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I'm working on some nonlinear partial differential equations and I have been led to the following puzzle. Let $W(x)$ be a symmetric positive semidefinite-valued function of $x \in \mathbb{R}^d$ (a "matrix weight"). I am interested in the Reverse Hölder coefficient, which I define as:

$$[W]_{RH(p)} := \sup_{K,u} {|K|^{1-{1 \over p}}}{\|u^TW(x)u\|_{L^p(K)} \over \|u^TW(x)u\|_{L^1(K)}}.$$

Here, $u \in \mathbb{R}^{d'}$ is some nonzero vector (not a function of $x$) and $K$ ranges over simplices of bounded aspect ratio, but for simplicity, one could consider to start, the usual axis-aligned cubes or balls. I'm defining division by $0$ as $0$. I want to understand better those weights that satisfy $[W]_{RH(p)} < \infty$, the cases $p = 1.5$ and $p = \infty$ are particularly interesting to me.

The problem is that all the literature I found on matrix weights, assumes that $W(x)$ is invertible at almost every $x$. I'm also interested in rank-deficient cases, e.g. $W(x) = v(x)v^T(x)$ of rank $1$. Standard arguments show this works, e.g. when $v(x)$ is a polynomial in $x$, but where could I find out more about this scenario?

Next-day edit: Maybe my question was "too short". What I am asking is for a theory for these $RH(p)$ classes. For example, there is this famous paper:

[1] Cruz-Uribe, David, and Christoph J. Neugebauer. "The structure of the reverse Hölder classes." Transactions of the American Mathematical Society 347.8 (1995): 2941-2960.

This is quite good, but it is for scalar weights. You can "do" matrix weights by expanding the matrix products and "dropping down" to scalar-valued functions, but then you are asking for a quantity $[u^TW(x)u]_{RH(p)}$ to be bounded uniformly in $u$, and the theory of scalar weights is not very good at this because it is often not quantitative (at least in this direction). Furthermore, the theory from Muckenhoupt classes gives very small values of $p$ in the $RH(p)$ estimates (I've seen $p=1.001$ in some papers), but I need much larger values of $p$.

For matrix weights, I have not found a lot for the Hölder inequality specifically (everyone seems to be working instead on Muckenhoupt classes). Here's an example that is related to reverse Hölder inequalities:

[2] Bownik, M. (2001). Inverse volume inequalities for matrix weights. Indiana University Mathematics Journal, 383-410.

However, the standard literature is almost always about invertible weights, my weight $W(x)$ can have degenerate rank.

I would like some systematic theory like [1], but for my possibly-singular matrix weights.

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  • $\begingroup$ Could you be a bit more specific about what exactly you want to know about such weights? I mean, unless I miss something, the definition itself can hardly be simplified but if you are interested in proving some particular properties/sufficient conditions/..., that can certainly be discussed. $\endgroup$
    – fedja
    Commented Jun 6, 2022 at 21:40
  • $\begingroup$ I'm going to impose a version of this as a "regularity condition", so I'd like $RH(p)$ functions to be "unexceptional". The rank 1 example is interesting. With $W(x) = \lambda(x)v(x)v^T(x)$, one can assume lower and upper bounds on $\lambda$ to reduce it to the unit vector case, and the function $g(x)=(v^T(x)u)^2$ parametrized by $u$. This is a "nontrivial" function because it vanishes when $v^T(x)u=0$ which is probably a manifold of codimension 1, and there should be some sort of reasonable hypothesis (e.g on $v'$) that yields $[g]_{RH(p)}<\infty$ but I can't figure it out. $\endgroup$
    – Sébastien Loisel
    Commented Jun 6, 2022 at 22:47

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