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Displaying similar documents to “On the variation of certain fractional part sequences”

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.

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 .

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 .

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.

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...

Lower bound for class numbers of certain real quadratic fields

Mohit Mishra (2023)

Czechoslovak Mathematical Journal

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Let d be a square-free positive integer and h ( d ) be the class number of the real quadratic field ( d ) . We give an explicit lower bound for h ( n 2 + r ) , where r = 1 , 4 . Ankeny and Chowla proved that if g > 1 is a natural number and d = n 2 g + 1 is a square-free integer, then g h ( d ) whenever n > 4 . Applying our lower bounds, we show that there does not exist any natural number n > 1 such that h ( n 2 g + 1 ) = g . We also obtain a similar result for the family ( n 2 g + 4 ) . As another application, we deduce some criteria for a class group of prime power order to be...

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...

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.

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...

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 note on the size Ramsey numbers for matchings versus cycles

Edy Tri Baskoro, Tomáš Vetrík (2021)

Mathematica Bohemica

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For graphs G , F 1 , F 2 , we write G ( F 1 , F 2 ) if for every red-blue colouring of the edge set of G we have a red copy of F 1 or a blue copy of F 2 in G . The size Ramsey number r ^ ( F 1 , F 2 ) is the minimum number of edges of a graph G such that G ( F 1 , F 2 ) . Erdős and Faudree proved that for the cycle C n of length n and for t 2 matchings t K 2 , the size Ramsey number r ^ ( t K 2 , C n ) < n + ( 4 t + 3 ) n . We improve their upper bound for t = 2 and t = 3 by showing that r ^ ( 2 K 2 , C n ) n + 2 3 n + 9 for n 12 and r ^ ( 3 K 2 , C n ) < n + 6 n + 9 for n 25 .

Copies of l p n ’s uniformly in the spaces Π 2 ( C [ 0 , 1 ] , X ) and Π 1 ( C [ 0 , 1 ] , X )

Dumitru Popa (2017)

Czechoslovak Mathematical Journal

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We study the presence of copies of l p n ’s uniformly in the spaces Π 2 ( C [ 0 , 1 ] , X ) and Π 1 ( C [ 0 , 1 ] , X ) . By using Dvoretzky’s theorem we deduce that if X is an infinite-dimensional Banach space, then Π 2 ( C [ 0 , 1 ] , X ) contains λ 2 -uniformly copies of l n ’s and Π 1 ( C [ 0 , 1 ] , X ) contains λ -uniformly copies of l 2 n ’s for all λ > 1 . As an application, we show that if X is an infinite-dimensional Banach space then the spaces Π 2 ( C [ 0 , 1 ] , X ) and Π 1 ( C [ 0 , 1 ] , X ) are distinct, extending the well-known result that the spaces Π 2 ( C [ 0 , 1 ] , X ) and 𝒩 ( C [ 0 , 1 ] , X ) are distinct.