Mittag-Leffler family for an integro-differential equation

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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)

and

\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}

and

\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}

where

\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).}

Finite time blowup for an averaged three-dimensional Navier-Stokes equation

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Originally posted on What's new:

I’ve just uploaded to the arXiv the paper “Finite time blowup for an averaged three-dimensional Navier-Stokes equation“, submitted to J. Amer. Math. Soc.. The main purpose of this paper is to formalise the “supercriticality barrier” for the global regularity problem for the Navier-Stokes equation, which roughly speaking asserts that it is not possible to establish global regularity by any “abstract” approach which only uses upper bound function space estimates on the nonlinear part of the equation, combined with the energy identity. This is done by constructing a modification of the Navier-Stokes equations with a nonlinearity that obeys essentially all of the function space estimates that the true Navier-Stokes nonlinearity does, and which also obeys the energy identity, but for which one can construct solutions that blow up in finite time. Results of this type had been previously established by Montgomery-Smith, Gallagher-Paicu, and Li-Sinai for variants of…

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Minlin-Hormander theorem for Morrey spaces

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In the last post we given a proof of the following theorem.

Theorem 1 (Minlin-Hormander theorem) Let {k>n/2} and {\sigma\in C^{k}(\mathbb{R}^{n}\backslash\{0\})}. 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 \ \ \ \ \ (1)

then {T_{\sigma}:=\mathcal{F}^{-1}\sigma(\xi)\mathcal{F}f} 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, if

\displaystyle \vert D^{\gamma}\sigma(\xi)\vert \leq L \vert \xi\vert ^{-\vert\gamma\vert},\;\xi\neq 0. \ \ \ \ \ (2)

In this post, we would like to extend this theorem for a class of Morrey-type spaces. In other words

Theorem 2 Let {1<p<\infty} and {0<\mu<n}. If {\sigma \in\Sigma_{1}^{0}(\mathbb{R}^{n})}, the Fourier multiplier {T_{\sigma}} on {L^{p}} can be extended to {\mathcal{M}_{p,\mu}}, that is, there exists {C>0} such that

\displaystyle \Vert T_{\sigma}f\Vert_{\mathcal{M}_{p,\mu}}\leq C\,L\Vert f\Vert_{\mathcal{M}_{p,\mu}}, \ \ \ \ \ (3)

for all {f\in\mathcal{M}_{p,\mu}}.

 Proof. A proof can be found in [Taylor] (see also [Kozono1Kozono2]. For reader convenience we give some details. This key theorem was important in my recent article in join with Ferreira, Lucas C.F.

We have that {T_{\sigma}} is a convolution operator with kernel {k_{\sigma}(z)=(\sigma(\xi))^{\vee}(z)}. As {\sigma\in \Sigma_1^0} it follows from [p.26, Stein] that

\displaystyle \vert \partial_{z}^{\gamma}k_{\sigma}(z)\vert\leq C L |z|^{-n-\vert\gamma\vert},\; z\neq 0, \ \ \ \ \ (4)

Based in (4) and Minlin-Hormander theorem we obtain a proof of the Theorem 2. Indeed, firstly one splits {f\in\mathcal{M}_{p,\mu}(\mathbb{R}^{n})} as

\displaystyle f=f_{0}+\sum_{j=1}^{\infty}g_{j},

where

\displaystyle f_{0}=\chi_{B_{2r}(x_{0})}f,\text{ }\,g_{j}=f\chi_{A_{rj}}\text{ and } A_{rj}=\{x\,:\,2^{j}r\leq|x_{0}-x|\leq2^{j+1}r]\}.

Defining {k_{j}(x,y)=\chi_{B_{r}(x_{0})}(x)k(x-y)\chi_{A_{rj}}(y)} and {T_{\sigma,j}(f)(x)=\int_{\mathbb{R}^{n}}k_{j}(x,y)f(y)dy} easily gets by Hölder inequality and estimate (4) the following

\displaystyle \begin{array}{rcl} \vert T_{\sigma,j}(f)(x)\vert^{p}&\leq& \left(\int_{\mathbb{R}^{n}}\vert k_{j}(x,y)f(y)\vert dy\right)^{p}\\ \\ &\leq& \left(\int_{\mathbb{R}^{n}}\vert k_{j}(x,y)\vert dy\right)^{p/q}\int_{\mathbb{R}^{n}}\vert k_{j}(x,y)\vert\, \vert f(y)\vert^{p} dy\\ \\ &=&\left(\int_{A_{rj}(x_0)}\chi_{_{D_{r}(x_0)}}(x)\vert k_{\sigma}(x-y)\vert dy\right)^{p/q}\int_{\mathbb{R}^{n}}\vert k_{j}(x,y)\vert\, \vert f(y)\vert^{p} dy\\ \\ &\leq& \left(CL(2^{j}r)^{-n} vol(A_{rj}(x_0))\right)^{p/q}\int_{\mathbb{R}^{n}}\vert k_{j}(x,y)\vert\, \vert f(y)\vert^{p} dy\\ \\ &\leq& C L^{p/q}\int_{\mathbb{R}^{n}}\vert k_{j}(x,y)\vert\, \vert f(y)\vert^{p} dy, \end{array}

where {q} denotes the conjugate exponent of {p}. Using Tonelli’s theorem and once more estimate (4) we have

\displaystyle \begin{array}{rcl} \int_{\mathbb{R}^{n}}\vert T_{\sigma,j}(f)(x)\vert^{p}dx &\leq& CL^{p/q} \int_{\mathbb{R}^{n}}\vert k_{j}(x,y)\vert dx \int_{\mathbb{R}^{n}}\vert f(y)\vert^{p}dy\\ \\ &=& CL^{p/q} \int_{D_{r}(x_0)}\vert k_{\sigma}(x-y)\vert\chi_{_{A_{rj}}}(y) dx \int_{\mathbb{R}^{n}}\vert f(y)\vert^{p}dy\\ \\ &\leq& CL^{p/q+1} 2^{-jn} \int_{\mathbb{R}^{n}}\vert f(y)\vert^{p}dy. \end{array}

Now we ready to proof the theorem. By Minlin-Hormander theorem and the last estimate one has

\displaystyle \begin{array}{rcl} \left\Vert T_{\sigma}f\right\Vert _{L^{p}(B_{r}(x_{0}))} & \leq&\left\Vert T_{\sigma}f_{0}\right\Vert _{L^{p}(\mathbb{R}^{n})}+\sum_{j=1}^{\infty}\left\Vert T_{\sigma}g_{j}\right\Vert _{L^{p}(B_{r}(x_{0}))}\nonumber\\ \\ & \leq& CL\left\Vert f_{0}\right\Vert _{L^{p}(\mathbb{R}^{n})}+\sum_{j=1}^{\infty}\left\Vert T_{\sigma,j}(\chi_{A_{rj}}f)\right\Vert_{L^{p}(\mathbb{R}^{n})}\nonumber\\ \\ &\leq& CL\left\Vert f_{0}\right\Vert _{L^{p}(\mathbb{R}^{n})}+\sum_{j=1}^{\infty}\left(\int_{\mathbb{R}^n} \vert T_{\sigma,j}(\chi_{A_{rj}}f(x))\vert^{p}\right)^{\frac{1}{p}}\nonumber\\ \\ & \leq& CL\left\Vert f\right\Vert _{L^{p}(B_{2r}(x_{0}))}+\sum_{j=1}^{\infty}CL^{(p/q+1)/p}2^{-jn/p}\left\Vert \chi_{A_{rj}}f\right\Vert_{L^{p}(\mathbb{R}^{n})}\nonumber\\ \\ & \leq&2^{\frac{\mu}{p}}CL\left\Vert f\right\Vert _{p,\mu}r^{\frac{\mu}{p}}+CL^{1/q+1/p}\sum_{j=1}^{\infty}2^{-jn/p}\left\Vert f\right\Vert _{p,\mu}(2^{j}r)^{\frac{\mu}{p}}\nonumber\\ \\ & \leq& 2C(1+\sum_{j=1}^{\infty}2^{-j(\frac{n-\mu}{p})})L\left\Vert f\right\Vert _{p,\mu}r^{\frac{\mu}{p}}, \end{array}

which yields (3), because the series above is convergent.

Minklin-Hormander

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1. Minklin-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 Minklin-Hormander theorem.

Theorem 1 (Minklin-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).

Heat-wave equation in Morrey spaces

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1. A Heat-Wave equation in Morrey spaces 

This notes is an exposition of the article written by me in join with full professor Lucas Catão de Freitas Ferreira and therefore no proofs will be given in general.

Throughout this notes, the symbol {\mathbb{N}} stands for the set of all natural numbers. For {n\in\mathbb{N}}, we always let {\mathbb{R}^{n}} be {n}-dimensional Euclidean space with Euclidean norm denoted by {\Vert x\Vert} and endowed with Lebesgue measure {dx}. Also {\mathbb{R}^{n+1}_{+}} stands for {n+1}-dimensional upper half-space {\{(x,t)\,:\, x\in\mathbb{R}^{n},\, t>0\}} with Lebesgue measure {dxdt}. Let {u:\mathbb{R} ^{n+1}_{+}\rightarrow\mathbb{R}^{n}} be a vector field on upper half-space {\mathbb{R}^{n+1}_{+}}, we denote {\vert u\vert=\max_{i=1,...,n}\vert u_{i}(x,t) \vert_{\mathbb{R}}}. Here {\vert u_{i}(x,t)\vert_{\mathbb{R}}} is so-called absolute value of {u_{i}(x,t)\in\mathbb{R}}.

Here we are interested in a semilinear integro-partial differential equation, which interpolates the semilinear heat and wave equations which reads as follows

\displaystyle u(x,t)=u_{0}(x)+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1} (\Delta_{x}u(x,s)+|u(x,s)|^{\rho}u(x,s))ds \ \ \ \ \ (1)

where {1<\alpha<2,} {0<\rho<\infty,} {\Gamma(\alpha)} denotes the Gamma function, {\Delta_{x}} is the Laplacian in the {x}-variable and {u=u(x,t)=(u_{1}(x,t),\cdots,u_{n}(x,t))} is a fluid at time {t\in [0,\infty)} and position {x\in\mathbb{R}^{n}} that assumes the given data (initial velocity) {u_{0}=u_{0}(x)}. This equation is formally equivalent to the initial value problem for the time-fractional partial differential equation (FPDE)

\displaystyle \begin{array}{rcl} \mathbf{\partial}_{t}^{\alpha}u =\Delta_{x}u+|u|^{\rho}u,\text{ in }\mathbb{R}^{n},\;t>0,\\ u(x,0)=u_{0}\text{ and }\partial_{t}u(x,0)=0\text{ in }\mathbb{R}^{n}, \end{array}

where {\mathbf{\partial}_{t}^{\alpha}u=\mathbf{D}_{t}^{\alpha-1}(\partial _{t}u)} and {\mathbf{D}_{t}^{\alpha-1}} stands for the Riemann-Liouville derivative of order {\alpha-1} given at (16).

Differential equations of time-fractional order appear naturally in several fields such as physics, chemistry and engineering by modelling phenomena in viscoelasticity, thermoelasticity, processes in media with fractal geometry, heat flow in material with memory and many others. The two most common types of fractional derivatives acting on time variable {t} are those of Riemann-Liouville and Caputo. We refer the surveys Kilbas and Povstenko in which the reader can find a good bibliography for applications on those fields. Models with fractional derivatives can naturally connect structurally different groups of PDEs and their mathematical analysis may give information about the transition (or loss) of basic properties from one to another. Two groups are the parabolic and hyperbolic PDEs whose well-posedness and asymptotic behavior theory presents a lot of differences. For instance, in {L^{p},} weak-{L^{p},} Besov-spaces, Morrey spaces and other ones, there is an extensive bibliography for global well-posedness and asymptotic behavior for semilinear heat equations (and other parabolic equations). On the other hand, for semilinear wave equations, although there exist results in {L^{p}}, weak{-L^{p}} and Besov spaces, there is no results in Morrey spaces. The main reason is the loss of decay of the semigroup (and its time-derivative) associated to the free wave equation, namely {(\frac {\sin(\left\vert \xi\right\vert t)}{\left\vert \xi\right\vert t})^{\vee}} and {(\cos(\left\vert \xi\right\vert t))^{\vee}.} So, it is natural to wonder what would be the behavior of the semilinear (FPDE) in the framework of Morrey spaces, which presents a mixed parabolic-hyperbolic structure.

