Mittag-Leffler family for an integro-differential equation

In a few months ago I post a note about a family of an integro-partial differential equations (2), there the main objective was to expose, without details, a first work to treat questions such as existence and symmetries of solutions in certain critical space, namely, homogeneous Morrey Spaces. On that work the Mittag-Leffler family below was central on the estimates,

\displaystyle \widehat{u}(t,\xi)=\mathbb{E}_{\alpha}(-t^{\alpha}\vert\xi\vert^2)\widehat{\varphi}(\xi) \ \ \ \ \ (1)

which is an integral form of the integro-partial differential equation

\displaystyle u_t(t,x)=\int_{0}^{t}r_{\alpha-1}(t-s) \Delta_{x}u(s,x)ds,\; u(0,x)=\varphi(x) \ \ \ \ \ (2)

where {1\leq\alpha<2}, {\Delta_{x}=\Sigma_{j=1}^{n}(\partial/\partial x_j)^2}, {u(t,x)=(u_{1}(t,x),\cdots,u_{n}(t,x))} for {t\in [0,\infty)} and {x\in\mathbb{R}^{n}} and {r_{\alpha}(s)=s^{\alpha-1}/\Gamma(\alpha)}, here {\Gamma(\alpha)} stands for Gamma function. In this post we will prove (1). Before, we need some properties of the Laplace transform. Given a “well” real-valued function {u_j(\cdot,x)} on {[0,\infty]}, we define the Laplace transform as

\displaystyle \mathcal{L}(u_j(\cdot,x))(s)=\int_0^{\infty}e^{-st}u_j(t,x)dt. \ \ \ \ \ (3)

Given {\lambda\in (0,\infty)}, let  {\partial_tu, u,v:[0,\lambda]\times\mathbb{R}^{n}\rightarrow\mathbb{R}^n} be certain  {\mathbb{R}^n}-valued functions for each {t\in(0,\lambda)} fixed, then

\displaystyle \mathcal{L}(\partial_tu)(s)=-u(0,x)+s\mathcal{L}(u)(s) \ \ \ \ \ (4)


\displaystyle \mathcal{L}(u\ast^{t}v)(s)=\mathcal{L}(u)(s)\mathcal{L}(v)(s) \ \ \ \ \ (5)

where {\ast^{t}} stands for convolution operator on variable {t}, that is, {u\ast^{t}v(t)=\int_0^t u(t-s,x)v(s,x)ds}. Hence, applying the Laplace transform in (2), we obtain

\displaystyle s\mathcal{L}(u)(s)-\varphi(x)=\mathcal{L}(r_{\alpha-1})(s)\mathcal{L}(\Delta_{x} u)(s). \ \ \ \ \ (6)

In view of

\displaystyle \begin{array}{rcl} \mathcal{L}(t^{d})(s)=\int_0^{\infty}e^{-ts}t^{d}dt=\frac{\Gamma(d+1)}{s^{d+1}}, \;\; (d>0) \end{array}


\displaystyle \begin{array}{rcl} \mathcal{L}(\frac{\partial^2}{\partial x_i^2}u)(s)=\frac{\partial^2}{\partial x_i^2}\mathcal{L}(u)(s) \end{array}

we have that the equation (6) can be written as

\displaystyle s\mathcal{L}(u)(s)-\varphi(x)=s^{1-\alpha}\Delta_{x}\mathcal{L}(u)(s). \ \ \ \ \ (7)

For {j=1,\cdots n}, let {u_j\in \mathcal{S}(\mathbb{R}^{n})} (Schwartz space) and let {\widehat{u_j}} be the Fourier transform of {u_j} given by

\displaystyle \widehat{u_j}(t,\xi)=\int_{\mathbb{R}^{n}}e^{-ix\cdot \xi}u_j(t,x)dx. \ \ \ \ \ (8)

If {u_j\in C^{\infty}(\mathbb{R}^{n})} with compact support, easily seems that

\displaystyle \widehat{\frac{\partial^{2} u_j}{\partial x_i^{2}}}(t,\xi)=-\xi_j^{2}\,\widehat{u_j}(t,\xi)\Rightarrow \widehat{\Delta_{x} u}(t,\xi)=-\vert \xi\vert^2\,\widehat{u}(t,\xi). \ \ \ \ \ (9)

