3 self)
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We study the existence andmultiplicity of solutions for a class of quasilinear elliptic problem in exterior domain with Neumann boundary conditions.
Characterization of the range of onedimensional fractional integration in the space with variable exponent
, 2008
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A new proof of the boundedness of maximal operators on variable Lebesgue spaces, submitted
"... Abstract. We give a new proof using the classic CalderónZygmund decomposition that the HardyLittlewood maximal operator is bounded on the variable Lebesgue space Lp(·) whenever the exponent function p(·) satisfies logHölder continuity conditions. We include the case where p(·) assumes the val ..."
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Cited by 6 (1 self)
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Abstract. We give a new proof using the classic CalderónZygmund decomposition that the HardyLittlewood maximal operator is bounded on the variable Lebesgue space Lp(·) whenever the exponent function p(·) satisfies logHölder continuity conditions. We include the case where p(·) assumes the value infinity. The same proof also shows that the fractional maximal operator Mα, 0 < α < n, maps Lp(·) into Lq(·), where 1/p(·) − 1/q(·) = α/n. 1.
Operators of Harmonic Analysis in Weighted Spaces with Nonstandard Growth
, 2008
"... Last years there was increasing an interest to the so called function spaces with nonstandard growth, known also as variable exponent Lebesgue spaces. For weighted such spaces on homogeneous spaces, we develop a certain variant of Rubio de Francia’s extrapolation theorem. This extrapolation theorem ..."
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Cited by 6 (1 self)
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Last years there was increasing an interest to the so called function spaces with nonstandard growth, known also as variable exponent Lebesgue spaces. For weighted such spaces on homogeneous spaces, we develop a certain variant of Rubio de Francia’s extrapolation theorem. This extrapolation theorem is applied to obtain the boundedness in such spaces of various operators of harmonic analysis, such as maximal and singular operators, potential operators, Fourier multipliers, dominants of partial sums of trigonometric Fourier series and others, in weighted Lebesgue spaces with variable exponent. There are also given their vectorvalued analogues.
Fractional and hypersingular operators in variable exponent spaces on metric measure spaces
 MEDITERRANEAN JOURNAL OF MATHEMATICS
, 2008
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Weighted Boundedness of the Maximal, Singular and Potential Operators in Variable Exponent Spaces
, 2008
"... We present a brief survey of recent results on boundedness of some classical operators within the frameworks of weighted spaces Lp(·) () with variable exponent p(x), mainly in the Euclidean setting and dwell on a new result of the boundedness of the HardyLittlewood maximal operator in the space Lp( ..."
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Cited by 5 (4 self)
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We present a brief survey of recent results on boundedness of some classical operators within the frameworks of weighted spaces Lp(·) () with variable exponent p(x), mainly in the Euclidean setting and dwell on a new result of the boundedness of the HardyLittlewood maximal operator in the space Lp(·) (X,) over a metric measure space X satisfying the doubling condition. In the case where X is bounded, the weight function satisfies a certain version of a general Muckenhoupttype condition For a bounded or unbounded X we also consider a class of weights of the form (x) = [1 + d(x0,x)] β ∞ ∏ m k=1 wk(d(x,xk)), xk ∈ X, where the functions wk(r) have finite upper and lower indices m(wk) and M(wk). Some of the results are new even in the case of constant p.