Lipschitz-type conditions on homogeneous Banach spaces of analytic functions

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• 作者：Oscar Blasco^a ; ¹ ; ^{oblasco@uv.es" class="auth_mail" title="E-mail the corresponding author} ; Georgios Stylogiannis^b ; ^{stylog@math.auth.gr" class="auth_mail" title="E-mail the corresponding author} ; ^{g.stylog@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Banach spaces ; Lipschitz-type conditions ; Approximation by partial sums
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：445
• 期：1
• 页码：612-630
• 全文大小：387 K

文摘

In this paper we deal with Banach spaces of analytic functions X   defined on the unit disk satisfying that d="mmlsi1" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=9e41b59e0d39d55b3e17c56f5a20b3f0" title="Click to view the MathML source">R_tf∈Xdden">de">mi>Rmi><mi>tmi><mi>fmi>∈<mi>Xmi> for any d="mmlsi2" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si2.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=653e962132a2a043da43c6f2e2e862fd" title="Click to view the MathML source">t>0dden">de">mi>tmi>>0 and d="mmlsi3" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si3.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=7cce2c1e426ebd5343134f03376773ba" title="Click to view the MathML source">f∈Xdden">de">mi>fmi>∈<mi>Xmi>, where d="mmlsi4" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si4.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=d3b24d9a9134cf0350e164af3f95adce" title="Click to view the MathML source">R_tf(z)=f(e^itz)dden">de">mi>Rmi><mi>tmi><mi>fmi>(<mi>zmi>)=<mi>fmi>(<mi>emi><mi>imi><mi>tmi><mi>zmi>). We study the space of functions in X   such that d="mmlsi5" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si5.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=df60d8b0f633c3b9ac9117bb542a8bb6">dth="174" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304012-si5.gif">dden">de">mi>Pmi><mi>rmi>(<mi>Dmi><mi>fmi>)‖<mi>Xmi>=<mi>Omi>(<mi>ωmi>(1&minus;<mi>rmi>)1&minus;<mi>rmi>), d="mmlsi6" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si6.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=2dfc7f7d4bd29e0478efef28a572f697" title="Click to view the MathML source">r→1^&minus;dden">de">mi>rmi>→1&minus; where d="mmlsi7" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si7.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=c49ccaa1c22b69a6d7af9a00baf09b9e">dth="193" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304012-si7.gif">dden">de">mi>Dmi><mi>fmi>(<mi>zmi>)=∑<mi>nmi>=0∞(<mi>nmi>+1)<mi>ami><mi>nmi><mi>zmi><mi>nmi> and ω is a continuous and non-decreasing weight satisfying certain mild assumptions. The space under consideration is shown to coincide with the subspace of functions in X   satisfying any of the following conditions: (a) d="mmlsi8" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si8.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=6c7de8a366cabfa20757bdd49ae6b3e7" title="Click to view the MathML source">‖R_tf&minus;f‖_X=O(ω(t))dden">de">mi>Rmi><mi>tmi><mi>fmi>&minus;<mi>fmi>‖<mi>Xmi>=<mi>Omi>(<mi>ωmi>(<mi>tmi>)), (b) d="mmlsi88" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si88.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=8fc6135e1c34e45209f79b0c4a94a141" title="Click to view the MathML source">‖P_rf&minus;f‖_X=O(ω(1&minus;r))dden">de">mi>Pmi><mi>rmi><mi>fmi>&minus;<mi>fmi>‖<mi>Xmi>=<mi>Omi>(<mi>ωmi>(1&minus;<mi>rmi>)), (c) d="mmlsi10" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si10.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=526fe1e29d061c1eaae46d74132ff9cd" title="Click to view the MathML source">‖Δ_nf‖_X=O(ω(2^&minus;n))dden">de">mi mathvariant="normal">Δmi><mi>nmi><mi>fmi>‖<mi>Xmi>=<mi>Omi>(<mi>ωmi>(2&minus;<mi>nmi>)), or (d) d="mmlsi11" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si11.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=33571ec83d8f9cdc78759394ed6a567c" title="Click to view the MathML source">‖f&minus;s_nf‖_X=O(ω(n^&minus;1))dden">de">mi>fmi>&minus;<mi>smi><mi>nmi><mi>fmi>‖<mi>Xmi>=<mi>Omi>(<mi>ωmi>(<mi>nmi>&minus;1)), where d="mmlsi12" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si12.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=8e19c1accea6fd7969f55a0b1095afb8" title="Click to view the MathML source">P_rf(z)=f(rz)dden">de">mi>Pmi><mi>rmi><mi>fmi>(<mi>zmi>)=<mi>fmi>(<mi>rmi><mi>zmi>), d="mmlsi13" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si13.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=c910845d630e4641062d2a5314de0384">dth="146" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304012-si13.gif">dden">de">mi>smi><mi>nmi><mi>fmi>(<mi>zmi>)=∑<mi>kmi>=0<mi>nmi><mi>ami><mi>kmi><mi>zmi><mi>kmi> and d="mmlsi14" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si14.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=8ace7f9b4b1464072a3a84b239ed393c" title="Click to view the MathML source">Δ_nf=s_2ⁿf&minus;s_2^n&minus;1fdden">de">mi mathvariant="normal">Δmi><mi>nmi><mi>fmi>=<mi>smi>2<mi>nmi><mi>fmi>&minus;<mi>smi>2<mi>nmi>&minus;1<mi>fmi>. Our results extend those known for Hardy or Bergman spaces and power weights d="mmlsi15" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si15.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=56e302821347adb809361e9a188ad623" title="Click to view the MathML source">ω(t)=t^αdden">de">mi>ωmi>(<mi>tmi>)=<mi>tmi><mi>αmi>.