Kabe Moen

Weighted Inequalities for Multilinear Fractional Integral Operators

A weighted theory for multilinear fractional integral operators and maximal functions is presented. Sufficient conditions for the two weight inequalities of these operators are found, including “power and logarithmic bumps” and anA∞ condition. For one weight inequalities a necessary and sufficient condition is then obtained as a consequence of the two weight inequalities. As an application, Poincaré and Sobolev inequalities adapted to the multilinear setting are presented.

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.

New weighted estimates for bilinear fractional integral operators

Rn f(x− t)g(x+ t) |t|n−α dt, 0 < α < n. When the target space has an exponent greater than one, many weighted estimates follow trivially from Hölder’s inequality and the known linear theory. We address the case where the target Lebesgue space is at most one and prove several interesting one and two weight estimates. As an application we formulate a bilinear version of the Stein-Weiss inequality for fractional integrals.

A panorama of sampling theory

By a sampling function we mean a member \(\varphi \) of a vector space V of, preferably, continuous, \(\mathbf{C}\)-valued functions on a topological space X for which there is an orbit \(G \cdot {x}_{0}\) of a countable abelian group G acting continuously on X, and each f∈V is the sum of the terms \(f(k \cdot {x}_{0})\varphi (k \cdot x)\), \(k \in G\). Such a recovery formula generalizes the well-known Shannon sampling formula. This chapter presents a general discussion of sampling theory and introduces several new classes of sampling functions \(\varphi : \mathbf{R} \rightarrow \mathbf{C}\) for sampling sets of the form \(\mathbf{Z} + {x}_{0}\). In Sect.2 we discuss the very close connection between general convolution idempotents and sampling functions. In Sect.3 we review the properties of the Zak transform and use it to construct a large family of continuous sampling functions \(\varphi \in {L}^{2}(\mathbf{R})\) where \(\{{T}_{k}\varphi : k \in \mathbf{Z}\}\) is a frame for the principal shift-invariant space \({V }_{\varphi } =\langle \varphi \rangle\) generated by \(\varphi \). This family includes all band-limited sampling functions as well as all continuous sampling functions \(\varphi \in {V }_{\psi }\), \(\psi \in {C}_{c}(\mathbf{R})\). In Sect.4 we look at a class of continuous functions \(\psi \) which do not generate (via the Z-transform) any square-integrable sampling functions and use the Laurent transform (or Z-transform) to show how \(\psi \) generates a possibly infinite family of non-square-integrable sampling functions. In Sect.5 we sketch the manner in which purely algebraic tools lead to construction of a very large class of convolution idempotents and associated sampling functions that cannot be obtained by Zak or Laurent transform methods.

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 (Adv Math 212(1):225–276, 2007) on limited range extrapolation. We then provide several applications of our extrapolation techniques. These applications include new results and new proofs of known results for two weight inequalities for linear and bilinear operators.

Two Weight Bump Conditions for Matrix Weights

In this paper we extend the theory of two weight,

Sharp weighted bounds without testing or extrapolation

A_p

Sharp norm inequalities for commutators of classical operators

Ap bump conditions to the setting of matrix weights. We prove two matrix weight inequalities for fractional maximal operators, fractional and singular integrals, sparse operators and averaging operators. As applications we prove quantitative, one weight estimates, in terms of the matrix