As {\widehat{\mathcal{L}(u)(s)}(\xi)=\mathcal{L}(\widehat{u}(\cdot,\xi))(s)} it follows from (7) and (9) that

\displaystyle s\mathcal{L}(\widehat{u}(\cdot,\xi))(s)=\widehat{\varphi}(\xi)-s^{1-\alpha}\vert \xi\vert^2 \mathcal{L}(\widehat{u}(\cdot,\xi))(s)\nonumber \ \ \ \ \ (10)

which yields

\displaystyle \mathcal{L}(\widehat{u}(\cdot,\xi))(s)=\frac{\widehat{\varphi}(\xi)}{s+s^{1-\alpha}\vert\xi\vert^2}=\frac{s^{\alpha-1}}{s^{\alpha}+\vert\xi\vert^2} \widehat{\varphi}(\xi). \ \ \ \ \ (11)

This motivates to define Mittag-Leffler function as a complex integral on certain curve. Indeed, firstly notice that the complex valued function {f(z)=\frac{e^zz^{\alpha-1}}{z^{\alpha}+\vert\xi\vert^2}} has singularities in

\displaystyle a_{\alpha}(\xi)=|\xi|^{\frac{2}{\alpha}}e^{\frac{i\pi}{\alpha}} ,\;\;\;b_{\alpha}(\xi)=|\xi|^{\frac{2}{\alpha}}e^{-\frac{i\pi}{\alpha}},\text{ for }\xi\in\mathbb{R}^{n}. \ \ \ \ \ (12)

Let {\gamma} be the standard Hankel’s curve in {\mathbb{C}}, positive oriented, such that {a_{\alpha}(\xi)\in Int(\gamma)} and {b_{\alpha}(\xi)\in Int(\gamma)}, that is, let {\gamma=r_1+r_2+C_r} be a parametrized curve given by {r_1(t)=te^{i\theta}}, {r_2(t)=te^{-i\theta}} for {t\in (r,\infty)} and {C_r(t)=r e^{it}} for {t\in (-\theta,\theta)}. As we want to chose {\gamma} such that {a_{\alpha}(\xi)\in Int(\gamma)} and {b_{\alpha}(\xi)\in Int(\gamma)}, we shall suppose that {r^{\alpha}>\vert \xi\vert^2>\epsilon^{\alpha}>0} and one defines Mittag-Leffler function as

\displaystyle \begin{array}{rcl} \mathbb{E}_{\alpha,\beta}(-\vert\xi\vert^2)=\frac{1}{2\pi i}\int_{\gamma}\frac{e^zz^{\alpha-\beta}}{z^{\alpha}+\vert\xi\vert^2}dz,\; (\alpha>0,\beta>0). \end{array}

Using residue theorem, a characterization very important of this definition was obtained in [Fujita] (see also [Hirata-Miao][Fujita2]).

Proposition 1 If {1<\alpha<2} and {\beta=1}, we have

\displaystyle \begin{array}{rcl} L^{1}(\mathbb{R}^{n})\ni\mathbb{E}_{\alpha}(-|\xi|^{2})=\frac{1}{\alpha}(\exp(a_{\alpha}(\xi))+\exp(b_{\alpha}(\xi)))+l_{\alpha}(\xi) \end{array}


\displaystyle l_{\alpha}(\xi)= \begin{cases} \frac{\sin(\alpha\pi)}{\pi}\int_{0}^{\infty}\frac{|\xi|^{2}s^{\alpha-1}e^{-s} }{s^{2\alpha}+2|\xi|^{2}s^{\alpha}\cos(\alpha\pi)+|\xi|^{4}}ds & \text{ if }\xi\neq0\\ 1-\frac{2}{\alpha}, & \text{ if }\xi=0. \end{cases} \ \ \ \ \ (13)

Some corollaries can be obtained by Lemma above. Indeed, taking {t=\vert\xi\vert^{\frac{2}{\alpha}}s^{\frac{1}{\alpha}}} one has

\displaystyle \begin{array}{rcl} \vert \mathbb{E}_{\alpha}(-|\xi|^{2})\vert&\leq& \frac{2}{\alpha}+\vert l_{\alpha}(\xi)\vert\\ &\leq&\frac{2}{\alpha}+\frac{\sin(\alpha\pi)}{\pi}\int_{0}^{\infty}\frac{e^{-\vert\xi\vert^{\frac{2}{\alpha}}s^{\frac{1}{\alpha}}}}{s^{2}+2s\cos(\alpha\pi)+1}ds\\ &\leq&\frac{2}{\alpha}+\frac{\sin(\alpha\pi)}{\pi}\int_{0}^{\infty}\frac{1}{s^{2}+2s\cos(\alpha\pi)+1}ds\\ &=&\frac{2}{\alpha} +(1-\frac{2}{\alpha})=1, \end{array}

more general (see de Almeida, Ferrreira, L.F.C).