Between interesting points obtained, let us comment about technical ones. Due to the semigroup property (21), further restrictions appear in our main theorems in comparison with classical nonlinear heat equations. Making the derivative index {\alpha} go from {\alpha=1} to {\alpha=2,} the estimates and corresponding restrictions become worse, and they are completely lost when {\alpha} reaches the endpoint {2} (see Lemmas 45). The proof of the pointwise estimate (23) shows that the worst parcel in (21) is the term {l_{\alpha}(\xi)} (see (21)). Then In particular, notice that for {\alpha=1} (heat semigroup) the upper bound on parameter {\delta} is not necessary, that is, one can take {\delta\in\lbrack0,\infty).} Finally, based on above observations, our results and estimates suggest the following: the semilinear wave equation ({\alpha=2}) in {\mathbb{R}^{n}} is not well-posed in Morrey spaces. The mathematical verification of this assertion seems to be an interesting open problem.

2. Morrey spaces 

In this section some basic properties about Morrey spaces are reviewed. For further details on theses spaces, the reader is referred to KatoPeetreTaylor. Let {\mathbb{D}_{r}} denote the open ball in {\mathbb{R}^{n}} centered in the origin and with radius {r>0}. For two parameters {1\leq p<\infty} and {0\leq\mu<n}, we define the Morrey spaces {\mathcal{M}_{p,\mu}=\mathcal{M} _{p,\mu}(\mathbb{R}^{n})} as the set of functions {f\in L^{p}(\mathbb{D}_{r})} such that

\displaystyle \Vert f\Vert_{L^{p}(\mathbb{D}_{r})}\leq C\,r^{\frac{\mu}{p}}, \ \ \ \ \ (2)

where {C>0} denotes a constant independent of {x_{0},r} and {f}. The space {\mathcal{M}_{p,\mu}} endowed with the norm

\displaystyle \Vert f\Vert_{p,\mu}=\sup_{\mathbb{D}_{r}}r^{-\frac{\mu}{p}}\Vert f\Vert_{L^{p}(\mathbb{D}_{r})} \ \ \ \ \ (3)

is a Banach space. For {s\in\mathbb{R}} and {1\leq p<\infty,} the homogeneous Sobolev-Morrey space {\mathcal{M}_{p,\mu}^{s}=(-\Delta)^{-s/2}\mathcal{M} _{p,\mu}} is Banach with norm

\displaystyle \left\Vert f\right\Vert _{\mathcal{M}_{p,\mu}^{s}}=\Vert (-\Delta )^{s/2}f\Vert _{p,\mu}. \ \ \ \ \ (4)

Taking {p=1}, {\Vert f\Vert_{L^{1}(\mathbb{D}_{r})}} stands for the total variation of {f} on {\mathbb{D}_{r}} and {\mathcal{M}_{1,\mu}} is a space of signed measures{.} In particular, when {\mu=0,} {\mathcal{M}_{1,0}} {=\mathcal{M}} is the space of finite measures. For {p>1,} {\mathcal{M} _{p,0}=L^{p}} and {\mathcal{M}_{p,0}^{s}=\dot{H}_{p}^{s}} is the homogeneous Sobolev space{.} With the natural adaptation in (3) for {p=\infty,} the space {L^{\infty}} corresponds to {\mathcal{M}_{\infty,\mu}}.

Morrey spaces present the following scaling

\displaystyle \Vert f(\lambda x)\Vert_{p,\mu}=\lambda^{-\frac{n-\mu}{p}}\Vert f\Vert_{p,\mu} \ \ \ \ \ (5)

and

\displaystyle \left\Vert f(\lambda x)\right\Vert _{\mathcal{M}_{p,\mu}^{s}}=\lambda ^{s-\frac{n-\mu}{p}}\left\Vert f(x)\right\Vert _{\mathcal{M}_{p,\mu}^{s} }\text{,} \ \ \ \ \ (6)

where the exponent {s-\frac{n-\mu}{p}} is called scaling index. We have that

\displaystyle (-\Delta)^{l/2}\mathcal{M}_{p,\mu}^{s}=\mathcal{M}_{p,\mu}^{s-l}. \ \ \ \ \ (7)

Let us define the closed subspace of {\mathcal{M}_{p,\mu}} (denoted by {\ddot{\mathcal{M}}_{p,\mu}}) by means of the property {f\in\ddot{\mathcal{M} }_{p,\mu}} if and only if

\displaystyle \Vert f(\cdot-y)-f(\cdot)\Vert_{p,\mu}\rightarrow0\text{ as }y\rightarrow0. \ \ \ \ \ (8)

This subspace is useful to deal with semigroup of convolution operators when the respective kernels present a suitable polynomial decay at infinity. In general, such semigroups are only weakly continuous at {t=0^{+}} in {\mathcal{M}_{p,\mu},} but they are {C_{0}}-semigroups in {\ddot{\mathcal{M} }_{p,\mu},} as it is the case of {\{G_{\alpha}(t)\}_{t\geq0}}. This property is important in order to derive local-in-time well-posedness for PDEs.

Morrey spaces contain Lebesgue and Marcinkiewicz spaces with the same scaling indexes. Precisely, we have the continuous proper inclusions

\displaystyle L^{q}(\mathbb{R}^{n})\varsubsetneq L^{q,\infty}(\mathbb{R}^{n})\varsubsetneq \mathcal{M}_{p,\mu}(\mathbb{R}^{n}) \ \ \ \ \ (9)

where {p<q} and {\mu=n(q-p)/q} (see e.g. Miyakawa or MF de Almeida and LFC Ferreira).

In the next lemma, we remember some important inequalities and inclusions in Morrey spaces, see e.g. KatoTaylor.

Lemma 1 Suppose that {s_{1},s_{2}\in\mathbb{R}}, {1\leq p,q,r<\infty} and {0\leq\lambda,\mu,\upsilon<n}.

  • (Inclusion) {\mathcal{M}_{p,\mu}} is decreasing in {p}, i.e., if {p\leq q} and {\frac{n-\mu}{p} =\frac{n-\lambda}{q}} then

    \displaystyle \mathcal{M}_{p,\mu}\supseteq\mathcal{M}_{q,\lambda} \ \ \ \ \ (10)

  • (Sobolev type embedding) If {p\leq q}, {s_{2}\geq s_{1}} and {s_{2}-\frac{n-\mu}{p}=s_{1}-\frac{n-\mu}{q}} then

\displaystyle \mathcal{M}_{p,\mu}^{s_{2}}\subset\mathcal{M}_{q,\mu}^{s_{1}} \ \ \ \ \ (11)

  • (Holder inequality) If {\;\frac{1}{r}=\frac{1} {p}+\frac{1}{q}} and {\frac{\upsilon}{r}=\frac{\lambda}{p}+\frac{\mu}{q}} then {fg\in\mathcal{M}_{r,\upsilon}} and   

\displaystyle\Vert fg\Vert_{r,\upsilon}\leqslant\Vert f\Vert_{p,\lambda}\Vert g\Vert _{q,\mu}. \ \ \ \ \ (12)

  • (Homogeneous function) Let {\Omega\in L^{\infty }(\mathbb{S}^{n-1})}, {0<d<n} and {1\leq r<n/d}. Then {\Omega(x/|x|)|x|^{-d} \in\mathcal{M}_{r,n-dr}}, for all {x\in{\mathbb{R}^{n}\backslash\{0\}}}.

We finish this section by recalling estimates for certain multiplier operators on {\mathcal{M}_{p,\mu}^{s}} see e.g. Kozono-YamazakiKozono-YamazakiTaylor for lemma below.

In the next posts, inspired in the work of Taylor, we given a proof based in the Theorem 7 given in the post A fractional Hörmander type multiplier theorem.

Lemma 2 Let {m,s\in\mathbb{R},} {1<p<\infty} and {0\leq\mu<n} and {F(\xi)\in C^{[n/2]+1}(\mathbb{R}^{n}\backslash\{0\}).} Assume that there is {A>0} such that

\displaystyle \left\vert \frac{\partial^{k}F}{\partial\xi^{k}}(\xi)\right\vert \leq A\left\vert \xi\right\vert ^{m-\left\vert k\right\vert },\text{ } \ \ \ \ \ (13)

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.} Then the multiplier operator {F(D)} on {\mathcal{S}^{\prime}/\mathcal{P}} is bounded from {\mathcal{M}_{p,\mu}^{s}} to {\mathcal{M}_{p,\mu}^{s-m}} and the following estimate hold true

\displaystyle \left\Vert F(D)f\right\Vert _{\mathcal{M}_{p,\mu}^{s-m}}\leq CA\left\Vert f\right\Vert _{\mathcal{M}_{p,\mu}^{s}}, \ \ \ \ \ (14)

where {\mathcal{S}^{\prime}/\mathcal{P}} is the set of equivalence classes in {\mathcal{S}^{\prime}} modulo polynomials with {n} variables.

3. Mittag-Leffler Function

Differential equations of time-fractional order is old, but even being old, the fractional calculus is little studied by mathematicians. Possibly, because many of them are unfamiliar with this topic and its applications in various sciences. After Liouville and Riemann, deep developments was obtained by many authors. Today, time-fractional order integral and derivative is known as Riemann-Liouville’s integral and derivative. More precisely, let {\varphi} be a Lebesgue integrable function in {\mathbb{R}} and {\alpha\geq 0}, Riemann-Liouville’s integral is defined by

\displaystyle \textbf{I}_{\alpha}\varphi(t)=\int_{0}^{t}R_{\alpha}(t-s)\varphi(s)ds \ \ \ \ \ (15)

and Riemann-Liouville’s derivative by

\displaystyle \textbf{D}^{\alpha}_{t}\varphi= \left(\frac{\partial}{\partial t}\right)^{k}\textbf{I}_{k-\alpha}\varphi(t), \ \ \ \ \ (16)

where {k=\lfloor\alpha\rfloor+1}, {R_{\alpha}(s)=\frac{s^{\alpha-1}}{\Gamma(\alpha)}}, being {\Gamma(\alpha)} denoted by so-called Gamma function.

Hardy and Littlewood give wider properties about this integrals. His showed, for instance, the boundedness of {\textbf{I}_{\alpha}} from {L^{p}(\mathbb{R})} to {L^{q}(\mathbb{R})}, for {1<p<q<\infty}. This result, is well known as Hardy-Littlewood-Sobolev theorem, Sobolev because of its importance in the theory of fractional Sobolev Spaces.

