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Maximal functions and Hilbert transforms associated to polynomials. (English) Zbl 0917.42022

Let \(p:\;{\mathbb{R}}\times{\mathbb{R}}\to {\mathbb{R}}\) be a polynomial mapping \(p:(x,t)\to p(x,t)\) and \(p\) have degree \(n\geq 1\) in \(t\) and \(p(x,0)=x.\) The authors studied the boundedness of the following maximal function and Hilbert transform associated to \(p\) as \[ M_pf(x)=\sup_{h>0}{1\over {2h}}\int^h_{-h}| f(p(x,t))| dt \] and \[ H_pf(x)=\text{ p. v.}\int^\infty_{-\infty}| f(p(x,t))| {dt\over t}. \] The authors showed that if \(p\) has degree \(n\) in \(t,\) then \(M_p\) and \(H_p\) are bounded on \(L^r(\mathbb{R})\) when \(r>n\) and they may not be bounded on \(L^n(\mathbb{R})\) for certain \(p\) of degree \(n\) in \(t\). The authors also proved that if \(p\) is quadratic in \(t\), then the mapping properties of \(M_p\) and \(H_p\) can be precisely given in terms of the behaviour of the coefficients of \(t\) and \(t^2\) in \(p\). Moreover, the authors investigated the boundedness of \(M_p\) and \(H_p\) when \(p(x,t)=x-p_1(t)\) and the supermaximal function and the superhilbert transform associated to those \(M_p\) and \(H_p\), where \(p_1\) is a polynomial of degree \(n\) of one real variable \(t\) satisfying \(p_1(0)=0.\)

MSC:

42B25 Maximal functions, Littlewood-Paley theory
44A15 Special integral transforms (Legendre, Hilbert, etc.)
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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