A maximal function characterization of 𝐻^{𝑝} on the space of homogeneous type
1980; American Mathematical Society; Volume: 262; Issue: 2 Linguagem: Inglês
10.1090/s0002-9947-1980-0586737-4
ISSN1088-6850
Autores Tópico(s)Mathematical Analysis and Transform Methods
ResumoLet ψ 0 ( x ) ∈ S ( R n ) {\psi _0}(x)\, \in \,{\mathcal {S(}}{R^n}{\text {)}} and let ∫ R n ψ 0 ( y ) d y ≠ 0 \int _{{R^n}} {{\psi _0}(y)\,dy\, \ne \,0} .For f ∈ S ′ R n f\, \in \,{\mathcal {S}’}{R^n} , x ∈ R n x\, \in \,{R^n} and M ⩾ 0 M\, \geqslant \,0 , let \[ f + ( x ) = sup t > 0 | f ∗ ψ 0 t ( x ) | {f^ + }(x)\, = \,\sup \limits _{t\, > \,0} \,\left | {f\,{\ast }\,{\psi _{0t}}(x)} \right | \] and let f ∗ M ( x ) = sup { | f ∗ ψ t ( x ) | : t > 0 {f^{{\ast }M}}(x)\, = \,\sup \{ \left | {f\,{\ast }\,{\psi _t}(x)} \right |:\,t\, > \,0 , ψ ( y ) ∈ S ( R n ) \psi (y)\, \in \,{\mathcal {S(}}{R^n}) , supp ψ ⊂ { y ∈ R n : | y | > 1 } \operatorname {supp} \,\psi \, \subset \,\{ y\, \in \,{R^n}:\,\left | y \right |\, > \,1\} , ‖ D α ψ ‖ L ∞ ⩽ 1 {\left \| {{D^\alpha }\psi } \right \|_{{L^\infty }}}\, \leqslant \,1 for any multi-index α = ( α 1 , … , α n ) \alpha \, = \,({\alpha _1},\, \ldots ,\,{\alpha _n}) such that Σ i = 1 n α i ⩽ M } \Sigma _{i = 1}^n\,{\alpha _i}\, \leqslant \,M\} where ψ t ( y ) = t − n ψ ( y / t ) {\psi _t}(y)\, = \,{t^{ - n}}\psi (y/t) . Fefferman-Stein [ 11 ] showed Theorem A. Let p > 0 p\, > \,0 . Then there exists M ( p , n ) M(p,\,n) , depending only on p and n, such that if M ⩾ M ( p , n ) M\, \geqslant \,M(p,\,n) , then \[ c ‖ f + ‖ L p ⩽ ‖ f ∗ M ‖ L p ⩽ C ‖ f + ‖ L p c\left \| {{f^ + }} \right \|{L^p}\, \leqslant \,\left \| {{f^{{\ast }M}}} \right \|{L^p}\, \leqslant \,{\textbf {C}}\left \| {{f^ + }} \right \|{L^p} \] for any f ∈ S ′ ( R n ) f\, \in \,{\mathcal {S}’}({R^n}) , where c and C are positive constants depending only on ψ 0 {\psi _0} , p, M and n . We investigate this on the space of homogeneous type with certain assumptions.
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