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).

Heat-wave equation in Morrey spaces


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)


\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)


\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}


\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)


\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


— 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}


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}


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}


\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}


\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)


\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)


\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.


\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}


\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.

Translation invariant operators

In this notes we show that bounded translation invariant operators from {L^{p}(\mathbb{R}^{d})} to {L^{q}(\mathbb{R}^{d})} are essentially convolution singular integral operators for certain p,q. More precisely, let {A} be a bounded linear operator from {L^{p}(\mathbb{R}^{d})} to {L^{q}(\mathbb{R}^{d})}, we say that {A} is a translation invariant operator, if

\displaystyle \tau_{y}A=A\tau_{y} \ \ \ \ \ (1)

where {(\tau_{y}\varphi)(x)=\varphi(x-y)} denotes the translation operator on , \mathbb{R}^{d} and \varphi is a  Schwartz function. The object of this notes is only find a tempered distribution {T\in \mathcal{S}'(\mathbb{R}^{d})} such that

\displaystyle (A\varphi)(x) =(\varphi\ast T)(x), \ \ \ \ \ (2)

if {A} is translation invariant operator and {A} is bounded on certain Lebesgue {L^{q}(\mathbb{R}^{d})} spaces. Firstly, let us recall some well know results.

  • Let {\alpha=(\alpha_1,\cdots,\alpha_d)\in (\mathbb{N}\cup\{0\}})^{d}, {x^{\alpha}=x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_d^{\alpha_d}}. Then there exist a positive constant {C_{d,\alpha}} such that

    \displaystyle \vert x^{\alpha}\vert \leq C_{d,\alpha}\vert x\vert^{\vert \alpha\vert}, \;\; \vert \alpha\vert =\alpha_1+\cdots+\alpha_{d}. \ \ \ \ \ (3)

    In fact, let {\mathbb{S}^{d-1}} be an unity sphere and let {f:\mathbb{S}^{d-1}\rightarrow \mathbb{R}} which is given by  {f(x_1,\cdots,x_d)=\vert x_{1}^{\alpha_1}\cdots x_{d}^{\alpha_d}\vert}. Then the function {f} is a continuous function in {\mathbb{R}^{d}} and for

    \displaystyle C_{d,\alpha}=\sup_{x\in \mathbb{S}^{d-1}}\vert f(x)\vert,\nonumber \ \ \ \ \ (4)

    we get {\vert y^{\alpha}\vert \leq C_{d,\alpha}}, for all {y=\frac{x}{\vert x\vert}\in \mathbb{S}^{d-1}} and {x\in\mathbb{R}^{d}}. Since

    \displaystyle y^{\alpha}=\left(\frac{x_{1}}{\vert x\vert}\right)^{\alpha_1}\cdots \left(\frac{x_{d}}{\vert x\vert}\right)^{\alpha_d}=\frac{x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_d^{\alpha_d}}{\vert x\vert^{\alpha_1+\cdots+\alpha_{d}}},\nonumber \ \ \ \ \ (5)

    we obtain (3).

  • Let {k\in \mathbb{Z}^{+}}, we have

    \displaystyle \vert x\vert^{k} \leq C_{d,k}\sum_{\vert \beta\vert=k}\vert x^{\beta}\vert, \ \ \ \ \ (6)

    for all {x\in \mathbb{R}^{d}}.

  • \displaystyle \frac{C_{d, d+1}}{(2\pi) ^{d+1}}\sum_{\vert \alpha\vert \leq d+1}\vert (-2\pi i x)^{\alpha}\vert \geq (1+\vert x\vert)^{-d-1}. \ \ \ \ \ (8)

    Indeed, by inequality (6) we have

    \displaystyle \begin{array}{rcl} \frac{C_{d, d+1}}{(2\pi)^{d+1}}\sum_{\vert \alpha\vert \leq d+1}\vert (-2\pi x)^{\alpha}\vert \geq \vert x\vert ^{d+1}\geq \left(\frac{1}{2}\right)^{d+1}\geq (1+\vert x\vert)^{-d-1}. \end{array}

Lemma 1 Let {u} be a tempered distribution. Suppose that {D^{\alpha} u\in L^{p}_{loc}(\mathbb{R}^{d})} for all {\vert \alpha\vert \leq d+1}. Then {u} is a function and there exists a positive constant {C} such that

\displaystyle \vert u(x)\vert \leq C\sum_{\vert \alpha\vert \leq d+1}\left(\int_{\mathbb{R}^{d}}\vert D^{\alpha}u(y)\vert^{p}dy\right)^{\frac{1}{p}}, \;\;\vert x\vert\geq 2. \ \ \ \ \ (9)