Related the theory of partial differential equations, much works is devoted to find an unified theory of Green functions associated to fractional problems. In this way, fractional diffusion-wave equation is a celebrity, and an unified theory of the heat equation and wave equation was obtained. FujitaSchneider and Wyss, for instance, founds the special function to represent the Green function associated to the linear part of the diffusion-wave problem (FPDE)

\displaystyle \begin{array}{rcl} \mathbf{\partial}_{t}^{\alpha}u =\Delta_{x}u+|u|^{\rho}u,\text{ in }\mathbb{R}^{n},\;t>0 \end{array}

such special function, is knew as Mittag-Leffler function. More precisely,

\displaystyle \mathbb{E}_{\alpha}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\alpha k +1)} \ \ \ \ \ (17)

and the Green function, defined via Fourier inversion, is given by

\displaystyle \mathcal{K}_{\alpha}(x,t)=\int_{\mathbb{R}^{n}}e^{ix\cdot \xi}\mathbb{E}_{\alpha}(-t^{\alpha}\vert \xi\vert^{2})d\xi, \ \ \ \ \ (18)

which generate the following semigroup of operators {\{G_{\alpha}(t)\}_{t\geq 0}},

\displaystyle G_{\alpha}(t)\varphi=\mathcal{K}_{\alpha}(\cdot,t)\ast\varphi. \ \ \ \ \ (19)

In what follows, we recall some functions which is useful to handle the symbol of the semigroup {G_{\alpha}(t)} (see (19)). For {1<\alpha<2,} let us set

\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}\text{,} \ \ \ \ \ (20)

and

\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} \ \ \ \ \ (21)

Lemma 3 Let {1<\alpha<2} and {\mathcal{K}_{\alpha}} be as in (18). We have that

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

and

\displaystyle \frac{\partial^{k}\mathcal{K}_{\alpha}}{\partial x_{i}^{k}}(x,t)=\lambda ^{n+k}\frac{\partial^{k}}{\partial x_{i}^{k}}\mathcal{K}_{\alpha}(\lambda x,\lambda^{\frac{2}{\alpha}}t), \ \ \ \ \ (22)

for all {\;\lambda>0.} Moreover, {\mathcal{K}_{\alpha}(x,t)\geq0}, {P_{\alpha }(|x|,1)=\alpha\mathcal{K}_{\alpha}(x,1)} is a probability measure, and

\displaystyle \begin{array}{rcl} \Vert\mathcal{K}_{\alpha}(\cdot,t)\Vert_{L^{1}(\mathbb{R}^{n})}=\frac {1}{\alpha},\text{ for all }t>0. \end{array}

Proof: Except for (22), all properties contained on the statement can be found in Fujita and Hirata-Miao when {n=1} and {n\geq2,} respectively.

In order to prove (22), we use Fourier inversion and (18) to obtain

\displaystyle \begin{array}{rcl} \frac{\partial^{k}\mathcal{K}_{\alpha}}{\partial x_{i}^{k}}(x,t) &=&\int_{\mathbb{R}^{n}}e^{ix\cdot\xi}(-i\xi_{i})^{k}\mathbb{E}_{\alpha}(-t^{\alpha}|\xi|^{2})d\xi\\ &=&t^{-n\frac{\alpha}{2}}\int_{\mathbb{R}^{n}}e^{ix\cdot\frac{y}{\sqrt{t^{\alpha}}}}(it^{-\frac{\alpha}{2}}y_{i})^{k}\mathbb{E}_{\alpha}(-|y|^{2})dy\\ &=&t^{-\frac{\alpha}{2}(n+k)}\int_{\mathbb{R}^{n}}e^{i\frac{x}{\sqrt{t^{\alpha}}}\cdot y}(iy_{i})^{k}\mathbb{E}_{\alpha}(-|y|^{2})dy\\ &=&t^{-\frac{\alpha}{2}(n+k)}\frac{\partial^{k}\mathcal{K}_{\alpha}}{\partial x_{i}^{k}}(\frac{x}{\sqrt{t^{\alpha}}},1). \end{array}

The desired identity follows by taking {\lambda=1/\sqrt{t^{\alpha}}} in the last equality. \Box

4. Some estimates

The aim of this section is to derive estimates for the semigroup {G_{\alpha}(t).} For that matter we will need pointwise estimates for the fundamental solution {\mathcal{K}_{\alpha}} in Fourier variables.

Lemma 4 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}E_{\alpha}(-|\xi|^{2})\right] \right\vert \leq C\left\vert \xi\right\vert ^{-\left\vert k\right\vert },\text{ } \ \ \ \ \ (23)

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.}

In the sequel we prove key estimates on Morrey spaces for the semigroup {\{G_{\alpha}(t)\}_{t\geq0}.}

Lemma 5 Let {1\leq\alpha<2}, {1<p\leq q<\infty} , {0\leq\mu<n,} and {\frac{n-\mu}{p}-\frac{n-\mu}{q}<2}. There exists {C>0} such that

\displaystyle \Vert G_{\alpha}(t)f\Vert_{q,\mu}\leq Ct^{-\frac{\alpha}{2}(\frac{n-\mu} {p}-\frac{n-\mu}{q})}\Vert f\Vert_{p,\mu}, \ \ \ \ \ (24)

for all {f\in\mathcal{M}_{p,\mu}.}

Proof: Let {\delta=\frac{n-\mu}{p}-\frac{n-\mu}{q},} {f_{\lambda}(x)=f(\lambda x)} and {h_{\alpha}(x,t)} defined through {\widehat{h_{\alpha}}(\xi,t)=} {\left\vert \xi\right\vert ^{\delta}E_{\alpha }(-|\xi|^{2}t^{\alpha}).} Consider the multiplier operators

\displaystyle \begin{array}{rcl} F(D)f & =&[(-\Delta)^{\frac{\delta}{2}}G_{\alpha}(1)]f=h_{\alpha} (\cdot,1)\ast f(\cdot),\nonumber\\ (\lbrack(-\Delta)^{\frac{\delta}{2}}G_{\alpha}(t)]f)(x) &=&t^{^{-\delta\frac{\alpha}{2}}}\left( h_{\alpha}(\cdot,1)\ast f_{t^{\alpha/2}}(\cdot)\right) _{t^{-\alpha/2}}(x), \end{array}

that is,

\displaystyle F(D)f=t^{^{-\delta\frac{\alpha}{2}}}\left( F(D)(f_{t^{\alpha/2}})\right) _{t^{-\alpha/2}}(x), \ \ \ \ \ (25)

where the symbol of {F(D)} is {\left\vert \xi\right\vert ^{\delta}E_{\alpha }(-|\xi|^{2}t^{\alpha})}. Lemma 4 implies that {F(\xi)} satisfies (13) with {m=0.} Then, we use (5) to obtain

\displaystyle \begin{array}{rcl} \left\Vert \left( F(D)(f_{t^{\alpha/2}})\right) _{t^{-\alpha/2}}\right\Vert _{p,\mu} & =&t^{-\frac{\alpha}{2}(\frac{n-\mu}{p})}\left\Vert F(D)(f_{t^{\alpha/2}})\right\Vert _{p,\mu}\\\ & \leq &Ct^{-\frac{\alpha}{2}(\frac{n-\mu}{p})}\left\Vert f_{t^{\alpha/2} }\right\Vert _{p,\mu}\\ & =&Ct^{-\frac{\alpha}{2}(\frac{n-\mu}{p})}t^{\frac{\alpha}{2}(\frac{n-\mu} {p})}\left\Vert f\right\Vert _{p,\mu} \end{array}

and therefore,

\displaystyle \left\Vert \left( F(D)(f_{t^{\alpha/2}})\right) _{t^{-\alpha/2}}\right\Vert_{p,\mu} \leq C\left\Vert f\right\Vert _{p,\mu}. \ \ \ \ \ (26)

Now, using Sobelev embedding (11), and afterwards (25), we obtain

\displaystyle \begin{array}{rcl} \left\Vert G_{\alpha}(t)f\right\Vert _{q,\mu} & \leq&\left\Vert G_{\alpha }(t)f\right\Vert _{\mathcal{M}_{p,\mu}^{\delta}}\\ & =&\left\Vert (-\Delta)^{\frac{\delta}{2}}G_{\alpha}(t)f\right\Vert _{p,\mu }\\ & =&t^{^{-\delta\frac{\alpha}{2}}}\left\Vert \left( F(D)(f_{t^{\alpha/2} })\right) _{t^{-\alpha/2}}\right\Vert _{p,\mu}\\ & \leq &Ct^{^{-\frac{\alpha}{2}(\frac{n-\mu}{p}-\frac{n-\mu}{q})}}\left\Vert f\right\Vert _{p,\mu}, \end{array}

because of (26). \Box

5. Self-similarity and symmetries for a fractional-wave equation

The problem (1) can be formally converted to the integral equation (see [Hirata-Miao])

\displaystyle u(x,t)={G}_{\alpha}(t)u_{0}(x)+B_{\alpha}(u), \ \ \ \ \ (27)

with

\displaystyle B_{\alpha}(u)(t)=\int_{0}^{t}G_{\alpha}(t-s)\int_{0}^{s}R_{\alpha-1} (s-\tau)|u(\tau)|^{\rho}u(\tau)d\tau ds, \ \ \ \ \ (28)

which should be understood in the Bochner sense in Morrey spaces, being {R_{\eta}(s)=s^{\eta-1}/\Gamma(\eta)}. Throughout this notes a mild solution for (1) is a function {u} satisfying (27). We shall employ the Kato-Fujita method (see [Kato]) to integral equation to get our results.

From now on, we perform a scaling analysis in order to choose the correct indexes for Kato-Fujita spaces. A simple computation by using

\displaystyle \mathbf{\partial}_{t}^{\alpha}u =\Delta_{x}u+|u|^{\rho}u,\text{ in }\mathbb{R}^{n},\;t>0 \ \ \ \ \ (29)

shows that the indexes {k_{1}=2/\rho} and {k_{2}=2/\alpha} are the unique ones such that the function {u_{\lambda}} given by

\displaystyle u_{\lambda}(x,t)=\lambda^{k_{1}}u(\lambda x,\lambda^{k_{2}}t) \ \ \ \ \ (30)

is a solution of that, for each {\lambda>0,} whenever {u} is also. The scaling map for (29) is defined by

\displaystyle u(x,t)\rightarrow u_{\lambda}(x,t). \ \ \ \ \ (31)

Making {t\rightarrow0^{+}} in (31) one obtains the following scaling for the initial condition

\displaystyle u_{0}(x)\rightarrow\lambda^{2/\rho}u_{\lambda}(x). \ \ \ \ \ (32)

Solutions invariant by (31), that is

\displaystyle u(x,t)=u_{\lambda}(x,t)\text{ for all }\lambda>0, \ \ \ \ \ (33)

are called self-similar ones. Since we are interested in such solutions, it is suitable to consider critical spaces for {u(x,t)} and {u_{0}}, i.e., the ones whose norms are invariant by (31) and (32), respectively.

Consider the parameters

\displaystyle \mu=n-2p/\rho\text{ and }\beta=\frac{\alpha}{\rho}-\alpha\frac{n-\mu}{2q}, \ \ \ \ \ (34)

and let {BC((0,\infty),X)} stands for the class of bounded and continuous functions from {(0,\infty)} to a Banach space {X.} We take {u_{0}} belonging to the critical space {\mathcal{M}_{p,\mu}} and study (27) in the Kato-Fujita type space

\displaystyle H_{q}=\{u(x,t)\in BC((0,\infty);\mathcal{M}_{p,\mu})\,:\,t^{\beta}u\in BC((0,\infty);\mathcal{M}_{q,\mu})\}, \ \ \ \ \ (35)

which is Banach with the norm

\displaystyle \Vert u\Vert_{H_{q}}=\sup_{t>0}\Vert u(\cdot,t)\Vert_{p,\mu}+\sup _{t>0}t^{\beta}\Vert u(\cdot,t)\Vert_{q,\mu}. \ \ \ \ \ (36)

Notice that the norm (36) is invariant by scaling transformation (31).

From Lemma 1, a typical data belonging to {\mathcal{M}_{p,\mu}} is the homogeneous function

\displaystyle u_{0}(x)=\,{\Omega(}\frac{x}{\left\vert x\right\vert }{)}/\left\vert x\right\vert {^{-\frac{2}{\rho}}} \ \ \ \ \ (37)

where {1\leq p<\frac{n\rho}{2}} and {\Omega} is a bounded function on sphere {\mathbb{S}^{n-1}.} We refer the book [Gigabook, chapter 3] for more details about self-similar solutions and PDE’s.

Our well-posedness result reads as follows.

