Trace inequalities for fractional integrals in grand Lebesgue spaces

Vakhtang Kokilashvili; Alexander Meskhi

Displaying similar documents to “Trace inequalities for fractional integrals in grand Lebesgue spaces”

Multi-Morrey spaces for non-doubling measures

Suixin He (2019)

Czechoslovak Mathematical Journal

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The spaces of multi-Morrey type for positive Radon measures satisfying a growth condition on $ℝ^{d}$ are introduced. After defining the spaces, we investigate the multilinear maximal function, the multilinear fractional integral operator and the multilinear Calderón-Zygmund operators, respectively, from multi-Morrey spaces to Morrey spaces.

Existence results for systems of conformable fractional differential equations

Bouharket Bendouma, Alberto Cabada, Ahmed Hammoudi (2019)

Archivum Mathematicum

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In this article, we study the existence of solutions to systems of conformable fractional differential equations with periodic boundary value or initial value conditions. where the right member of the system is $L_{α}^{1}$ -carathéodory function. We employ the method of solution-tube and Schauder’s fixed-point theorem.

Generalized fractional integral operators on weak Choquet spaces over quasi-metric measure spaces

Toshihide Futamura, Tetsu Shimomura (2024)

Czechoslovak Mathematical Journal

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We prove the boundedness of the generalized fractional maximal operator $M_{α}$ and the generalized fractional integral operator $I_{α}$ on weak Choquet spaces with respect to Hausdorff content over quasi-metric measure spaces.

Generalized $H_{m, m}^{m, 0}$ function fractional integration operators in some classes of analytic functions

V. Kiryakova (1988)

Matematički Vesnik

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Pointwise inequalities and approximation in fractional Sobolev spaces

David Swanson (2002)

Studia Mathematica

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We prove that a function belonging to a fractional Sobolev space $L^{α, p} (ℝ ⁿ)$ may be approximated in capacity and norm by smooth functions belonging to $C^{m, λ} (ℝ ⁿ)$ , 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].

Multiplicity results for a class of fractional boundary value problems

Nemat Nyamoradi (2013)

Annales Polonici Mathematici

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We prove the existence of at least three solutions to the following fractional boundary value problem: ⎧ $- d / d t (1 / 2_{0} D_{t}^{- σ} (u^{'} (t)) + 1 / 2_{t} D_{T}^{- σ} (u^{'} (t))) - λ β (t) f (u (t)) - μ γ (t) g (u (t)) = 0$ , a.e. t ∈ [0, T], ⎨ ⎩ u (0) = u (T) = 0, where $_{0} D_{t}^{- σ}$ and $_{t} D_{T}^{- σ}$ are the left and right Riemann-Liouville fractional integrals of order 0 ≤ σ < 1 respectively. The approach is based on a recent three critical points theorem of Ricceri [B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal. 74 (2011), 7446-7454].

Boundary value problems for Caputo-Hadamard fractional differential inclusions in Banach spaces

Amouria Hammou, Samira Hamani, Johnny Henderson (2022)

Archivum Mathematicum

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In this article, we study the existence of solutions in a Banach space of boundary value problems for Caputo-Hadamard fractional differential inclusions of order $r \in (0, 1]$ .

Fractional Langevin equation with α-stable noise. A link to fractional ARIMA time series

M. Magdziarz, A. Weron (2007)

Studia Mathematica

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We introduce a fractional Langevin equation with α-stable noise and show that its solution $Y_{κ} (t), t \geq 0$ is the stationary α-stable Ornstein-Uhlenbeck-type process recently studied by Taqqu and Wolpert. We examine the asymptotic dependence structure of $Y_{κ} (t)$ via the measure of its codependence r(θ₁,θ₂,t). We prove that $Y_{κ} (t)$ is not a long-memory process in the sense of r(θ₁,θ₂,t). However, we find two natural continuous-time analogues of fractional ARIMA time series with long memory in the framework of...

