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

V. Kiryakova

Displaying similar documents to “Generalized $H_{m, m}^{m, 0}$ function fractional integration operators in some classes of analytic functions”

Multiplicity results for a class of fractional boundary value problems

Nemat Nyamoradi (2013)

Annales Polonici Mathematici

Similarity:

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].

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

Hussein A.H. Salem (2008)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Similarity:

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.

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

Ali Akbulut, Amil Hasanov (2016)

Open Mathematics

Similarity:

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...

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

M. Magdziarz, A. Weron (2007)

Studia Mathematica

Similarity:

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...

Commutators with fractional integral operators

Irina Holmes, Robert Rahm, Scott Spencer (2016)

Studia Mathematica

Similarity:

We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $μ, λ \in A_{p, q}$ and α/n + 1/q = 1/p, the norm $| | [b, I_{α}] : L^{p} (μ^{p}) \to L^{q} (λ^{q}) | |$ is equivalent to the norm of b in the weighted BMO space BMO(ν), where $ν = μ λ^{- 1}$ . This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey,...

Density of some sequences modulo 1

Artūras Dubickas (2012)

Colloquium Mathematicae

Similarity:

Recently, Cilleruelo, Kumchev, Luca, Rué and Shparlinski proved that for each integer a ≥ 2 the sequence of fractional parts ${a ⁿ / n}_{n = 1}^{\infty}$ is everywhere dense in the interval [0,1]. We prove a similar result for all Pisot numbers and Salem numbers α and show that for each c > 0 and each sufficiently large N, every subinterval of [0,1] of length $c N^{- 0.475}$ contains at least one fractional part Q(αⁿ)/n, where Q is a nonconstant polynomial in ℤ[z] and n is an integer satisfying 1 ≤ n ≤ N.

On Sobolev spaces of fractional order and ε-families of operators on spaces of homogeneous type

A. Gatto, Stephen Vági (1999)

Studia Mathematica

Similarity:

We introduce Sobolev spaces $L_{α}^{p}$ for 1 < p < ∞ and small positive α on spaces of homogeneous type as the classes of functions f in $L^{p}$ with fractional derivative of order α, $D^{α} f$ , as introduced in [2], in $L^{p}$ . We show that for small α, $L_{α}^{p}$ coincides with the continuous version of the Triebel-Lizorkin space $F_{p}^{α, 2}$ as defined by Y. S. Han and E. T. Sawyer in [4]. To prove this result we give a more general definition of ε-families of operators on spaces of homogeneous type, in which the identity...

Fractional integral operators on $B^{p, λ}$ with Morrey-Campanato norms

Katsuo Matsuoka, Eiichi Nakai (2011)

Banach Center Publications

Similarity:

We introduce function spaces $B^{p, λ}$ with Morrey-Campanato norms, which unify $B^{p, λ}$ , $C M O^{p, λ}$ and Morrey-Campanato spaces, and prove the boundedness of the fractional integral operator $I_{α}$ on these spaces.

Stability of nonlinear $h$ -difference systems with $n$ fractional orders

Małgorzata Wyrwas, Ewa Pawluszewicz, Ewa Girejko (2015)

Kybernetika

Similarity:

In the paper we study the subject of stability of systems with $h$ -differences of Caputo-, Riemann-Liouville- and Grünwald-Letnikov-type with $n$ fractional orders. The equivalent descriptions of fractional $h$ -difference systems are presented. The sufficient conditions for asymptotic stability are given. Moreover, the Lyapunov direct method is used to analyze the stability of the considered systems with $n$ -orders.

Pointwise inequalities and approximation in fractional Sobolev spaces

David Swanson (2002)

Studia Mathematica

Similarity:

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].

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

Similarity:

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...

