{"id":154,"date":"2015-11-23T23:25:16","date_gmt":"2015-11-23T15:25:16","guid":{"rendered":"https:\/\/blog.vanabel.info\/?p=154"},"modified":"2016-12-07T05:28:53","modified_gmt":"2016-12-07T05:28:53","slug":"campanato-kong-jian-lpn-yu-bmokong-jian-de-deng-jia-xing","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=154","title":{"rendered":"Campanato\u7a7a\u95f4$L^{p,n}$\u4e0eBMO\u7a7a\u95f4\u7684\u7b49\u4ef7\u6027"},"content":{"rendered":"

\u5047\u8bbe $\\Omega\\subset\\RR^n$ \u662f\u4e00\u4e2a\u6709\u754c\u533a\u57df. \u6211\u4eec\u79f0\u5176\u4e3a$(A)$-\u578b\u57df, \u5982\u679c\u5b58\u5728\u5e38\u6570$A$, \u4f7f\u5f97
\n\\[
\n|\\Omega\\cap B_r(x)|\\geq A r^n.
\n\\]
\n\u5b9a\u4e49Campanato\u7a7a\u95f4 $L^{p,\\lambda}(\\Omega)$ \u5982\u4e0b: \u5bf9 $1\\leq p< +\\infty$, $\\lambda >0$, \u79f0 $f\\in L^p(\\Omega)$ \u5c5e\u4e8e $L^{p,\\lambda}(\\Omega)$ \u5982\u679c
\n\\[[f]_{L^{p,\\lambda};\\Omega}\\eqdef\\sup_{x\\in\\Omega, r>0}\\dkf{\\frac{1}{r^\\lambda}\\int_{\\Omega_r(x)}|f-f_{x,r}|^p}^{1\/p}<+\\infty, \\]
\n
\n\u5176\u4e2d$\\Omega_r(x)=\\Omega\\cap B_r(x)$, $f_{x,r}$ \u5b9a\u4e49\u4e3a \\[ f_{x,r}\\eqdef\\frac{1}{|\\Omega_r(x)|}\\int_{\\Omega_r(x)}fdx. \\]
\n\u5b9a\u4e49BMO(\u6709\u754c\u5e73\u5747\u9707\u8361)\u7a7a\u95f4\u5982\u4e0b: \u5047\u8bbe $f\\in L^1_{\\loc}(\\Omega)$, \u79f0 $f\\in\\BMO(\\Omega)$, \u5982\u679c
\n\\[ [f]_{\\BMO;\\Omega}\\eqdef\\sup_{x\\in\\Omega, r>0}\\frac{1}{|\\Omega_r(x)|}\\int_{\\Omega_r(x)}|f-f_{x,r}|dx< +\\infty. \\]
\n\u7531\u4e8e\u5728Caleron-Zygmund\u5206\u89e3\u4e2d\u5e38\u7528\u7684\u662f\u65b9\u4f53, \u6211\u4eec\u9996\u5148\u6765\u770b\u4e0a\u8ff0\u5b9a\u4e49\u4e2d, \u65b9\u4f53\u548c\u7403\u662f\u7b49\u4ef7\u7684.

\u547d\u9898 1<\/span>.<\/span> \u5047\u8bbe$\\Omega\\subset\\RR^N$\u662f$(A)$-\u578b\u6709\u754c\u57df, \u5219\u5b58\u5728\u5e38\u6570$C(n,A)>0$, \u4f7f\u5f97
\n\\[
\n\\frac{1}{C(n,A)}\\|f\\|_{\\BMO(\\Omega);\\mathrm{balls}}\\leq\\|f\\|_{\\BMO(\\Omega);\\mathrm{cubes}}\\leq C(n,A)\\|f\\|_{\\BMO(\\Omega);\\mathrm{balls}}.
\n\\]
\n<\/div>
\n\u4e8b\u5b9e\u4e0a, \u6211\u4eec\u53ea\u9700\u89c2\u5bdf\u5230\u5bf9\u4efb\u4e00\u65b9\u4f53$Q_r$, \u7403$B_{r}\\subset Q\\subset B_{\\sqrt nr}$,
\n\\begin{align*}
\n\\frac{1}{|Q_r\\cap\\Omega|}\\int_{Q_r\\cap\\Omega}|f-f_{x,\\sqrt nr}|&\\leq\\frac{|B_{\\sqrt nr}\\cap\\Omega|}{|Q_r\\cap\\Omega|}\\frac{1}{|B_{\\sqrt nr}\\cap\\Omega|}\\int_{B_{\\sqrt nr}\\cap\\Omega}|f-f_{x,\\sqrt nr}|\\\\
\n&\\leq\\frac{|B_1|(\\sqrt n)^n}{A}\\frac{1}{|B_{\\sqrt nr}\\cap\\Omega|}\\int_{B_{\\sqrt nr}\\cap\\Omega}|f-f_{x,\\sqrt nr}|\\\\
\n&\\leq C(n,A)\\|f\\|_{\\BMO(\\Omega);\\mathrm{balls}},
\n\\end{align*}
\n\u5176\u4e2d
\n\\[
\nf_{x,\\sqrt nr}\\eqdef\\frac{1}{|B_{\\sqrt nr}(x)\\cap\\Omega|}\\int_{B_{\\sqrt nr}(x)\\cap\\Omega}f(y)dy.
\n\\]
\n\u4ee4
\n\\[
\nf_{x,Q_r}\\eqdef\\frac{1}{|Q_r\\cap\\Omega|}\\int_{Q_r\\cap\\Omega}f(y)dy.
\n\\]
\n\u7ed3\u5408
\n\\begin{align*}
\n\\frac{1}{|Q_r\\cap\\Omega|}\\int_{Q_r\\cap\\Omega}|f-f_{x,Q_r}|&\\leq
\n\\frac{1}{|Q_r\\cap\\Omega|}\\int_{Q_r\\cap\\Omega}|f-f_{x,\\sqrt nr}|+|f_{x,\\sqrt nr}-f_{x,Q_r}|\\\\
\n&\\leq C(n,A)\\|f\\|_{\\BMO(\\Omega);\\mathrm{balls}}\\\\
\n&\\qquad+\\frac{1}{|Q_r\\cap\\Omega|}\\int_{Q_r\\cap\\Omega}|f_{x,\\sqrt nr}-f(y)|dy\\\\
\n&\\leq C(n,A)\\|f\\|_{\\BMO(\\Omega);\\mathrm{balls}},
\n\\end{align*}
\n\u77e5
\n\\[
\n\\|f\\|_{\\BMO(\\Omega);\\mathrm{cubes}}\\leq C(n,A)\\|f\\|_{\\BMO(\\Omega);\\mathrm{balls}}.
\n\\]
\n\u53cd\u8fc7\u6765,
\n\\begin{align*}
\n\\frac{1}{|\\Omega_{x,r}|}\\int_{\\Omega_{x,r}}|f-f_{Q_r}|
\n&\\leq \\frac{|Q_r\\cap\\Omega|}{|\\Omega_{x,r}|}\\frac{1}{|Q_r\\cap\\Omega|} \\int_{Q_r\\cap\\Omega}|f-f_{Q_r}|\\\\
\n&\\leq \\frac{2^n}{A}\\|f\\|_{\\BMO(\\Omega);\\mathrm{cubes}}\\\\
\n&\\leq C'(n,A)\\|f\\|_{\\BMO(\\Omega);\\mathrm{cubes}},
\n\\end{align*}
\n\u7136\u540e\u5229\u7528\u4e0a\u9762\u540c\u6837\u7684\u6280\u5de7\u8bf4\u660e
\n\\[
\n\\|f\\|_{\\BMO(\\Omega);\\mathrm{balls}}\\leq C'(n,A)\\|f\\|_{\\BMO(\\Omega);\\mathrm{cubes}}.
\n\\]
\n\u5bf9\u4e00\u822c\u7684$L^{p,\\lambda}(\\Omega)$, \u6211\u4eec\u4e5f\u6709\u4e0a\u8ff0\u7b49\u4ef7\u5173\u7cfb.
\n
\u547d\u9898 2<\/span>.<\/span> \u5047\u8bbe$\\Omega\\subset\\RR^n$\u662f\u6709\u754c\u57df, $1\\leq p<+\\infty$, $\\lambda>0$, \u5b58\u5728\u5e38\u6570$C(p)=2^p>0$, \u4f7f\u5f97, \u5bf9\u4efb\u610f\u7684 $f\\in L^p(\\Omega)$,
\n\\[
\n\\frac{1}{C(p)}\\|f\\|_{L^{p,\\lambda}(\\Omega);\\mathrm{balls}}\\leq\\|f\\|_{L^{p,\\lambda}(\\Omega);\\mathrm{cubes}}\\leq C(p)\\|f\\|_{L^{p,\\lambda}(\\Omega);\\mathrm{balls}}.
