Sobolev's inequality for Riesz potentials of functions in Morrey spaces of integral form (Q2786309)

If \(n\) is a positive integer, the Euclidean \(n\)-dimensional spaces are denoted by \(\mathbb{R}^n\), and the ball \(\{y= (y_1,\dots, y_n)\in \mathbb{R}^n:|x- y|\leq r\}\) with centre \(x= (x_1,\dots, x_n)\in\mathbb{R}^n\) is denoted by \(B(x,r)\) for \(r>0\), where \(|x|\) denotes the norm \((x^2_1+ \cdots+ x^2_n)^{1/2}\).NEWLINENEWLINEIf \(1<p<\infty\), \(0< v< n\) and \(G\) is an open subset of \(\mathbb{R}^n\) with diameter \(d(G)\), and if \(\phi_1\), \(\phi_2\) are positive functions on \((0,\infty)\) such thatNEWLINENEWLINE(*1) (i) \(r^{-v}\phi_1(r)\), \(r> 0\), is non-increasing; (ii) \(\phi_2(r)\), \(r> 0\), is non-decreasing; (iii) there is a constant \(c> 0\), such that \(c^{-1}\phi_j(r)\leq \phi_j(r^2)\leq c\phi_j(r)\), \(r> 0\), \(j= 1,2\), then the generalized Morrey space, \(L^p(G)= L^{p,v,\phi_1,\phi_2}(G)\), and the generalized Morrey space of integral form, \({\mathcal L}^p(G)= {\mathcal L}^{p,v,\phi_1,\phi_2}(G)\), are defined byNEWLINENEWLINE(*2) (i) \(L^p(G)= \{f\) locally integrable on \(G:\| f\|_{p,G}=\| f\|_{p,v,\phi_1, \phi_2 G}<\infty\}\), (ii) \({\mathcal L}^p(G)= \{f\) locally integrable on \(G:|||f|||_{p,G}= |||f|||_{p,v,\phi_1,\phi_2 G}< \infty\}\), where NEWLINENEWLINE\[NEWLINE(\| f\|_{p,G})^p= \sup_{x\in G,0< r< d(G)}r^{-v}\phi_1(r) \int_{B(x,r)}|f(y)|^p \phi_2(|f(y)|)\,dy,NEWLINE\]NEWLINE NEWLINEand NEWLINENEWLINE\[NEWLINE|||f|||_{p, G}= \sup_{x\in G}\Biggl(\int^{d(G)}_0 r^{-v-1} \phi_1(r)\Biggl(\int_{B(x, r)} |f(y)|^p\phi_2(|f(y)|)\, dy\Biggr)\, dr\Biggr)^{1/p}.NEWLINE\]NEWLINE NEWLINEThe main results of this paper relate especially to estimates involving the Riesz potential operator \({\mathcal U}_\alpha\), \(0<\alpha< n\), on Morrey spaces, where NEWLINENEWLINE\[NEWLINE{\mathcal U}_\alpha(f)(x)= \int_{\mathbb{R}^n} |x-y|^{\alpha-n} f(y)\,dy.NEWLINE\]NEWLINENEWLINEIn particular, Sobolev estimates in \(L^p\)-spaces are indicated as applicable in Morrey spaces in forms including:NEWLINENEWLINE(*3) (i) if \(v= n-ap> 0\) and \(G\) is a bounded open set, then there is a finite constant \(c> 0\), such that NEWLINENEWLINE\[NEWLINE|||\Psi_{p,v,\phi, G}(c|{\mathcal U}_\alpha(f)|)|||_{p,v,\phi_1,\phi_2 G}\leq 1,NEWLINE\]NEWLINE NEWLINEfor \(|||f|||_{p, v,\phi_1,\phi_2 G}\leq 1\), where NEWLINENEWLINE\[NEWLINE\Psi_{p,v,\phi, G}(r)= \{(\Phi_{p,v, \phi,G})^{-1}(r)\}^{(v-n)/p} \phi(r)^{-1/p};NEWLINE\]NEWLINENEWLINENEWLINE\[NEWLINE\Phi_{p,v,\phi, G}(r^{-1})= \Biggl(\int^{d(G)}_r |t^{v-ap-n} \phi(t^{-1})|^{-p'/p} t^{-1} dt\Biggr)^{1,p'},NEWLINE\]NEWLINENEWLINE\(\phi(r)= \phi_1(r^{-1})\phi_2(r)\), \((1/p)+ (1/p')= 1\).NEWLINENEWLINEThere are references in the main text to earlier results extended in the paper.