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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 |{|x|<R}| 1/q-1/p . A central (p,q)-atom is a central (p,q)-block having integral zero. The block space K q p consists of those distributions that can be expressed as superpositions k λ k a k of central (p,q)-blocks such that k |λ k | p < and is p-normed by f K q p p =inf k |λ k | p , with infimum taken over block representations of f. The space HK q p 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,ε)-block is a (p,q)-block a supported in {|x|<R} for some R1 such that |a||{|x|<R}| ε-1/p . This notion allows the author to define a new block space K p 1,ε just as above, but in terms of (p,q,ε) blocks, and to extend boundedness of certain singular integrals to these spaces.

Specifically, a (q,θ) t -central singular integral is a linear operator T:𝒟𝒟 ' that is bounded on L q ( n ) and has integral kernel K satisfying

sup R1 sup |y|<R R n(q-1) 2 j R<|x|<2 j+1 R |K(x,y)-K(x,0)| q dx<e j suchthat j=1 2 jθ e j <·

The author’s main result says the following: Suppose that n/(n+1)<p1<q<, q/(q-1)s, λε-1, and T is a (q,θ) t -central singular integral with θ>n(1/p-1/q). If T t (1)CMO s,λ ( n ) then T is bounded from HK p q ( n ) to K p q,ε ( n ).

Here, the finite central oscillation space CMO s,λ consists of those f such that

sup R1 1 R n(1+λq) |x|<R |f(x)-ave(f,{|x|<R})| q dx 1/q

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:
42B20Singular and oscillatory integrals, several variables
42B30H p -spaces (Fourier analysis)
42B35Function spaces arising in harmonic analysis