Fully measurable small Lebesgue spaces

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• 作者：Giuseppina Anatriello^a ; ^{anatriello@unina.it" class="auth_mail" title="E-mail the corresponding author} ; Maria Rosaria Formica^b ; ^{mara.formica@uniparthenope.it" class="auth_mail" title="E-mail the corresponding author} ; Raffaella Giova^b ; ^{raffaella.giova@uniparthenope.it" class="auth_mail" title="E-mail the corresponding author}
• 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 mathmlsrc">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_p(x)mathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll">ρp(x)math> denotes the norm of the Lebesgue space of exponent mathmlsrc">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)mathContainer hidden">mathCode"><math altimg="si227.gif" overflow="scroll">p(x)math> (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 mathmlsrc">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)^αmathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll">L(mathvariant="normal">log⁡L)αmath>, mathmlsrc">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>0mathContainer hidden">mathCode"><math altimg="si166.gif" overflow="scroll">α>0math>.

Furthermore we prove the following Hölder-type inequality

where mathmlsrc">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_{p[⋅]),δ[⋅]}(f)mathContainer hidden">mathCode"><math altimg="si7.gif" overflow="scroll">ρp[⋅]),δ[⋅](f)math> is the norm of fully measurable grand Lebesgue spaces introduced by Anatriello and Fiorenza in [2]. For suitable choices of mathmlsrc">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)mathContainer hidden">mathCode"><math altimg="si227.gif" overflow="scroll">p(x)math> and mathmlsrc">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)mathContainer hidden">mathCode"><math altimg="si8.gif" overflow="scroll">δ(x)math> it reduces to the classical Hölder's inequality for the spaces mathmlsrc">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_1/αmathContainer hidden">mathCode"><math altimg="si9.gif" overflow="scroll">EXP1/αmath> and mathmlsrc">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)^αmathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll">L(mathvariant="normal">log⁡L)αmath>, mathmlsrc">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>0mathContainer hidden">mathCode"><math altimg="si166.gif" overflow="scroll">α>0math>.