Proof: Let {C_{0}^{\infty}(\mathbb{R}^{d})} be the space of all bump functions {\varphi_{r}:\mathbb{R}^{d}\rightarrow\mathbb{R}} with support on  euclidean sphere {\mathbb{S}_r} of radius {r}. Since {u\in L^{p}(\mathbb{R}^{d})}, it follows from Hölder inequality that {w:=\varphi_{r}u\in L^{1}(\mathbb{R}^{d})}. Moreover, we have that  {\hat{w}} lives in {L^{1}(\mathbb{R}^{d})} too. This follows by means of the inequality (8) and some properties of Fourier transform on euclidean spaces. Indeed,

\displaystyle \begin{array}{rcl} \vert\widehat{w}(x)\vert &\leq & C_{d}(1+\vert x\vert)^{-(d+1)}\sum_{\vert\alpha\vert\leq d+1}\vert (-2\pi i x)^{\alpha}\widehat{w}(x)\vert\\    \\    &\leq &C_{d}(1+\vert x\vert)^{-(d+1)}\sum_{\vert\alpha\vert\leq d+1}\Vert\widehat{D^{\alpha}w}\Vert_{\infty}\\    \\    &\leq &C_{d}(1+\vert x\vert)^{-(d+1)}\sum_{\vert\alpha\vert\leq d+1}\Vert D^{\alpha}w\Vert_{L^{1}(\mathbb{R}^{d})}\\    \\    &\leq &C_{d}(1+\vert x\vert)^{-(d+1)}\sum_{\vert\alpha\vert\leq d+1}\Vert \sum_{\gamma\leq \alpha}\binom{\alpha}{\gamma}(D^{\gamma}u)(D^{\alpha-\gamma}\varphi_{r})\Vert_{L^{1}}\\    \\    &\leq &C_{d,\alpha, p'}(1+\vert x\vert)^{-(d+1)}\sum_{\vert\alpha\vert\leq d+1}\Vert D^{\alpha}u\Vert_{L^{p}}. \end{array}

Integrating the last inequality, we obtain

\displaystyle \Vert \widehat{w}\Vert_{L^{1}}\leq C\sum_{\vert\alpha\vert\leq d+1}\Vert D^{\alpha}u\Vert_{L^{p}(\mathbb{R}^{d})}.\nonumber \ \ \ \ \ (10)

It follows that Fourier inverse of {\widehat{w}} agrees, almost everywhere, with an uniformly continuous function (see Folland p. 239). Since {\varphi_{r}(x)=1} on {\mathbb{S}_r}, so the tempered distribution {u} can be re-defined as an uniformly continuous function, namely {\psi}, which is defined on {B_{r}(y)} for all {r>0}. Since the choice of {r} is not relevant, we get

\displaystyle \begin{array}{rcl} \Vert u\Vert_{\infty}=\Vert \psi \Vert_{L^{\infty}}\leq \Vert \widehat{\psi}\Vert_{L^{1}}= \Vert \widehat{w}\Vert_{L^{1}}\leq C\sum_{\vert\alpha\vert\leq d+1}\Vert D^{\alpha}u\Vert_{L^{p}(\mathbb{R}^{d})}. \end{array}


Lemma 2 Let {u\in\mathcal{S}'(\mathbb{R}^{d})} and let  {D^{\alpha}u\in L^{p}(\mathbb{R}^{d})}. If {A:L^{p}(\mathbb{R}^{d})\rightarrow L^q(\mathbb{R}^{d})} is a bounded linear operator which is translation invariant, then {A} commutes with derivatives, that is,

\displaystyle A(D^{\alpha}u)=D^{\alpha}(Au), \ \ \ \ \ (11)

for all multi-index {\alpha}.

Proof: For readers convenience, let {\alpha=e_j:=(0,\cdots,0, 1,0,\cdots,0)}. The general case, follows by interactions. Firstly, let {h\in\mathbb{R}} and set

\displaystyle u_{h}(x):=\tau_{-he_j}(x)=u(x_1,\cdots, x_{j}+h,\cdots,x_{d}) .\nonumber \ \ \ \ \ (12)

Let v=Au and note that Au_{h}=A\tau_{-he_j}=\tau_{-he_j}A=v_{h}. It follows that,

