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)
Notes on singular integrals on some inhomogeneous Herz spaces. (English) Zbl 1090.42008
A central $(p,q)$ block is a function $a$ supported in $\{|x|<R\}$ such that $\|a\|_{L^q}\le |\{|x|<R\}|^{1/q-1/p}$. A central $(p,q)$-atom is a central $(p,q)$-block having integral zero. The block space $K^p_q$ consists of those distributions that can be expressed as superpositions $\sum_k \lambda_k a_k$ of central $(p,q)$-blocks such that $\sum_{k} |\lambda_k|^p<\infty$ and is $p$-normed by $\|f\|_{K^p_q}^p=\inf \sum_k |\lambda_k|^p$, with infimum taken over block representations of $f$. The space $HK^p_q$ is defined in exactly the same way with blocks replaced by atoms. The $p$-norm then makes sense when $p>n/(n+1)$. The author introduces a notion intermediate to that of a block and an atom, namely, a $(p,q,\varepsilon)$-block is a $(p,q)$-block $a$ supported in $\{|x|<R\}$ for some $R\geq 1$ such that $|\int a|\leq |\{|x|<R\}|^{\varepsilon -1/p}$. This notion allows the author to define a new block space $K_p^{1,\varepsilon}$ just as above, but in terms of $(p,q,\varepsilon)$ blocks, and to extend boundedness of certain singular integrals to these spaces. Specifically, a $(q,\theta)^t$-central singular integral is a linear operator $T:\Cal{D}\to \Cal{D}'$ that is bounded on $L^q(\Bbb{R}^n)$ and has integral kernel $K$ satisfying $$\sup_{R\geq 1} \sup_{|y|<R} R^{n(q-1)} \int_{2^jR<|x|<2^{j+1}R}\, |K(x,y)-K(x,0)|^q\, dx < e_j \text{ such that }\sum_{j=1}^\infty 2^{j\theta} e_j<\infty .$$ The author’s main result says the following: Suppose that $n/(n+1)<p\leq 1<q<\infty$, $q/(q-1)\leq s$, $\lambda\leq \varepsilon -1$, and $T$ is a $(q,\theta)^t$-central singular integral with $\theta> n(1/p-1/q)$. If $T^t(1)\in {\mathrm CMO}^{s,\lambda}(\Bbb{R}^n)$ then $T$ is bounded from $HK_p^q (\Bbb{R}^n)$ to $K_p^{q,\varepsilon}(\Bbb{R}^n)$. Here, the finite central oscillation space ${\mathrm CMO}^{s,\lambda}$ consists of those $f$ such that $$\sup_{R\geq 1} \left(\frac{1}{R^{n(1+\lambda q)}} \int_{|x|<R} |f(x)-{\mathrm ave}(f,\{|x|<R\})|^q\, dx\right)^{1/q}$$ is finite. The result extends previous work of {\it J. Alvarez, J. Lakey} and {\it M. Guzmán-Partida} [Collect. Math. 51, No. 1, 1--47 (2000; Zbl 0948.42013)] concerning boundedness of operators from block spaces into Herz-Hardy spaces. The author also corrects a minor error in that work.

42B20Singular and oscillatory integrals, several variables
42B30$H^p$-spaces (Fourier analysis)
42B35Function spaces arising in harmonic analysis