文摘
In this paper we consider conical square functions in the Bessel, Laguerre and Schrödinger settings where the functions take values in UMD Banach spaces. Following a recent paper of Hytönen, van Neerven and Portal [36], in order to define our conical square functions, we use γ -radonifying operators. We obtain new equivalent norms in the Lebesgue–Bochner spaces class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305923&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305923&_rdoc=1&_issn=0022247X&md5=78048556e6010c0c9ce14e0190f66b8a" title="Click to view the MathML source">Lp((0,∞),B)class="mathContainer hidden">class="mathCode"> and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305923&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305923&_rdoc=1&_issn=0022247X&md5=e7cb7d8ef0fbdc068f3f46fdc7eae96b" title="Click to view the MathML source">Lp(Rn,B)class="mathContainer hidden">class="mathCode">, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305923&_mathId=si22.gif&_user=111111111&_pii=S0022247X16305923&_rdoc=1&_issn=0022247X&md5=11d4508e796370f7c892b12163739bd3" title="Click to view the MathML source">1<p<∞class="mathContainer hidden">class="mathCode">, in terms of our square functions, provided that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305923&_mathId=si4.gif&_user=111111111&_pii=S0022247X16305923&_rdoc=1&_issn=0022247X&md5=59459551365f1d02c5b08a4436a15270" title="Click to view the MathML source">Bclass="mathContainer hidden">class="mathCode"> is a UMD Banach space. Our results can be seen as Banach valued versions of known scalar results for square functions.