zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Almost-everywhere summability of Fourier integrals. (English) Zbl 0631.42004
Let $$ (T\sp{\lambda}\sb{{\bbfR}}f){\hat{\ }}(x)=(1-\vert x\vert \sp 2/R\sp 2)\sb +\sp{\lambda}\hat f(x) $$ denote the Bochner-Riesz operator of order $\lambda$ $\ge 0$. We investigate the almost everywhere convergence of $T\sb R\sp{\lambda}f(x)$ as $R\to \infty.$ Theorem 1. Let $\lambda >0$ and $n\ge 2$. For all $f\in L\sp p({\bbfR}\sp n)$ with $2\le p<2n/(n-1-2\lambda)$, $\lim \sb{R\to \infty}T\sb R\sp{\lambda}f(x)=f(x)$ almost everywhere. Theorem 2. For all $f\in L\sp p({\bbfR}\sp n)$ with $2\le p<2n/(n-1)$, $\lim \sb{k\to \infty}T\sp 0\sb{R\sb k}f(x)=f(x)$ a.e. if $\{R\sb k\}$ is a lacunary sequence. The proofs proceed via domination of the maximal function by square functions; however instead of seeking $L\sp p$ bounds for the square function we examine weighted $L\sp 2$ inequalities, which, by duality, correspond to studying pointwise multipliers of Sobolev spaces.

42A45Multipliers, one variable
42B25Maximal functions, Littlewood-Paley theory
42A55Lacunary series of trigonometric and other functions; Riesz products