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Kabe Moen

Publications and source records attributed to Kabe Moen.

At least 37 records · Page 2Linked to original sources
arXiv2017

The boundedness of multilinear Calderón-Zygmund operators on weighted and variable Hardy spaces

We establish the boundedness of the multilinear Calderón-Zygmund operators from a product of weighted Hardy spaces into a weighted Hardy or Lebesgue space. Our results generalize to the weighted setting results obtained by Grafakos and Kalton (Collect. Math. 2001) and recent work by the third author, Grafakos, Nakamura, and Sawano. As part of our proof we provide a finite atomic decomposition theorem for weighted Hardy spaces, which is interesting in its own right. As a consequence of our weighted results, we prove the corresponding estimates on variable Hardy spaces. Our main tool is a multilinear extrapolation theorem that generalizes a result of the first author and Naibo (Differential Integral Equations 2016).

math.CA↗
arXiv2017

Extrapolation in the scale of generalized reverse Hölder weights

We develop a theory of extrapolation for weights that satisfy a generalized reverse Hölder inequality in the scale of Orlicz spaces. This extends previous results by Auscher and Martell [2] on limited range extrapolation. As an application, we show that a number of weighted norm inequalities for linear and bilinear operators can be proved using our extrapolation theorem.

math.CA↗
arXiv2017

On the harmonic and geometric maximal operators

We examine the harmonic and geometric maximal operators defined for a general basis of open sets in $\R^n$. We prove two weight norm inequalities for the harmonic maximal operator assuming testing conditions over characteristic functions of unions of sets from the basis. We also prove a that a bumped two weight $A_p$-like condition is sufficient for the two weight boundedness of the harmonic maximal operator.

math.CA↗
arXiv2017

Muckenhoupt-Wheeden conjectures for sparse operators

We provide an example of a pair of weights $(u,v)$ for which the Hardy-Littlewood maximal function is bounded from $L^p(v)$ to $L^p(u)$ and from $L^{p'}(u^{1-p'})$ to $L^{p'}(v^{1-p'})$ while a dyadic sparse operator is not bounded on the same domain and range. Our construction also provides an example of a single weight for which the weak-type endpoint does not hold for sparse operators.

math.CA↗
arXiv2016

Weighted estimates for bilinear fractional integral operators and their commutators

In this paper we prove several weighted estimates for bilinear fractional integral operators and their commutators with BMO functions. We also prove maximal function control theorem for these operators, that is, we prove the weighted $L^p$ norm is bounded by the weighted $L^p$ norm of a natural maximal operator when the weight belongs to $A_\infty$. As a corollary we are able to obtain new weighted estimates for the bilinear maximal function associated to the bilinear Hilbert transform.

math.CA↗
arXiv2015

Matrix $A_p$ weights, degenerate Sobolev spaces, and mappings of finite distortion

We study degenerate Sobolev spaces where the degeneracy is controlled by a matrix $A_p$ weight. This class of weights was introduced by Nazarov, Treil and Volberg, and degenerate Sobolev spaces with matrix weights have been considered by several authors for their applications to PDEs. We prove that the classical Meyers-Serrin theorem, H = W, holds in this setting. As applications we prove partial regularity results for weak solutions of degenerate p-Laplacian equations, and in particular for mappings of finite distortion.

math.AP↗
arXiv2014

Unions of Lebesgue spaces and $A_1$ majorants

We study two questions. When does a function belong to the union of Lebesgue spaces and when does a function have an $A_1$ majorant? We show these questions are fundamentally related. For functions restricted to a fixed cube we prove that the following are equivalent: a function belongs to $L^p$ for some $p>1$; the function has an $A_1$ majorant; for any $p>1$ the function belongs to $L^p_w$ for some $A_p$ weight $w$. We also examine the case of functions defined on ${\mathbb R}^n$ and give characterizations of the union of $L^p_w$ over $w$ in $A_p$ and when a function has an $A_1$ majorant on all of ${\mathbb R}^n$.

