Displaying similar documents to “Fractional cartesian products of sets”

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.

On the variation of certain fractional part sequences

Michel Balazard, Leila Benferhat, Mihoub Bouderbala (2021)

Communications in Mathematics

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Let b > a > 0 . We prove the following asymptotic formula n 0 | { x / ( n + a ) } - { x / ( n + b ) } | = 2 π ζ ( 3 / 2 ) c x + O ( c 2 / 9 x 4 / 9 ) , with c = b - a , uniformly for x 40 c - 5 ( 1 + b ) 27 / 2 .

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.

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.

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.

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 .

A spatially sixth-order hybrid L 1 -CCD method for solving time fractional Schrödinger equations

Chun-Hua Zhang, Jun-Wei Jin, Hai-Wei Sun, Qin Sheng (2021)

Applications of Mathematics

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We consider highly accurate schemes for nonlinear time fractional Schrödinger equations (NTFSEs). While an L 1 strategy is employed for approximating the Caputo fractional derivative in the temporal direction, compact CCD finite difference approaches are incorporated in the space. A highly effective hybrid L 1 -CCD method is implemented successfully. The accuracy of this linearized scheme is order six in space, and order 2 - γ in time, where 0 < γ < 1 is the order of the Caputo fractional derivative...

Limiting behaviour of intrinsic seminorms in fractional order Sobolev spaces

Rémi Arcangéli, Juan José Torrens (2013)

Studia Mathematica

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We collect and extend results on the limit of σ 1 - k ( 1 - σ ) k | v | l + σ , p , Ω p as σ → 0⁺ or σ → 1¯, where Ω is ℝⁿ or a smooth bounded domain, k ∈ 0,1, l ∈ ℕ, p ∈ [1,∞), and | · | l + σ , p , Ω is the intrinsic seminorm of order l+σ in the Sobolev space W l + σ , p ( Ω ) . In general, the above limit is equal to c [ v ] p , where c and [·] are, respectively, a constant and a seminorm that we explicitly provide. The particular case p = 2 for Ω = ℝⁿ is also examined and the results are then proved by using the Fourier transform.

Two-weighted estimates for generalized fractional maximal operators on non-homogeneous spaces

Gladis Pradolini, Jorgelina Recchi (2018)

Czechoslovak Mathematical Journal

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Let μ be a nonnegative Borel measure on d satisfying that μ ( Q ) l ( Q ) n for every cube Q n , where l ( Q ) is the side length of the cube Q and 0 < n d . We study the class of pairs of weights related to the boundedness of radial maximal operators of fractional type associated to a Young function B in the context of non-homogeneous spaces related to the measure μ . Our results include two-weighted norm and weak type inequalities and pointwise estimates. Particularly, we give an improvement of a two-weighted result...

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...

Existence and uniqueness of solutions of the fractional integro-differential equations in vector-valued function space

Bahloul Rachid (2019)

Archivum Mathematicum

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The aim of this work is to study the existence and uniqueness of solutions of the fractional integro-differential equations d d t [ x ( t ) - L ( x t ) ] = A [ x ( t ) - L ( x t ) ] + G ( x t ) + 1 Γ ( α ) - t ( t - s ) α - 1 ( - s a ( s - ξ ) x ( ξ ) d ξ ) d s + f ( t ) , ( α > 0 ) with the periodic condition x ( 0 ) = x ( 2 π ) , where a L 1 ( + ) . Our approach is based on the R-boundedness of linear operators L p -multipliers and UMD-spaces.

Existence Results for a Fractional Boundary Value Problem via Critical Point Theory

A. Boucenna, Toufik Moussaoui (2015)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In this paper, we consider the following boundary value problem D T - α ( D 0 + α ( D T - α ( D 0 + α u ( t ) ) ) ) = f ( t , u ( t ) ) , t [ 0 , T ] , u ( 0 ) = u ( T ) = 0 D T - α ( D 0 + α u ( 0 ) ) = D T - α ( D 0 + α u ( T ) ) = 0 , where 0 < α 1 and f : [ 0 , T ] × is a continuous function, D 0 + α , D T - α are respectively the left and right fractional Riemann–Liouville derivatives and we prove the existence of at least one solution for this problem.

Existence and multiplicity of solutions for a fractional p -Laplacian problem of Kirchhoff type via Krasnoselskii’s genus

Ghania Benhamida, Toufik Moussaoui (2018)

Mathematica Bohemica

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We use the genus theory to prove the existence and multiplicity of solutions for the fractional p -Kirchhoff problem - M Q | u ( x ) - u ( y ) | p | x - y | N + p s d x d y p - 1 ( - Δ ) p s u = λ h ( x , u ) in Ω , u = 0 on N Ω , where Ω is an open bounded smooth domain of N , p > 1 , N > p s with s ( 0 , 1 ) fixed, Q = 2 N ( C Ω × C Ω ) , λ > 0 is a numerical parameter, M and h are continuous functions.

L p - L q boundedness of analytic families of fractional integrals

Valentina Casarino, Silvia Secco (2008)

Studia Mathematica

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We consider a double analytic family of fractional integrals S z γ , α along the curve t | t | α , introduced for α = 2 by L. Grafakos in 1993 and defined by ( S z γ , α f ) ( x , x ) : = 1 / Γ ( z + 1 / 2 ) | u - 1 | z ψ ( u - 1 ) f ( x - t , x - u | t | α ) d u | t | γ d t / t , where ψ is a bump function on ℝ supported near the origin, f c ( ² ) , z,γ ∈ ℂ, Re γ ≥ 0, α ∈ ℝ, α ≥ 2. We determine the set of all (1/p,1/q,Re z) such that S z γ , α maps L p ( ² ) to L q ( ² ) boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel K - 1 + i θ i ϱ , α is a product kernel on ℝ², adapted to the curve t | t | α ; as a consequence, we show...

Results of nonexistence of solutions for some nonlinear evolution problems

Medjahed Djilali, Ali Hakem (2019)

Commentationes Mathematicae Universitatis Carolinae

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In the present paper, we prove nonexistence results for the following nonlinear evolution equation, see works of T. Cazenave and A. Haraux (1990) and S. Zheng (2004), u t t + f ( x ) u t + ( - Δ ) α / 2 ( u m ) = h ( t , x ) | u | p , posed in ( 0 , T ) × N , where ( - Δ ) α / 2 , 0 < α 2 is α / 2 -fractional power of - Δ . Our method of proof is based on suitable choices of the test functions in the weak formulation of the sought solutions. Then, we extend this result to the case of a 2 × 2 system of the same type.

Inverse source problem in a space fractional diffusion equation from the final overdetermination

Amir Hossein Salehi Shayegan, Reza Bayat Tajvar, Alireza Ghanbari, Ali Safaie (2019)

Applications of Mathematics

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We consider the problem of determining the unknown source term f = f ( x , t ) in a space fractional diffusion equation from the measured data at the final time u ( x , T ) = ψ ( x ) . In this way, a methodology involving minimization of the cost functional J ( f ) = 0 l ( u ( x , t ; f ) | t = T - ψ ( x ) ) 2 d x is applied and shown that this cost functional is Fréchet differentiable and its derivative can be formulated via the solution of an adjoint problem. In addition, Lipschitz continuity of the gradient is proved. These results help us to prove the monotonicity and...

Some applications of subordination theorems associated with fractional q -calculus operator

Wafaa Y. Kota, Rabha Mohamed El-Ashwah (2023)

Mathematica Bohemica

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Using the operator 𝔇 q , ϱ m ( λ , l ) , we introduce the subclasses 𝔜 q , ϱ * m ( l , λ , γ ) and 𝔎 q , ϱ * m ( l , λ , γ ) of normalized analytic functions. Among the results investigated for each of these function classes, we derive some subordination results involving the Hadamard product of the associated functions. The interesting consequences of some of these subordination results are also discussed. Also, we derive integral means results for these classes.

Optimal estimates for the fractional Hardy operator

Yoshihiro Mizuta, Aleš Nekvinda, Tetsu Shimomura (2015)

Studia Mathematica

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Let A α f ( x ) = | B ( 0 , | x | ) | - α / n B ( 0 , | x | ) f ( t ) d t be the n-dimensional fractional Hardy operator, where 0 < α ≤ n. It is well-known that A α is bounded from L p to L p α with p α = n p / ( α p - n p + n ) when n(1-1/p) < α ≤ n. We improve this result within the framework of Banach function spaces, for instance, weighted Lebesgue spaces and Lorentz spaces. We in fact find a ’source’ space S α , Y , which is strictly larger than X, and a ’target’ space T Y , which is strictly smaller than Y, under the assumption that A α is bounded from X into Y and the Hardy-Littlewood...

Sobolev type inequalities for fractional maximal functions and Riesz potentials in Morrey spaces of variable exponent on half spaces

Yoshihiro Mizuta, Tetsu Shimomura (2023)

Czechoslovak Mathematical Journal

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Our aim is to establish Sobolev type inequalities for fractional maximal functions M , ν f and Riesz potentials I , α f in weighted Morrey spaces of variable exponent on the half space . We also obtain Sobolev type inequalities for a C 1 function on . As an application, we obtain Sobolev type inequality for double phase functionals with variable exponents Φ ( x , t ) = t p ( x ) + ( b ( x ) t ) q ( x ) , where p ( · ) and q ( · ) satisfy log-Hölder conditions, p ( x ) < q ( x ) for x , and b ( · ) is nonnegative and Hölder continuous of order θ ( 0 , 1 ] .

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.