Weighted norm inequalities for singular integrals over a domain (Q1311201)

Let \(D\) be a domain in \(\mathbb{R}^ m\), \(r(x)= r(x,\partial D)\) be the distance between a point \(x\) and the boundary \(\partial D\) of the domain \(D\), \(1< p<\infty\). Moreover, let \(\varphi\) and \(\omega\) be monotone positive functions on \((0,D)\). If \[ Tu(x)= \lim_{\varepsilon\to 0} \int_{D\backslash\{| y-x|<\varepsilon\}} f(x,y-x)| y- x|^{-m} u(y) dy \] is a singular integral defined on \(\mathbb{R}^ m\), then one can consider the weighted inequalities \[ \int_ D | Tu(x)|^ p \varphi(r(x)) dx\leq C\int_ D | u(x)|^ p \omega(r(x))dx,\tag{1} \] where the constant \(C>0\) does not depend on the measurable function \(u\). B. Muckenhoupt (1972) introduced the following condition: A non-negative measurable function \(V\) on \(\mathbb{R}^ m\) is said to satisfy the \(A_ p\)- condition if there exists a positive constant \(C\) such that \[ | J|^{-p} \int_ J V\left(\int_ J V^{-{1\over p-1}}\right)^{p- 1}\leq C \] for any cube \(J\) with edges being parallel to the coordinate axes. Let us denote by \(A_ p(D)\) the set of all weight functions restricted on the domain \(D\) satisfying the \(A_ p\)-condition in \(\mathbb{R}^ m\) and by \(B_ p(D)\) the set of all weight functions defined in the domain \(D\) of the form \(\omega(r(x,\partial D))\) with monotone function \(\omega\) such that the inequality (1) is valid for \(\omega=\varphi\). The following proposition is proved in the paper: There exists a domain \(D\) in \(\mathbb{R}^ m\) such that \(B_ p(D)\not\subset A_ p(D)\) \((1<p<\infty)\). The author also formulates a sufficient condition of validity of the inequality (1) in terms of monotone functions \(\varphi\) and \(\omega\).