Essential normality for certain finite linear combinations of linear-fractional composition operators on the Hardy space $H^{2}$

Mahsa Fatehi, Bahram Khani Robati (2012)

Czechoslovak Mathematical Journal

Similarity:

In 1999 Nina Zorboska and in 2003 P. S. Bourdon, D. Levi, S. K. Narayan and J. H. Shapiro investigated the essentially normal composition operator $C_{ϕ}$ , when $ϕ$ is a linear-fractional self-map of $𝔻$ . In this paper first, we investigate the essential normality problem for the operator $T_{w} C_{ϕ}$ on the Hardy space $H^{2}$ , where $w$ is a bounded measurable function on $\partial 𝔻$ which is continuous at each point of $F (ϕ)$ , $ϕ \in 𝒮 (2)$ , and $T_{w}$ is the Toeplitz operator with symbol $w$ . Then we use these results and characterize the essentially...

Regularity of solutions of the fractional porous medium flow

Luis Caffarelli, Fernando Soria, Juan Luis Vázquez (2013)

Journal of the European Mathematical Society

Similarity:

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_{t} = \nabla \cdot (u \nabla {(- Δ)}^{- s} u), 0 < s < 1$ . The problem is posed in ${x \in ℝ^{n}, t \in ℝ}$ with nonnegative initial data $u (x, 0)$ that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. As main results we establish the boundedness and $C^{α}$ regularity of such weak solutions. Finally, we extend...

Weighted estimates for the iterated commutators of multilinear maximal and fractional type operators

Qingying Xue (2013)

Studia Mathematica

Similarity:

The following iterated commutators $T_{*, Π b}$ of the maximal operator for multilinear singular integral operators and $I_{α, Π b}$ of the multilinear fractional integral operator are introduced and studied: $T_{*, Π b} (f ⃗) (x) = s u p_{δ > 0} | [b ₁, [b ₂, \dots {[b_{m - 1}, [b ₘ, T_{δ}] ₘ]}_{m - 1} \dots] ₂] ₁ (f ⃗) (x) |$ , $I_{α, Π b} (f ⃗) (x) = [b ₁, [b ₂, \dots {[b_{m - 1}, [b ₘ, I_{α}] ₘ]}_{m - 1} \dots] ₂] ₁ (f ⃗) (x)$ , where $T_{δ}$ are the smooth truncations of the multilinear singular integral operators and $I_{α}$ is the multilinear fractional integral operator, $b_{i} \in B M O$ for i = 1,…,m and f⃗ = (f1,…,fm). Weighted strong and L(logL) type end-point estimates for the above iterated commutators associated with two classes of multiple...

Generalized fractional integrals on central Morrey spaces and generalized λ-CMO spaces

Katsuo Matsuoka (2014)

Banach Center Publications

Similarity:

We introduce the generalized fractional integrals $I ̃_{α, d}$ and prove the strong and weak boundedness of $I ̃_{α, d}$ on the central Morrey spaces $B^{p, λ} (ℝ ⁿ)$ . In order to show the boundedness, the generalized λ-central mean oscillation spaces $Λ_{p, λ}^{(d)} (ℝ ⁿ)$ and the generalized weak λ-central mean oscillation spaces $W Λ_{p, λ}^{(d)} (ℝ ⁿ)$ play an important role.

Fractional Laplacian with singular drift

Tomasz Jakubowski (2011)

Studia Mathematica

Similarity:

For α ∈ (1,2) we consider the equation $\partial_{t} u = Δ^{α / 2} u + b \cdot \nabla u$ , where b is a time-independent, divergence-free singular vector field of the Morrey class $M ₁^{1 - α}$ . We show that if the Morrey norm ${| | b | |}_{M ₁^{1 - α}}$ is sufficiently small, then the fundamental solution is globally in time comparable with the density of the isotropic stable process.

Regular fractional iteration of convex functions

Marek Kuczma (1980)

Annales Polonici Mathematici

Similarity:

The existence of a unique $C^{1}$ solution φ of equation (1) is proved under the condition that f: I → I is convex or concave and of class $C^{1}$ in I, 0 < f(x) < x in I*, and f’(x) > 0 in I. Here I = [0, a] or [0, a), 0 < a ≤ ∞, and I* = I 0.

