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Международная конференция по функциональным пространствам и теории приближения функций, посвященная 110-летию со дня рождения академика С. М. Никольского
29 мая 2015 г. 17:55, Функциональные пространства, г. Москва, МИАН
 


Sharp Pitt inequality and logarithmic uncertainty principle for Dunkl transform in $L^{2}$

D. V. Gorbachev, V. I. Ivanov, S. Yu. Tikhonov
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Аннотация: Let $\Gamma(t)$ be the gamma function, $\mathbb{R}^{d}$ be the real space of $d$ dimensions, equipped with a scalar product $(x, y)$ and a norm $|x|=\sqrt{(x, x)}$. Denote by $S(\mathbb{R}^{d})$ the Schwartz space on $\mathbb{R}^{d}$ and by $L^{2}(\mathbb{R}^{d})$ the Hilbert space of complex-valued functions endowed with a norm $\|f\|_{2}=(\int_{\mathbb{R}^{d}}|f(x)|^{2} dx)^{1/2}$. The Fourier transform is defined by
$$ \widehat{f}(y)=(2\pi)^{-n/2}\int_{\mathbb{R}^{d}}f(x)e^{-i(x, y)} dx. $$

W. Beckner [N449:Bec] proved the Pitt inequality for the Fourier transform
\begin{equation}\label{N449:eq1} \||y|^{-\beta}\widehat{f}(y)\|_{2}\le C(\beta)\||x|^{\beta}f(x)\|_{2},\qquad f\in S(\mathbb{R}^{d}),\quad 0<\beta<\frac d2\mspace{2mu}, \end{equation}
with sharp constant
$$ C(\beta)=2^{-\beta} \frac{\Gamma(\frac{1}{2}(\frac{d}{2}-\beta))}{\Gamma(\frac{1}{2}(\frac{d}{2}+\beta))}\mspace{2mu}. $$
Noting that $\||y|^{-\beta}\widehat{f}(y)\|_{2}= (2\pi)^{-\beta} \| |(-\Delta)^{\beta/2}f| \|_{2}$, Pitt's inequality can be viewed as a Hardy–Rellich inequality; see the papers by D. Yafaev [N449:Yaf] and S. Eilertsen [N449:Eil] for alternative proofs and extensions of \eqref{N449:eq1}.
For $\beta=0$, \eqref{N449:eq1} is the Plancherel theorem. If $\beta>0$ there is no extremiser in inequality \eqref{N449:eq1} and its sharpness can be obtained on the set of radial functions.
The proof of \eqref{N449:eq1} in [N449:Bec] is based on an equivalent integral realization as a Stein-Weiss fractional integral on $\mathbb{R}^d$. D. Yafaev in [N449:Yaf] used the following decomposition
\begin{equation}\label{N449:eq2} L^{2}(\mathbb{R}^{d})=\sum_{n=0}^{\infty}\oplus \mathfrak{R}_{n}^{d}, \end{equation}
where $\mathfrak{R}_{0}^{d}$ is the space of radial function, and $\mathfrak{R}_{n}^{d}=\mathfrak{R}_{0}^{d}\otimes \mathfrak{H}_{n}^{d}$ is the space of functions in $\mathbb{R}^d$ that are products of radial functions and spherical harmonics of degree $n$. Thanks to this decomposition it is enough to study inequality \eqref{N449:eq1} on the subsets of $\mathfrak{R}_{n}^{d}$ which are invariant under the Fourier transform.
Following [N449:Yaf] and using similar decomposition of the space $L^{2}(\mathbb{R}^{d})$ with the Dunkl weight, we prove sharp Pitt's inequality for the Dunkl transform.
Let $R\subset \mathbb{R}^{d}$ be a root system, $R_{+}$ be the positive subsystem of $R$, and $k\colon R\to \mathbb{R}_{+}$ be a multiplicity function with the property that $k$ is $G$-invariant. Here $G(R)\subset O(d)$ is a finite reflection group generated by reflections $\{\sigma_{a}: a\in R\}$, where $\sigma_{a}$ is a reflection with respect to a hyperplane $(a,x)=0$.
Let
$$ v_{k}(x)=\prod_{a\in R_{+}}|(a,x)|^{2k(a)} $$
be the Dunkl weight, $d\mu_{k}(x)=c_{k}v_{k}(x)dx$, where
$$ c_{k}^{-1}=\int_{\mathbb{R}^{d}}e^{-|x|^{2}/2}v_{k}(x) dx $$
is the Macdonald–Mehta–Selberg integral. Let $L^{2}(\mathbb{R}^{d},d\mu_{k})$ be the Hilbert space of complex-valued functions endowed with a norm
$$ \|f\|_{2,d\mu_{k}}=(\int_{\mathbb{R}^{d}}|f(x)|^{2} d\mu_{k}(x))^{1/2}. $$

Introduced by C. F. Dunkl, a family of differential–difference operators (Dunkl's operators) associated with $G$ and $k$ are given by
$$ D_{j}f(x)=\frac{\partial f(x)}{\partial x_{j}}+ \sum_{a\in R_{+}}k(a)(a,e_{j}) \frac{f(x)-f(\sigma_{a}x)}{(a,x)}\mspace{2mu},\qquad j=1,…,d. $$
The Dunkl kernel $e_{k}(x, y)=E_{k}(x, iy)$ is the unique solution of the joint eigenvalue problem for the corresponding Dunkl operators:
$$ D_{j}f(x)=iy_{j}f(x),\quad j=1,…,d,\qquad f(0)=1. $$
Let us define the Dunkl transforms as follows
$$ \mathcal{F}_{k}(f)(y)=\int_{\mathbb{R}^{d}}f(x)\overline{e_{k}(x,y)} d\mu_{k}(x), \qquad \mathcal{F}_{k}^{-1}(f)(x)=\mathcal{F}_{k}(f)(-x), $$
where $\mathcal{F}_{k}(f)$ and $\mathcal{F}_{k}^{-1}(f)$ are the direct and inverse transforms correspondingly (see, e.g., [N449:Ros]). For $k\equiv0$ we have $\mathcal{F}_{0}(f)=\widehat{f}$.
Our goal is to study Pitt's inequality for the Dunkl transform
\begin{equation}\label{N449:eq3} \||y|^{-\beta}\mathcal{F}_{k}(f)(y)\|_{2,d\mu_{k}}\le C(\beta,k)\||x|^{\beta}f(x)\|_{2,d\mu_{k}},\quad f\in S(\mathbb{R}^{d}), \end{equation}
with the sharp constant $C(\beta,k)$.
Let us first recall some known results on Pitt's inequality for the Hankel transform. Let $\lambda\ge -1/2$. Denote by $J_{\lambda}(t)$ the Bessel function of degree $\lambda$ and by $j_{\lambda}(t)=2^{\lambda}\Gamma(\lambda+1)t^{-\lambda}J_{\lambda}(t)$ the normalized Bessel function. Setting
$$ b_{\lambda}=(\int_{0}^{\infty}e^{-t^{2}/2}t^{2\lambda+1} dt)^{-1}= \frac{1}{2^{\lambda}\Gamma(\lambda+1)} $$
and $d\nu_{\lambda}(r)=b_{\lambda}r^{2\lambda+1} dr$, we define $\|f\|_{2,d\nu_{\lambda}}=(\int_{\mathbb{R}_{+}}|f(r)|^{2} d\nu_{\lambda}(r))^{1/2}$.
The Hankel transform is defined by
$$ \mathcal{H}_{\lambda}(f)(\rho)=\int_{\mathbb{R}_{+}}f(r)j_{\lambda}(\rho r) d\nu_{\lambda}(r). $$
Note that $\mathcal{H}_{\lambda}^{-1}=\mathcal{H}_{\lambda}$.
Pitt's inequality for the Hankel transform is written as
\begin{equation}\label{N449:eq4} \|\rho^{-\beta}\mathcal{H}_{\lambda}(f)(\rho)\|_{2,d\nu_\lambda}\le c(\beta,\lambda)\|r^{\beta}f(r)\|_{2,d\nu_\lambda},\qquad f\in S(\mathbb{R}_{+}), \end{equation}
where $c(\beta,\lambda)$ is the sharp constant in (\ref{N449:eq4}) and $S(\mathbb{R}_{+})$ is the the Schwartz space on $\mathbb{R}_{+}$. Note that if $f\in \mathfrak{R}_{0}^{d}$, a study of the Hankel transform is of special interest since the Fourier transform of a radial function can be written as the Hankel transform.
L. De Carli [N449:Car] proved that $c(\beta,\lambda)$ is finite only if $0\le \beta<\lambda+1$. For $\lambda=d/2-1$, $d\in\mathbb{N}$, the constant $c(\beta,\lambda)$ was calculated by D. Yafaev [N449:Yaf], and in the general case by S. Omri [N449:Omr]. The proof of Pitt's inequality in [N449:Omr] is rather technical and uses the Stein–Weiss type estimate for the so-called B-Riesz potential operator. Following [N449:Yaf], we give a direct and simple proof of inequality \eqref{N449:eq4}.
