Some transplantation theorems for the generalized Mehler transform and related asymptotic expansions

Abstract:

Let $P_{ - 1/2 + ix}^m(z)$ be the associated Legendre function of order $m$ and degree $- 1/2 + ix$. We give, here, two integral transforms ${G^m}$ and ${H^m}$, arising naturally from the generalized Mehler transform, which is induced by $P_{ - 1/2 + ix}^m(\cosh y)$, such thatb ${H^m}{G^m}$ = Identity (formally). We show that if $1 < p < \infty , - 1/p < \alpha < 1 - 1/p,m \leqq 1/2$ or $m = 1,2, \ldots ,$ then $||{G^m}f|{|_{p,\alpha }} \leqq A_{p,\alpha }^m||\hat f|{|_{p,\alpha }}$ and $||{H^m}f|{|_{p,\alpha }} \leqq A_{p,\alpha }^m||\hat f|{|_{p,\alpha }}$, where $^ \wedge$ denotes the Fourier cosine transform. We also prove that ${G^m}f,{H^m}f$ exist as limits in ${L^{p,\alpha }}$ of partial integrals, and we prove inequalities equivalent to the above pair: $||{G^m}\hat f|{|_{p,\alpha }} \leqq A_{p,\alpha }^m||f|{|_{p,\alpha }}$ and $||{H^m}\hat f|{|_{p,\alpha }} \leqq A_{p,\alpha }^m||f|{|_{p,\alpha }}$. These we dualize to $||{({H^m}f)^ \wedge }|{|_{p,\alpha }} \leqq A_{p,\alpha }^m||f|{|_{p,\alpha }}$, and $||{({G^m}f)^ \wedge }|{|_{p,\alpha }} \leqq A_{p,\alpha }^m||f|{|_{p,\alpha }}$. ${G^m}$ and ${H^m}$ are given by ${G^m}(f;y) = \int _0^\infty {f(x){K^m}(x,y)dx}$ and ${H^m}(f;x) = \int _0^\infty {f(y){K^m}(x,y)dy\;} (0 \leqq y < \infty )$, where \[ {K^m}(x,y) = |\Gamma (1/2 - m - ix)/\Gamma ( - ix)/{(\sinh y)^{1/2}}P_{ - 1/2 + ix}^m(\cosh y).\] The principal method of proving the inequalities involves getting asymptotic expansions for ${K^m}(x,y)$; these are in terms of sines and cosines for large $y$, and in terms of Bessel functions for $y$ small. Then we can use Fourier and Hankel multiplier theorems. The main consequences of our results are the typical ones for transplantation theorems: mean convergence and multiplier theorems. They can easily be restated in terms of the more usual Mehler transform pair \[ g(y) = \int _0^\infty {f(x){P_{ - 1/2 + ix}}(y)dx} \] and $f(x) = {\pi ^{ - 1}}x\sinh \pi x \cdot \Gamma (1/2 - m + ix)\Gamma (1/2 - m - ix)\int _0^\infty {g(y){P_{ - 1/2 + ix}}(y)dy.}$