Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A non-homogeneous local $Tb$ theorem for Littlewood-Paley
  $g_{\lambda}^{*}$-function with $L^p$-testing condition   In this paper, we present a local $Tb$ theorem for the non-homogeneous
Littlewood-Paley $g_{\lambda}^{*}$-function with non-convolution type kernels
and upper power bound measure $\mu$. We show that, under the assumptions $\supp
b_Q \subset Q$, $|\int_Q b_Q d\mu| \gtrsim \mu(Q)$ and $||b_Q||^p_{L^p(\mu)}
\lesssim \mu(Q)$, the norm inequality $\big\| g_{\lambda}^{*}(f)
\big\|_{L^p(\mu)} \lesssim \big\| f \big\|_{L^p(\mu)}$ holds if and only if the
following testing condition holds : $$\sup_{Q : cubes \ in \ \Rn}
\frac{1}{\mu(Q)}\int_Q \bigg(\int_{0}^{\ell(Q)} \int_{\Rn}
\Big(\frac{t}{t+|x-y|}\Big)^{m\lambda}|\theta_t(b_Q)(y,t)|^2 \frac{d\mu(y)
dt}{t^{m+1}}\bigg)^{p/2} d\mu(x) < \infty.$$ This is the first time to
investigate $g_\lambda^*$-function in the simultaneous presence of three
attributes : local, non-homogeneous and $L^p$-testing condition. It is
important to note that the testing condition here is $L^p$ type with $p \in
(1,2]$.

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The categories of this paper are: acc-phys
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The categories of this paper are: econ
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: chao-dyn
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The categories of this paper are: quant-ph
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: physics
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The categories of this paper are: nlin
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The categories of this paper are: q-alg
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The categories of this paper are: solv-int
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The categories of this paper are: plasm-ph