The Hausdorff-Young Inequality for n-dimensional Hermite Expansions

La desigualdad de Hausdorff-Young para las expansiones de Hermite n-dimensionales

This note concerns the sharpened Hausdorff–Young inequality in the context of n-dimensional Hermite expansions. The corresponding 1-dimensional result was considered in [4].

The Hermite functions constitute an ONS in R with respect to the Lebesgue measure there, and are defined as follows [5, 12, 14]. Szegö introduced the Hermite polynomials, Hm(.), in Chapter V of [12]. Earlier, Hille had also considered the Hermite polynomials, and proved some remarkable formulas and estimates [5, 14]. In particular, Hille considered the generating formula

The n–dimensional Hermite functions are obtained as products of the 1–dimensional Hermite functions [10, 14], and constitute an ONS in R. with respect to the Lebesgue measure there. To the point, given x= (x1, · · · , xn) in $\textcolor{red}{}{R}^n$ and an n–tuple of nonnegative integers m= (m1, . . . , mn), let the Hermite function Hm(x) be given by

Hm(x) = Hm1 (x1) · · · Hmn (xn) .

Now, for a function f(x) defined on $R{\textcolor{red}{}}^n$ , the Hermite expansion of is given by

and by the completeness of the Hermite expansion in R, f(x1, x2) = 0 a.e. x2 in R whenever x1 ∈ R \ E. Then, by Tonelli’s theorem, on account of the above observations it follows that

Given a function f defined on$\textcolor{red}{}{R}^n$, with $\textcolor{red}{}\upsilon$ the Lebesgue measure on $\textcolor{red}{}{R}^n$, let m(f, λ) denote the distributionfunction of f,

m(f, λ) = v({x∈$R^n$ : | f(x)| > λ}) , λ > 0 .

m(f, λ) is nonincreasing and right continuous, and the nonincreasing rearrangement $f{}^{*}$of defined for t > 0 by

f ∗(t) = inf{λ : m(f, λ) ≤ t} , inf ∅ = 0 ,

is informally its inverse (this statement is made precise in [9, p. 43]). f∗ is nonincreasing and right continuous and, at its points of continuity t, f ∗(t) = λ is equivalent to m(f, λ) = t.

The Lorentz space Lp,q($R{\textcolor{red}{}}^n$) = L(p, q), 0 < p < ∞, 0 < q ≤ ∞, consists of those measurable functions f with finite quasinorm ${‖f‖}_{\mathrm{p,q}}$ given by

and, with $\textcolor{red}{}\mu$ the atomic measure concentrated on the lattice of n-tuples of nonnegative integer atoms m taking the value $\textcolor{red}{}\mu$(m) = 1 on each such atom,

Finally, an operator T of a class of functions f on $R{\textcolor{red}{}}^n$ into a linear class of functions is said to be linear provided that, if T is defined for f0, f1, and λ ∈ R, then T is defined for f0 + λf1, and T (f0 + λf1)(x) = T (f0)(x) + λ T (f1)(x).

A linear operator T defined for f ∈ LA($\textcolor{red}{}{R}^n$) and taking values T(f) = {cm} in lB is said to be bounded if there is a constant K>0 such that

The sharpened Hausdorff-Young inequality for n=1 proved in [4, Theorem 4.1] rests on a remarkable estimate for the Hermite functions established by Hille [5, p. 436], [12, p. 240], to wit,

Let $\mu$ denote the atomic measure concentrated on the lattice of 2-tuples of integer atoms m=(m1, m2) with m1, m2=0,1,2,..., taking the value $\textcolor{red}{}\mu$(m)=1 on each atom.

Given λ > 0, let I$\textcolor{red}{}\lambda$= {m : |Cm|> λ} . Now, if m= (m1, m2) is in I$\textcolor{red}{}\lambda$. and m1 · m2$\textcolor{red}{}\ne$ 0, by (3.7) we have

and T is of type (p; q) . This conclusion also follows letting A(t) = $t^p$ in (3.12) above. This proves (3.3), and we have finished.

A companion result to the Hausdorff-Young inequality addresses under what conditions fcmg is the sequence of Fourier coefficients of a function f in the Hausdorff-Young range [2], [15, Vol.2, Theorem 2.3, p, 101]. For the Hermite expansions in R, this is done in [4, Theorem 4.2].

In our context, for the Hermite expansions in n dimensions we have,