## 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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