Theorem 6 (Well-posedness) Let {0<\rho<\infty}, {1<\alpha<2}, {1<p<\frac {n\rho}{2}}, and {\mu=n-{2p}/{\rho}}. Suppose that {\frac{n-\mu}{p} -\frac{n-\mu}{q}<2} and

\displaystyle 1-\frac{1}{\alpha}\,\frac{\rho}{\rho+1}<\frac{p}{q}<\frac{1}{\alpha}\;\;\text{ and }\;\frac{\rho p}{p-1}<q<\infty. \ \ \ \ \ (38)

  • (i) (Existence and uniqueness) There exist {\ \varepsilon>0} and {\delta=\delta(\varepsilon)} such that if {\Vert u_{0}\Vert_{p,\mu}\leq\delta} then the equation (1) has a mild solution {u\in H_{q}}, which is the unique one in the ball {D_{2\varepsilon}=\{u\in H_{q};\left\Vert u\right\Vert _{H_{q}}\leq2\varepsilon\}}. Moreover, {u\rightharpoonup u_{0}} in {\mathcal{D}^{\prime}(\mathbb{R}^{n})} as {t\rightarrow0^{+}}
  • (ii) (Continuous dependence on data) Let {\mathcal{I}_{0}=\{u_{0} \in\mathcal{M}_{p,\mu};\left\Vert u\right\Vert _{p,\mu}\leq\delta\}.\,\ }The data-solution map is Lipschitz continuous from {\mathcal{I}_{0}} to {D_{2\varepsilon}}.

Remark 1

  • (i) With a slight adaptation of the proof of Theorem 6, we could treat more general nonlinearities. Precisely, one could consider (1) and (29) with {f(u)} instead of {u\left\vert u\right\vert ^{\rho}}, where {f\in C(\mathbb{R})}, {f(0)=0} and there is {C>0} such that

    \displaystyle |f(a)-f(b)|\leq C|a-b|(|a|^{\rho}+|b|^{\rho}),\text{ for all }a,b\in \mathbb{R}.

  • (ii) (Local-in-time well-posedness) A local version of Theorem 6 holds true by replacing the smallness condition on initial data by a smallness one on existence time {T>0}. Here we should consider the local-in-time space\displaystyle H_{q,T}=\{u(x,t)\in BC((0,T);\mathcal{M}_{p,\mu})\,:\,\lim\sup_{t\rightarrow 0^{+}}t^{\beta}\Vert u(\cdot,t)\Vert_{q,\mu}=0\}, and {u_{0}\in\mathcal{M}_{p,\mu}} such that {\lim\sup_{t\rightarrow0^{+} }t^{\beta}\Vert G_{\alpha}(t)u_{0}\Vert_{q,\mu}=0.} In particular, this condition is verified when {u_{0}} belongs to the closed subspace {\ddot{\mathcal{M}}_{p,\mu}} (see (8)).

Let {O(n)} be the orthogonal matrix group in {\mathbb{R}^{n}} and let {\mathcal{G}} be a subset of {O(n).} A function {h} is said symmetric and antisymmetric under the action of {\mathcal{G}} when {h(x)=h(Mx)} and {h(x)=-h(M^{-1}x)}, respectively, for every {M\in\mathcal{G}}.

Theorem 7 Under the hypotheses of Theorem 6.

  • (i) (Self-similarity) If {u_{0}} is a homogeneous function of degree {-\frac{2}{\rho}}, then the mild solution given in Theorem 6 is self-similar.
  • (ii) (Symmetry and antisymmetry) The solution {u(x,t)} is antisymmetric (resp. symmetric) for {t>0}, when {u_{0}} is antisymmetric (resp. symmetric) under {\mathcal{G}.}
  • (iii) (Positivity) If {u_{0}\not \equiv 0} and {u_{0}(x)\geq0} (resp. {u_{0}(x)\leq0}) then {u} is positive (resp. negative).

Remark 2 (Special examples of symmetry and antisymmetry)

  • (i) The case {\mathcal{G}=O(n)} corresponds to radial symmetry. Therefore, it follows from Theorem 7 (ii) that if {u_{0}} is radially symmetric then {u(x,t)} is radially symmetric for {t>0}.
  • (ii) Let {Mx=-x} be the reflection over the origin and let {I_{\mathbb{R}^{n}}} be the identity map. The case {\mathcal{G} =\{I_{\mathbb{R}^{n}},M\}} corresponds to parity of functions, that is, {h(x)} is even and odd when {h(x)=h(-x)} and {h(x)=-h(-x)}, respectively. So, from Theorem 7 (ii), we have that the solution {u(x,t)} is even (resp. odd) for {t>0,} when {u_{0}(x)} is even (resp. odd).

A fractional Hörmander type multiplier theorem

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— 1. A fractional Hörmander type multiplier theorem —

In a few months ago we deal with translation invariant operators {T} in {L^{p}(\mathbb{R}^{d})}, see previous post. More precisely, was shown that there exists a tempered distribution {K\in\mathcal{S}'(\mathbb{R}^{d})} such that {Tu=u\ast K}. However, what does happens with {\widehat{K}} to get {L^{p}(\mathbb{R}^{d})}-boundedness of {T}? Such questions belongs to a fruitiful area of Fourier analysis which many personages work in that, such as Marcinkiewicz, Minhklin, Hörmander, Lizorkin and more recentily Fefferman’s work about ball multiplier conjecture. In this post will be initiated the saga related to {L^{p}(\mathbb{R}^{d})} spaces so far. Before, we recall some important definitions. Let {K\in\mathcal{S}'(\mathbb{R}^{d})}, we set

\displaystyle \widehat{K}(\varphi)=K(\widehat{\varphi}), \; \; \widehat{\varphi}(\xi)=(\mathcal{F}\varphi)(\xi):=(2\pi)^{-d}\int_{\mathbb{R}^{d}}e^{-ix\cdot\xi}\varphi(x)dx \ \ \ \ \ (1)

for any {\varphi\in\mathcal{S}(\mathbb{R}^{d})}. Also, recall that

\displaystyle (u\ast K)(\varphi)=K(Ru\ast\varphi). \ \ \ \ \ (2)

Then via Theorem 3 the operator {T} may be written in {\mathcal{S}'(\mathbb{R}^{d})} as

\displaystyle Tu(x)=\mathcal{F}^{-1}[\sigma(\xi)\mathcal{F}u(\xi)](x) \ \ \ \ \ (3)

where {\sigma=\widehat{K}(\varphi)}. In fact, by observing that

\displaystyle \mathcal{F}(ab)(\xi)=\mathcal{F}a(\xi)\mathcal{F}b(\xi)

we have from (2) that

\displaystyle \begin{array}{rcl} \mathcal{F}(u\ast K)(\varphi)=u\ast K(\mathcal{F}\varphi)=K(Ru\ast \mathcal{F}\varphi)=K(\mathcal{F}(\check{Ru}\,\varphi))=\widehat{K}(\varphi)\mathcal{F}u \end{array}

being {\check{a}} the inverse Fourier transform of {a}. We call the operator {T} in (3) of multiplier operator associated to symbol {\sigma} and we will written it as {T_{\sigma}}. Also, we say that {T_{\sigma}} is a multiplier on {L^{p}(\mathbb{R}^{d})}, if for all {u\in L^{2}\cap L^{p}} the map {u\mapsto T_{\sigma}u} is bounded on {L^{p}(\mathbb{R}^{d})}.

Example 1 The Fourier transform in classical sense of

\displaystyle \begin{array}{rcl} f(x)=\frac{1}{\vert x\vert^{2}}, \text{ for } x\in\mathbb{R}^{3}\backslash\{0\} \end{array}

is not possible, because {f} not belong to any {L^{p}(\mathbb{R}^{3})} space. However, Fourier transform of {f} in distribution sense is given by

\displaystyle \begin{array}{rcl} \widehat{\frac{1}{\vert x\vert^{2}}}(\xi)=\frac{\pi}{\vert \xi\vert}, \end{array}

for more details, see notes of Iannis Parissis.

Let {s\in\mathbb{R}}, the Bessel space {L_{s}^{p}(\mathbb{R}^{d})} is the space of {g} functions such that {(1+\vert \xi\vert^{2})^{s/2}\widehat{g}(\xi)\in L^{p}(\mathbb{R}^{d})} which endowed with norm

\displaystyle \Vert g\Vert_{L_{s}^{p}(\mathbb{R}^{d})}=\Vert (1+\vert \cdot\vert^{2})^{s/2}\widehat{g}\Vert_{L^{p}(\mathbb{R}^{d})}, \text{ for } 1\leq p\leq\infty

is a Banach space. The following proposition give a condition on symbol {\sigma} for which {T_{\sigma}} is a multiplier on {L^{p}(\mathbb{R}^{d})}.

Proposition 1 Let {s>d/2} and {\sigma\in L^{2}_{s}(\mathbb{R}^{d})}. Then {T_{\sigma}} is a multiplier on {L^{p}(\mathbb{R}^{d})}, for {1\leq p\leq \infty}.

Proof: In view of {T_{\sigma}f(x)=(K\ast f)(x)}, where {\widehat{K}(\xi)=\sigma(\xi)}. From Young’s inequality, just we need to show {K\in L^{1}(\mathbb{R}^{d})}. To do this, let {\widehat{h}(\xi)=(1+\vert \xi\vert^{2})^{s/2}\sigma(\xi)\in L^2(\mathbb{R}^{d})}. As {\Vert h\Vert_{L^2}=\Vert \widehat{h}\Vert_{L^2}=\Vert \sigma\Vert_{L_{s}^{2}}}, from Cauchy-Schwartz inequality one gets

\displaystyle \begin{array}{rcl} \int_{\mathbb{R}^{d}}\vert K(x)\vert dx&=&\int_{\mathbb{R}^{d}}\vert \check{\sigma}(x)\vert dx\\ \\ &=& \left (\int_{\mathbb{R}^{d}}(1+\vert x\vert^{2})^{-s} dx\right)^{\frac{1}{2}} \left(\int_{\mathbb{R}^{d}} \vert h(x)\vert^{2} dx\right)^{\frac{1}{2}}\\ \\ &\leq& C \Vert \sigma\Vert_{L_{s}^{2}}. \end{array}

\Box

Example 2 Let {\sigma} be bounded, that is, {\sigma\in L^{\infty}(\mathbb{R}^{d})}. We claim that

\displaystyle \Vert\sigma\Vert_{L^{\infty}(\mathbb{R}^{d})}=\Vert T_{\sigma}\Vert_{L^2(\mathbb{R}^{d})\rightarrow L^2(\mathbb{R}^{d})}.

It’s quite easy to get {\Vert T_{\sigma}\Vert_{L^2(\mathbb{R}^{d})\rightarrow L^2(\mathbb{R}^{d})}\leq \Vert\sigma\Vert_{L^{\infty}(\mathbb{R}^{d})}} via Plancherel’s theorem. It remains to get the other inequality. Let {D_{r}(x)} be a ball in {\mathbb{R}^{d}} with radius {r}. Let {\varphi_r\in C^{\infty}_c(\mathbb{R}^{d})} such that {supp\,(\varphi_r)\subset D_{2r}(x)} and {\varphi_r\equiv1} on {D_r(x)}. Note that

\displaystyle \begin{array}{rcl} \sigma(\xi)\varphi_r=\mathcal{F}(T_{\sigma}\ast \mathcal{F}^{-1}\varphi_r) &\Rightarrow& \Vert \sigma\varphi_r\Vert_{L^2(\mathbb{R}^{d})}=\\ \\ &=&\Vert T_{\sigma}\ast\mathcal{F}^{-1}\varphi_r\Vert_{L^2(\mathbb{R}^{d})}=\Vert T_{\sigma}\mathcal{F}^{-1}\varphi_r\Vert_{L^2}\\ \\ &\Rightarrow& \sigma\in L^{2}(D_r(x))\\ \\ &\Rightarrow& f\sigma\in L^{2}(\mathbb{R}^{d}),\,f(x)=\vert D_r(x)\vert^{-1}\chi_{D_r(x)}\\ \\ &\Rightarrow& \int_{\mathbb{R}^{d}}\vert f\sigma\vert^2d\xi=\int_{\mathbb{R}^{d}}\vert T_{\sigma}f\vert^2d\xi\leq \Vert T_{\sigma}\Vert_{L^2\rightarrow L^2}^{2} \int_{\mathbb{R}^{d}}\vert f\vert^2d\xi. \end{array}

The Lebesgue differentiation’s theorem yields

\displaystyle \begin{array}{rcl} \Vert T_{\sigma}\Vert_{L^2\rightarrow L^2}^{2}-\vert\sigma(x)\vert^{2} &=&\lim_{r\rightarrow0}\frac{1}{\vert D_{r}(x)\vert}\int_{D_r(x)}(\Vert T_{\sigma}\Vert_{L^2\rightarrow L^2}^{2}-\vert\sigma(\xi)\vert^{2})d\xi\\ \\ &=&\lim_{r\rightarrow0}\int_{\mathbb{R}^{d}}(\Vert T_{\sigma}\Vert_{L^2\rightarrow L^2}^{2}-\vert\sigma(\xi)\vert^{2})\vert f(\xi)\vert^{2}d\xi\geq 0, \end{array}

in other words, {\Vert\sigma\Vert_{L^{\infty}(\mathbb{R}^{d})}\leq \Vert T_{\sigma}\Vert_{L^2(\mathbb{R}^{d})\rightarrow L^2(\mathbb{R}^{d})}}.

