We prove fractional Leibniz rules and related commutator estimates in the settings of weighted and variable Lebesgue spaces. Our main tools are uniform weighted estimates for sequences of square-function-type operators and a bilinear extrapolation theorem. We also give applications of the extrapolation theorem to the boundedness on variable Lebesgue spaces of certain bilinear multiplier operators and singular integrals.

Kato-Ponce inequalities on weighted and variable Lebesgue spaces

This paper is devoted to studying the Rubio de Francia extrapolation for multilinear compact operators. It allows one to extrapolate the compactness of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from just one space to the full range of weighted spaces, whenever an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-linear operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is bounded on weighted Lebesgue spaces. This result is indeed established in terms of the multilinear Muckenhoupt weights <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript ModifyingAbove p With right-arrow comma ModifyingAbove r With right-arrow"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>p</mml:mi> <mml:mo stretchy="false">→</mml:mo> </mml:mover> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>r</mml:mi> <mml:mo stretchy="false">→</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">A_{\vec {p}, \vec {r}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the limited range of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> scale. To show extrapolation theorems above, by means of a new weighted Fréchet-Kolmogorov theorem, we present the weighted interpolation for multilinear compact operators. To prove the latter, we also need to build a weighted interpolation theorem in mixed-norm Lebesgue spaces. As applications, we obtain the weighted compactness of commutators of many multilinear operators, including multilinear <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi>ω</mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Calderón-Zygmund operators, multilinear Fourier multipliers, bilinear rough singular integrals and bilinear Bochner-Riesz means. Beyond that, we establish the weighted compactness of higher order Calderón commutators, and commutators of Riesz transforms related to Schrödinger operators.

Extrapolation for multilinear compact operators and applications

Let $T$ be a multilinear Calder\\'on-Zygmund operator of type $\\omega$ with $\\omega(t)$ being nondecreasing and satisfying a kind of Dini's type condition. Let $T_{\\Pi\\vec{b}}$ be the iterated commutators of $T$ with $BMO$ functions. The weighted strong and weak $L(\\log{L})$-type endpoint estimates for $T_{\\Pi\\vec{b}}$ with multiple weights are established. Some boundedness properties on weighted variable exponent Lebesgue spaces are also obtained. \nAs applications, multiple weighted estimates for iterated commutators of paraproducts and bilinear pseudo-differential operators with mild regularity are given.

/pdf/commutators-of-multilinear-calderon-zygmund-operators-with-2kxvjdiyhi.pdf

Commutators of multilinear Calderón-Zygmund operators with kernels of Dini's type and applications

In this paper, the authors study a class of multilinear singular integral operators with generalized kernels and their multilinear commutators with BMO functions. By establishing the sharp maximal estimates, the boundedness on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces is obtained, respectively. Moreover, the endpoint estimate of this class of mutilinear singular integral operators is also established. These results can improve the corresponding known results of classical multilinear Calderon–Zygmund operators and multilinear Calderon–Zygmund operators with Dini type kernels.

Multilinear singular integral operators with generalized kernels and their multilinear commutators

Abstract The main purpose of this paper is to establish a Calderón–Vaillancourt-type theorem on L p ${L^{p}}$ estimates for bi-parameter and bilinear pseudo-differential operators with subcritical order.

Bi-parameter and bilinear Calderón–Vaillancourt theorem with subcritical order

The main purpose of this paper is to establish a number of results concerning boundedness of multi-linear Calderon–Zygmund operators with kernels of mild regularity. Let T be a multilinear Calderon–Zygmund operator of type ω ( t ) with ω being nondecreasing and ω ∈ Dini ( 1 ) , but without assuming ω to be concave. We obtain the end-point weak-type estimates for multilinear operator T . The multiple-weighted norm inequalities for multilinear operator T and multilinear commutators of T with B M O functions are also established. As applications, multiple-weighted norm estimates for para-products and bilinear pseudo-differential operators with mild regularity and their commutators are obtained. Moreover, some boundedness properties of the multilinear operators are also established on variable exponent Lebesgue spaces. Our results improve most of the earlier ones in the literature by removing the assumption of concavity of ω ( t ) and weakening the assumption of ω ∈ Dini ( 1 / 2 ) to ω ∈ Dini ( 1 ) .

Multilinear Calderón–Zygmund operators with kernels of Dini’s type and applications

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Multilinear Calderón–Zygmund operators with kernels of Dini’s type and applications