Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Partial spectral multipliers and partial Riesz transforms for degenerate
  operators   We consider degenerate differential operators $A =
\displaystyle{\sum_{k,j=1}^d \partial_k (a_{kj} \partial_j)}$ on
$L^2(\mathbb{R}^d)$ with real symmetric bounded measurable coefficients. Given
a function $\chi \in C_b^\infty(\mathbb{R}^d)$ (respectively, $\Omega$ a
bounded Lipschitz domain) and suppose that $(a_{kj}) \ge \mu > 0$ a.e.\ on $
\supp \chi$ (resp., a.e.\ on $\Omega$). We prove a spectral multiplier type
result: if $F\colon [0, \infty) \to \mathbb{C}$ is such that $\sup_{t > 0} \|
\varphi(.) F(t .) \|_{C^s} < \infty$ for some non-trivial function $\varphi \in
C_c^\infty(0,\infty)$ and some $s > d/2$ then $M_\chi F(I+A) M_\chi$ is weak
type $(1,1)$ (resp.\ $P_\Omega F(I+A) P_\Omega$ is weak type $(1,1)$). We also
prove boundedness on $L^p$ for all $p \in (1,2]$ of the partial Riesz
transforms $M_\chi \nabla (I + A)^{-1/2}M_ \chi$. The proofs are based on a
criterion for a singular integral operator to be weak type $(1,1)$.

1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
25
The categories of this paper are: patt-sol
37
The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
18
The categories of this paper are: econ
9
The categories of this paper are: nlin
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th
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The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int