Displaying similar documents to “Fractional integral operators on B p , λ with Morrey-Campanato norms”

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...

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...

Density of smooth maps for fractional Sobolev spaces W s , p into simply connected manifolds when s 1

Pierre Bousquet, Augusto C. Ponce, Jean Van Schaftingen (2013)

Confluentes Mathematici

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Given a compact manifold N n ν and real numbers s 1 and 1 p < , we prove that the class C ( Q ¯ m ; N n ) of smooth maps on the cube with values into N n is strongly dense in the fractional Sobolev space W s , p ( Q m ; N n ) when N n is s p simply connected. For s p integer, we prove weak sequential density of C ( Q ¯ m ; N n ) when N n is s p - 1 simply connected. The proofs are based on the existence of a retraction of ν onto N n except for a small subset of N n and on a pointwise estimate of fractional derivatives of composition of maps in W s , p W 1 , s p .

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...

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.

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 .

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...

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.

Approximate and L p Peano derivatives of nonintegral order

J. Marshall Ash, Hajrudin Fejzić (2005)

Studia Mathematica

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Let n be a nonnegative integer and let u ∈ (n,n+1]. We say that f is u-times Peano bounded in the approximate (resp. L p , 1 ≤ p ≤ ∞) sense at x m if there are numbers f α ( x ) , |α| ≤ n, such that f ( x + h ) - | α | n f α ( x ) h α / α ! is O ( h u ) in the approximate (resp. L p ) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or L p sense at each point of a bounded measurable set E. Then for every ε > 0 there is a perfect set Π ⊂ E and a smooth function g such that the Lebesgue measure of E∖Π is less than ε and...

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 ] .

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.

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.

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.

Semigroups generated by certain pseudo-differential operators on the half-space 0 + n + 1

Victoria Knopova (2004)

Colloquium Mathematicae

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The aim of the paper is two-fold. First, we investigate the ψ-Bessel potential spaces on 0 + n + 1 and study some of their properties. Secondly, we consider the fractional powers of an operator of the form - A ± = - ψ ( D x ' ) ± / ( x n + 1 ) , ( x ' , x n + 1 ) 0 + n + 1 , where ψ ( D x ' ) is an operator with real continuous negative definite symbol ψ: ℝⁿ → ℝ. We define the domain of the operator - ( - A ± ) α and prove that with this domain it generates an L p -sub-Markovian semigroup.

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 .

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.

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.

H calculus and dilatations

Andreas M. Fröhlich, Lutz Weis (2006)

Bulletin de la Société Mathématique de France

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We characterise the boundedness of the H calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if - A generates a bounded analytic C 0 semigroup ( T t ) on a UMD space, then the H calculus of A is bounded if and only if ( T t ) has a dilation to a bounded group on L 2 ( [ 0 , 1 ] , X ) . This generalises a Hilbert space result of C.LeMerdy. If X is an L p space we can choose another L p space in place of L 2 ( [ 0 , 1 ] , X ) .

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...

Generalized versions of Ilmanen lemma: Insertion of C 1 , ω or C loc 1 , ω functions

Václav Kryštof (2018)

Commentationes Mathematicae Universitatis Carolinae

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We prove that for a normed linear space X , if f 1 : X is continuous and semiconvex with modulus ω , f 2 : X is continuous and semiconcave with modulus ω and f 1 f 2 , then there exists f C 1 , ω ( X ) such that f 1 f f 2 . Using this result we prove a generalization of Ilmanen lemma (which deals with the case ω ( t ) = t ) to the case of an arbitrary nontrivial modulus ω . This generalization (where a C l o c 1 , ω function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

C * -points vs P -points and P -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

A localization property for B p q s and F p q s spaces

Hans Triebel (1994)

Studia Mathematica

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Let f j = k a k f ( 2 j + 1 x - 2 k ) , where the sum is taken over the lattice of all points k in n having integer-valued components, j∈ℕ and a k . Let A p q s be either B p q s or F p q s (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on n . The aim of the paper is to clarify under what conditions f j | A p q s is equivalent to 2 j ( s - n / p ) ( k | a k | p ) 1 / p f | A p q s .

On square functions associated to sectorial operators

Christian Le Merdy (2004)

Bulletin de la Société Mathématique de France

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We give new results on square functions x F = 0 F ( t A ) x 2 d t t 1 / 2 p associated to a sectorial operator A on L p for 1 &lt; p &lt; . Under the assumption that A is actually R -sectorial, we prove equivalences of the form K - 1 x G x F K x G for suitable functions F , G . We also show that A has a bounded H functional calculus with respect to . F . Then we apply our results to the study of conditions under which we have an estimate ( 0 | C e - t A ( x ) | 2 d t ) 1 / 2 q M x p , when - A generates a bounded semigroup e - t A on L p and C : D ( A ) L q is a linear mapping.

The boundedness of two classes of integral operators

Xin Wang, Ming-Sheng Liu (2021)

Czechoslovak Mathematical Journal

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The aim of this paper is to characterize the L p - L q boundedness of two classes of integral operators from L p ( 𝒰 , d V α ) to L q ( 𝒰 , d V β ) in terms of the parameters a , b , c , p , q and α , β , where 𝒰 is the Siegel upper half-space. The results in the presented paper generalize a corresponding result given in C. Liu, Y. Liu, P. Hu, L. Zhou (2019).