Multivariate Hörmander-Type Multiplier Theorem for the Hankel transform

Abstract

Let $\mathcal{H}(f)(x)=\int_{(0,\infty)^{d}} f(\lambda) E_{x}(\lambda) d\nu(\lambda )$, be the multivariate Hankel transform, where $E_{x}(\lambda)=\prod_{k=1}^{d} (x_{k} \lambda_{k})^{-\alpha _{k}+1/2}J_{\alpha_{k}-1/2}(x_{k} \lambda_{k})$, with dν(λ)=λ ^2α dλ, α=(α ₁,…,α _d ). We give sufficient conditions on a bounded function m(λ) which guarantee that the operator $\mathcal{H}(m\mathcal{H} f)$ is bounded on L ^p(dν) and of weak-type (1,1), or bounded on the Hardy space H ¹((0,∞)^d,dν) in the sense of Coifman-Weiss.

Appendix

Proof of (1.14)

The proof of Theorem 1.3 actually shows that

(5.1)

Now, using (1.13) we write n _u (λ)=Ξ(λ)(λ ₁+⋯+λ _d )^iu, so that $n_{u}(\lambda _{1}^{2},\ldots,\lambda_{d}^{2})=m_{u}(\lambda)$. We claim that

$$ \sup_{j\in\mathbb{Z}}\big\|\eta(\cdot)n_u \bigl(2^j\cdot\bigr)\big\|_{\mathcal{L}_{s}^\infty(\mathbb{R}^d)}\leq C_{s}\big(1+|u|\big)^{s}, \quad s>0. $$

(5.2)

Using (5.2) and combining it with (5.1), we get $\|L^{iu}\|_{L^{1}(X)\to L^{1,\infty}(X) }\leq C_{\varepsilon }(1+|u|)^{Q/2+\varepsilon}$. Since $\|L^{iu}\|_{L^{2}(X)\to L^{2}(X)}=1$, using the Marcinkiewicz interpolation theorem, see, e.g. [8, (2.2) p. 30], together with a duality argument, we obtain (1.14) for all 1<p<∞.

Now, we sketch the proof of (5.2). Let $B_{p,q}^{s}$, 1≤p,q≤∞, s≥0, be the Besov space, as defined in [2, p. 141]. It is known, see [2, Theorem 6.2.4 (10), p.142], that $B_{\infty, q}^{s}$ is the real interpolation space of the spaces $\mathcal{L}_{s}^{\infty}=\mathcal{L}_{s}^{\infty}(\mathbb{R}^{d})$, precisely

$$ \bigl(\mathcal{L}_{s_0}^\infty,\mathcal{L}_{s_1}^\infty \bigr)_{\theta, q}=B_{\infty, q}^s, \quad s=(1-\theta)s_0+\theta s_1,\quad0<\theta<1, \quad1\leq q\leq\infty. $$

(5.3)

Moreover, from [2, Theorem 6.2.4 (9), p.142], we have

$$ \|f\|_{\mathcal{L}_{s}^\infty}\leq C_s \|f\|_{B_{\infty,1}^s}, \quad f\in B_{\infty,1}^s,\quad s>0. $$

(5.4)

For general Banach spaces X and Y, one has

$$ \|x\|_Z\leq C_{\theta,q}\|x\| _X^{1-\theta}\|y\|_Y^{\theta},\quad Z=(X,Y)_{\theta,q},\quad 0<\theta<1, $$

(5.5)

see [24, Theorem (g), p. 25]. Now, it is straightforward to check that (5.2) is true for s=2n, $n\in\mathbb{N}\cup\{0\}$. Using the latter observation, (5.4), (5.3) with q=1, and (5.5) with $X=\mathcal{L}_{2n}^{\infty}$, $Y=\mathcal{L}_{2n+2}^{\infty}$, for s=(1−θ)2n+θ(2n+2), 0<θ<1, we obtain

 □

The following example shows that in the multivariable case for functions n(λ) supported in A _1/2,2 the Sobolev norms $\| n\|_{W_{2}^{s}(\mathbb{R}^{d})}$ do not control the Sobolev norms $\| m\|_{W_{2}^{s'}(\mathbb{R}^{d})}$ (even for certain range s′ smaller than s) where n and m are related by (1.8).

