O’Neil Inequality for Multilinear Convolutions and Some Applications

Acta Mathematica Sinica, English Series, 2013

In this paper, we study integral operators of the form T α f (x) = R n |x − A 1 y| −α 1 • • • |x − A m y| −α m f (y)dy, where A i are certain invertible matrices, α i > 0, 1 ≤ i ≤ m, α 1 + • • • + α m = n − α, 0 ≤ α < n. For 1 q = 1 p − α n , we obtain the L p (R n , w p) − L q (R n , w q) boundedness for weights w in A(p, q) satisfying that there exists c > 0 such that w(A i x) ≤ cw(x), a.e. x ∈ R n , 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.

Revista de la Unión Matemática Argentina, 2008

In this note we present weighted Coifman type estimates, and twoweight estimates of strong and weak type for general fractional operators. We give applications to fractional operators given by an homogeneous function, and by a Fourier multiplier. The complete proofs of these results appear in the work [5] done jointly with Ana L. Bernardis and María Lorente.

Acta Mathematica Sinica, English Series, 2010

In this paper, the weak type LlogL estimate for the multilinear fractional commutator is obtained by introducing a new kind of maximal operator of the multilinear fractional order associated with the mean Luxumburg norm and using the technique of sharp function.

Journal of the Mathematical Society of Japan, 2018

We prove some Sawyer-type characterizations for multilinear fractional maximal function for the upper triangle case. We also provide some two-weight norm estimates for this operator. As one of the main tools, we use an extension of the usual Carleson Embedding that is an analogue of the P. L. Duren extension of the Carleson Embedding for measures.

Acta Mathematica Sinica, English Series, 2007

We obtain weighted distributional inequalities for multilinear commutators of the fractional integral on spaces of homogeneous type. The techniques developed in this work involve the behavior of some fractional maximal functions. In relation to these operators, as a main tool, we prove a weighted weak type boundedness result, which is interesting in itself.

Communications in Mathematics and Applications, 2018

In this article, we originate some new Opial-type inequalities on fractional calculus involving generalized Riemann-Liouville fractional integral, the Riemann-Liouville \(k\)-fractional integral, the \((k,r)\) fractional integral of the Riemann-type and the generalized fractional integral operator involving Hypergeometric function in its kernel. As special case of our general results we obtain the results of Farid et al. [7].

Open Mathematics, 2014

In this paper we study integral operators with kernels $$K(x,y) = k_1 (x - A_1 y) \cdots k_m \left( {x - A_m y} \right),$$ $$k_i \left( x \right) = {{\Omega _i \left( x \right)} \mathord{\left/ {\vphantom {{\Omega _i \left( x \right)} {\left| x \right|}}} \right. \kern-\nulldelimiterspace} {\left| x \right|}}^{{n \mathord{\left/ {\vphantom {n {q_i }}} \right. \kern-\nulldelimiterspace} {q_i }}}$$ where Ωi: ℝn → ℝ are homogeneous functions of degree zero, satisfying a size and a Dini condition, A i are certain invertible matrices, and n/q 1 +…+n/q m = n−α, 0 ≤ α &lt; n. We obtain the appropriate weighted L p-L q estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.

arXiv (Cornell University), 2022

We deal with the boundedness properties of higher order commutators related to some generalizations of the multilinear fractional integral operator of order m, I m α , from a product of weighted Lebesgue spaces into adequate weighted Lipschitz spaces, extending some previous estimates for the linear case. Our study includes two different types of commutators and sufficient conditions on the weights in order to guarantee the continuity properties described above. We also exhibit the optimal range of the parameters involved. The optimality is understood in the sense that the parameters defining the corresponding spaces belong to a certain region, being the weights trivial outside of it. We further show examples of weights for the class which cover the mentioned area.

Journal of the Indonesian Mathematical Society, 2022

In 2019, Salim et al proved the vector-valued inequality for maximal operator with rough kernel on Lebesgue spaces and Morrey spaces. This results extend Fefferman-Stein inequality (1971). In 1970's, Adams introduced another variant of Morrey spaces, which called as Morrey-Adams spaces. In this article, we prove vector-valued inequality for maximal operator and fractional integral operator with rough kernel on Morrey-Adams spaces.

Bulletin des Sciences Mathématiques, 2013

Let T be a multilinear operator which is bounded on certain products of unweighted Lebesgue spaces of R n . We assume that the associated kernel of T satisfies some mild regularity condition which is weaker than the usual Hölder continuity of those in the class of multilinear Calderón-Zygmund singular integral operators. We then show the boundedness for T and the boundedness of the commutator of T with BMO functions on products of weighted Lebesgue spaces of R n . As an application, we obtain the weighted norm inequalities of multilinear Fourier multipliers and of their commutators with BMO functions on the products of weighted Lebesgue spaces when the number of derivatives of the symbols is the same as the best known result for the multilinear Fourier multipliers to be bounded on the products of unweighted Lebesgue spaces.