Let {\Sigma_{0}^{\beta}(\mathbb{R}^{d})} be the set of all tempered distributions {\sigma} away from origin of {\mathbb{R}^{d}} satisfying the estimate

\displaystyle \sup_{\xi\neq0}|\xi|^{|\alpha|+\beta}|\partial_{\xi}^{\alpha}\sigma(\xi)|\leq L, \;\beta\geq 0 \ \ \ \ \ (4)

for all multi-index {\alpha} with {\vert \alpha\vert\leq k}. In the next sections we give a much weaker hypotheses on symbol {\sigma}, via Littlewood-Paley theory, which imply such as so-called symbol-H\”{o}rmader condition and {\sigma\in\Sigma_{0}^{\beta}(\mathbb{R}^{d})}. In all that follows, the class {OP\Sigma_{0}^{\beta}(\mathbb{R}^{d})} denotes the set of multipliers operators with symbol {\sigma(T_{\sigma})=\sigma(\xi)} belonging to {\Sigma_{0}^{\beta}(\mathbb{R}^{d})}.

— 1.1. Littlewood-Paley theory —

Let {\varphi} be radial Schwartz function supported in {\{\xi\in\mathbb{R}^{d}\,:\, 0<\vert\xi\vert\leq 2\}} which is identically {1} on {\{\xi\in\mathbb{R}^{d}\,:\, 0\leq\vert\xi\vert\leq 1\}}. To get that, recall of the topological Urysohn’s lemma. For all {\xi\in\mathbb{R}^{d}}, set

\displaystyle \begin{array}{rcl} \widehat{\psi}(\xi)=\varphi(\xi)-\varphi(2\xi). \end{array}

Notice that {supp(\widehat{\psi})\subset \{\xi\in\mathbb{R}^{d}\,:\, 1/2\leq \vert\xi\vert\leq 2\}}, because {\varphi(\xi)=\varphi(2\xi)=0} and {\varphi(\xi)=\varphi(2\xi)=1} for, respectively, {\vert \xi\vert >2} and {\vert \xi\vert <1/2}. Moreover, the family of functions {\{\widehat{\psi}(\xi/2^{j})\}_{j\in\mathbb{Z}}} forms a partition of unity

\displaystyle \begin{array}{rcl} \sum_{j\in\mathbb{Z}} \widehat{\psi}(\xi/2^{j}) =1, \; \xi\in\mathbb{R}^{d}\backslash\{0\}. \end{array}

Next, for {j\in\mathbb{Z}} we define the cut-off operator or Littlewood-Palay operator as

\displaystyle \begin{array}{rcl} \widehat{\Delta_j(f)}(\xi)=\widehat{\psi}_j(\xi)\widehat{f}(\xi):=\widehat{\psi}(\xi/2^{j})\widehat{f}(\xi), \end{array}

where {f\in\mathcal{S}'(\mathbb{R}^{d})}. Notice that {\Delta_j} is supported in {\{x\in\mathbb{R}^{d}\,:\, 2^{j-1}\leq \vert x\vert\leq 2^{j+1}\}}. Indeed, by Fourier property

\displaystyle \begin{array}{rcl} \widehat{\textnormal{Dil}_{2^{-j}}^{1}\psi}(\xi)=\int_{\mathbb{R}^{d}}e^{-ix\cdot\xi}2^{jd}\psi(2^{j}x)dx=\widehat{\psi}(2^{-j}\xi) \end{array}

it follows that {\psi(2^{-j}x)=\textnormal{Dil}_{2^{-j}}^{1}\psi(x)=2^{jd}\psi(2^{j}x)}. Therefore, if {\widehat{\psi}} is supported in the annulus {\{\xi\in\mathbb{R}^{d}\,:\, 1/2\leq \vert\xi\vert\leq 2\}} then the inverse Fourier transform is supported in the annulus

\displaystyle \vert x\vert\simeq2^{j}:=\{x\in\mathbb{R}^{d}\,:\, 2^{j-1}\leq \vert x\vert\leq 2^{j+1}\}.

In other words, the operator {\Delta_j} isolate the part of a function being each one concetrated near annulus {\vert x\vert\simeq2^{j}}.

Proposition 2 The cut-off operator is a self-adjoint operator,

\displaystyle \begin{array}{rcl} \Delta_j\phi(\varphi)=\phi(\Delta_j\varphi) \end{array}

for any {\varphi\in\mathcal{S}(\mathbb{R}^{d})} and {\phi\in\mathcal{S}'(\mathbb{R}^{d})}.

Proof: The prove is a consequence of the radial property of {\psi}. For that, firstly, shall be observed that

\displaystyle \begin{array}{rcl} \Delta_j\phi(\varphi) &=&\mathcal{F}(\Delta_j\phi)(\mathcal{F}^{-1}{\varphi})\notag\\ \\ &=&\widehat{\psi}_j\widehat{\phi}(\mathcal{F}^{-1}{\varphi})\notag\\ \\ &=&\phi(\mathcal{F}(\psi_j\mathcal{F}^{-1}{\varphi})), \end{array}

when we have in mind (1). Now, radial property of {\psi} and (4) implies that

\displaystyle \begin{array}{rcl} \Delta_j\phi(\varphi) &=&\phi\left(\int e^{-i\xi\cdot x} \psi_{j}(x)\widehat{\varphi}(-x)dx\right)\\ \\ &=&\phi\left(\int e^{i\xi\cdot x} \psi_{j}(-x)\widehat{\varphi}(x)dx\right)\\ \\ &=&\phi(\mathcal{F}^{-1}(\psi_{j}(\cdot)\widehat{\varphi})(\xi))\\ \\ &=&\phi(\Delta_j\varphi). \end{array}

\Box

The decomposition of a tempered distribution {f} into a sum of “long pieces” {\Delta_j(f)} supported in the annulus {\vert x\vert \simeq 2^{j}} is known as Littlewood-Paley decomposition, because of the famous works de Littlewood and Paley related to Fourier and Power series, see xx. More precisely, Littlewood-Paley decomposition of a tempered distribution {f\in\mathcal{S}'(\mathbb{R}^{d})} is defined as follows

\displaystyle f=\sum_{j\in\mathbb{Z}} \Delta_j(f) \ \ \ \ \ (5)

However, there is tempered distributions for which Littlewood-Paley decomposition fail on {\mathcal{S}'(\mathbb{R}^{d})}. For instance take the distribution {f(x)=1} and note by from Proposition 2 and {\widehat{1}(\varphi)=\varphi(0)} that

\displaystyle \begin{array}{rcl} \Delta_jf(\varphi)=f(\Delta_j\varphi)=f(\mathcal{F}^{-1}(\widehat{\Delta_j\varphi}))=\widehat{f}(\widehat{\Delta_j\varphi})=\widehat{\Delta_j\varphi}(0)=0 \end{array}

that is, {\Delta_jf=0}, {j\in\mathbb{Z}}. To overcome this discomfort, we introduce in {\mathcal{S}'(\mathbb{R}^{d})} the equivalence class {\sim} given by

\displaystyle \begin{array}{rcl} u\sim v\Longleftrightarrow u-v\in\mathcal{P} \end{array}

where {\mathcal{P}} denotes the set of polynomials {p},

\displaystyle \begin{array}{rcl} p(x)=\sum_{\vert\alpha\vert\leq m}c_{\beta}(x)x^{\alpha} \end{array}

with complex coefficients {c_{\alpha}} and {m} a non-negative integer. The space {\mathcal{S}'(\mathbb{R}^{d})} endowed with {\sim} will be denoted by {\mathcal{S'}/\mathcal{P}}.

Proposition 3 Let {\mathcal{S}_{\infty}(\mathbb{R}^{d})} be the space of Schwartz functions such that

\displaystyle \begin{array}{rcl} \int_{\mathbb{R}^{d}} x^{\alpha}\varphi(x)dx=0. \end{array}

Then {\mathcal{S}_{\infty}(\mathbb{R}^{d})} is a subspace of {\mathcal{S}(\mathbb{R}^{d})} which has

\displaystyle \mathcal{S}_{\infty}'(\mathbb{R}^{d})=\mathcal{S'}/\mathcal{P} \ \ \ \ \ (6)

in sense of isomophism.

Proof: The prove is standard, consider the identification map {\mathcal{J}} which goes an element {\mathcal{S}'(\mathbb{R}^{d})} into the equivalence class {\mathcal{S'}/\mathcal{P}} that contains it. The kernel of the map {\mathcal{J}} is {\mathcal{P}}. \Box

The aim is that, for any {u\in\mathcal{S}'_{\infty}(\mathbb{R}^{d})} we have

\displaystyle \begin{array}{rcl} (\psi_{j}\widehat{u})^{\vee}\rightarrow 0 \text{ in } \mathcal{S}'_{\infty} \text{ as } j\rightarrow+\infty. \end{array}

This follows by observing that

\displaystyle \varphi(2^{-j}\cdot)\widehat{u}\rightarrow\widehat{u} \text{ in } \mathcal{S}'_{\infty}(\mathbb{R}^{d}) \text{ as } j\rightarrow+\infty \ \ \ \ \ (7)

where {\varphi} is that radial Schwartz function suported into a compact set and equal to {1} in {D_{1}(0)}. To show (7), recall that a distribution {u} is a tempered distribution if

\displaystyle \begin{array}{rcl} \lim_{k\rightarrow\infty}u(\lambda_k) = 0 \text{ whenever } \lambda_k\rightarrow0 \text{ in } \mathcal{S} \text{ as } k\rightarrow\infty. \end{array}

Easily, via identification (6), this indue in nature’s way the convergence in {\mathcal{S}'_{\infty}(\mathbb{R}^{d})}. Thus, it is quite easy to show {\varphi_{j}\widehat{u}\in \mathcal{S}'_{\infty}(\mathbb{R}^{d})}. It remains to show

\displaystyle \begin{array}{rcl} \lim_{j\rightarrow+\infty}\varphi(2^{-j}\cdot)\widehat{u}(\lambda)=\widehat{u}(\lambda). \end{array}

Let us fixe {i\in\mathbb{N}} such that {2^{i-1}\leq \vert \xi\vert\leq 2^{i}} and write

\displaystyle \begin{array}{rcl} \lim_{j\rightarrow+\infty}\varphi(2^{-j}\xi)\widehat{u}(\lambda)=\lim_{j\geq i,i\rightarrow+\infty}\varphi(2^{-j}\xi)\widehat{u}(\lambda)=\widehat{u}(\lambda) \end{array}

because for {j\geq i} we have {1/2\leq \vert\xi\vert \leq 1} and {\varphi \equiv1} on ball {D_{1}(0)}. Thus, given {p\in\mathcal{P}(\mathbb{R}^{d})} one has {p=0} and the its Littlewood-Paley decomposition is null in {\mathcal{S}'_{\infty}(\mathbb{R}^{d})}, because of

\displaystyle \begin{array}{rcl} \lim_{j\rightarrow+\infty}\psi_{j}\widehat{p}(\lambda)=0 \end{array}

and

\displaystyle \begin{array}{rcl} \lim_{j\rightarrow-\infty}\psi_{j}\widehat{p}(\lambda)=0. \end{array}

The last equality is obtained via

\displaystyle \begin{array}{rcl} \lim_{j\rightarrow-\infty}\varphi(2^{-j}\cdot)\widehat{p}(\lambda) &=&\sum_{\vert\alpha\vert\leq m}c_{\alpha}\lim_{j\rightarrow-\infty}\varphi(2^{-j}\cdot)\partial^{\alpha}\delta_{0}(\lambda)\\ \\ &=&\sum_{\vert\alpha\vert\leq m}c_{\alpha}\lim_{j\rightarrow-\infty}\varphi(2^{-j}\cdot)\delta_{0}(\partial^{\alpha}\lambda)=0, \end{array}

where {\delta_0} is the Dirac mass and don’t forget that {\mathcal{F}^{-1}(\partial^{\beta}\delta_{0})(x)=(-2\pi i x)^{\beta}}, which is consequence of properties

  • (i){(\partial_{x}^{\alpha}\varphi)^{\wedge}(\xi)=\xi^{\alpha}\widehat{\varphi}(\xi)} for every multi-index {\alpha\in (\mathbb{N}\cup\{0\})^{d}}
  • (ii){(\partial^{\alpha}_{\xi}\widehat{\varphi})(\xi)=((-x)^{\alpha}\varphi(x))^{\wedge}(\xi)} for every multi-index {\alpha\in (\mathbb{N}\cup\{0\})^{d}}

for all {\varphi\in\mathcal{S}(\mathbb{R}^{d})}.

