Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Boundedness properties of maximal operators on Lorentz spaces   We study mapping properties of the centered Hardy--Littlewood maximal
operator $\mathcal{M}$ acting on Lorentz spaces. Given $p \in (1,\infty)$ and a
metric measure space $\mathfrak{X}$ we let $\Omega^p_{\rm HL}(\mathfrak{X})
\subset [0,1]^2$ be the set of all pairs $(\frac{1}{q},\frac{1}{r})$ such that
$\mathcal{M}$ is bounded from $L^{p,q}(\mathfrak{X})$ to
$L^{p,r}(\mathfrak{X})$. For each fixed $p$ all possible shapes of
$\Omega^p_{\rm HL}(\mathfrak{X})$ are characterized. Namely, we show that the
boundary of $\Omega^p_{\rm HL}(\mathfrak{X})$ either is empty or takes the form
$$\{ \delta \} \times [0, \lim_{u \rightarrow \delta} F(u)] \ \cup \ \{(u,
F(u)) : u \in (\delta, 1] \},$$ where $\delta \in [0,1]$ and $F \colon [\delta,
1] \rightarrow [0,1]$ is concave, non-decreasing, and satisfying $F(u) \leq u$.
Conversely, for each such $F$ we find $\mathfrak{X}$ such that $\mathcal{M}$ is
bounded from $L^{p,q}(\mathfrak{X})$ to $L^{p,r}(\mathfrak{X})$ if and only if
the point $(\frac{1}{q}, \frac{1}{r})$ lies on or under the graph of $F$, that
is, $\frac{1}{q} \geq \delta$ and $\frac{1}{r} \leq F\big(\frac{1}{q}\big)$.

4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
8
The categories of this paper are: hep-ex
15
The categories of this paper are: nucl-ex
32
The categories of this paper are: atom-ph
14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas