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Multiparameter operators and sharp weighted inequalities. (English) Zbl 0877.42004
The authors consider multiparameter operators, and are especially interested in operators whose behavior is not a direct consequence of reduction to products of one-parameter operators. Most of the article considers $${\mathbf R}^{3}$$ with the 2-parameter dilation $$\rho_{s,t}(x,y,z)=(sx,ty,stz)$$. The type of operators considered here are of the form, $T_{\mathfrak z} f=[\sum_{k,j\in \mathbf Z}2^{-2(k+j)}\psi_{k,j}(2^{-k}x,2^{-j}y,2^{-(k+j)}z)]*f=K*f$ where $$\psi_{k,j}$$ are uniformly in the Schwartz class $${\mathcal S}_{N}$$ which consists of the functions in $$C^{\infty}({\mathbf R}^{3})$$ with finite $$|\cdot|_{{\mathcal S}_{N}}$$ norm, where $|\psi|_{{\mathcal S}_{N}}=\sup_{(x,y,z)\in {\mathbf R}^{3}}(1+|(x,y,z)|^{N})\sum_{\alpha,\beta,\gamma=0}^{N} |\partial_{x}^{\alpha}\partial_{y}^{\beta}\partial_{z}^{\gamma}\psi(x,y,z)|.$ In the main result it is also required that the following cancellation properties hold, $\int_{{\mathbf R}^{2}}y^{\alpha}z^{\beta}\psi_{k,j}(x,y,z)dydz=0,\quad \int_{{\mathbf R}^{2}}x^{\alpha}z^{\beta}\psi_{k,j}(x,y,z)dxdz=0,$ for the first integral this should hold for all fixed $$x$$ , for the second for all fixed $$y$$ and both for all $$\alpha,\beta\leq N$$. The weight class $$A^{p}(\mathfrak z)$$ is defined as the functions $$w\geq 0$$ in $${\mathbf R}^{3}$$, for which the following expression is finite $\sup_{R\in {\mathcal R}_{\mathfrak z}}\biggl({1 \over m(R)}\int_{R}wdm)({1 \over m(r)}\int_{R}w^{-1\over p-1}dm\biggr)^{p-1},$ where $${\mathcal R}_{\mathfrak z}$$ are the rectangles with sides parallel to the axes and with sides $$(s,t,st)$$.
One of the main result is that the operator $$T_{\mathfrak z}f=K*f$$, where $$K$$ satisfies the above conditions, is bounded on $$L^{p}(w)$$ if $$w\in A^{p}(\mathfrak z)$$, $$1<p<\infty$$.
They also use the concept of “sharp weighted inequalities” to obtain some result in multiparameter theory.
Reviewer: O.Svensson (Lulea)

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory
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