Hardy-type inequalities for the generalized Mehler transform

We establish Hardy-type inequalities for the generalized Mehler transform on the real Hardy space H^p, 0 < p < 1

Hardy\u27s inequalities for Hermite and Laguerre expansions

金沢大学大学院自然科学研究科機能開発システム金沢大学工学部The well-known inequality of Hardy for Fourier coefficients of functions f(t) ∼ ∑∞n=-∞ bneint in the real Hardy space is ∑∞n=-∞ |bn|/(|n| + 1) < ∞. We shall establish analogues of this inequality for the Hermite function expansions and also for the Laguerre function expansions

Negative powers of Laguerre operators

We study negative powers of Laguerre differential operators in $\R$, $d\ge1$. For these operators we prove two-weight $L^p-L^q$ estimates, with ranges of $q$ depending on $p$. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here these results are applied in certain Laguerre settings. The procedure is fairly direct for Laguerre function expansions of Hermite type, due to some monotonicity properties of the kernels involved. The case of Laguerre function expansions of convolution type is less straightforward. For half-integer type indices $\alpha$ we transfer the desired results from the Hermite setting and then apply an interpolation argument based on a device we call the {\sl convexity principle} to cover the continuous range of $\alpha\in[-1/2,\infty)^d$. Finally, we investigate negative powers of the Dunkl harmonic oscillator in the context of a finite reflection group acting on $\R$ and isomorphic to $\mathbb Z^d_2$. The two weight $L^p-L^q$ estimates we obtain in this setting are essentially consequences of those for Laguerre function expansions of convolution type.Comment: 30 page

Laguerre and Disk Polynomial Expansions with Nonnegative Coefficients

We establish Wiener type theorems and Paley type theorems for Laguerre polynomial expansions and disk polynomial expansions with nonnegative coefficients. © 2013 Springer Science+Business Media New York