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\(L^ p\) weighted inequalities for the dyadic square function. (English) Zbl 0842.42010

As the main result of the paper the author proves the inequality \[ \int_{\mathbb{R}^n} (S_d f(x))^p V(x) dx\leq C_{n, p} \int_{\mathbb{R}^n} |f(x)|^p M_d^{([p/2]+ 2)} V(x)dx,\qquad f\in \bigcup_{1\leq q< \infty} L^q(\mathbb{R}^n).\tag{1} \] Here \(V(x)\geq 0\) is a weighted function, \(2< p< \infty\), the symbol \([p/2]\) denotes the greatest integer not exceeding \(p/2\), the constant \(C_{n, p}\) depends only on \(p\) and \(n\), and the symbols \(S_d f\) and \(M^{(k)}_d f\) denote dyadic square function of \(f\) and \(k\)-fold dyadic maximal function of \(f\), respectively. More precisely, for \(k\in \mathbb{Z}\) (integer numbers) let \(D_k\) be the set of all cubes in \(\mathbb{R}^n\) of the form \[ [2^{- k} j_1, 2^{- k} (j_1+1)) \times\cdots\times [2^{-k} j_n, 2^{- k}(j_n+ 1)), \] where \(j_1,\dots, j_n\in \mathbb{Z}\). For \(k\in \mathbb{Z}\) and \(x\in \mathbb{R}^n\) take \(I(x, k)\in D_k\) such that \(x\in I(x, k)\) and for \(f\in L^1_{\text{loc}}(\mathbb{R}^n)\) denote \(E_k f(x)= 2^{kn} \int_{I(x, k)} f(y) dy\). Then \[ \begin{aligned} S_d f(x) &:= \Biggl(\sum_{k\in \mathbb{Z}} (E_k f(x)- E_{k- 1} f(x))^2\Biggr)^{1/2},\\ M_d f(x) &:= \sup_{k\in \mathbb{Z}} E_k |f|(x)\end{aligned} \] and \[ M^{(1)}_d f= M_d f,\quad M^{(k+ 1)}_d f= M_d (M^{(k)}_d f)\quad (k= 1,2,\dots). \] The inequality (1) is closely related with a problem pointed out by J. M. Wilson [J. Lond. Math. Soc., II. Ser. 41, 283-294 (1990; Zbl 0712.42032)]; see also W. R. Derrick [J. Integral Equations Appl. 5, No. 1, 23-28 (1993; Zbl 0773.42011)].
Reviewer: P.Gurka (Praha)

MSC:

42B25 Maximal functions, Littlewood-Paley theory
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