Orlicz boundedness for certain classical operators

E. Harboure, O. Salinas, B. Viviani (2002)

Colloquium Mathematicae

Similarity:

Let ϕ and ψ be functions defined on [0,∞) taking the value zero at zero and with non-negative continuous derivative. Under very mild extra assumptions we find necessary and sufficient conditions for the fractional maximal operator $M_{Ω}^{α}$ , associated to an open bounded set Ω, to be bounded from the Orlicz space $L^{ψ} (Ω)$ into $L^{ϕ} (Ω)$ , 0 ≤ α < n. For functions ϕ of finite upper type these results can be extended to the Hilbert transform f̃ on the one-dimensional torus and to the fractional integral operator...

Fractional global domination in graphs

Subramanian Arumugam, Kalimuthu Karuppasamy, Ismail Sahul Hamid (2010)

Discussiones Mathematicae Graph Theory

Similarity:

Let G = (V,E) be a graph. A function g:V → [0,1] is called a global dominating function (GDF) of G, if for every v ∈ V, $g (N [v]) = \sum_{u \in N [v]} g (u) \geq 1$ and $g (\bar{N (v)}) = \sum_{u \notin N (v)} g (u) \geq 1$ . A GDF g of a graph G is called minimal (MGDF) if for all functions f:V → [0,1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V, f is not a GDF. The fractional global domination number $γ_{f g} (G)$ is defined as follows: $γ_{f g} (G)$ = min|g|:g is an MGDF of G where $| g | = \sum_{v \in V} g (v)$ . In this paper we initiate a study of this parameter.

New characterizations and applications of inhomogeneous Besov and Triebel-Lizorkin spaces on homogeneous type spaces and fractals

Yongsheng Han, Dachun Yang

Similarity:

Let d > 0 and θ ∈ (0,1]. We consider homogeneous type spaces, ${(X, ϱ, μ)}_{d, θ}$ , which are variants of the well known homogeneous type spaces in the sense of Coifman and Weiss. We introduce fractional integrals and derivatives, and prove that the Besov spaces $B_{p q}^{s} (X)$ and Triebel-Lizorkin spaces $F_{p q}^{s} (X)$ have the lifting properties for |s| < θ. Moreover, we give explicit representations for the inverses of these fractional integrals and derivatives. By using these representations, we prove that the fractional...

Finite time asymptotics of fluid and ruin models: multiplexed fractional Brownian motions case

Krzysztof Dębicki, Grzegorz Sikora (2011)

Applicationes Mathematicae

Similarity:

Motivated by applications in queueing fluid models and ruin theory, we analyze the asymptotics of $ℙ (s u p_{t \in [0, T]} (\sum_{i = 1}^{n} λ_{i} B_{H_{i}} (t) - c t) > u)$ , where $B_{H_{i}} (t) : t \geq 0$ , i = 1,...,n, are independent fractional Brownian motions with Hurst parameters $H_{i} \in (0, 1]$ and λ₁,...,λₙ > 0. The asymptotics takes one of three different qualitative forms, depending on the value of $m i n_{i = 1, . . ., n} H_{i}$ .

A uniform dimension result for two-dimensional fractional multiplicative processes

Xiong Jin (2014)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

Given a two-dimensional fractional multiplicative process ${(F_{t})}_{t \in [0, 1]}$ determined by two Hurst exponents $H_{1}$ and $H_{2}$ , we show that there is an associated uniform Hausdorff dimension result for the images of subsets of $[0, 1]$ by $F$ if and only if $H_{1} = H_{2}$ .

Displaying similar documents to “Generalized H m , m m , 0 function fractional integration operators in some classes of analytic functions”

Displaying similar documents to “Generalized $H_{m, m}^{m, 0}$ function fractional integration operators in some classes of analytic functions”