Example 5.1

Let F(x,y) be a function defined on $\mathbb{R}\times\mathbb{R}^{\ell}$ and s>0. Observe that

$$ \| F\|_{W_2^s(\mathbb{R}^{1+\ell})}^2 \sim\| \hat{F}\| ^2_{L^2(\mathbb{R}^{1+\ell})} + \int_{\mathbb{R}^\ell}\int _{\mathbb{R}} \big|\hat{F}(\xi, \eta)\big|^2 \bigl(| \xi|^{2s}+|\eta|^{2s} \bigr) d\xi d\eta, $$

(5.6)

Moreover, it can be shown that for every r>0 there is a constant C>0 such that for f supported in the interval $(\frac{1}{2}, 2 )$ one has

$$ C^{-1} \| \tilde{f} \|_{ W^r_2(\mathbb{R})} \leq\| f\| _{W^r_2(\mathbb{R})} \leq C \| \tilde{f}\|_{ W^r_2(\mathbb{R})}, $$

(5.7)

where $\tilde{f}(x)=f(x^{2})$.

Let $\varphi\in C_{c}^{\infty}(\frac{1}{2},\frac{3}{2} )$ and $\psi\in C_{c}^{\infty}(\mathbb{R}^{\ell})$, ψ(y)=0 for $|y|>\frac{1}{2}$, $\varphi, \psi\not\equiv0$. Fix ε∈(0,1), R>1 and define the functions n _R (x,y) on $\mathbb{R}\times\mathbb{R}^{\ell}$ by

$$n(x,y)=n_R(x,y)= \cos(Rx)\varphi(x)\psi\bigl(R^{1-\varepsilon}y \bigr)=f(x)g(y). $$

The functions n(x,y) are supported in A _1/2,2, near the vector e ₁. Moreover,

$$ \hat{f} (\xi)=c \bigl(\hat{\varphi}(\xi-R)+\hat{\varphi}(\xi+R) \bigr) \quad \text{and} \quad \hat {g}(\eta)=R^{\ell(\varepsilon-1)} \hat{\psi}\bigl(R^{\varepsilon-1} \eta\bigr). $$

(5.8)

From (5.6) and (5.8) we conclude that

$$ \| n\|_{W^s_2(\mathbb{R}^{1+\ell})}\leq C_s R^{s-(1-\varepsilon)\ell /2}. $$

Set $m(x,y)=m_{R}(x,y)=n_{R}(x^{2}, y_{1}^{2},\ldots,y_{\ell}^{2})=f(x^{2})g(y_{1}^{2}, y_{2}^{2},\ldots,y_{\ell}^{2})= \tilde{f}(x)\tilde{g}(y)$. The functions m _R are supported near the vectors ±e ₁. By (5.7) for s′>0 and R large we have

$$ \| \tilde{f}\|_{W_2^{s'}(\mathbb{R})} \sim\| f\|_{W_2^{s'}(\mathbb{R})} \sim R^{s'}. $$

(5.9)

Clearly,

$$ \| \tilde{g}\|_{L^2(\mathbb{R}^\ell)}=\| \hat{\tilde{g}}\| _{L^2(\mathbb{R}^\ell)} = c R^{-(1-\varepsilon) \ell/4}. $$

(5.10)

Now, (5.6) combined with (5.9) and (5.10) imply that

$$ \| m\|_{W_2^{s'}(\mathbb{R}^{1+\ell})} \geq c R^{s'-(1-\varepsilon )\ell/4} \quad \text{for large } R. $$

Summarizing,

$$ \frac{\| m\|_{W_2^{s'}(\mathbb{R}^{1+\ell})}}{\| n\|_{W_2^{s}(\mathbb{R}^{1+\ell})}} \geq c R^{s'-s+(1-\varepsilon)\ell/4}, $$

which clearly tends to infinity as R→∞ provided that s′>s−(1−ε)ℓ/4.