Proposition 4 If {\sigma\in \Sigma_{0}^{1}(\mathbb{R}^{d})}, then {k_{\sigma}} agrees with {C^{k}(\mathbb{R}^{d})} function away from the origin and satisfies the following pointwise estimate

\displaystyle \vert \partial_{z}^{\beta}k_{\sigma}(z)\vert\leq C L |z|^{-d-\vert\beta\vert},\; z\neq 0 \ \ \ \ \ (8)

for all multi-index {\beta\in(\mathbb{N}\cup\{0\})^{d}}, where {k_{\sigma}(z)=(2\pi)^{-d}\int_{\mathbb{R}^{d}}e^{iz\cdot\xi}\sigma(\xi)d\xi}.

Proof: Let {\sigma_j(\xi)=\psi_j(\xi)\sigma(\xi)}, then for all {\xi\neq 0} this give us

\displaystyle \begin{array}{rcl} \sum_{j\in\mathbb{Z}}\sigma_j \,=\,\sigma \text{ in } \mathcal{S}'/\mathcal{P} \end{array}

where each peace {\sigma_j} is supported on annulus {\vert\xi\vert \simeq 2^{j}:=\{\xi\in\mathbb{R}^{d}\,:\, 2^{j-1}\leq\vert\xi\vert\leq 2^{j+1}\}}. So makes sense to define,

\displaystyle \begin{array}{rcl} k_j(x)=(2\pi)^{-d}\int_{\mathbb{R}^{d}}e^{ix\cdot\xi}\sigma_j(\xi)d\xi. \end{array}

If {\sigma\in\Sigma_{0}^{1}(\mathbb{R}^{d})}, for {\alpha\in (\mathbb{N}\cup\{0\})^{d}} one has

\displaystyle \vert \partial_{\xi}^{\alpha}\sigma_j(\xi)\vert \leq\vert \sum_{j\in\mathbb{Z}}\partial_{\xi}^{\alpha}\sigma_j(\xi)\vert\leq \sup_{\vert\xi\vert \simeq 2^{j}}\vert \partial_{\xi}^{\alpha}\sigma(\xi)\vert\leq L\, 2^{-j\vert\alpha\vert}. \ \ \ \ \ (9)

It follows that

\displaystyle \begin{array}{rcl} \vert (-2\pi iz)^{\alpha}\partial_{z}^{\beta}k_{j}(z)\vert &=&\vert\int_{\mathbb{R}^{d}}e^{2\pi i\xi\cdot z} \partial_{\xi}^{\alpha}(\xi^{\beta}\sigma_{j}(\xi))d\xi\vert\nonumber\\ \notag\\ &=&\vert\int_{\vert\xi\vert \simeq 2^{j}}e^{2\pi i\xi\cdot z} \partial_{\xi}^{\alpha}(\xi^{\beta}\sigma_{j}(\xi))d\xi\vert\nonumber\\ \notag\\ &=&\vert\sum_{\gamma\leq\alpha}\binom{\alpha}{\gamma}\int_{\vert\xi\vert \simeq 2^{j}}e^{ 2\pi i\xi\cdot z}(\partial_{\xi}^{\gamma}\xi^{\beta})(\partial_{\xi}^{\alpha-\gamma}\sigma_{j}(\xi))d\xi\vert\\ \notag\\ &=&C_{\alpha,\beta,\gamma}\int_{\vert\xi\vert \simeq 2^{j}}e^{2\pi i\xi\cdot z} (\xi^{\beta-\gamma})(\partial_{\xi}^{\alpha-\gamma}\sigma_{j}(\xi))d\xi\vert\\ \notag\\ &\leq& C \,2^{jd}2^{j(\vert\beta\vert-\vert\gamma\vert)}L\,2^{j(-\vert\alpha\vert+\vert\gamma\vert)}\\ \notag\\ &=&C L\,2^{j(d+\vert\beta\vert -\vert\alpha\vert)}. \end{array}

If we put {\vert \alpha\vert = m} into the last equality above, for all multi-indices {\beta} and non-negative {m} we have

\displaystyle \vert \partial_{z}^{\beta}k_{j}(z)\vert\leq CL\,\min\{2^{j(d+\vert\beta\vert)}, \vert z\vert^{-m}2^{j(d+\vert\beta\vert -m)}\}. \ \ \ \ \ (10)

We split {\sum_{2^{j}}\vert \partial_{z}^{\beta}k_{j}(z)\vert} into two sums as

\displaystyle \begin{array}{rcl} \sum_{2^{j}}\vert \partial_{z}^{\beta}k_{j}(z)\vert=\sum_{2^j\leq \vert z\vert^{-1}}\vert \partial_{z}^{\beta}k_{j}(z)\vert+\sum_{2^j\geq \vert z\vert^{-1}}\vert \partial_{z}^{\beta}k_{j}(z)\vert. \end{array}

For {m=0} and {m>d+\vert\beta\vert} into (10), respectively, we get

\displaystyle \begin{array}{rcl} \sum_{2^j\leq \vert z\vert^{-1}}\vert \partial_{z}^{\beta}k_{j}(z)\vert&\leq& CL\, \sum_{2^j\leq \vert z\vert^{-1}}2^{j(d+\vert\beta\vert)}\leq CL\, \vert z\vert^{-(d+\vert\beta\vert)} \end{array}

and

\displaystyle \begin{array}{rcl} \sum_{2^j\geq \vert z\vert^{-1}}\vert \partial_{z}^{\beta}k_{j}(z)\vert&\leq& CL\, \vert z\vert^{-m}\sum_{2^j\geq \vert z\vert^{-1}}2^{j(d+\vert\beta\vert -m)}\notag\\ \\ &\leq&CL\, \vert z\vert^{-m}\vert z\vert^{-(d+\vert\beta\vert -m)}=CL\,\vert z\vert^{-(d+\vert\beta\vert)}. \end{array}

This inequalities says us that {\sum_{j}\partial_{z}^{\beta}k_{j}(z)} converges absolutely and uniforms in {z\neq 0}. It follows that {\sum_{j}k_{j}(z)} converges in {C^{k}(\mathbb{R}^{d}\backslash\{0\})} to a function {\tilde{k}} which also satisfies the estimate

\displaystyle \vert \partial_{z}^{\beta}\tilde{k}(z)\vert\leq CL\, \vert z\vert^{-(d+\vert\beta\vert)}. \ \ \ \ \ (11)

As {\sum_{j}\sigma_j=\sum_{j}\widehat{k}_{\sigma}} converges to {\sigma=\widehat{k}_{\sigma}}, then {k_{\sigma}=\tilde{k}} and we obtain the desired estimate. \Box

In what follows, {L^{p}(\mathbb{R}^{d}\rightarrow X)} denotes the space of {p-}Bochner integrable functions {g} from {\mathbb{R}^{d}} to a Banach space {X} and we write

\displaystyle \Vert g\Vert_{L^{p}(\mathbb{R}^{d}\rightarrow X)}=\left(\int_{\mathbb{R}^{d}}\Vert g(x)\Vert_{X}^{p}dx\right)^{\frac{1}{p}}. \ \ \ \ \ (12)

Theorem 5 (Littlewood-Paley Theorem) Let {\Delta_{j}} the Littlewood-Paley operator. If {1<p<\infty}, there exists a positive constant {C} such that

\displaystyle \Vert \Delta_{j}f\Vert_{L^p(\mathbb{R}^{d}\rightarrow l^{2}(\mathbb{Z}))}\leq C \Vert f\Vert_{L^p(\mathbb{R}^{d})}. \ \ \ \ \ (13)

Moreover,

\displaystyle \Vert f\Vert_{L^p(\mathbb{R}^{d})}\leq C \Vert \Delta_{j}f\Vert_{L^p(\mathbb{R}^{d}\rightarrow l^{2}(\mathbb{Z}))}. \ \ \ \ \ (14)

Proof: The inequality (13) follows from Theorem 5.4 in [1]. In fact,

\displaystyle \begin{array}{rcl} \Vert (\Sigma_{j\in\mathbb{Z}}\vert \Delta_{j}f\vert^{2})^{\frac{1}{2}}\Vert_{L^2(\mathbb{R}^{d})}^{2} &=&\int_{\mathbb{R}^{d}}\Sigma_{j}\vert \Delta_{j}f(x)\vert^{2}dx=\sum_{j}\int_{\mathbb{R}^{d}}\vert \Delta_{j}f(x)\vert^{2}dx\nonumber\\ \\ &=&\sum_{j}\int_{\mathbb{R}^{d}}\vert \widehat{\psi}_{j}(\xi)\widehat{f}(\xi)\vert^{2} d\xi\nonumber\\ \\ &=&\int_{\mathbb{R}^{d}}\Sigma_{j}\vert \widehat{\psi}_{j}(\xi)\vert^{2}\vert \widehat{f}(\xi)\vert^{2} d\xi\nonumber\\ \\ &=&\int_{\mathbb{R}^{d}}\vert \widehat{f}(\xi)\vert^{2} d\xi=\Vert f\Vert_{L^2}^{2}, \end{array}

in other words, the map {f\mapsto \Delta_j f} is bounded from {L^{2}(\mathbb{R}^{d})} to {L^{2}(\mathbb{R}^{d}\rightarrow l^{2}(\mathbb{Z}))}. It remains to show that the kernel {\psi_{j}} of the map {f\mapsto \Delta_j f} satisfies the {l^2(\mathbb{Z})-}H\”{o}rmander condition, which can be obtained from estimate

\displaystyle \begin{array}{rcl} \Vert \nabla \psi_j(x)\Vert_{l^{2}(\mathbb{Z})}\leq C\vert x\vert^{-d-1}. \end{array}

Indeed, by mean value theorem one has

\displaystyle \begin{array}{rcl} \int_{\vert x\vert\geq 2\vert y\vert }\Vert \psi_{j}(y-x) -\psi_j(x)\Vert_{l^{2}}dx &\leq& \int_{\vert x\vert\geq 2\vert y\vert }\vert y\vert\Vert\nabla\psi_j(x-\theta y)\Vert_{l^{2}}dx,\,0<\theta <1\\ \\ &\leq& C\int_{\vert x\vert\geq 2\vert y\vert }\vert y\vert\vert x-\theta y\vert^{-d-1} dx\\ \\ &\leq& C \int_{\vert x\vert\geq 2\vert y\vert }\vert x\vert^{-d} dx,\,\vert x-\theta y\vert\geq (1-\theta)\frac{1}{2}\vert x\vert\\ \\ &\leq& C. \end{array}

The inequalities above show us that the kernel {K(x,y):=\{\psi_j(x-y)\}_{j\in\mathbb{Z}}} is a singular kernel from {\mathbb{R}} to {l^{2}(\mathbb{Z})} and the inequality (13) follows from Theorem 5.4 in [1].

From now on, will be showed the estimate (14). To this end, we remember of {l^1(\mathbb{Z})\subset l^2(\mathbb{Z})} which give us

\displaystyle (\Sigma_j\vert \nabla \psi_j(x)\vert ^{2})^{\frac{1}{2}}\leq \Sigma_j\vert \nabla \psi_j(x)\vert. \ \ \ \ \ (15)

Recall that {\psi_j(x)=2^{jd}\psi(2^{j}x)}, where {\widehat{\psi}} is a Schwartz function. Therefore, {\vert\xi\vert^{\vert\alpha\vert}\vert\partial_{\xi}^{\alpha}\widehat{\psi}(\xi)\vert \leq C} for all multi-index {\alpha}, that is, {\widehat{\psi}\in\Sigma_{0}^{1}(\mathbb{R}^{d})}. And by Proposition 4 follows that there exists a positive constant {C>0} for which

\displaystyle \vert \nabla \psi(x)\vert \leq C\vert x\vert^{-d-1}. \ \ \ \ \ (16)

Setting

\displaystyle \begin{array}{rcl} \sum_j\vert \nabla \psi_j(x)\vert= \sum_{j\geq l }\vert 2^{j(d+1)}\nabla \psi(2^{j}x)\vert+\sum_{j\leq l}\vert 2^{j(d+1)}\nabla \psi(2^{j}x)\vert \text{ for a fixed } l>0 \end{array}

and recalling that {supp(\psi_j(x))\subset \vert x\vert \simeq 2^{j}}, we have via inequalities (15) and (16), respectively, that

\displaystyle \begin{array}{rcl} (\sum_j\vert \nabla \psi_j(x)\vert ^{2})^{\frac{1}{2}} &\leq& C\sum_{j\geq l}2^{j(d+1)-2j(d+1)}+ C\sum_{j\leq l}2^{j(d+1)-j(d+1)}\vert x\vert^{-d-1}\\ \\ &=&C\sum_{j\geq l}2^{-j(d+1)}+ C\sum_{j\leq l}\vert x\vert^{-d-1}\\ \\ &\leq& C \vert x\vert ^{-d-1}. \end{array}

Let {f} be in {L^{p'}(\mathbb{R}^{d})/\mathcal{P}}, where {1=\frac{1}{p}+\frac{1}{p'}}. For each {x\in\mathbb{R}^{d}}, let {g(x)=\{g_{j}(x)\}_{j}\in l^{2}(\mathbb{Z})} and {g\in L^{p}(\mathbb{R}^{d}\rightarrow l^2(\mathbb{Z}))}. From self-adjointness of {\Delta_{j}}, follows that

\displaystyle \begin{array}{rcl} \int_{\mathbb{R}^{d}}\langle \Delta_{j}(f), g\rangle_{l^2(\mathbb{Z})}(x)dx=\int_{\mathbb{R}^{d}}\langle f, \Delta_{j}(g_{j})\rangle_{l^2(\mathbb{Z})}(x)dx:= \int_{\mathbb{R}^{d}}f(x)\Sigma_{j}\bar{\Delta_j}(g_{j})(x)dx. \end{array}

By duality of {L^{p}(\mathbb{R}^{d})} and Cauchy-Schwartz inequality, respectively, one has

\displaystyle \begin{array}{rcl} \Vert \Sigma_j\Delta_{j}(g_j)\Vert_{L^{p}} &=&\sup_{\Vert f\Vert_{{p'}}\leq 1}\left\vert \int_{\mathbb{R}^{d}}\langle \Delta_{j}(f), g\rangle_{l^2(\mathbb{Z})}(x)dx\right\vert\\ \\ &\leq& \sup_{\Vert f\Vert_{{p'}}\leq 1}\int_{\mathbb{R}^{d}}\Vert \Delta_{j}(f)(x)\Vert_{l^{2}} \Vert g(x)\Vert_{l^2}dx\\ \\ &\leq& \sup_{\Vert f\Vert_{{p'}}\leq 1}\Vert \Delta_{j}(f)\Vert_{L^{p'}(\mathbb{R}^{d}\rightarrow l^{2})} \Vert g(x)\Vert_{{L^{p}(\mathbb{R}^{d}\rightarrow l^{2})}}\\ \\ &\leq& C\sup_{\Vert f\Vert_{{p'}}\leq 1}\Vert f\Vert_{L^{p'}} \Vert g(x)\Vert_{{L^{p}(\mathbb{R}^{d}\rightarrow l^{2})}}\\ \\ &\leq& C\Vert g(x)\Vert_{{L^{p}(\mathbb{R}^{d}\rightarrow l^{2})}} \end{array}

where we use the Hölder inequality and first Littlewood-Paley inequality (13) above. If we replace {\Delta_j} by {\tilde{\Delta}_j} in (16) such that {\tilde{\Delta}_{j}\Delta_j=\Delta_j} and we put {g=\{\Delta_j(f)\}_{j\in\mathbb{Z}}}, it follows that

\displaystyle \Vert f\Vert_{L^{p}}=\Vert \Sigma_j\Delta_j(f)\Vert_{L^{p}}(\mathbb{R}^{d})\leq C\, \Vert \Delta_{j}(f)\Vert_{L^p(\mathbb{R}^{d}\rightarrow l^{2} )}, \ \ \ \ \ (17)

as desired. It remains to get {\tilde{\Delta}_{j}}. To do this, let

\displaystyle \widehat{\tilde{\psi}}(\xi)=\varphi(\xi/4)-\varphi(4\xi) \ \ \ \ \ (18)

and set

\displaystyle \widehat{\tilde{\Delta}}_{j}(f)(\xi)=\widehat{\tilde{\psi}}_{j}(\xi)\widehat{f}(\xi)=(\varphi(\xi/2^{j+2})-\varphi(\xi/2^{j-2}))\widehat{f}(\xi). \ \ \ \ \ (19)

Then {\widehat{\tilde{\psi}}_{j}\widehat{\psi}_{j}=\widehat{\psi}_{j}} on annulus {\{\xi\,:\, 2^{j-1}<\vert\xi\vert<2^{j+1}\}}, that is, {\tilde{\Delta}_{j}{\Delta}_{j}={\Delta}_{j}}. Indeed, {\varphi(\xi/2^{j+2})=1} since {\vert\xi\vert/2^{k-2}> 2} and {\varphi(\xi/2^{j-2})=0} since {\vert\xi\vert/2^{k+2}<1/2}. This finish our prove. \Box

Lemma 6 Let {\sigma\in L^2_s(\mathbb{R}^{d})} and {2s>d\geq1}. Set {\widehat {T_{\sigma(2^{j}\cdot)}f}(\xi)=\sigma(2^{j}\xi)\widehat{f}(\xi)}. If {u\in L^{1}_{loc}(\mathbb{R}^{d})}, then there exists a positive constant {C>0} (independent of {\Vert\sigma\Vert_{L^{2}_s}}, {f} and {u}) such that

\displaystyle \int_{\mathbb{R}^{d} } \vert T_{\sigma(2^{j}\cdot)}f(x)\vert^{2}u(x)dx\leq C \Vert \sigma \Vert_{L^{2}_{s}}^{2}\int_{\mathbb{R}^{d}}\vert f(y)\vert^{2}Mu(y)dy \ \ \ \ \ (20)

where {Mu} denotes the Hardy-Littlewood maximal function,

\displaystyle Mu(x)=\sup_{r>0}\frac{1}{\vert D_{r}(x_0)\vert}\int_{D_{r}(x_0)}\vert u(x)\vert dx. \ \ \ \ \ (21)

Proof: Let {\widehat{k}_{\sigma}(\xi)=\sigma(\xi)}, then the kernel of {T_{\sigma(2^{j}\cdot)}} satisfy {k_{\sigma}(2^{j}x)=\check{\sigma}(2^{j}x)=2^{-jd}k_{\sigma}(2^{-j}x)}. Therefore, by Hölder inequality and Plancharel theorem we obtain

\displaystyle \begin{array}{rcl} \vert T_{\sigma(2^{j}\cdot)}f(x)\vert &\leq& 2^{-jd}\int_{\mathbb{R}^{d}}\vert k_{\sigma}(2^{-j}y)f(x-y)\vert dy\\ \\ &=&2^{-jd}\int_{\mathbb{R}^{d}}\vert (1+\vert 2^{-j}y\vert^{2})^{\frac{s}{2}}k_{\sigma}(2^{-j}y)\frac{f(x-y)}{(1+\vert 2^{-j}y\vert^{2})^{\frac{s}{2}}}\vert dy \\ \\ &\leq& 2^{-jd}\Vert (1+\vert 2^{-j}\cdot \vert^{2})^{\frac{s}{2}}k_{\sigma}(2^{-j}\cdot )\Vert_{L^{2}}\left(\int_{\mathbb{R}^{d}}\frac{\vert f(x-y)\vert^{2}}{(1+\vert 2^{-j}y\vert^{2})^{s}}dy\right)^{\frac{1}{2}}\\ \\ &=&2^{-jd}2^{jd/2}\Vert (1+\vert\cdot \vert^{2})^{\frac{s}{2}}k_{\sigma}(\cdot )\Vert_{L^{2}}\left(\int_{\mathbb{R}^{d}}\frac{\vert f(x-y)\vert^{2}}{(1+\vert 2^{-j}y\vert^{2})^{s}}dy\right)^{\frac{1}{2}}\\ \\ &=&\Vert \sigma \Vert_{L^{2}_{s}(\mathbb{R}^{d})}\left(\int_{\mathbb{R}^{d}}\varphi_{2^{j}}(y)\vert f(x-y)\vert^{2}dy\right)^{\frac{1}{2}} \end{array}

where {\varphi_{t}(x)=t^{-d}\varphi(x/t)} and {\varphi(x)=(1+\vert x\vert^{2})^{s}}. It follows by Fubini theorem that

\displaystyle \begin{array}{rcl} \int_{\mathbb{R}^{d} } \vert T_{\sigma(2^{j}\cdot)}f(x)\vert^{2}u(x)dx &\leq& \Vert \sigma \Vert_{L^{2}_{s}}^{2}\int_{\mathbb{R}^{d}}\vert f(y)\vert^{2}\int_{\mathbb{R}^{d}}\varphi_{2^{j}}(x-y)u(x)dxdy\\ \\ &\leq& \Vert \sigma \Vert_{L^{2}_{s}}^{2}\int_{\mathbb{R}^{d}}\vert f(y)\vert^{2}\varphi_{2^{j}}\ast u(y)dy\\ \\ &\leq& C \Vert \sigma \Vert_{L^{2}_{s}}^{2}\int_{\mathbb{R}^{d}}\vert f(y)\vert^{2}Mu(y)dy. \end{array}

The last inequality is obtaind from splitting {\mathbb{R}^{d}} into lattices {I_{q}} of length {\sqrt{\lambda}}, see figure below.

Indeed,

\displaystyle \begin{array}{rcl} \varphi_{\sqrt{\lambda}}\ast u(y) &=&\int_{\mathbb{R}^{d}}\varphi_{\sqrt{\lambda}}(x-y) u(y)dy\\ \\ &=&\sum_{q\in\mathbb{Z}^{d}}\int_{I_{q}}\varphi_{\sqrt{\lambda}}(x-y)u(y)dy\\ \\ &=&\sum_{q\in\mathbb{Z}^{d}}\lambda^{-\frac{d}{2}}\int_{I_{q}}\varphi(\frac{x-y}{\sqrt{\lambda}})u(y)dy\\ \\ &=&\sum_{q\in\mathbb{Z}^{d}}\frac{1}{vol(I_{q})}\int_{I_{q}}\varphi(\frac{x-y}{\sqrt{\lambda}})u(y)dy \end{array}

where {q_i} is a integer such that {y_i=x_i+\sqrt{\lambda}q_i}. Setting {I_{q}=\prod_{i=1}^{d}[q_i\sqrt{\lambda},(1+q_i)\sqrt{\lambda}]} it leads us to {vol(I_q)=\lambda^{\frac{d}{2}}}. By definition of {\varphi} and (21), it follows that

\displaystyle \begin{array}{rcl} \int_{\mathbb{R}^{d}}\varphi_{\sqrt{\lambda}}(x-y) u(y)dy&\leq& \sum_{q\in\mathbb{Z}^{d}}\frac{1}{(1+\vert q\vert ^{2})^{s}}Mu(y)\\ \\ &\leq& \sum_{q\in\mathbb{Z}^{d}}\frac{1}{\vert q\vert^{2s}}Mu(y) \end{array}

the last {2s-}series converges, because of {2s>d\geq 1}. \Box

The next theorem is a fractional variant of Hörmander multiplier theorem.

