Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Weighted $L_q(L_p)$-estimate with Muckenhoupt weights for the
  diffusion-wave equations with time-fractional derivatives   We present a weighted $L_{q}(L_{p})$-theory ($p,q\in(1,\infty)$) with
Muckenhoupt weights for the equation $$ \partial_{t}^{\alpha}u(t,x)=\Delta
u(t,x) +f(t,x), \quad t>0, x\in \mathbb{R}^d. $$ Here, $\alpha\in (0,2)$ and
$\partial_{t}^{\alpha}$ is the Caputo fractional derivative of order $\alpha$.
In particular we prove that for any $p,q\in (1,\infty)$, $w_{1}(x)\in A_p$ and
$w_{2}(t)\in A_q$, $$ \int^{\infty}_0\left(\int_{\mathbb{R}^d} |u_{xx}|^p
\,w_{1} dx \right)^{q/p}\,w_{2}dt \leq N
\int^{\infty}_0\left(\int_{\mathbb{R}^d} |f|^p \,w_{1} dx
\right)^{q/p}\,w_{2}dt, $$ where $A_p$ is the class of Muckenhoupt $A_p$
weights. Our approach is based on the sharp function estimates of the
derivatives of solutions.

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The categories of this paper are: stat
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The categories of this paper are: dg-ga
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The categories of this paper are: math-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ex
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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: supr-con
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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: nlin
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The categories of this paper are: bayes-an
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The categories of this paper are: physics
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The categories of this paper are: ao-sci