Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hardy spaces for semigroups with Gaussian bounds   Let T_t=e^{-tL} be a semigroup of self-adjoint linear operators acting on
L^2(X,mu), where (X,d mu) is a space of homogeneous type. We assume that T_t
has an integral kernel T_t(x,y) which satisfies the upper and lower Gaussian
bounds: \frac{C_1}{mu(B(x,\sqrt{t}))} \exp(-c_1d(x,y)^2/t)\leq T_t(x,y) \leq
\frac{C_2}{\mu(B(x,\sqrt{t}))} \exp(-c_2 d(x,y)^2/t). By definition, f belongs
to H^1_L if \| f\|_{H^1_L}=\|\sup_{t>0}|T_t f(x)|\|_{L^1(X,\mu)} <\infty. We
prove that there is a function \omega(x), 0<c \leq \omega(x) \leq C, such that
H^1_L admits an atomic decomposition with atoms satisfying: supp a \subset B,
\|a\|_{L^\infty} \leq mu(B)^{-1}, and the weighted cancellation condition \int
a(x)\omega(x) dmu(x)=0.

15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
17
The categories of this paper are: q-fin
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
8
The categories of this paper are: hep-ex
1
The categories of this paper are: cs
22
The categories of this paper are: chao-dyn
23
The categories of this paper are: solv-int
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The categories of this paper are: econ
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas