Displaying similar documents to “Pointwise convergence to the initial data for nonlocal dyadic diffusions”

Regularity of solutions of the fractional porous medium flow

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

Journal of the European Mathematical Society

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We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is u t = · ( u ( - Δ ) - s u ) , 0 < s < 1 . The problem is posed in { x n , t } 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...

Commutators with fractional integral operators

Irina Holmes, Robert Rahm, Scott Spencer (2016)

Studia Mathematica

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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 A p , q and α/n + 1/q = 1/p, the norm | | [ b , I α ] : L p ( μ p ) 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,...

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...

Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball

Vladimir Varlamov (2000)

Studia Mathematica

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The nonlinear heat equation with a fractional Laplacian [ u t + ( - Δ ) α / 2 u = u 2 , 0 < α 2 ] , is considered in a unit ball B . Homogeneous boundary conditions and small initial conditions are examined. For 3/2 + ε₁ ≤ α ≤ 2, where ε₁ > 0 is small, the global-in-time mild solution from the space C ( [ 0 , ) , H κ ( B ) ) with κ < α - 1/2 is constructed in the form of an eigenfunction expansion series. The uniqueness is proved for 0 < κ < α - 1/2, and the higher-order long-time asymptotics is calculated.

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...

Limiting Behaviour of Dirichlet Forms for Stable Processes on Metric Spaces

Katarzyna Pietruska-Pałuba (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Supposing that the metric space in question supports a fractional diffusion, we prove that after introducing an appropriate multiplicative factor, the Gagliardo seminorms | | f | | W σ , 2 of a function f ∈ L²(E,μ) have the property 1 / C ( f , f ) l i m i n f σ 1 ( 1 σ ) | | f | | W σ , 2 l i m s u p σ 1 ( 1 σ ) | | f | | W σ , 2 C ( f , f ) , where ℰ is the Dirichlet form relative to the fractional diffusion.

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 - i = 1 n - 1 a i D α i ) x ( t ) F ( t , x ( φ ( t ) ) ) , a.e. on (0,1), I 1 - α 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.

Density of some sequences modulo 1

Artūras Dubickas (2012)

Colloquium Mathematicae

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Recently, Cilleruelo, Kumchev, Luca, Rué and Shparlinski proved that for each integer a ≥ 2 the sequence of fractional parts a / n n = 1 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.

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].

Fractional integral operators on B p , λ with Morrey-Campanato norms

Katsuo Matsuoka, Eiichi Nakai (2011)

Banach Center Publications

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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.

Fractional global domination in graphs

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

Discussiones Mathematicae Graph Theory

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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 ] ) = u N [ v ] g ( u ) 1 and g ( N ( v ) ¯ ) = u N ( v ) g ( u ) 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 | = v 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

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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...

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 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...

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

Katsuo Matsuoka (2014)

Banach Center Publications

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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 ( ) . In order to show the boundedness, the generalized λ-central mean oscillation spaces Λ p , λ ( d ) ( ) and the generalized weak λ-central mean oscillation spaces W Λ p , λ ( d ) ( ) play an important role.

Trace inequalities for fractional integrals in grand Lebesgue spaces

Vakhtang Kokilashvili, Alexander Meskhi (2012)

Studia Mathematica

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rning the boundedness for fractional maximal and potential operators defined on quasi-metric measure spaces from L p ) , θ ( X , μ ) to L q ) , q θ / p ( X , ν ) (trace inequality), where 1 < p < q < ∞, θ > 0 and μ satisfies the doubling condition in X. The results are new even for Euclidean spaces. For example, from our general results D. Adams-type necessary and sufficient conditions guaranteeing the trace inequality for fractional maximal functions and potentials defined on so-called s-sets in ℝⁿ follow. Trace...

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

Qingying Xue (2013)

Studia Mathematica

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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 , [ b , [ b m - 1 , [ b , T δ ] ] m - 1 ] ] ( f ) ( x ) | , I α , Π b ( f ) ( x ) = [ b , [ b , [ b m - 1 , [ b , I α ] ] m - 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 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...

Fractional Laplacian with singular drift

Tomasz Jakubowski (2011)

Studia Mathematica

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

Heat kernel estimates for the Dirichlet fractional Laplacian

Zhen-Qing Chen, Panki Kim, Renming Song (2010)

Journal of the European Mathematical Society

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We consider the fractional Laplacian - ( - Δ ) α / 2 on an open subset in d with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such a Dirichlet fractional Laplacian in C 1 , 1 open sets. This heat kernel is also the transition density of a rotationally symmetric α -stable process killed upon leaving a C 1 , 1 open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

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 ( ) may be approximated in capacity and norm by smooth functions belonging to C m , λ ( ) , 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].

A uniform dimension result for two-dimensional fractional multiplicative processes

Xiong Jin (2014)

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

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Given a two-dimensional fractional multiplicative process ( F t ) t [ 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 .

Stability of nonlinear h -difference systems with n fractional orders

Małgorzata Wyrwas, Ewa Pawluszewicz, Ewa Girejko (2015)

Kybernetika

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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.