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Weak type (1,1) bounds for rough operators. II. (English) Zbl 0695.47052

Let T be a (possibly nonlinear) operator mapping \(L^ 1({\mathbb{R}}^ n)\) to functions on \({\mathbb{R}}^ n\). We say that “T is of weak type (1,1)” if there is a constant C such that for all \(f\in L^ 1({\mathbb{R}}^ n)\) and all \(\alpha >0\), \[ meas\{x\in {\mathbb{R}}^ n:\quad | Tf(x)| >\alpha \}\leq C\| f\|_ 1\alpha^{-1}. \] This property was important in the work of Calderón and Zygmund on singular integral operators; the present paper establishes it for some operators even more singular.
If \(\Omega \in L^ 1(S^{n-1})\), consider the maximal function \[ M_{\Omega}f(x)=\sup_{r>0}r^{-n}\int_{| y| \leq r}| f(x-y)\Omega (y/| y|)| dy. \] In [Part I, Ann. Math., II. Ser. 128, No.1, 19-42 (1988; Zbl 0666.47027)] it was shown that \(M_{\Omega}\) is of weak type (1,1) if \(n=2\) and \(\Omega \in L^ q\) for some \(q>1\). The principal result of this paper is that if \(n\geq 2\) and \(\Omega\in L(\log L)(S^{n-1})\), then \(M_{\Omega}\) is of weak type (1,1).
Let \(S\subset {\mathbb{R}}^ n\) be a measurable set, star-shaped with respect to the origin, with finite measure; equivalently, in polar coordinates, \(S=\{(r,\theta):\) \(0\leq r<\rho (\theta)\}\) where \(\rho\) is non-negative in \(L^ n(S^{n-1})\). Define \({\mathcal M}_ Sf(x)=\sup_{r>0}\int_{y\in S}| f(x-ry)| dy.\) As a corollary of the preceding theorem, if \(\rho\) satisfies \(\int_{S^{n-1}}\rho^ n\log^+\rho d\theta <\infty\) then \({\mathcal M}_ S\) is of weak type (1,1).
Consider the integral \[ Tf(x)=pv\int f(x-y)\Omega (y/| y|)| y|^{-n}dy. \] Let \(n=2\). If \(\Omega\in L(\log L)(S^ 1)\) and \(\int \Omega =0\) then T is of weak type (1,1). The authors state that they have proved the analogue for \(n\leq 5\), but only the two-dimensional case is given here.
The fourth theorem concerns an operator T, bounded on \(L^ 2({\mathbb{R}}^ n)\), such that \[ <g,Tf>=\int g(x)f(y)K(x-y)dydx \] for all f, g in \(C^{\infty}_ 0\) with disjoint supports; here K is locally integrable away from the origin, and in polar coordinates \(K(r,\theta)=h(r)r^{- n}\Omega_ r(\theta)\) where \(\| \Omega_ r\|_{\infty}+\| \partial_{\theta}\Omega_ r\|_{\infty}\leq C<\infty \quad for\quad all\quad r\) and, uniformly in \(j\in {\mathbb{Z}}\), \[ \int_{2^ j\leq r\leq 2^{j+1}}| h(r)| \log^+| h(r)| dr/r\leq C. \] Then T is of weak type (1,1). The proofs of these four theorems are formally quite similar.
Reviewer: C.J.Henrich

MSC:

47Gxx Integral, integro-differential, and pseudodifferential operators
47B38 Linear operators on function spaces (general)
45Exx Singular integral equations

Citations:

Zbl 0666.47027
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References:

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