\n\\]
\n<\/div>
\n
\u8bc1\u660e<\/span>.<\/span> \u8bc1\u660e\u4eff\u7167$\\BMO$ ($\\BMO=L^{1,n}$). \u975e\u5e38\u5bb9\u6613\u5f97\u5230\u4f30\u8ba1
\n\\begin{align*}
\n\\frac{1}{r^\\lambda}\\int_{Q_r\\cap\\Omega}|f-f_{x,\\sqrt nr}|^pdx&\\leq
\n\\frac{1}{r^\\lambda}\\int_{B_{\\sqrt nr}\\cap\\Omega}|f-f_{x,\\sqrt nr}|^pdx\\\\
\n&\\leq \\|f\\|^p_{L^{p,\\lambda}(\\Omega);\\mathrm{balls}},
\n\\end{align*}
\n\u4ee5\u53ca
\n\\begin{align*}
\n\\frac{1}{r^\\lambda}\\int_{B_r\\cap\\Omega}|f-f_{x,Q_r}|^pdx
\n&\\leq\\frac{1}{r^\\lambda}\\int_{Q_r\\cap\\Omega}|f-f_{x,Q_r}|^pdx\\\\
\n&\\leq \\|f\\|^p_{L^{p,\\lambda}(\\Omega);\\mathrm{cubes}}.
\n\\end{align*}
\n\u63a5\u4e0b\u6765\u7684\u63d2\u503c\u6280\u5de7\u53d8\u5316\u4e3a(\u5229\u7528H\\”older\u4e0d\u7b49\u5f0f),
\n\\begin{align*}
\n\\frac{1}{r^{\\lambda}}\\int_{Q_r\\cap\\Omega}|f-f_{x,Q_r}|^pdx
\n&\\leq 2^{p-1}\\Biggl\\{\\frac{1}{r^{\\lambda}}\\int_{Q_r\\cap\\Omega}|f-f_{x,\\sqrt nr}|^pdx\\\\
\n&\\qquad\\qquad+\\frac{1}{r^{\\lambda}}|Q_r\\cap\\Omega||f_{x,\\sqrt nr}-f_{x,Q_r}|^p\\Biggr\\}\\\\
\n&\\leq 2^{p-1}\\xkf{\\|f\\|^p_{L^{p,\\lambda}(\\Omega);\\mathrm{balls}}
\n+r^{-\\lambda}\\|f-f_{x,\\sqrt nr}\\|^p_{L^p(Q_r\\cap\\Omega)}}\\\\
\n&\\leq 2^p\\|f\\|^p_{L^{p,\\lambda}(\\Omega);\\mathrm{balls}},
\n\\end{align*}
\n\u53ef\u89c1
\n\\[
\n\\|f\\|^p_{L^{p,\\lambda}(\\Omega);\\mathrm{cubes}}\\leq 2^p\\|f\\|^p_{L^{p,\\lambda}(\\Omega);\\mathrm{balls}}.
\n\\]
\n\u5b8c\u5168\u7c7b\u4f3c\u8bc1\u660e
\n\\[
\n2^{-p}\\|f\\|^p_{L^{p,\\lambda}(\\Omega);\\mathrm{balls}}
\n\\leq\\|f\\|^p_{L^{p,\\lambda}(\\Omega);\\mathrm{cubes}}.
\n\\]
\n<\/div>
\n\u63a5\u4e0b\u6765, \u6211\u4eec\u5c06\u8bc1\u660e
\n
\u5b9a\u7406 3<\/span>.<\/span> \u5047\u8bbe $\\Omega\\subset\\RR^n$ \u662f $(A)$-\u578b\u6709\u754c\u57df. \u5bf9\u6240\u6709\u7684 $1\\leq p<+\\infty$, \u5b58\u5728\u5e38\u6570 $C(n,p,A)>0$, \u4f7f\u5f97
\n\\[
\n\\frac{1}{C(n,p,A)}\\|f\\|_{\\BMO(\\Omega)}\\leq\\|f\\|_{L^{p,n}(\\Omega)}\\leq C(n,p,A)\\|f\\|_{\\BMO(\\Omega)}.
\n\\]
\n<\/div>
\n
\u8bc1\u660e<\/span>.<\/span> \u9996\u5148\u56de\u5fc6 $\\|f\\|_{L^p(\\Omega)}$ \u7684\u5206\u522b\u5e03\u5b9a\u4e49. \u7531Fubini\u5b9a\u7406\u6613\u77e5,
\n\\begin{align*}
\n\\int_0^\\infty|\\set{x\\in\\Omega:|f(x)|>t}|dt&=\\int_0^\\infty\\int_\\Omega\\chi_{|f(x)|>t}(x)dxdt\\\\
\n&=\\int_{\\Omega}\\int_0^\\infty\\chi_{t<|f(x)|}(t)dt dx\\\\ &=\\int_\\Omega\\int_0^{|f(x)|}dx\\\\ &=\\int_{\\Omega}|f(x)|dx. \\end{align*} \u73b0\u5728, \u5bb9\u6613\u5f97\u5230 \\begin{align*} \\int_\\Omega|f(x)|^pdx&=\\int_0^\\infty|\\set{x\\in\\Omega:|f(x)|^p>t}|dt\\\\
\n&=\\int_0^\\infty|\\set{x\\in\\Omega:|f(x)|>\\alpha}|d(\\alpha^p)\\\\
\n&=p\\int_0^\\infty\\alpha^{p-1}|\\set{x\\in\\Omega:|f(x)|>\\alpha}|d\\alpha.