Proposition 2 Let {1\leq\alpha<2} and {0\leq\delta<2.} There is {C>0} such that

\displaystyle \left\vert \frac{\partial^{k}}{\partial\xi^{k}}\left[ \left\vert \xi\right\vert ^{\delta}\mathbb{E}_{\alpha}(-|\xi|^{2})\right] \right\vert \leq C\left\vert \xi\right\vert ^{-\left\vert k\right\vert },\text{ } \ \ \ \ \ (14)

for all {k\in(\mathbb{N}\cup\{0\})^{n}} with {\left\vert k\right\vert \leq\lbrack n/2]+1} and for all {\xi\neq0.}

Using an exercise,

\displaystyle \int_{0}^{\infty}e^{-t}\mathbb{E}_{\alpha}(at^{\alpha})dt=\frac{1}{1-a},\;\; (r^{\alpha}>\vert a\vert>0)\text{ and } (1< \alpha< 2),\nonumber \ \ \ \ \ (15)

we have

\displaystyle \mathcal{L}(\mathbb{E}_{\alpha}(-t^{\alpha}\vert\xi\vert^2))(s)=\frac{s^{\alpha-1}}{s^{\alpha}+\vert\xi\vert^2}. \ \ \ \ \ (16)

Taking in (11) the inverse Laplace transform and using (16), we obtain {\widehat{u}(t,\xi)=\mathbb{E}_{\alpha}(-t^{\alpha}\vert\xi\vert^2)\widehat{\varphi}(\xi).}




1. Mikhlin-Hormander type symbols

Let {X,Y} be vector-spaces of measurable functions from {\mathbb{R}^{n}} to itself and let {T} be a bounded linear operator from {X} to {Y}. Recall that {T} is called translation invariant if {T(\tau_{y}f)=\tau_{y}(T(f))} for all {y\in\mathbb{R}^{n}} and {f\in X}. Let {X=L^p} and {Y=L^q} with {1\leq q\leq p<\infty}, we found that each such operator {T} is determined by a certain tempered distribution {K} such that {Tf=K\ast f} for every {f\in\mathcal{S}} (Schwartz space). So taking Fourier transform {\mathcal{F}} into {Tf} we have {\mathcal{F}(Tf)=\mathcal{F}(K)\mathcal{F}(f)}. This motivate us to define a Fourier multiplier as a map {T_{\sigma}:\mathcal{S}(\mathbb{R}^n)\rightarrow\mathcal{S}'(\mathbb{R}^n)} given by

\displaystyle \begin{array}{rcl} \widehat{T_{\sigma}f}(\xi)=\sigma(\xi) \widehat{f}(\xi), \end{array}

where {\sigma} is a tempered distribution {\mathcal{S}'(\mathbb{R}^n)} and {\,\widehat{}\,} denotes the Fourier transform {\mathcal{F}}. We refer to {\sigma} as symbol of {T_{\sigma}}, sometimes one writes {T_{\sigma}} as {\sigma(D)} to relate it with more general operators {\sigma(D,X)} so-called pseudo-differential operators. No standard example of such symbols is, for {\delta>0},

\displaystyle \sigma_{\delta}(\xi)=(1-\vert\xi\vert^2)^{\delta}\text{ if }\vert \xi\vert\leq 1\text{ and }\;\sigma_{\delta}(\xi) =0 \text{ otherwise}. \ \ \ \ \ (1)