Let $|k|=\sum_{a\in R_{+}}k(a)$ and $\lambda_{k}= d/2-1+|k|$. For a radial function $f(r)$, $r=|x|$, Pitt's inequality for the Dunkl transform \eqref{N449:eq3} corresponds to Pitt's inequality for the Hankel transform \eqref{N449:eq4} with $\lambda=\lambda_{k}$. Therefore the condition
\begin{equation}\label{N449:eq5} 0\le \beta<\lambda_{k}+1 \end{equation}
is necessary for $C(\beta,k)<\infty$. Our goal is to show that in fact $C(\beta,k)=c(\beta,\lambda_{k})$ if condition \eqref{N449:eq5} holds.
Note that for the one-dimensional Dunkl weight
$$ v_{\lambda}(t)=|t|^{2\lambda+1}, \qquad d\mu_{\lambda}(t)=\frac{v_{\lambda}(t) dt}{2^{\lambda+1}\Gamma(\lambda+1)}, \qquad \lambda\ge -\frac12\mspace{2mu}, $$
and the corresponding Dunkl transform
$$ \mathcal{F}_{\lambda}(f)(s)= \int_{\mathbb{R}}f(t)\overline{e_{\lambda}(st)} |t|^{2\lambda+1} d\mu_{\lambda}(t),\qquad e_{\lambda}(t)=j_{\lambda}(t)-ij_{\lambda}'(t), $$
F. Soltani [N449:Sol1] proved Pitt's inequality that can be equivalently written as
\begin{equation}\label{N449:eq6} \||s|^{-\beta}\mathcal{F}_{\lambda}(f)(s)\|_{2,d\mu_\lambda}\le \max \{c(\beta,\lambda),c(\beta,\lambda+1)\}\||t|^{\beta}f(t)\|_{2,d\mu_\lambda} \end{equation}
for $f\in S(\mathbb{R})$ and $0\le \beta<\lambda+1$. Since $c(\beta,\lambda)\ge c(\beta,\lambda+1)$ (see [N449:Yaf]), then in fact \eqref{N449:eq6} holds with the constant $c(\beta,\lambda)$ and therefore, we have in this case $C(\beta,k)=c(\beta,\lambda_{k})$.
Finally, we remark that Pitt's inequality in $L^{2}$ for the multi-dimensional Dunkl transform has been recently established in [N449:Sol2] in the case of $\lambda_{k}-1/2<\beta<\lambda_{k}+1$. The obtained constant is not sharp.
Let $\mathbb{S}^{d-1}$ be the unit sphere in $\mathbb{R}^{d}$, $x'\in \mathbb{S}^{d-1}$, and $dx'$ be the Lebesgue measure on the sphere. Set $a_{k}^{-1}=\int_{\mathbb{S}^{d-1}}v_{k}(x') dx'$, $d\omega_{k}(x')=a_{k}v_{k}(x') dx'$, and $\|f\|_{2,d\omega_{k}}=(\int_{\mathbb{S}^{d-1}}|f(x')|^{2} d\omega_{k}(x'))^{1/2}$.
Let us denote by $\mathfrak{H}_{n}^{d}(v_{k})$ the subspace of $k$-spherical harmonics of degree $n\in \mathbb{Z}_{+}$ in $L^{2} (\mathbb{S}^{d-1},d\omega_{k})$. Let $\Delta_{k}f(x)=\sum_{j=1}^{d}D_{j}^{2}f(x)$ be the Dunkl Laplacian and $\mathfrak{P}_{n}^{d}$ be the space of homogeneous polynomials of degree $n$ in $\mathbb{R}^{d}$. Then $\mathfrak{H}_{n}^{d}(v_{k})$ is the restriction of $\ker \Delta_{k}\cap \mathfrak{P}_{n}^{d}$ to the sphere $\mathbb{S}^{d-1}$.
If $l_{n}$ is the dimension of $\mathfrak{H}_{n}^{d}(v_{k})$, we denote by $\{Y_{n}^{j}\colon j=1,\ldots,l_{n}\}$ the real-valued orthonormal basis $\mathfrak{H}_{n}^{d}(v_{k})$ in $L^{2}(\mathbb{S}^{d-1},d\omega_{k})$. A union of these bases forms orthonormal basis in $L^{2}(\mathbb{S}^{d-1},d\omega_{k})$ consisting of $k$-spherical harmonics, i.e., we have
\begin{equation}\label{N449:eq7} L^{2}(\mathbb{S}^{d-1},d\omega_{k})=\sum_{n=0}^{\infty}\oplus \mathfrak{H}_{n}^{d}(v_{k}). \end{equation}

