By introducing a kind of maximal operator of fractional order associated with the mean Luxemburg norm and using the technique of Sharp function,multilinear commutators of fractional integral operator with vector symbo...By introducing a kind of maximal operator of fractional order associated with the mean Luxemburg norm and using the technique of Sharp function,multilinear commutators of fractional integral operator with vector symbol b=(b 1,...,b m)defined byI b αf(x)=∫ R∏mj=1(b j(x)-b j(y))1｜x-y｜ n-αf(y)dyare considered.The following priori estimates are proved.For 1<p<∞,there exists a constant c such that‖I b αf‖ Lp(Rn)≤c‖b‖‖M L(logL) 1r,α(f)‖ Lp(Rn),So‖I b αf‖ Lq(Rn)≤c‖b‖‖f‖ Lp(Rn),where 1<p<nα,1q=1p-αn,0<α<n,and supt>01Φ1t｜｛y∈Rn:｜I b αf(y)｜>t｝｜ 1q≤ c supt>01Φ1t｜｛y∈Rn:M L(logL) 1r,α(‖b‖f)(y)>t｝｜ 1q,where ‖b‖=∏ m j=1‖b j‖ Osc expL r j,Φ(t)=t(1+log+t) 1r,1r=1r 1+...+1r m, M L(logL) 1r,α is an Orlicz type maximal operator.展开更多
文摘By introducing a kind of maximal operator of fractional order associated with the mean Luxemburg norm and using the technique of Sharp function,multilinear commutators of fractional integral operator with vector symbol b=(b 1,...,b m)defined byI b αf(x)=∫ R∏mj=1(b j(x)-b j(y))1｜x-y｜ n-αf(y)dyare considered.The following priori estimates are proved.For 1<p<∞,there exists a constant c such that‖I b αf‖ Lp(Rn)≤c‖b‖‖M L(logL) 1r,α(f)‖ Lp(Rn),So‖I b αf‖ Lq(Rn)≤c‖b‖‖f‖ Lp(Rn),where 1<p<nα,1q=1p-αn,0<α<n,and supt>01Φ1t｜｛y∈Rn:｜I b αf(y)｜>t｝｜ 1q≤ c supt>01Φ1t｜｛y∈Rn:M L(logL) 1r,α(‖b‖f)(y)>t｝｜ 1q,where ‖b‖=∏ m j=1‖b j‖ Osc expL r j,Φ(t)=t(1+log+t) 1r,1r=1r 1+...+1r m, M L(logL) 1r,α is an Orlicz type maximal operator.