In the limit case, {\delta=0}, the above symbol can be written as {\mathcal{X}_{\mathbb{D}}}, where {\mathcal{X}_{\mathbb{D}}} denotes the characteristic function of unit disk {\mathbb{D}}. It’s well-known that the condition {n\geq2} and {2n/(n+1)<p<2n/(n-1)} is necessary for {T_{\sigma_{0}}} be a Fourier multiplier on {L^p(\mathbb{R}^{n})} (see e.g., [1]). However, Fefferman (see [2],[3]) gave an intricate proof which show us that this condition is not sufficient, that is, he showed that the operator {T_{\sigma_{0}}} does not extend to a bounded operator on {L^p(\mathbb{R}^{n})} for any {p\neq 2} and {n\geq2}. This result give us a negative answer to the famous disk conjecture which states that {T_{\sigma_0}} is bounded on {L^p(\mathbb{R}^{2})} for {4/3\leq p\leq 4}. In this post we will work with symbols more regular than (1) such as Minklin symbols {\Sigma_{1}^{0}(\mathbb{R}^{n})=\{\sigma\in C^{k}(\mathbb{R}^n\backslash\{0\}); \vert D^{\gamma}\sigma(x)\vert \leq C \vert x\vert ^{-\vert\gamma\vert}, \vert\gamma\vert\leq k\}}.

In a few months ago, based on Littlewood-Paley theorem, we gave a proof that the operator {T_{\sigma}} is a Fourier multiplier from {L^{p}(\mathbb{R}^{n})} to itself (see Theorem 7) provided that {1<p<\infty} and {\sigma} satisfies

\displaystyle \sup_{j\in\mathbb{Z}}\Vert \widehat{\psi}_{j}\sigma\Vert_{L^2_{s}(\mathbb{R}^{n})}\leq L \ \ \ \ \ (2)

for {s>n/2} and {n\geq1}. In this post will be showed that if {\sigma\in \Sigma_{1}^0}, then its satisfies the inequality (2). Hence, by Theorem 7 one has the following classical Mikhlin-Hormander theorem.

Theorem 1 (Mikhlin-Hormander theorem) Let {k>n/2} and {\sigma\in C^{k}(\mathbb{R}^{n})} away from the origin. If for {\vert\gamma\vert\leq k} we have

\displaystyle \sup_{r>0}r^{\vert \gamma\vert}\left(\frac{1}{r^n}\int_{\frac{r}{2}<\vert\xi\vert<2r}\vert (D^{\gamma}_{\xi}\sigma)(\xi)\vert^{2}d\xi\right)^{\frac{1}{2}}\leq L \ \ \ \ \ (3)

then {T_{\sigma}} is a Fourier multiplier on {L^p}, {1<p<\infty}. In particular, {T_{\sigma}} is a Fourier multiplier on {L^p} if {\sigma\in\Sigma_1^0(\mathbb{R}^n)}, that is,

\displaystyle \vert D^{\gamma}\sigma(\xi)\vert \leq C \vert \xi\vert ^{-\vert\gamma\vert}. \ \ \ \ \ (4)

Let us recall some important definitions. Let {1\leq p\leq \infty} and {k\in\mathbb{Z}_{+}}, a function {f} lies in Sobolev spaces {L^{p}_k(\mathbb{R}^{n})} if for every {\gamma\in (\mathbb{N}\cup\{0\})^n} with {\vert \gamma\vert\leq k} there exists {g_{\gamma}\in L^{p}(\mathbb{R}^{n}) } such that

\displaystyle \int_{\mathbb{R}^{n}}f(x)D^{\gamma}\varphi(x)dx= (-1)^{\vert \gamma\vert}\int_{\mathbb{R}^{n}}g_{\gamma}(x)\varphi(x)dx, \;\;\; \forall\varphi\in C^{\infty}_{0}(\mathbb{R}^{n}). \ \ \ \ \ (5)

Here, we use the standard notations

\displaystyle \begin{array}{rcl} \vert \gamma\vert=\sum_{i=1}^{n}\gamma_i \text{ and } D^{\gamma}\varphi=\frac{\partial^{\vert\gamma\vert} \varphi}{\partial^{\gamma_1} _{x_1}\partial^{\gamma_2} _{x_2}\cdots \partial^{\gamma_n} _{x_n}} \end{array}

and we say that {g_{\gamma}} is the derivative of {f} in distribution sense (more precisely in {\mathcal{D}'(\mathbb{R}^{n})}) and we write {D^{\gamma}f=g_{\gamma}} in {\mathcal{D}'(\mathbb{R}^{n})} to mean (5). The space {L^p_k} equipped with the norm

