Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $\ell^p$-improving inequalities for Discrete Spherical Averages   Let $ \lambda ^2 \in \mathbb N $, and in dimensions $ d\geq 5$, let $
A_{\lambda } f (x)$ denote the average of $ f \;:\; \mathbb Z ^{d} \to \mathbb
R $ over the lattice points on the sphere of radius $\lambda$ centered at $x$.
We prove $ \ell ^{p}$ improving properties of $ A_{\lambda }$.
\begin{equation*} \lVert A_{\lambda }\rVert_{\ell ^{p} \to \ell ^{p'}} \leq
C_{d,p, \omega (\lambda ^2 )} \lambda ^{d ( 1-\frac{2}p)}, \qquad
\tfrac{d-1}{d+1} < p \leq \frac{d} {d-2}. \end{equation*} It holds in dimension
$ d =4$ for odd $ \lambda ^2 $. The dependence is in terms of $ \omega (\lambda
^2 )$, the number of distinct prime factors of $ \lambda ^2 $. These
inequalities are discrete versions of a classical inequality of Littman and
Strichartz on the $ L ^{p}$ improving property of spherical averages on $
\mathbb R ^{d}$, in particular they are scale free, in a natural sense. The
proof uses the decomposition of the corresponding multiplier whose properties
were established by Magyar-Stein-Wainger, and Magyar. We then use a proof
strategy of Bourgain, which dominates each part of the decomposition by an
endpoint estimate.

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The categories of this paper are: q-bio
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The categories of this paper are: solv-int
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The categories of this paper are: physics
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The categories of this paper are: ao-sci
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The categories of this paper are: cmp-lg
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: funct-an
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The categories of this paper are: stat
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The categories of this paper are: hep-lat
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The categories of this paper are: mtrl-th