Fully measurable small Lebesgue spaces

详细信息    查看全文

• 作者：Giuseppina Anatriello<sup>asup><sup> ; sup><sup>anatriello@unina.it" class="auth_mail" title="E-mail the corresponding authorsup> ; Maria Rosaria Formica<sup>bsup><sup> ; sup><sup>mara.formica@uniparthenope.it" class="auth_mail" title="E-mail the corresponding authorsup> ; Raffaella Giova<sup>bsup><sup> ; sup><sup>raffaella.giova@uniparthenope.it" class="auth_mail" title="E-mail the corresponding authorsup>
• 关键词：Banach function spaces ; Grand and small Lebesgue spaces ; Measurable exponent ; Hö ; lder-type inequality
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：447
• 期：1
• 页码：550-563
• 全文大小：364 K

文摘

We build a new class of Banach function spaces, whose function norm is where <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si2.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=288bfcb2563e0ce0f2027172f281363f" title="Click to view the MathML source">ρ<sub>p(x)sub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>ρpstretchy="false">(xstretchy="false">)sub>span>span>span> denotes the norm of the Lebesgue space of exponent <span id="mmlsi227" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si227.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=eda3d5d66f75fe3877f812ff840c9fc3" title="Click to view the MathML source">p(x)span><span class="mathContainer hidden"><span class="mathCode">227.gif" overflow="scroll">pstretchy="false">(xstretchy="false">)span>span>span> (assumed measurable and possibly infinite), constant with respect to the variable of f, and δ   is measurable, too. Such class contains some known Banach spaces of functions, among which are the classical and the small Lebesgue spaces, and the Orlicz space <span id="mmlsi4" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si4.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=06d1119e119156ecdc74f00154d4a162" title="Click to view the MathML source">L(log⁡L)<sup>αsup>span><span class="mathContainer hidden"><span class="mathCode">scroll">Lsup>stretchy="false">(log⁡Lstretchy="false">)αsup>span>span>span>, <span id="mmlsi166" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si166.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=1e37bfbf7e131042375db0855073c0ae" title="Click to view the MathML source">α>0span><span class="mathContainer hidden"><span class="mathCode">scroll">α>0span>span>span>.

sp0020">Furthermore we prove the following Hölder-type inequality

where <span id="mmlsi7" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si7.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=7b9024614e3681ac08d9e06ff1401e1f" title="Click to view the MathML source">ρ<sub>p[&sdot;]),δ[&sdot;]sub>(f)span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>ρpstretchy="false">[&sdot;stretchy="false">]stretchy="false">),δstretchy="false">[&sdot;stretchy="false">]sub>stretchy="false">(fstretchy="false">)span>span>span> is the norm of fully measurable grand Lebesgue spaces introduced by Anatriello and Fiorenza in <span id="bbr0020">[2]span>. For suitable choices of <span id="mmlsi227" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si227.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=eda3d5d66f75fe3877f812ff840c9fc3" title="Click to view the MathML source">p(x)span><span class="mathContainer hidden"><span class="mathCode">227.gif" overflow="scroll">pstretchy="false">(xstretchy="false">)span>span>span> and <span id="mmlsi8" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si8.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=5f19f2b710b401c4c884827be4da994c" title="Click to view the MathML source">δ(x)span><span class="mathContainer hidden"><span class="mathCode">scroll">δstretchy="false">(xstretchy="false">)span>span>span> it reduces to the classical Hölder's inequality for the spaces <span id="mmlsi9" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si9.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=33f6ababbdbb44aae401e2ccfb570ba2" title="Click to view the MathML source">EXP<sub>1/αsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">EXsub>P1stretchy="false">/αsub>span>span>span> and <span id="mmlsi4" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si4.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=06d1119e119156ecdc74f00154d4a162" title="Click to view the MathML source">L(log⁡L)<sup>αsup>span><span class="mathContainer hidden"><span class="mathCode">scroll">Lsup>stretchy="false">(log⁡Lstretchy="false">)αsup>span>span>span>, <span id="mmlsi166" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306308&_mathId=si166.gif&_user=111111111&_pii=S0022247X16306308&_rdoc=1&_issn=0022247X&md5=1e37bfbf7e131042375db0855073c0ae" title="Click to view the MathML source">α>0span><span class="mathContainer hidden"><span class="mathCode">scroll">α>0span>span>span>.