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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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