Theorem 7 Let {\sigma_{j}(\xi)=\widehat{\psi}_{j}(\xi)\sigma(\xi)} be such that

\displaystyle \sup_{j\in\mathbb{Z}}\Vert \sigma_{j}\Vert_{L^2_{s}(\mathbb{R}^{d})}<\infty \ \ \ \ \ (22)

for {s>d/2} and let {1<p<\infty}. Then the operator {T_{\sigma}} is a multiplier on {L^{p}(\mathbb{R}^{d})}.

Proof: From Littlewood-Paley inequality (14),

\displaystyle \begin{array}{rcl} \Vert T_{\sigma}f\Vert_{L^{p}(\mathbb{R}^{d})} &\leq& C\Vert \Delta_{j}T_{\sigma}f\Vert_{L^p(\mathbb{R}^{d}\rightarrow l^{2}(\mathbb{Z}))}\\ \\ &=&C\Vert \tilde{\Delta}_{j}\Delta_{j}T_{\sigma}f\Vert_{L^p(\mathbb{R}^{d}\rightarrow l^{2}(\mathbb{Z}))}\\ \\ &=&C\Vert \Delta_{j}T_{\sigma}\tilde{\Delta}_{j}f\Vert_{L^p(\mathbb{R}^{d}\rightarrow l^{2}(\mathbb{Z}))}. \end{array}

Thus,

\displaystyle \Vert T_{\sigma}f\Vert_{L^{p}(\mathbb{R}^{d})}\leq C\Vert \Delta_{j}T_{\sigma}g_j\Vert_{L^p(\mathbb{R}^{d}\rightarrow l^{2}(\mathbb{Z}))} \ \ \ \ \ (23)

where {g_j=\tilde{\Delta}_{j}f} and {\tilde{\Delta}_{j}} is a Littlewood-Paley operator such that {\Delta_j\tilde{\Delta}_{j}=\tilde{\Delta}_{j}\Delta_j=\Delta_j} given in paragraphs above. Now, we write

\displaystyle \Vert \Delta_{j}T_{\sigma}g_j\Vert_{L^p(\mathbb{R}^{d}\rightarrow l^{2}(\mathbb{Z}))}^{2}=\Vert \left(\Sigma_{j}\vert \Delta_{j}T_{\sigma}g_j\vert^{2}\right)^{\frac{1}{2}}\Vert_{L^p(\mathbb{R}^{d})}^{2}=\Vert \Sigma_{j}\vert \Delta_{j}T_{\sigma}g_j\vert^{2}\Vert_{L^{\frac{p}{2}}(\mathbb{R}^{d})}. \ \ \ \ \ (24)

If {p>2}, by Riesz representation theorem there exists {r} satisfying {\frac{1}{r}+\frac{2}{p}=1} and {u\in L^{r}(\mathbb{R}^{d})} for which

\displaystyle \Vert \Sigma_{j}\vert \Delta_{j}T_{\sigma}g_j\vert^{2}\Vert_{L^{\frac{p}{2}}(\mathbb{R}^{d})}=\sup_{\Vert u\Vert_{L^r}\leq 1}\int_{\mathbb{R}^{d}} \Sigma_{j}\vert \Delta_{j}T_{\sigma}g_j(x)\vert^{2}\, u(x)dx. \ \ \ \ \ (25)

As the symbol of {\Delta_{j}T_{\sigma}} is {\sigma_{j}=\psi_{j}(\xi)\sigma(\xi)=\psi(\tilde{\xi})\sigma(2^{j}\tilde{\xi})} and {\sigma_j\in L^2_s(\mathbb{R}^{d})}, for all {j\in\mathbb{Z}}. The Lemma 6 gives

\displaystyle \Vert \Sigma_{j}\vert \Delta_{j}T_{\sigma}g_j\vert^{2}\Vert_{L^{\frac{p}{2}}(\mathbb{R}^{d})}\leq C\Vert \sigma_j\Vert_{L^2_{s}(\mathbb{R}^{d})}\sup_{\Vert u\Vert_{L^r}\leq 1}\int_{\mathbb{R}^{d}} \Sigma_{j}\vert g_j(x)\vert^{2}\, Mu(x)dx. \ \ \ \ \ (26)

As {1<r<\infty}, by Hardy-Littlewood theorem and Hölder’s inequality we get

\displaystyle \begin{array}{rcl} \Vert \Sigma_{j}\vert \Delta_{j}T_{\sigma}g_j\vert^{2}\Vert_{L^{\frac{p}{2}}(\mathbb{R}^{d})} &\leq& C\Vert \sigma_j\Vert_{L^2_{s}(\mathbb{R}^{d})}\sup_{\Vert u\Vert_{L^r}\leq 1}\Vert \Sigma_{j}\vert g_j\vert^{2}\Vert_{L^{\frac{p}{2}}(\mathbb{R}^{d})} \Vert Mu\Vert_{L^{r}(\mathbb{R}^{d})}\\ \\ &\leq& C\Vert \sigma_j\Vert_{L^2_{s}}\Vert \Sigma_{j}\vert \tilde{\Delta}_{j}f\vert^{2}\Vert_{L^{\frac{p}{2}}}\\ \\ &=&C\Vert \sigma_j\Vert_{L^2_{s}}\Vert \tilde{\Delta}_{j}f\Vert_{L^p(\mathbb{R}^{d}\rightarrow l^{2}(\mathbb{Z}))}^{2}\\ \\ &\leq& C\Vert \sigma_j\Vert_{L^2_{s}}\Vert f\Vert_{L^p(\mathbb{R}^{d})}^{2}, \end{array}

because the Littlewood-Paley inequality (13) still hold with {\tilde{\Delta}_{j}} in place of {\Delta_j}. Inserting the last inequality into (23) we have

\displaystyle \Vert T_{\sigma}f\Vert_{L^{p}(\mathbb{R}^{d})}\leq C\Vert \sigma_j\Vert_{L^2_{s}(\mathbb{R}^{d})}\Vert f\Vert_{L^p(\mathbb{R}^{d})} \ \ \ \ \ (27)

for all {p>2}. The case {p<2} follows by standard duality argument and shall be remarked only that the symbol of form adjuint operator {T^{\ast}_{\lambda}} is {\lambda(\xi)=\sigma(-\xi)} and hence is bounded by arguments above. The case {p=2} it’s a small exercise to reader. \Box

In the next post shall be showed that fractional condition (22) implies so-called symbol Hörmander condition in Theorem 2.5 and also Minhklin condition for {d>1}. Some small applications to PDEs equations will be mentioned finally.

Why global regularity for Navier-Stokes is hard

Originally posted on What's new:

It is always dangerous to venture an opinion as to why a problem is hard (cf. Clarke’s first law ), but I’m going to stick my neck out on this one, because (a) it seems that there has been a lot of effort expended on this problem recently, sometimes perhaps without full awareness of the main difficulties, and (b) I would love to be proved wrong on this opinion :-) .

The global regularity problem for Navier-Stokes is of course a Clay Millennium Prize problem and it would be redundant to describe it again here. I will note, however, that it asks for existence of global smooth solutions to a Cauchy problem for a nonlinear PDE. There are countless other global regularity results of this type for many (but certainly not all) other nonlinear PDE; for instance, global regularity is known for Navier-Stokes in two spatial dimensions rather than three…

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Polar vortices and climate change

Originally posted on cartesian product:

Satellite Image Shows Entry of the Polar Vorte...

Satellite Image Shows Entry of the Polar Vortex into the Northern U.S. (Photo credit: NASA Goddard Photo and Video)

 

The recent cold snap in the United States was, apparently, seen by many as undermining the claims of a majority of climate scientists that human activity was gradually – but potentially catastrophically – warming the climate.

But, according to this article from the Associated Press, the most remarkable things about the cold was that it was not all that cold (merely the 55th coldest day since 1900) and that cold days appear, now, to be relatively rare (though, of course, a random element of statistical fluctuation may also have contributed to the length of the period – 17 years – since the last very cold period.)

One other thought – the cold in the US seems to have been matched by a very mild (in temperature, if not much else)…

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Introduction to Cambridge IA Analysis I 2014

Marcelo de Almeida:

Cambridge’s course Analysis I

Originally posted on Gowers's Weblog:

This term I shall be giving Cambridge’s course Analysis I, a standard first course in analysis, covering convergence, infinite sums, continuity, differentiation and integration. This post is aimed at people attending that course. I plan to write a few posts as I go along, in which I will attempt to provide further explanations of the new concepts that will be covered, as well as giving advice about how to solve routine problems in the area. (This advice will be heavily influenced by my experience in attempting to teach a computer, about which I have reported elsewhere on this blog.)

I cannot promise to follow the amazing example of Vicky Neale, my predecessor on this course, who posted after every single lecture. However, her posts are still available online, so in some ways you are better off than the people who took Analysis I last year, since you will have…

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Week 1-An Introduction to Flows

Marcelo de Almeida:

In 8 weeks you will learn a little bit about fluid dynamics

Originally posted on math3402ellisrooney:

So after considering what fluids are present in my day-to-day life I have not only thought of all the straight forward scenarios like running water through a pipe, aerodynamics of a moving object but the more complicated cases like pressure build up in heavy machinery and car engines. Even planes have to consider the impact of a fluid (air) to ensure they are able to fly, so put simply the majority of things we do are impacted by fluid flow.

During week one we were introduced to the concept of fluid dynamics. This was done in a rather interactive way of teaching, we were shown various easy to replicate models that questioned the normal way of thinking about scenarios. For example we saw a ping-pong ball stay afloat by the use of a straw being blown through.

If there is one quote that will stay in my head from week…

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Is the Product of Measurable Spaces the Categorical Product?

Originally posted on Graduated Understanding:

This post requires some knowledge of measure theory.

Today I’m going to show that the product of two measurable spaces $latex {(X, \mathcal{B}_X)}&fg=000000$ and $latex {(Y, \mathcal{B}_Y)}&fg=000000$, is actually the product in the category of measurable spaces. See Product (category theory).

The category of measurable spaces, $latex {\mathbf{Measble}}&fg=000000$, is the collection of objects $latex {(X,\mathcal{B}_X)}&fg=000000$, where $latex {X}&fg=000000$ is a set and $latex {\mathcal{B}_X}&fg=000000$ is a $latex {\sigma}&fg=000000$-algebra on $latex {X}&fg=000000$, and the collection of morphisms $latex {\phi : (X, \mathcal{B}_X) \rightarrow (Y, \mathcal{B}_Y)}&fg=000000$ such that

  1. $latex {\phi : X \rightarrow Y}&fg=000000$;
  2. $latex {\phi^{-1}(E) \in \mathcal{B}_X}&fg=000000$ for all $latex {E \in \mathcal{B}_Y}&fg=000000$.

Such a function $latex {\phi}&fg=000000$ is called a measurable morphism, and $latex {(X, \mathcal{B}_X)}&fg=000000$ is called a measurable space.

Given two measurable spaces $latex {(X, \mathcal{B}_X)}&fg=000000$ and $latex {(Y, \mathcal{B}_Y)}&fg=000000$ we can define their product $latex {(X\times Y, \mathcal{B}_X \times \mathcal{B}_Y)}&fg=000000$, where $latex {X\times Y}&fg=000000$ is the…

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