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Notes on singular integrals on some inhomogeneous Herz spaces. (English) Zbl 1090.42008

A central $\left(p,q\right)$ block is a function $a$ supported in $\left\{|x| such that ${\parallel a\parallel }_{{L}^{q}}\le {|\left\{|x|. A central $\left(p,q\right)$-atom is a central $\left(p,q\right)$-block having integral zero. The block space ${K}_{q}^{p}$ consists of those distributions that can be expressed as superpositions ${\sum }_{k}{\lambda }_{k}{a}_{k}$ of central $\left(p,q\right)$-blocks such that ${\sum }_{k}{|{\lambda }_{k}|}^{p}<\infty$ and is $p$-normed by ${\parallel f\parallel }_{{K}_{q}^{p}}^{p}=inf{\sum }_{k}{|{\lambda }_{k}|}^{p}$, with infimum taken over block representations of $f$. The space $H{K}_{q}^{p}$ is defined in exactly the same way with blocks replaced by atoms. The $p$-norm then makes sense when $p>n/\left(n+1\right)$.

The author introduces a notion intermediate to that of a block and an atom, namely, a $\left(p,q,\epsilon \right)$-block is a $\left(p,q\right)$-block $a$ supported in $\left\{|x| for some $R\ge 1$ such that $|\int a|\le {|\left\{|x|. This notion allows the author to define a new block space ${K}_{p}^{1,\epsilon }$ just as above, but in terms of $\left(p,q,\epsilon \right)$ blocks, and to extend boundedness of certain singular integrals to these spaces.

Specifically, a ${\left(q,\theta \right)}^{t}$-central singular integral is a linear operator $T:𝒟\to {𝒟}^{\text{'}}$ that is bounded on ${L}^{q}\left({ℝ}^{n}\right)$ and has integral kernel $K$ satisfying

$\underset{R\ge 1}{sup}\underset{|y|

The author’s main result says the following: Suppose that $n/\left(n+1\right), $q/\left(q-1\right)\le s$, $\lambda \le \epsilon -1$, and $T$ is a ${\left(q,\theta \right)}^{t}$-central singular integral with $\theta >n\left(1/p-1/q\right)$. If ${T}^{t}\left(1\right)\in {\mathrm{C}MO}^{s,\lambda }\left({ℝ}^{n}\right)$ then $T$ is bounded from $H{K}_{p}^{q}\left({ℝ}^{n}\right)$ to ${K}_{p}^{q,\epsilon }\left({ℝ}^{n}\right)$.

Here, the finite central oscillation space ${\mathrm{C}MO}^{s,\lambda }$ consists of those $f$ such that

$\underset{R\ge 1}{sup}{\left(\frac{1}{{R}^{n\left(1+\lambda q\right)}}{\int }_{|x|

is finite.

The result extends previous work of J. Alvarez, J. Lakey and 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.

##### MSC:
 42B20 Singular and oscillatory integrals, several variables 42B30 ${H}^{p}$-spaces (Fourier analysis) 42B35 Function spaces arising in harmonic analysis
##### Keywords:
Herz space; Hardy space; singular integral; commutator; boundedness