\displaystyle A(\frac{u_h-u}{h})=\frac{v_h-v}{h}\nonumber \ \ \ \ \ (14)

which converges to {D_{x_j}v} in {L^q(\mathbb{R}^{d})} as {h\rightarrow 0}. The continuity of {A} in join with {D_{x_j} u\in L^{p}(\mathbb{R}^{d})} give us

\displaystyle \begin{array}{rcl} \Vert \frac{v_h-v}{h} -A(D_{x_j} u)\Vert_{L^q(\mathbb{R}^{d})}&=&\Vert A(\frac{u_h-u}{h} -D_{x_j} u)\Vert_{L^q(\mathbb{R}^{d})}\\\\&\leq & \Vert A\Vert\, \Vert \frac{u_h-u}{h} -D_{x_j} u\Vert_{L^p(\mathbb{R}^{d})}. \end{array}

Therefore, by uniqueness of limit in {L^q(\mathbb{R}^{d})} one has

\displaystyle D_{x_j}Au =A(D_{x_j} u),\nonumber \ \ \ \ \ (15)

as we desired. \Box

Let us to introduce the reflection operator in {\mathcal{S}(\mathbb{R}^{d})} which is given by,

\displaystyle (R\varphi)(x)=\varphi(-x), \ \ \ \ \ (16)

for all {\varphi\in\mathcal{S}(\mathbb{R}^{d})}. Notice that, for {\varphi\in L^{1}_{loc}(\mathbb{R}^{d})} the integral

\displaystyle \begin{array}{rcl} (\varphi\ast u)(x)&=&\int u(y) \varphi(-(y-x)) dy\\\\ &=&\int u(y) (R\varphi)(y-x) dy\\\\ &=&\int u(y) (\tau_{x}R\varphi)(y) dy \end{array}

for all {u\in C_{0}^{\infty}(\mathbb{R}^{d})}. This motivates to  define convolution operator between a Schwartz function {\varphi\in\mathcal{S}(\mathbb{R}^{d})} and a tempered distribution {u\in\mathcal{S}'(\mathbb{R}^{d})} as follows

\displaystyle (\varphi\ast u )(x)= u(\tau_{x}R\varphi). \ \ \ \ \ (17)


Theorem 3 Let {A: L^p(\mathbb{R}^{d})\rightarrow L^q(\mathbb{R}^{d})} be a bounded linear operator. If {A} is translation invariant, there exists {T\in \mathcal{S}'(\mathbb{R}^{d})} such that (2) holds.

Proof: Let {\varphi\in\mathcal{S}(\mathbb{R}^{d})} be a Schwartz function such that {D^{\alpha}\varphi\in L^{p}(\mathbb{R}^{d})}. In view of Lemma 1 and Lemma 2, {A\varphi} is a continuous function which satisfy the estimate

\displaystyle \begin{array}{rcl} \vert (A\varphi)(0)\vert &\leq& C \sum_{\vert \alpha\vert\leq d+1} \Vert D^{\alpha} (A\varphi)\Vert_{L^{q}(\mathbb{R}^{d})}\\ \\&\leq & C\,\Vert A\Vert \sum_{\vert \alpha\vert\leq d+1} \Vert D^{\alpha}\varphi\Vert_{L^{p}(\mathbb{R}^{d})}\\ \\&\leq & C\,\Vert A\Vert \left(\int (1+\vert x\vert^{2})^{-Np}\right)^{1/p}\sum_{\vert \alpha\vert\leq d+1} \Vert \varphi\Vert_{(N,\alpha)}\\ \\&\leq & C\,\Vert A\Vert \sum_{\vert \alpha\vert\leq d+1} \Vert \varphi\Vert_{(N,\alpha)} \end{array}

for {N>d/p}. Here {\Vert \cdot\Vert_{(N,\alpha)}} stands for Schwartz norm

\displaystyle \Vert \varphi\Vert_{(N,\alpha)}=\sup_{x\in \mathbb{R}^{d}}\,(1+\vert x\vert^{2})^{N}\vert D^{\alpha}\varphi(x)\vert. \ \ \ \ \ (18)

In other words, there exists a tempered distribution {T} such that \displaystyle (A\varphi)(0)=T(R\varphi). As  {A} is translation invariance, in view of (17) we have

\displaystyle \begin{array}{rcl} (A\varphi)(x)&=&\tau_{-x}(A\varphi)(0)=(A\tau_{-x}\varphi)(0)\\ \\&=&T(R\tau_{-x}\varphi)=T(\tau_{x}R\varphi)=(\varphi\ast T)(x) \end{array}

because of {(R\tau_{x}\varphi)(z)=(\tau_{-x}R\varphi)(z)}. \\\\\\\Box