arXiv subjects

Robert Rahm

Publications and source records attributed to Robert Rahm.

4 recordsLinked to original sources

arXiv2018

Weighted Alpert Wavelets

In this paper we construct a wavelet basis in weighted L^2 of Euclidean space possessing vanishing moments of a fixed order for a general locally finite positive Borel measure. The approach is based on a clever construction of Alpert in the case of Lebesgue measure that is appropriately modified to handle the general measures considered here. We then use this new wavelet basis to study a two-weight inequality for a general Calder\'on-Zygmund operator on the real line and show that under suitable natural conditions, including a weaker energy condition, the operator is bounded from one weighted L^2 space to another if certain stronger testing conditions hold on polynomials. An example is provided showing that this result is logically different than existing results in the literature.

math.CA↗

arXiv2015

Two-Weight Inequalities for Commutators with Fractional Integral Operators

In this paper we investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $μ,λ\in A_{p,q}$ and $α/n+1/q=1/p$, the norm $\| [b,I_α]:L^p(μ^p)\to L^q(λ^q) \|$ is equivalent to the norm of $b$ in the weighted BMO space $BMO(ν)$, where $ν=μλ^{-1}$. This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.

math.CA↗

arXiv2015

Some Entropy Bump Conditions for Fractional Maximal and Integral Operators

We investigate weighted inequalities for fractional maximal operators and fractional integral operators. We work within the innovative framework of "entropy bounds" introduced by Treil--Volberg. Using techniques developed by Lacey and the second author, we are able to efficiently prove the weighted inequalities.

math.CA↗

arXiv2014

A Reproducing Kernel Thesis for Operators on $\ell^2$--valued Bergman-type Function Spaces

In this paper we consider the reproducing kernel thesis for boundedness and compactness for operators on $\ell^2$--valued Bergman-type spaces. This paper generalizes many well--known results about classical function spaces to their $\ell^2$--valued versions. In particular, the results in this paper apply to the weighted $\ell^2$--valued Bergman space on the unit ball, the unit polydisc and, more generally to weighted Fock spaces.

math.CA↗