On a Hardy type general weighted inequality in spaces \(L ^{p(\cdot )}\)
A Hardy type two-weighted inequality \[ \|v^{1/q(x)}Hf(x)\|_{q(\cdot)}\leq C\|\omega^{1/p(x)}f(x)\|_{p(\cdot)} \] is studied in the norms of generalized Lebesgue spaces \(L^{p(\cdot)}(\mathbb R^n)\) for the multidimensional Hardy operator \(Hf(x)=\int_{\{t\in \mathbb R^n;|t|<|x|\}} f(t) \,dt\). Equivalent necessary and sufficient conditions are found for the \({L^{p(\cdot)} \rightarrow L^{q(\cdot)}}\) boundedness of the Hardy operator when the exponents satisfy \(q(0) < p(0), q(\infty ) < p(\infty )\). It is proved that the condition for such an inequality to hold coincides with the condition for the validity of two-weighted Hardy inequalities with constant exponents if we require of the exponents to be regular near zero and at infinity.
scientific article; zbMATH DE number 5774005
scientific article