Fractional Type Operators Research Articles

Weighted inequalities for fractional type operators with some homogeneous kernels

In this paper, we study integral operators of the form $$T_\alpha f(x) = \int_{\mathbb{R}^n } {\left| {x - A_1 y} \right|^{ - \alpha _1 } \cdots \left| {x - A_m y} \right|^{ - \alpha _m } f(y)dy,}$$ , where Ai are certain invertible matrices, αi > 0, 1 ≤ i ≤ m, α1 + … + αm = n − α, 0 ≤ α 0 such that w(Aix) ≤ cw(x), a.e. x ∈ ℝn, 1 ≤ i ≤ m. Moreover, we obtain the appropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman type estimate for these operators.

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

The following iterated commutators $T_{*,\Pi b}$ of the maximal operator for multilinear singular integral operators and $I_{\alpha, \Pi b}$ of the multilinear fractional integral operator are introduced and studied: $$\eqalign{ T_{*,\Pi b}(\vec{f}\,)(x

About integral operators of fractional type on variable Lp spaces

If A1;:::;Am are orthogonal matrices, in this paper we obtain strong and weak estimates for certain integral operators with kernels of the form k1(x A1y):::km(x Amy); in the context of Lebesgue spaces with variable exponents.

Weighted Trudinger inequality associated with rough multilinear fractional type operators

Let I Ω , α Θ Open image in new window be the multilinear fractional type operator defined by I Ω , α Θ ( f → ) ( x ) = ∫ R n Ω ( y ) ∏ j = 1 m f j ( x − θ j y ) | y | ( α − n ) d y Open image in new window. In this paper, we study the weighted estimates for the Trudinger inequality associated to I Ω , α Θ Open image in new window with rough homogeneous kernels, which improve some known results significantly. A similar Trudinger inequality holds for another type of fractional integral defined by I ¯ Ω , α ( f → ) ( x ) = ∫ ( R n ) m ∏ j = 1 m | f j ( y j ) | | Ω j ( x − y j ) | | ( x − y 1 , x − y 2 , … , x − y m ) | m n − α d y → Open image in new window, where d y → = d y 1 ⋯ d y m Open image in new window.

Weighted endpoint estimates for commutators of multilinear fractional integral operators

Let m be a positive integer, 0 < α < mn, \(\overrightarrow b \) = (b 1, …, b m ) ε BMOm. We give sufficient conditions on weights for the commutators of multilinear fractional integral operators \({\rm I}_a^{\overrightarrow b }\) to satisfy a weighted endpoint inequality which extends the result in D. Cruz-Uribe, A. Fiorenza: Weighted endpoint estimates for commutators of fractional integrals, Czech. Math. J. 57 (2007), 153–160. We also give a weighted strong type inequality which improves the result in X. Chen, Q. Xue: Weighted estimates for a class of multilinear fractional type operators, J. Math. Anal. Appl., 362, (2010), 355–373.

Mountain Pass solutions for non-local elliptic operators

The purpose of this paper is to study the existence of solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions. These equations have a variational structure and we find a non-trivial solution for them using the Mountain Pass Theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. We prove this result for a general integrodifferential operator of fractional type and, as a particular case, we derive an existence theorem for the fractional Laplacian, finding non-trivial solutions of the equation{(−Δ)su=f(x,u)in Ω,u=0in Rn∖Ω. As far as we know, all these results are new.

Weighted estimates for a class of multilinear fractional type operators

In this paper, a class of multiple fractional type weights A ( p → , q ) is defined as sup Q ( 1 | Q | ∫ Q ν ω → q ) 1 q ∏ i = 1 m ( 1 | Q | ∫ Q ω i − p i ′ ) 1 p i ′ < ∞ , where ν ω → = ∏ i = 1 m ω i . And the necessary condition for the characterization of A ( p → , q ) is also obtained. Strong ( L p 1 ( ω 1 p 1 ) × ⋯ × L p m ( ω m p m ) , L q ( ν ω → q ) ) estimates when each p i > 1 and weighted endpoint estimates ( L p 1 ( ω 1 p 1 ) × ⋯ × L p m ( ω m p m ) , L q , ∞ ( ν ω → q ) ) when there exists p i = 1 for some multilinear fractional type operators (e.g. fractional maximal operator, fractional integral, commutators of fractional integral operators) are obtained. As applications of these results, we give some weighted estimates for the above operators with rough homogeneous kernels when suitable conditions were assumed on the kernels. Weighted strong and L ( log L ) type endpoint estimates for commutators of multilinear fractional integral operators are also obtained. Similar results for multilinear Calderón–Zygmund singular integral can be found in A.K. Lerner et al. (in press) [12].

The higher order commutators of the fractional integrals on Hardy spaces

AbstractIn this paper we investigate the boundedness on Hardy spaces for the higher order commutator Tb, m generated by the BMO function b and fractional integral type operator Tτ, and establish the boundness theorems for Tτb, m from Hp1.q1.sb, m to Lp2 and to Hp2 (0 &lt; p1 ≤ 1), and from H Ka. p1.sq1, b, m to Ka.p2q2 and to H Ka. p2q2, respectively, for certain ranges of α, p1, q1, p2, q2 and s.

A Characterization of two Weight Norm Inequalities for One-Sided Operators of Fractional Type

AbstractIn this paper we give a characterization of the pairs of weights (ω, v) such that T maps Lp(v) into Lq(ω),where T is a general one-sided operator that includes as a particular case theWeyl fractional integral. As an applicationwe solve the following problem: given a weight v, when is there a nontrivial weight ω such that T maps Lp(v) into Lq(ω)?

Potential operators and multipliers on locally compact Vilenkin groups

We study, under the setting of a locally compact Vilenkin group G, a weighted norm inequality for the potential operators of Riesz type and its applications to multipliers on G. We also consider the maximal operators of fractional type.

Fractional Type Operators in Weighted Generalized Hölder Spaces

Abstract Weighted Zygmund type estimates are obtained for the continuity modulus of some convolution type integrals. In the case of fractional integrals this is strengthened to a result on isomorphism between certain weighted generalized Hölder type spaces.

Fractional type operators in weighted generalized Hölder spaces

Weighted Zygmund type estimates are obtained for the continuity modulus of some convolution type integrals. In the case of fractional integrals this is strengthened to a result on isomorphism between certain weighted generalized Holder type spaces.

Sequences of weak solutions for fractional equations

This work is devoted to study the existence of infin- itely many weak solutions to nonlocal equations involving a gen- eral integrodifferential operator of fractional type. These equations have a variational structure and we find a sequence of nontrivial weak solutions for them exploiting the Z2-symmetric version of the Mountain Pass Theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. As a particular case, we derive an existence theorem for the frac- tional Laplacian, finding nontrivial solutions of the equation � (−�) s u = f(x,u) in u = 0 in IR n . As far as we know, all these results are new and represent a frac- tional version of classical theorems obtained working with Lapla- cian equations.