Weighted estimates in a limited range with applications to the Bochner-Riesz operators (Q2851023)

The authors define a subset of the Muckenhoupt \(A_p\) class by NEWLINE\[NEWLINEA_{p;(\alpha_0, \alpha_1)}:= \{w= w_0^{\alpha_0} w_1^{\alpha_1 (1-p)}: w_0, w_1 \in A_1\},NEWLINE\]NEWLINE for \(0\leq \alpha_0 \leq 1\) and \(0\leq \alpha_1 \leq 1\). Based on inequalities for the distribution function, alternative proofs for extrapolation theorems are shown by the authors as follows: Let \((f,g)\) be a pair of positive functions such that, for some \(1\leq p <\infty\), NEWLINE\[NEWLINE\|g\|_{L^{p,\infty}(w)} \leq C \|f\|_{L^p(w)}\qquad \forall\;w\in A_{p;(\alpha_0, \alpha_1)}\;\text{with}\;\alpha_0 >0,NEWLINE\]NEWLINE where the constant \(C\) depends only on the \(A_p\) characteristic constant \([w]_{A_p}\). Then, for every \(q\in (\frac{p'}{p'-1+\alpha_1}, \frac p{1-\alpha_0})\), \(0< \alpha < 1- \frac {q(1-\alpha_0)}p \), \(v\in A_1\) and every weight \(u\), NEWLINE\[NEWLINE\int_{\{z| g(z)>y\}} u(x) dx \leq \frac 1{y^q} \int_{\mathbb R^d} f^q(x) M_\alpha(uv^{q/p_0-1} \chi_{\{g>y\}})(x) v(x)^{q/p_0-1} dx,\tag{\(\ast\)} NEWLINE\]NEWLINE where \(M\) is the usual Hardy-Littlewood maximal operator and \(M_\alpha f(x)= M(|f|^{1/\alpha})(x)^\alpha\).NEWLINENEWLINEAs main consequences of the inequality \((\ast)\), the authors deduce from the weighted inequalities of an operator \(T\) for weights in \(A_{p;(\alpha_0, \alpha_1)}\) two kinds of weighted estimates (by considering \(g=Tf\)): \(T: L^p(u)\mapsto L^{q,\infty}(v)\), \(T: \Lambda^p(w)\mapsto \Lambda^{q,\infty}(w)\), where \(\Lambda^p(w)\) stands for the weighted Lorentz space and \(\Lambda^{q,\infty}(w)\) denotes the weak version of \(\Lambda^p(w)\).NEWLINENEWLINEThe applications include the Bochner-Riesz operators \(T_\beta^1\) and other operators.