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Zbl 1090.42008
Komori, Yasuo
Notes on singular integrals on some inhomogeneous Herz spaces.
(English)
[J] Taiwanese J. Math. 8, No. 3, 547-556 (2004). ISSN 1027-5487

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 $p>n/(n+1)$. \par 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. \par 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 $K_p^{q,\varepsilon}(\Bbb{R}^n)$. \par 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. \par 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.
[Joseph Lakey (Las Cruces)]
MSC 2000:
*42B20 Singular integrals, several variables
42B30 Hp-spaces (Fourier analysis)
42B35 Function spaces arising in harmonic analysis

Keywords: Herz space; Hardy space; singular integral; commutator; boundedness

Citations: Zbl 0948.42013

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