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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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