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Oscillation and variation for semigroups associated with Bessel operators

Research paper by Huoxiong Wu, Dongyong Yang, Jing Zhang

Indexed on: 01 Jun '16Published on: 30 May '16Published in: Journal of Mathematical Analysis and Applications



Abstract

Let λ>0λ>0 and <img height="23" border="0" style="vertical-align:bottom" width="141" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16301974-si2.gif">△λ:=−d2dx2−2λxddx be the Bessel operator on R+:=(0,∞)R+:=(0,∞). The authors show that the oscillation operator <img height="22" border="0" style="vertical-align:bottom" width="55" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16301974-si100.gif">O(P⁎[λ]) and variation operator <img height="22" border="0" style="vertical-align:bottom" width="60" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16301974-si112.gif">Vρ(P⁎[λ]) of the Poisson semigroup <img height="22" border="0" style="vertical-align:bottom" width="67" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16301974-si108.gif">{Pt[λ]}t>0 associated with ΔλΔλ are both bounded on Lp(R+,dmλ)Lp(R+,dmλ) for p∈(1,∞)p∈(1,∞), BMO(R+,dmλ)BMO(R+,dmλ), from L1(R+,dmλ)L1(R+,dmλ) to <img height="18" border="0" style="vertical-align:bottom" width="112" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16301974-si12.gif">L1,∞(R+,dmλ), and from H1(R+,dmλ)H1(R+,dmλ) to L1(R+,dmλ)L1(R+,dmλ), where ρ∈(2,∞)ρ∈(2,∞) and <img height="18" border="0" style="vertical-align:bottom" width="125" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16301974-si16.gif">dmλ(x):=x2λdx. As an application, an equivalent characterization of H1(R+,dmλ)H1(R+,dmλ) in terms of <img height="22" border="0" style="vertical-align:bottom" width="60" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16301974-si112.gif">Vρ(P⁎[λ]) is also established. All these results hold if <img height="22" border="0" style="vertical-align:bottom" width="67" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16301974-si108.gif">{Pt[λ]}t>0 is replaced by the heat semigroup <img height="22" border="0" style="vertical-align:bottom" width="72" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16301974-si102.gif">{Wt[λ]}t>0.