We calculate the sharp bounds for some q-analysis variants of Hausdorff type inequalities of the form As applications, we obtain several sharp q-analysis inequalities of the classical positive integral operators, incl...We calculate the sharp bounds for some q-analysis variants of Hausdorff type inequalities of the form As applications, we obtain several sharp q-analysis inequalities of the classical positive integral operators, including the Hardy operator and its adjoint operator, the Hilbert operator, and the Hardy-Littlewood-P61ya operator.展开更多
Let x=(x',x'')with x'∈Rk and x''∈R^N-k andΩbe a x'-symmetric and bounded domain in R^N(N≥2).We show that if 0≤a≤k-2,then there exists a positive constant C>0 such that for all x...Let x=(x',x'')with x'∈Rk and x''∈R^N-k andΩbe a x'-symmetric and bounded domain in R^N(N≥2).We show that if 0≤a≤k-2,then there exists a positive constant C>0 such that for all x'-symmetric function u∈C0^∞(Ω)with∫Ω|■u(x)|^N-a|x'|^-adx≤1,the following uniform inequality holds1/∫Ω|x|^-adx∫Ωe^βa|u|N-a/N-a-1|x'|^-adx≤C,whereβa=(N-a)(2πN/2Γ(k-a/2)Γ(k/2)/Γ(k/2)r(N-a/2))1/N-a-1.Furthermore,βa can not be replaced by any greater number.As the application,we obtain some weighted Trudinger–Moser inequalities for x-symmetric function on Grushin space.展开更多
文摘We calculate the sharp bounds for some q-analysis variants of Hausdorff type inequalities of the form As applications, we obtain several sharp q-analysis inequalities of the classical positive integral operators, including the Hardy operator and its adjoint operator, the Hilbert operator, and the Hardy-Littlewood-P61ya operator.
基金Supported by the National Natural Science Foundation of China(Grant No.11201346)。
文摘Let x=(x',x'')with x'∈Rk and x''∈R^N-k andΩbe a x'-symmetric and bounded domain in R^N(N≥2).We show that if 0≤a≤k-2,then there exists a positive constant C>0 such that for all x'-symmetric function u∈C0^∞(Ω)with∫Ω|■u(x)|^N-a|x'|^-adx≤1,the following uniform inequality holds1/∫Ω|x|^-adx∫Ωe^βa|u|N-a/N-a-1|x'|^-adx≤C,whereβa=(N-a)(2πN/2Γ(k-a/2)Γ(k/2)/Γ(k/2)r(N-a/2))1/N-a-1.Furthermore,βa can not be replaced by any greater number.As the application,we obtain some weighted Trudinger–Moser inequalities for x-symmetric function on Grushin space.
