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Keywords:
Hardy-Sobolev integrability; generalized Riesz potential; weighted Morrey-Orlicz space; double phase functional
Hardy-Sobolev integrability; generalized Riesz potential; weighted Morrey-Orlicz space; double phase functional
Summary:
We are concerned with weighted Hardy-Sobolev type integrabilities for generalized Riesz potentials $$ R_{\alpha ,m}f(x) = \int _{{\mathbb R}^n} R_{\alpha ,m} (x,y) f(y) {\rm d} y $$ of functions $f$ in weighted Morrey-Orlicz spaces, where $R_{\alpha }(x) = |x|^{\alpha -n}$ and $$ R_{\alpha ,m}(x,y) = R_{\alpha }(x-y) - \sum _{|\ell | \le m-1} \frac {y^{\ell }}{\ell !} (D^{\ell }R_{\alpha })(-x). $$ Those potentials are used to give a representation of $C^1$-functions on the punctured space ${\mathbb R} ^n\setminus \{0\}$. As an application, we obtain Hardy-Sobolev type integrabilities for general double phase functionals given by $$ \varphi (x,t) =\varphi _1(t) + \varphi _2(b(x)t). $$
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