Mixed inequalities for commutators with multilinear symbol

Acta Mathematica Scientia, 2011

The aim of this paper is to establish a sufficient condition for certain weighted norm inequalities for singular integral operators with non-smooth kernels and for the commutators of these singular integrals with BMO functions. Our condition is applicable to various singular integral operators, such as the second derivatives of Green operators associated with Dirichlet and Neumann problems on convex domains, the spectral multipliers of non-negative self-adjoint operators with Gaussian upper bounds, and the Riesz transforms associated with magnetic Schrödinger operators.

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. Contents

Indiana University Mathematics Journal, 2016

We improve on several mixed weak type inequalities both for the Hardy-Littlewood maximal function and for Calderón-Zygmund operators. These type of inequalities were considered by Muckenhoupt and Wheeden and later on by Sawyer estimating the L 1,∞ (uv) norm of v -1 T (f v) for special cases. The emphasis is made in proving new and more precise quantitative estimates involving the A p or A ∞ constants of the weights involved.

Journal of the Mathematical Society of Japan, 2005

Let X be a space of homogeneous type. Assume that L has a bounded holomorphic functional calculus on L 2 (Ω) and L generates a semigroup with suitable upper bounds on its heat kernels where Ω is a measurable subset of X. For appropriate bounded holomorphic functions b, we can define the operators b(L) on L p (Ω), 1 ≤ p ≤ ∞. We establish conditions on positive weight functions u, v such that for each p, 1 < p < ∞, there exists a constant cp such that Z Ω |b(L)f (x)| p u(x)dµ(x) ≤ cp b p ∞ Z Ω |f (x)| p v(x)dµ(x) for all f ∈ L p (vdµ). Applications include two-weight L p inequalities for Schrödinger operators with non-negative potentials on R n and divergence form operators on irregular domains of R n .

Journal of Mathematical Analysis and Applications, 2012

In this work we obtain weighted L p , 1 < p < ∞, and weak L log L estimates for the commutator of the Riesz transforms associated to a Schrödinger operator −∆ + V , where V satisfies some reverse Hölder inequality. The classes of weights as well as the classes of symbols are larger than Ap and BM O corresponding to the classical Riesz transforms.

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.

arXiv: Classical Analysis and ODEs, 2015

Let $T$ be a multilinear {integral} operator which is bounded on certain products of Lebesgue spaces on $\mathbb R^n$. We assume that its associated kernel satisfies some mild regularity condition which is weaker than the usual H\&quot;older continuity of those in the class of multilinear Calder\&#39;on-Zygmund singular integral operators. In this paper, given a suitable multiple weight $\vec{w}$, we obtain the bound for the weighted norm of multilinear operators $T$ in terms of $\vec{w}$. As applications, we exploit this result to obtain the weighted bounds {for} certain singular integral operators such as linear and multilinear Fourier multipliers and the Riesz transforms associated to Schr\&quot;odinger operators on $\mathbb{R}^n$ and these results are new in the literature

Proceedings of the American Mathematical Society, 1995

Integral Equations and Operator Theory, 1992

If P is a positive operator on a Hilbert space H whose range is dense, then a theorem of Foias, Ong, and Rosenthal says that: II[qo(P)]-lT[tp(P)]ll < 12 max{llTII, IIp-1TPII} for any bounded operator T on H, where q~ is a continuous, concave, nonnegative, nondecreasing function on [0, IIPII]. This inequality is extended to the class of normal operators with dense range to obtain the inequality II[tp(N)]-lT[tp(N)]ll < 12c 2 max{llTII, IIN-ITNII} where tp is a complex valued function in a class of functions called vase-like, and c is a constant which is associated with q~ by the definition of vase-like. As a corollary, it is shown that the reflexive lattice of operator ranges generated by the range NH of a normal operator N consists of the ranges of all operators of the form tp(N), where q0 is vase-like. Similar results are obtained for scalar-type spectral operators on a Hilbert space,

Advances in Pure and Applied Mathematics

We establish a weighted inequality for fractional maximal and convolution type operators, between weak Lebesgue spaces and Wiener amalgam type spaces on {\mathbb{R}} endowed with a measure which needs not to be doubling.