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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

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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