Norm inequalities for linear and multilinear singular integrals on weighted and variable exponent Hardy spaces

Abstract: I will discuss a (relatively) new approach to norm inequalities in the weighted and variable exponent Hardy spaces. The weighted Hardy spaces $H^p(w)$, $p>0$, where $w$ is a Muckenhoupt weight, were first considered by Stromberg and Torchinsky in the 1980s. The variable Lebesgue space $L^{p(\cdot)}$ is, intuitively, a classical Lebesgue space with the constant exponent $p$ replaced by an exponent function $p(\cdot)$. They have been studied extensively for the last 30 years. The corresponding variable Hardy spaces $H^{p(\cdot)}$ were introduced by me and Li-An Wang and independently by Nakai and Sawano.

We give inter-related conditions on a Calderon-Zygmund singular integral operator $T$, a weight $w$, and an exponent $p(\cdot)$ for $T$ to satisfy estimates of the form$$ T : H^p(w) \rightarrow L^p(w), \qquad T: H^{p(\cdot)} \rightarrow L^{p(\cdot)}.$$ Some of our results were known for convolution type singular integrals, but we give new and simpler proofs and give extensions to non-convolution type operators. Our proofs depend very heavily on three tools: atomic decompositions of the Hardy spaces, vector-valued inequalities, and the Rubio de Francia theory of extrapolation.

We will also discuss generalizations of these results to the bilinear setting, where we prove norm inequalities of the form $$ T: H^{p_1}(w_1) \times H^{p_2}(w_2) \rightarrow L^p(w), $$ where $T$ is a bilinear Calder\'on-Zygmund singular integral operator, $p,\,p_1,\,p_2>0$ and $$ \frac{1}{p_1}+\frac{1}{p_2} = \frac{1}{p}, $$ and the weights $w,\,w_1,\,w_2$ are Muckenhoupt weights. We also consider norm inequalities of the form $$ T: H^{p_1(\cdot)}\times H^{p_2(\cdot)}\rightarrow L^{p(\cdot)}. $$

This is joint work with Kabe Moen and Hanh Nguyen of the University of Alabama.

classical analysis and ODEs

Audience: learners

( paper )

Trends in Mathematical Research