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Maximal operators, Riesz transforms and Littlewood-Paley functions associated with Bessel operators on BMO. (English) Zbl 1184.42015

The authors analyze several classical estimates for operators arising in Harmonic Analysis but in the setting of the Bessel operator \(\Delta_\lambda=-x^{-\lambda}Dx^{2\lambda}Dx^{-\lambda}\) for \(\lambda>0\). In this context the Poisson kernel associated to this operator becomes \(P^\lambda(t,x,y)=\int_0^\infty e^{-tz}\phi_x(z)\phi_y(z)dz, t,x,y>0\) where \(\phi_y(x)=\sqrt{xy}J_{\lambda -1/2}(xy)\) is an eigenfunction of \(\Delta_\lambda\) and \(J_\nu\) denotes the Bessel function of first kind and order \(\nu\). As usual they denote \(P^\lambda_t(f)(x)=\int_0^\infty P^\lambda(t,x,y)f(y)dy\) and \(P^\lambda_*(f)=\sup_{t>0}|P^\lambda_t(f)(x)|\). These are known to be bounded on \(L^p(0,\infty)\) for \(1\leq p<\infty\). Similarly \(W_t^\lambda\) and \(W^\lambda_*\) stand for the heat kernel and its maximal operator associated to \(\Delta_\lambda\) and the corresponding Littlewood-Paley \(g\)-functions associated to the Poisson semigroups and the heat semigroups for the Bessel operator are defined, as usual, by \(g_{P,\lambda}(f)(x)=(\int_0^\infty (t \frac{\partial}{ \partial t}P^\lambda_t(f)(x))^2\frac{dt}{t})^{1/2}\) and \(g_{h,\lambda}(f)(x)=(\int_0^\infty (t \frac{\partial}{ \partial t}W^\lambda_t(f)(x))^2\frac{dt}{t})^{1/2}\). Finally the Riesz transform \(R_\lambda\) is defined by \(R_\lambda f=D_\lambda\Delta_\lambda^{-1/2}f\) where \(D_\lambda=x^\lambda Dx^{-\lambda}\) and \(\Delta_\lambda\) admits a factorization \(\Delta_\lambda=D_\lambda D'_\lambda\). The paper is devoted to show that the operators \(P^*_\lambda, W^*_\lambda, g_{P,\lambda}, g_{h,\lambda}\) and \(R_\lambda\) map \(BMO_+\) into \(BMO_+\) where \(BMO_+\) stands for the space of locally integrable functions such that its odd extension belongs to \(BMO\). As an important intermediate step the authors study the action of the Maximal operator \(M_0f(x)=\sup_{x\in I}\frac{1}{|I|}\int_I |f(y)|dy, I\subset (0,\infty)\) on the space \(BMO_+\).

MSC:

42B25 Maximal functions, Littlewood-Paley theory
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