Critical Kirchhoff $$ p(\cdot ) \&  q(\cdot )$$-fractional variable-order systems with variable exponent growth

AbstractWe show that the fractional operator {I^{\alpha(\,\cdot\,)}}, of variable order on a bounded open set in Ω, in a quasimetric measure space {(X,d,\mu)} in the case {\alpha(x)p(x)\equiv n} (where n comes from the growth condition on the measure μ), is bounded from the variable exponent Lebesgue space {L^{p(\,\cdot\,)}(\Omega)} into {\mathrm{BMO}(\Omega)} under certain assumptions on {p(x)} and {\alpha(x)}.

Download Full-text

Maximal and Potential Operators in Variable Exponent Morrey Spaces

Georgian Mathematical Journal ◽

10.1515/gmj.2008.195 ◽

2008 ◽

Vol 15 (2) ◽

pp. 195-208 ◽

Cited By ~ 2

Author(s):

Alexandre Almeida ◽

Javanshir Hasanov ◽

Stefan Samko

Keyword(s):

Variable Exponent ◽

Morrey Spaces ◽

Bmo Space ◽

Sobolev Type ◽

Potential Operator ◽

Variable Order ◽

Potential Operators ◽

Open Set ◽

Limiting Case ◽

Littlewood Maximal Operator

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 𝐼 α(·), also of variable order. In the case of constant α, the limiting case is also studied when the potential operator 𝐼 α acts into BMO space.

Download Full-text

On the Riesz potential operator of variable order from variable exponent Morrey space to variable exponent Campanato space

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.6187 ◽

2020 ◽

Vol 43 (16) ◽

pp. 9337-9344

Author(s):

Humberto Rafeiro ◽

Stefan Samko

Keyword(s):

Variable Exponent ◽

Morrey Space ◽

Riesz Potential ◽

Potential Operator ◽

Variable Order ◽

Campanato Space ◽

Riesz Potential Operator

Download Full-text

Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-15156 ◽

2010 ◽

Vol 107 (2) ◽

pp. 285 ◽

Cited By ~ 49

Author(s):

Vagif S. Guliyev ◽

Javanshir J. Hasanov ◽

Stefan G. Samko

Keyword(s):

Maximal Operator ◽

Variable Exponent ◽

Morrey Spaces ◽

Variable Order ◽

General Function ◽

Singular Operators ◽

Potential Operators ◽

Type Integral ◽

Bounded Sets ◽

Littlewood Maximal Operator

We consider generalized Morrey spaces ${\mathcal M}^{p(\cdot),\omega}(\Omega)$ with variable exponent $p(x)$ and a general function $\omega (x,r)$ defining the Morrey-type norm. In case of bounded sets $\Omega \subset {\mathsf R}^n$ we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev-Adams type ${\mathcal M}^{p(\cdot),\omega} (\Omega)\rightarrow {\mathcal M}^{q(\cdot),\omega} (\Omega)$-theorem for the potential operators $I^{\alpha(\cdot)}$, also of variable order. The conditions for the boundedness are given it terms of Zygmund-type integral inequalities on $\omega(x,r)$, which do not assume any assumption on monotonicity of $\omega(x,r)$ in $r$.

Download Full-text

Weighted Hardy-Type Inequalities in Variable Exponent Morrey-Type Spaces

Journal of Function Spaces and Applications ◽

10.1155/2013/716029 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11

Author(s):

Dag Lukkassen ◽

Lars-Erik Persson ◽

Stefan Samko ◽

Peter Wall

Keyword(s):

Variable Exponent ◽

Morrey Space ◽

Variable Order ◽

Generalized Morrey Space ◽

Hardy Type Inequalities ◽

Hardy Type ◽

Radial Weight ◽

Type Integral ◽

Type Spaces ◽

Hardy Type Operators

We study thep·→q·boundedness of weighted multidimensional Hardy-type operatorsHwα·andℋwα·of variable orderαx, with radial weightwx, from a variable exponent locally generalized Morrey spaceℒp·,φ·ℝn,wto anotherℒq·,ψ·ℝn,w. The exponents are assumed to satisfy the decay condition at the origin and infinity. We construct certain functions, defined byp,α, andφ, the belongness of which to the resulting spaceℒq·,ψ·ℝn,wis sufficient for such a boundedness. Under additional assumptions onφ/w, this condition is also necessary. We also give the boundedness conditions in terms of Zygmund-type integral inequalities for the functionsφandφ/w.

Download Full-text