math.CA↗
arXiv2013

Compact bilinear commutators: the weighted case

Commutators of bilinear Calderón-Zygmund operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be compact on appropriate products of weighted Lebesgue spaces.

math.CA↗
arXiv2013

Compactness properties of commutators of bilinear fractional integrals

Commutators of a large class of bilinear operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be jointly compact. Under a similar commutation, fractional integral versions of the bilinear Hilbert transform yield separately compact operators.

math.CA↗
arXiv2013

Mixed $A_p$-$A_\infty$ estimates with one supremum

We establish several mixed $A_p$-$A_\infty$ bounds for Calderón-Zygmund operators that only involve one supremum. We address both cases when the $A_\infty$ part of the constant is measured using the exponential-logarithmic definition and using the Fujii-Wilson definition. In particular, we answer a question of the first author and provide an answer, up to a logarithmic factor, of a conjecture of Hytönen and Lacey. Moreover, we give an example to show that our bounds with the logarithmic factors can be arbitrarily smaller than the previously known bounds (both one supremum and two suprema).

math.CA↗
arXiv2013

Logarithmic bump conditions for Calderón-Zygmund Operators on spaces of homogeneous type

We establish two-weight norm inequalities for singular integral operators defined on spaces of homogeneous type. We do so first when the weights satisfy a double bump condition and then when the weights satisfy separated logarithmic bump conditions. Our results generalize recent work on the Euclidean case, but our proofs are simpler even in this setting. The other interesting feature of our approach is that we are able to prove the separated bump results (which always imply the corresponding double bump results) as a consequence of the double bump theorem.

math.CA↗
arXiv2013

A fractional Muckenhoupt-Wheeden theorem and its consequences

In the 1970s Muckenhoupt and Wheeden made several conjectures relating two weight norm inequalities for the Hardy-Littlewood maximal operator to such inequalities for singular integrals. Using techniques developed for the recent proof of the $A_2$ conjecture we prove a related pair of conjectures linking the Riesz potential and the fractional maximal operator. As a consequence we are able to prove a number of sharp one and two weight norm inequalities for the Riesz potential.

math.CA↗
arXiv2013

The sharp weighted bound for multilinear maximal functions and Calderón-Zygmund operators

We investigate the weighted bounds for multilinear maximal functions and Calderón-Zygmund operators from $L^{p_1}(w_1)\times...\times L^{p_m}(w_m)$ to $L^{p}(v_{\vec{w}})$, where $1<p_1,...,p_m<\infty$ with $1/{p_1}+...+1/{p_m}=1/p$ and $\vec{w}$ is a multiple $A_{\vec{P}}$ weight. We prove the sharp bound for the multilinear maximal function for all such $p_1,..., p_m$ and prove the sharp bound for $m$-linear Calderón-Zymund operators when $p\geq 1$.

math.CA↗
arXiv2012

Regularity of solutions to degenerate $p$-Laplacian equations

We prove regularity results for solutions of the equation \[div(< AXu,X u>^{(p-2)/2} AX u) = 0,\] $1 \leq Λw(x)^{2/p}|ξ|^2,\] $w \in A_p$, then we show that solutions are locally Hölder continuous. If the degeneracy is of the form \[ k(x)^{-2/p'}|ξ|^2\leq < A(x)ξ,ξ>\leq k(x)^{2/p}|ξ|^2, \] $k\in A_{p'}\cap RH_τ$,where $τ$ depends on the homogeneous dimension, then the solutions are continuous almost everywhere, and we give examples to show that this is the best result possible. We give an application to maps of finite distortion.

math.AP↗
arXiv2012

One and two weight norm inequalities for Riesz potentials

We consider weighted norm inequalities for the Riesz potentials $I_α$, also referred to as fractional integral operators. First we prove mixed $A_p$-$A_\infty$ type estimates in the spirit of [13, 15, 17]. Then we prove strong and weak type inequalities in the case $p<q$ using the so-called log bump conditions. These results complement the strong type inequalities of Pérez [30] and answer a conjecture from [3]. For both sets of results our main tool is a corona decomposition adapted to fractional averages.

math.CA↗