\displaystyle \begin{array}{rcl} \vert f\vert_{L^p_k}=\sum_{\vert\gamma\vert\leq k}\vert D^{\gamma}f\vert_{L^p} \end{array}

is a Banach space. Also, notice that Sobolev spaces {L^{2}_k(\mathbb{R}^{n})} coincide with inhomogeneous fractional Sobolev spaces {L^{2}_a(\mathbb{R}^{n})} because of the norm equivalence

\displaystyle \begin{array}{rcl} \vert f\vert_{L^2_k}^2&\approx&\sum_{\vert\gamma\vert\leq k}\vert D^{\gamma}f\vert_{L^2}^2=\sum_{\vert\gamma\vert\leq k}\vert \xi^{\gamma}\widehat{f}\vert_{L^2}^2\\ &=&\int_{\mathbb{R}^{n}} \sum_{\vert\gamma\vert\leq k}\vert\xi^{2\gamma}\vert \vert\widehat{f}(\xi)\vert^2d\xi\\ &\approx& \int_{\mathbb{R}^{n}} (1+\vert\xi\vert^2)^{k}\vert\widehat{f}(\xi)\vert^2d\xi= \Vert f\Vert _{L^2_k}^{2}. \end{array}

It follows that

\displaystyle \begin{array}{rcl} \Vert \widehat{\psi}_{j}\sigma\Vert_{L^2_k}=\Vert \sigma(2^{j}\cdot)\widehat{\psi}\Vert_{L^2_k}=\sum_{\vert\gamma\vert \leq k}\vert D^{\gamma}(\sigma(2^{j}\cdot)\widehat{\psi})\vert_{L^2}. \end{array}

Using Leibniz’s formula we written the term {D^{\gamma}} as

\displaystyle \begin{array}{rcl} D^{\gamma}(\sigma(2^{j}\cdot)\widehat{\psi})=\sum_{\vert \nu\vert\leq\vert \gamma\vert}\binom{\nu}{\gamma}(D^{\nu}\sigma(2^{j}\cdot))(D^{\gamma-\nu}\widehat{\psi}). \end{array}

By observing that {\vert D^{\gamma-\nu}\widehat{\psi}\vert\leq C} on {supp(\widehat{\psi})\subset \{\xi\,:\, 1/2\leq \vert\xi\vert\leq2\}} and zero otherwise, we get

\displaystyle \Vert \widehat{\psi}_{j}\sigma\Vert_{L^2_k}\leq C\sum_{\vert\gamma\vert \leq k}\sum_{\vert \nu\vert\leq\vert \gamma\vert}C_{\gamma,\nu}\vert D^{\nu}\sigma(2^{j}\cdot)\vert_{L^2(\{\xi\,:\,\frac{1}{2}< \vert\xi\vert<2\})}. \ \ \ \ \ (6)

Now making the change of variable {\xi\mapsto r\xi} one has {D^{\nu}_{\xi}\sigma(r\cdot) =r^{\vert\nu\vert}(D_{\xi}^{\nu}\sigma)(r\xi)}. Hence, by (3) it follows that

\displaystyle \sup_{r>0}\left(\int_{\frac{1}{2}<\vert\xi\vert<2}\vert D^{\nu}_{\xi}\sigma(r\cdot)\vert^{2}d\xi\right)^{\frac{1}{2}}\leq L. \ \ \ \ \ (7)

Let {r=2^{j}}. Therefore, inserting (7) into (6) easily gets

\displaystyle \sup_{j}\Vert \widehat{\psi}_{j}\sigma\Vert_{L^2_k}\leq C\sum_{\vert\gamma\vert \leq k}\sum_{\vert \nu\vert\leq\vert \gamma\vert}C_{\gamma,\nu} L=C_{k}L \ \ \ \ \ (8)

and Theorem 1 is a consequence of the Theorem 7 as we desired. Notice that if {\vert D_{\xi}^{\gamma}\sigma (\xi)\vert\leq L\vert\xi\vert^{-\vert\gamma\vert}},

\displaystyle \begin{array}{rcl} \vert D^{\nu}_{\xi}\sigma(2^j\cdot)\vert = 2^{j\vert\nu\vert} \vert (D^{\nu}_{\sigma}\sigma )(2^j\xi)\vert \leq L\vert\xi\vert^{-\vert\nu\vert}. \end{array}

Hence {\vert D^{\nu}\sigma(2^{j}\cdot)\vert_{L^2(\{\xi\,:\,\frac{1}{2}< \vert\xi\vert<2\})}\leq C} which implies (8).