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Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. (English) Zbl 0783.42011
The aim of the paper is to deal with following weighted inequalities: \[ \left( \int_{\mathbb{R}^ n}\bigl(I_ \alpha f(x)\bigr)^ q w(x)dx\right)^{1/q}\leq C\left( \int_{\mathbb{R}^ n} | f(x)|^ p v(x)dx\right)^{1/p},{(*)} \] \(\forall f\), where \(1< p\leq q< \infty\), \(n>\alpha> 0\), and \(I_ \alpha g(x)=\int_{\mathbb{R}^ n} | x-y|^{\alpha-n} g(y)dy\), and \(v(x)\), \(w(x)\) are two weights. Five theorems are obtained. The necessary and sufficient conditions given in Theorem 1 are very close to each other and can be verified easily: The condition \[ \text{``}| Q|^{1-\alpha/n}\left(\int S^ q_ Q w dx\right)^{1/q}\left(\int S^{p'}_ Q v^{1-p'} dx\right)^{1/p'}\leq C,\quad \forall\text{ cubes }Q\text{''} \] is necessary for \((*)\), where \(S_ Q(x)= \bigl(| Q|^{1/n}+ | x- x_ Q|\bigr)^{\alpha-n}\), with \(x_ Q\) the centre of \(Q\), and the condition “for some \(r>1\), \[ | Q|^{\alpha/n+1/q- 1/p}\left({1\over | Q|}\int_ Q w^ r dx\right)^{1/qr}\left({1\over | Q|}\int_ Q v^{(1- p')r}dx\right)^{1/p'r}\leq C_ r,\quad\forall Q\text{''} \] is sufficient for \((*)\).
If some further mild conditions are imposed on \(v\), \(w\), or on \(v^{1- p'}\), \(w\) (for example, the reverse doubling condition (RD), or \(\beta\)- dimensional \(A_ \infty\) condition \(A^ \beta_ \infty\)) then Theorems 1 and 2 show also that the condition \[ A^ \alpha_{p,q}: | Q|^{\alpha/n-1}\left(\int_ Q w dx\right)^{1/q}\left(\int_ Q v^{1-p'}dx\right)^{1/p'}\leq C,\quad\forall Q, \] is necessary and sufficient for \((*)\).
The Theorems 3 and 4 are some versions of these results in the homogeneous type space setting. Theorem 5 is devoted to application of these results to the Poincaré inequality: \[ \left(\int_{Q_ 0}| f(x)|^ q w(x)dx\right)^{1/q}\leq C_{v,w,Q_ 0}\left(\int_{Q_ 0}|\nabla f(x)|^ p v(x)dx\right)^{1/p}, \] with \(\nabla\) gradient operator.

MSC:
42B25 Maximal functions, Littlewood-Paley theory
26D15 Inequalities for sums, series and integrals
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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