Pointwise convergence to the initial data for nonlocal dyadic diffusions

Marcelo Actis, Hugo Aimar (2016)

Czechoslovak Mathematical Journal

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We solve the initial value problem for the diffusion induced by dyadic fractional derivative $s$ in $ℝ^{+}$ . First we obtain the spectral analysis of the dyadic fractional derivative operator in terms of the Haar system, which unveils a structure for the underlying “heat kernel”. We show that this kernel admits an integrable and decreasing majorant that involves the dyadic distance. This allows us to provide an estimate of the maximal operator of the diffusion by the Hardy-Littlewood dyadic...

Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type

Juan Luis Vázquez (2014)

Journal of the European Mathematical Society

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We establish the existence, uniqueness and main properties of the fundamental solutions for the fractional porous medium equation introduced in [51]. They are self-similar functions of the form $u (x, t) = t^{- α} f (| x | t^{- β})$ with suitable and $β$ . As a main application of this construction, we prove that the asymptotic behaviour of general solutions is represented by such special solutions. Very singular solutions are also constructed. Among other interesting qualitative properties of the equation we prove an Aleksandrov...

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

Ali Akbulut, Amil Hasanov (2016)

Open Mathematics

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In this paper, we study the boundedness of fractional multilinear integral operators with rough kernels [...] TΩ,αA1,A2,…,Ak, $T_{Ω, α}^{A_{1}, A_{2}, ..., A_{k}},$ which is a generalization of the higher-order commutator of the rough fractional integral on the generalized weighted Morrey spaces Mp,ϕ (w). We find the sufficient conditions on the pair (ϕ1, ϕ2) with w ∈ Ap,q which ensures the boundedness of the operators [...] TΩ,αA1,A2,…,Ak, $T_{Ω, α}^{A_{1}, A_{2}, ..., A_{k}},$ from [...] Mp,φ1wptoMp,φ2wq $M_{p, ϕ_{1}} (w^{p}) to M_{p, ϕ_{2}} (w^{q})$ for 1 < p < q < ∞. In all cases the conditions...

Functions of finite fractional variation and their applications to fractional impulsive equations

Dariusz Idczak (2017)

Czechoslovak Mathematical Journal

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We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak $σ$ -additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a $σ$ -additive term—we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e. equations containing the Dirac measures.

Suitable domains to define fractional integrals of Weyl via fractional powers of operators

Celso Martínez, Antonia Redondo, Miguel Sanz (2011)

Studia Mathematica

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We present a new method to study the classical fractional integrals of Weyl. This new approach basically consists in considering these operators in the largest space where they make sense. In particular, we construct a theory of fractional integrals of Weyl by studying these operators in an appropriate Fréchet space. This is a function space which contains the $L^{p} (ℝ)$ -spaces, and it appears in a natural way if we wish to identify these fractional operators with fractional powers of a suitable...

Initial value problems for fractional functional differential inclusions with Hadamard type derivative

Nassim Guerraiche, Samira Hamani, Johnny Henderson (2016)

Archivum Mathematicum

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We establish sufficient conditions for the existence of solutions of a class of fractional functional differential inclusions involving the Hadamard fractional derivative with order $α \in (0, 1]$ . Both cases of convex and nonconvex valued right hand side are considered.

Set-valued fractional order differential equations in the space of summable functions

Hussein A.H. Salem (2008)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type $(D^{α ₙ} - \sum_{i = 1}^{n - 1} a_{i} D^{α_{i}}) x (t) \in F (t, x (φ (t)))$ , a.e. on (0,1), $I^{1 - α ₙ} x (0) = c$ , αₙ ∈ (0,1), where F(t,·) is lower semicontinuous from ℝ into ℝ and F(·,·) is measurable. The corresponding single-valued problem will be considered first.

Bilinear fractional Hardy-type operators with rough kernels on central Morrey spaces with variable exponents

Hongbin Wang, Chenchen Niu (2024)

Czechoslovak Mathematical Journal

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We introduce a type of $n$ -dimensional bilinear fractional Hardy-type operators with rough kernels and prove the boundedness of these operators and their commutators on central Morrey spaces with variable exponents. Furthermore, the similar definitions and results of multilinear fractional Hardy-type operators with rough kernels are obtained.