Using \eqref{N449:eq7} and the following Funk-Hecke formula for $k$-spherical harmonic $Y\in \mathfrak{H}_{n}^{d}(v_{k})$
\begin{equation*} \int_{§^{d-1}}Y(y')\overline{e_{k}(x,y')} d\omega_{k}(y')= \frac{(-i)^{n}\Gamma(\lambda_{k}+1)}{2^{n}\Gamma(n+\lambda_{k}+1)} Y(x')r^{n} j_{n+\lambda_{k}}(r),\qquad x=rx', \end{equation*}
similarly to \eqref{N449:eq2} we have the direct sum decomposition of $L^{2}(\mathbb{R}^{d},d\mu_{k})$:
\begin{equation*} L^{2}(\mathbb{R}^{d},d\mu_{k})=\sum_{n=0}^{\infty}\oplus \mathfrak{R}_{n}^{d}(v_{k}),\qquad \mathfrak{R}_{n}^{d}(v_{k})=\mathfrak{R}_{0}^{d}\otimes \mathfrak{H}_{n}^{d}(v_{k}), \end{equation*}
and that the space $\mathfrak{R}_{n}^{d}(v_{k})$ is invariant under the Dunkl transform.
The next result provides a sharp constant in the Pitt inequality for the Dunkl transform \eqref{N449:eq3}.
\begin{etheorem}\label{N449:t2} Let $\lambda_{k}= d/2-1+|k|$ and $0\le\beta<\lambda_{k}+1$, then for $f\in S(\mathbb{R}^{d})$ we have
$$ C(\beta,k)=2^{-\beta} \frac{\Gamma(\frac{1}{2}(\lambda_{k}+1-\beta))}{\Gamma(\frac{1}{2}(\lambda_{k}+1+\beta))}. $$
Sharpness of this inequality can be seen by considering radial functions. \end{etheorem}
W. Beckner in [N449:Bec] proved the logarithmic uncertainty principle for the Fourier transform using Pitt's inequality \eqref{N449:eq1}: if $f\in S(\mathbb{R}^{d})$, then
$$ \int_{\mathbb{R}^{d}}\ln(|x|)|f(x)|^{2} dx+ \int_{\mathbb{R}^{d}}\ln(|y|)|\widehat{f}(y)|^{2} dy\ge (\psi (\frac{d}{4})+\ln 2)\int_{\mathbb{R}^{d}}|f(x)|^{2} dx, $$
where $\psi(t)=d\ln \Gamma(t)/dt$ is the $\psi$-function.
For the Hankel transform the logarithmic uncertainty principle reads as follows (see [N449:Omr]): if $f\in S(\mathbb{R}_+)$ and $\lambda\ge -1/2$, then
\begin{align*} &\int_{\mathbb{R}_+}\ln(t)|f(t)|^{2}t^{2\lambda+1} dt+\int_{\mathbb{R}_+} \ln(s)|\mathcal{H}_{\lambda}(f)(s)|^{2}s^{2\lambda+1} ds
&\qquad \ge (\psi (\frac{\lambda+1}{2})+\ln 2)\int_{\mathbb{R}_+}|f(t)|^{2}t^{2\lambda+1} dt. \end{align*}

