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Weak type (1,1) bounds for rough operators. (English) Zbl 0666.47027
Let B be a Banach space of functions on $${\mathbb{R}}^ n$$. Let T be a (possibly nonlinear) operator mapping B to functions on $${\mathbb{R}}^ n$$. We say that “T maps B to weak $$L^ 1$$” if there is a constant C such that for all $$f\in B$$ and all $$\alpha >0$$, $$meas\{x\in {\mathbb{R}}^ n:\quad | Tf(x)| >\alpha \}\leq C\| f\|_ 1\alpha^{-1}.$$ When $$B=L^ 1({\mathbb{R}}^ n)$$ we say that “T is of weak type (1,1)”. Four kinds of operators are considered in this paper:
(1) T is the Bochner-Riesz mean over $${\mathbb{R}}^ n$$, characterized by $(Tf){\hat{\;}}(\xi)=f(\xi)(1-| \xi |^ 2)_+^{(n-1)/2}.$ Then T is of weak type (1,1).
(2) If $$\Omega\geq 0$$ is defined on $$S^ 1\subset {\mathbb{R}}^ n$$, let $M_{\Omega}f(x)=\sup_{r>0}\int_{| y| <1, y\in {\mathbb{R}}^ 2}| f(x-r^{-1}y)| \Omega (y/| y|)dy.$ If $$\Omega \in L^ q(S^ 1)$$, then $$M_{\Omega}$$ is of weak type (1,1).
(3) Let $$\phi$$ be a Schwartz function on the real line; in $${\mathbb{R}}^ 2$$ set $Mf(x)=\sup_{r>0}| \int f(x_ 1-t,x_ 2-t^ 2)\phi (r^{- 1}t)r^{-1} dt|.$ Let $$H^ 1$$ be the “parabolic real-variable Hardy space” [A. P. Calderón and A. Torchinsky, Adv. Math. 16, 1-64 (1975; Zbl 0315.46037)]. Then M maps $$H^ 1$$ to weak $$L^ 1.$$
(4) Let $$\sigma$$ denote surface measure on $$S^{n-1}\subset {\mathbb{R}}^ n$$. Define $Uf=\sup_{j\in Z}| \int f(x-2^{-j}y)d\sigma (y)|.$ Then U maps $$H^ 1$$ to weak $$L^ 1$$, if $$n>1$$.
Reviewer: C.J.Henrich

MSC:
 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 47B38 Linear operators on function spaces (general) 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Zbl 0315.46037
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