\n\\end{align*}
\n<\/div>
\n\u4e0b\u9762, \u6211\u4eec\u9700\u8981\u5982\u4e0b\u91cd\u8981\u7684John-Nirenberg\u5f15\u7406.
\n
\u5f15\u7406 4<\/span>.<\/span> \u5047\u8bbe $\\Omega=Q_0$ \u662f\u4e00\u4e2a\u65b9\u4f53, \u5219\u5bf9\u4efb\u610f\u7684 $f\\in\\BMO(\\Omega)$, \u5b58\u5728\u5e38\u6570 $C_1(n)=e^{2^ne}$, $C_2(n)=(2^ne)^{-1}$, \u4f7f\u5f97
\n\\[
\n|\\set{x\\in Q:|f-f_Q|>\\alpha}|\\leq C_1(n)|Q|e^{-C_2(n)\\alpha\/\\|f\\|_{\\BMO(\\Omega)}},\\quad\\forall Q\\subset\\Omega.
\n\\]
\n\u5176\u4e2d$f_Q=\\frac{1}{|Q|}\\int_Qf(x)dx$.
\n<\/div>
\n\u5176\u8bc1\u660e\u53ef\u53c2\u8003(Han Qin \\& Lin Fang-hua, p. 53ff).<\/p>\n

\u5229\u7528\u4e0a\u8ff0\u5f15\u7406, \u6211\u4eec\u5f97\u5230
\n\\begin{align*}
\n\\frac{1}{|Q|}\\int_Q|f(x)-f_Q|^pdx&=\\frac{p}{|Q|}\\int_0^\\infty\\alpha^{p-1}|\\set{x\\in Q:|f(x)-f_Q|>\\alpha}|d\\alpha\\\\
\n&\\leq \\frac{C_1(n)p}{|Q|}|Q|\\int_0^\\infty\\alpha^{p-1}e^{-C_2(n)\\alpha\/\\|f\\|_{BMO(\\Omega)}}d\\alpha\\\\
\n&=C_1(n)p\\Gamma(p)\\frac{\\|f\\|_{\\BMO(\\Omega)}^p}{C_2(n)^p}.
\n\\end{align*}
\n\u5176\u4e2d, $\\Gamma(p)=\\int_0^\\infty\\alpha^{p-1}e^{-\\alpha}d\\alpha$ \u662fGamma\u51fd\u6570(\u5bf9$p>0$\u7edd\u5bf9\u6536\u655b). \u5229\u7528$L^{p,\\lambda}$\u4e0e$\\BMO$\u5b9a\u4e49\u4e2d\u65b9\u4f53\u4e0e\u7403\u7ed9\u51fa\u7684\u5b9a\u4e49\u7684\u7b49\u4ef7\u6027, \u5bb9\u6613\u5f97\u5230
\n\\[
\n\\|f\\|_{L^{p,n}(\\Omega)}\\leq C(n,p,A)\\|f\\|_{\\BMO(\\Omega)}.
\n\\]
\n\u4e0d\u7b49\u5f0f\u7684\u53e6\u4e00\u8fb9\u662fH\\”older\u4e0d\u7b49\u5f0f\u7684\u76f4\u63a5\u63a8\u8bba.<\/p>\n","protected":false},"excerpt":{"rendered":"

\u5047\u8bbe $\\Omega\\subset\\RR^n$ \u662f\u4e00\u4e2a\u6709\u754c\u533a\u57df. \u6211\u4eec\u79f0\u5176\u4e3a$(A)$-\u578b\u57df, \u5982\u679c\u5b58\u5728\u5e38\u6570$… \u7ee7\u7eed\u9605\u8bfbCampanato\u7a7a\u95f4$L^{p,n}$\u4e0eBMO\u7a7a\u95f4\u7684\u7b49\u4ef7\u6027<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[5,6,13,44],"class_list":["post-154","post","type-post","status-publish","format-standard","hentry","category-math","tag-bmo","tag-campanato","tag-john-nirenberg","tag-44","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/154","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=154"}],"version-history":[{"count":5,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/154\/revisions"}],"predecessor-version":[{"id":278,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/154\/revisions\/278"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=154"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=154"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=154"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}