For the one-dimensional Dunkl transform of functions $f\in S(\mathbb{R})$, F. Soltani [N449:Sol1] has recently proved that
\begin{align*} &\int_{\mathbb{R}}\ln(|t|)|f(t)|^{2}|t|^{2\lambda+1} dt+ \int_{\mathbb{R}}\ln(|s|)|\mathcal{F}_{\lambda}(f)(s)|^{2}|s|^{2\lambda+1} ds
&\qquad \ge (\psi (\frac{\lambda+1}{2}) +\ln 2)\int_{\mathbb{R}}|f(t)|^{2}|t|^{2\lambda+1} dt. \end{align*}

Using Pitt's inequality \eqref{N449:eq3} we obtain the logarithmic uncertainty principle for the multi-dimensional Dunkl transform.
\begin{etheorem}\label{N449:t3} Let $\lambda_{k}= d/2-1+|k|$ and $f\in S(\mathbb{R}^{d})$. We have
\begin{align*} &\int_{\mathbb{R}^{d}}\ln(|x|)|f(x)|^{2} d\mu_{k}(x) +\int_{\mathbb{R}^{d}}\ln(|y|)|\mathcal{F}_{k}(f)(y)|^{2} d\mu_{k}(y)
&\qquad \ge (\psi (\frac{\lambda_{k}+1}{2})+\ln 2)\int_{\mathbb{R}^{d}}|f(x)|^{2} d\mu_{k}(x). \end{align*}
\end{etheorem}
The work was supported by grants RFBR № 13-01-00043, № 13-01-00045, Ministry of education and science of Russian Federation № 5414{\selectlanguage{russian}ГЗ}, № 1.1333.2014{\selectlanguage{russian}К}, Dmitry Zimin's Dynasty Foundation, MTM 2011-27637, 2014 SGR 289.

Материалы: abstract.pdf (193.2 Kb)

Язык доклада: английский

Список литературы
  1. W. Beckner, “Pitt's inequality and uncertainty principle”, Proc. Amer. Math. Soc., 123 (1995), 1897–1905  mathscinet  zmath  isi
  2. D. R. Yafaev, “Sharp constants in the Hardy–Rellich inequalities”, J. Funct. Anal., 168:1 (1999), 121–144  crossref  mathscinet  zmath  isi  scopus
  3. S. Eilertsen, “On weighted fractional integral inequalities”, J. Funct. Anal., 185:1 (2001), 342–366  crossref  mathscinet  zmath  isi  scopus
  4. M. Rösler, “Dunkl operators. Theory and applications”, Orthogonal Polynomials and Special Functions, Lecture Notes in Math., 1817, Springer-Verlag, 2002, 93–135  mathscinet
  5. L. De Carli, “On the $L^{p}$$L^{q}$ norm of the Hankel transform and related operators”, J. Math. Anal. Appl., 348 (2008), 366–382  crossref  mathscinet  zmath  isi  scopus
  6. S. Omri, “Logarithmic uncertainty principle for the Hankel transform”, Int. Trans. Spec. Funct., 22:9 (2011), 655–670   crossref  mathscinet  zmath  scopus
  7. F. Soltani, “Pitt's inequality and logarithmic uncertainty principle for the Dunkl transform on $\mathbb{R}$”, Acta Math. Hungar., 143:2 (2014), 480–490  crossref  mathscinet  zmath  isi  scopus
  8. F. Soltani, “Pitt's inequalities for the Dunkl transform on $\mathbb{R}^{d}$”, Int. Trans. Spec. Funct., 25:9 (2014), 686–696  crossref  mathscinet  zmath  scopus


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