Area integrals associated to self-adjoint operators on vector-valued generalized weighted Morrey spaces

Abstract In this paper, we mainly investigate the vector-valued analogs of area integrals associated with self-adjoint operators satisfying Gaussian upper bounds, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>S</m:mi> <m:mi>P</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mo maxsize="260%" minsize="260%">(</m:mo> <m:mrow> <m:mstyle displaystyle="true"> <m:msubsup> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mn>0</m:mn> <m:mi mathvariant="normal">∞</m:mi> </m:msubsup> </m:mstyle> <m:mrow> <m:mstyle displaystyle="true"> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mrow> <m:mi>B</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msub> </m:mstyle> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>⁢</m:mo> <m:msqrt> <m:mi>L</m:mi> </m:msqrt> <m:mo>⁢</m:mo> <m:msup> <m:mi>e</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>⁢</m:mo> <m:msqrt> <m:mi>L</m:mi> </m:msqrt> </m:mrow> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mstyle displaystyle="true"> <m:mfrac> <m:mrow> <m:mi>d</m:mi> <m:mo>⁢</m:mo> <m:mpadded width="+1.7pt"> <m:mi>y</m:mi> </m:mpadded> <m:mo>⁢</m:mo> <m:mi>d</m:mi> <m:mo>⁢</m:mo> <m:mi>t</m:mi> </m:mrow> <m:msup> <m:mi>t</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:mfrac> </m:mstyle> </m:mrow> </m:mrow> </m:mrow> <m:mo maxsize="260%" minsize="260%">)</m:mo> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:msup> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> see text \displaystyle S_{P}f(x)=\Biggl{(}\int_{0}^{\infty}\int_{B(x,t)}|t\sqrt{L}e^{-t% \sqrt{L}}f(y)|^{2}\frac{dy\,dt}{t^{n+1}}\Biggr{)}^{\frac{1}{2}}, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>S</m:mi> <m:mi>H</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mo maxsize="260%" minsize="260%">(</m:mo> <m:mrow> <m:mstyle displaystyle="true"> <m:msubsup> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mn>0</m:mn> <m:mi mathvariant="normal">∞</m:mi> </m:msubsup> </m:mstyle> <m:mrow> <m:mstyle displaystyle="true"> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m: