1 Introduction

Let X and Y be smooth manifolds of dimension n. In this work, we analyse the sharpness of the order \(-(n-1)/2\) for the weak (1,1) inequality of elliptic Fourier integral operators \(T\in I^\mu _1(X,Y;\Lambda )\) with order \(\mu \le -(n-1)/2.\) For the general aspects of the theory of Fourier integral operators, we refer the reader to Hörmander [5], Duistermaat and Hörmander [1], and Melin and Sjöstrand [8, 9].

First, let us review the mapping properties of Fourier integral operators. By following [1, 5], these classes are denoted by \(I^\mu _\rho (X,Y;\Lambda ),\) with \(\Lambda \) being considered locally a graph of a symplectomorphism from \(T^{*}X\setminus 0\) to \(T^{*}Y\setminus 0,\) which are equipped with the canonical symplectic forms \(d\sigma _X\) and \(d\sigma _Y,\) respectively. Such Fourier integral operators are called non-degenerate. The symplectic structure of \(\Lambda \) is determined by the symplectic 2-form \(\omega \) on \(X\times Y,\) \(\omega =\sigma _X\oplus -\sigma _Y .\) Let \(\pi _{X\times Y}\) be the canonical projection from \(T^*X\times T^*Y\) into \(X\times Y.\) As in the case of pseudo-differential operators, non-degenerate Fourier integral operators of order zero are bounded on \(L^2.\) The fundamental work of Segger, Sogge, and Stein [12] establishes the boundedness of \(T\in I^\mu _1(X,Y;\Lambda )\) from \(L^p_{\text {comp}}(Y)\) into \(L^p_{\text {loc}}(X)\) with the order

$$\begin{aligned} \mu \le -(n-1)|1/2-1/p| \end{aligned}$$
(1.1)

if \(1<p<\infty ,\) and from \(H^1_{\text {comp}}(Y)\) into \(L^1_{\text {loc}}(X)\) if \(p=1.\) Also, for \(p=1\) in (1.1) and as a consequence of the weak (1,1) estimate in Tao [14], an operator T of order \(-(n-1)/2\) is locally of weak (1,1) type.

The critical Seeger–Sogge–Stein order (1.1) is sharp if \(d\pi _{X\times Y}|_{\Lambda }\) has full rank equal to \(2n-1\) somewhere and if T is an elliptic operator. When \(d\pi _{X\times Y}|_{\Lambda }\) does not attain the maximal rank \(2n-1,\) the upper bound for the order (1.1) is not sharp and may depend on the singularities of \(d\pi _{X\times Y}|_{\Lambda }.\) In conclusion, as it was observed in [12], the mapping properties of the classes \( I^\mu _\rho (X,Y;\Lambda )\) of Fourier integral operators depend on the singularities and on the maximal rank of the canonical projection. So, an additional condition on the canonical relation \(\Lambda \) was introduced in [12], the so called factorisation condition for \(\pi _{X\times Y}\). Roughly speaking, it can be introduced as follows. Assume that there exists \(k\in {\mathbb {N}},\) with \(0\le k\le n-1,\) such that for any \(\lambda _0=(x_0,\xi _0,y_0,\eta _0)\in \Lambda ,\) there is a conic neighborhood \(U_{\lambda _0}\subset \Lambda \) of \(\lambda _0,\) and a smooth homogeneous of order zero map \(\pi _{\lambda _0}:U_{\lambda _0}\rightarrow \Lambda ,\) such that

$$\begin{aligned} \text {(RFC): } \text {rank}(d\pi _{\lambda _0})=n+k,\text { and }\pi _{X\times Y}|_{U_{\lambda _0}}=\pi _{X\times Y}|_{\Lambda }\circ \pi _{\lambda _0}. \end{aligned}$$
(1.2)

Under the real factorisation condition (RFC) in (1.2), Seeger, Sogge, and Stein in [12] proved that the order

$$\begin{aligned} \mu \le -(k+(n-k)(1-\rho ))|1/2-1/p| \end{aligned}$$
(1.3)

guarantees that any Fourier integral operator \(T\in I^\mu _\rho (X,Y;\Lambda ),\) with \(\rho \in [1/2,1],\) is bounded from \(L^p_{\text {comp}}(Y)\) into \(L^p_{\text {loc}}(X),\) and for \(p=1,\) from \(H^1_{\text {comp}}(Y)\) into \(L^1_{\text {loc}}(X).\) For a set A,  \(1_A\) denotes its characteristic function. We recall that T is locally of weak (1,1) type if, for any pair of compact subsets \(K\subset Y,\) and \(K'\subset X,\) the localised operator \(1_{K'}T1_K:L^{1}(Y)\rightarrow L^{1,\infty }(X)\) is bounded.

If \(\Lambda \) satisfies the factorisation condition (RFC) in (1.2), because \(\text {rank}(d\pi _{\lambda _0})=n+k \), we have that \( \text {rank}(d{\pi _{X\times Y}}|_{U_{\lambda _0}})\le n+k, \) in an open neighborhood \(U_{\lambda _0}\) of \(\lambda _0.\) The following Theorem 1.1 proves that if the rank \(n+k\) is attained somewhere, the order in (1.3) for \(\rho =1,\) that is \(\mu \le -k/2\), is sharp for \(0\le k\le n-1\). The sharpness of the order \(\mu \le -k/2\) for the \(H^1_{\text {loc}}(Y)\)-\(L^{1}_{\text {loc}}(X)\)-boundedness of elliptic Fourier integral operators has been proved in [10], with the case \(k=n-1\) known from [12]. Here, we observe that the geometric construction in [10] implies also the following sharp Theorem 1.1, proving for \(k=n-1,\) the converse of the weak (1,1)-inequality in Tao [14] for elliptic Fourier integral operators. More specifically, we have:

Theorem 1.1

Let the real canonical relation \(\Lambda \) be a local canonical graph such that the inequality \(\text {rank}(d{\pi _{X\times Y}}|_{\Lambda })\le n+k \) holds with \(0\le k\le n-1,\) and the rank \(n+k\) is attained at some point. Then elliptic operators \(T\in I_{1}^{\mu }(X,Y;\Lambda )\) are not locally of weak (1,1) type provided that \(\mu >-k/2.\)

Remark 1.2

In the endpoint case \(k=n-1,\) \(\text {rank}(d{\pi _{X\times Y}}|_{\Lambda })\le 2n-1,\) the elliptic operators \(T\in I_{1}^{\mu }(X,Y;\Lambda )\) are not locally of weak (1,1) type provided that \(\mu >-(n-1)/2.\) In view of the weak (1,1) estimate in Tao [14] for the class \(I_{1}^{-(n-1)/2}(X,Y;\Lambda ),\) when the real canonical relation has full rank \(\text {rank}(d{\pi _{X\times Y}}|_{\Lambda })\le 2n-1,\) the main [14, Theorem 1.1] together with Theorem 1.1 imply the following result.

Corollary 1.3

Let the real canonical relation \(\Lambda \) be a local canonical graph such that (RFC) in (1.2) is satisfied for \(k=n-1.\) Let \(T\in I_{1}^{\mu }(X,Y;\Lambda )\) be an elliptic Fourier integral operator. Then T is locally of weak (1,1) type if and only if \(\mu \le -(n-1)/2.\)

2 Preliminaries

2.1 Basics on symplectic geometry

Let M,  X, and Y be (paracompact) smooth real manifolds of dimension n. So, using partitions of the unity, the spaces \(L^{1}(Y)\) and \(L^{1,\infty }(X)\) are defined by the set of functions that under any changes of coordinates belong to \(L^{1}({\mathbb {R}}^n)\) and \(L^{1,\infty }({\mathbb {R}}^n),\) respectively. For instance, we can take \(M=X,Y\) or \(M=X\times Y.\) In this section, we will follow [11, Chapter I].

A 2-form \(\omega \) is called symplectic on M if \(d\omega =0,\) and for all \(x\in M,\) the bilinear form \(\omega _x\) is antisymmetric and non-degenerate on \(T_{x}M\). The canonical symplectic form \(\sigma _M\) on M is defined as follows. Let \(\pi :=\pi _M:T^*M\rightarrow M\) be the canonical projection. For any \((x,\xi )\in T^*M,\) let us consider the linear mappings

$$\begin{aligned} d\pi _{(x,\xi )}:T_{(x,\xi )}(T^*M)\rightarrow T_{x}M\text { and }\xi :T_{x}M\rightarrow {\mathbb {R}}. \end{aligned}$$

The composition \(\alpha _{(x,\xi )}:=\xi \circ d\pi _{(x,\xi )}\in T^{*}_{(x,\xi )}(T^*M),\) that is \( \xi \circ d\pi _{(x,\xi )}:T_{(x,\xi )}(T^*M)\rightarrow {\mathbb {R}}, \) defines a 1-form \(\alpha \) on \(T^*M.\) Then the canonical symplectic form \(\sigma _M\) on M is defined by

$$\begin{aligned} \sigma _M:=d\alpha . \end{aligned}$$
(2.1)

Because \(\sigma _M\) is an exact form, it follows that \(d\sigma _M=0\) and then that \(\sigma _M\) is symplectic. If \(M=X\times Y,\) it follows that \(\sigma _{X\times Y}=\sigma _X\oplus -\sigma _Y .\) Now we record the kind of submanifolds that are necessary when one defines the canonical relations.

  • A submanifold \(\Lambda \subset T^*M\) of dimension n is called Lagrangian if

    $$\begin{aligned} T_{(x,\xi )}\Lambda =( T_{(x,\xi )}\Lambda )^{\sigma }:=\{y\in T_{(x,\xi )}(T^*M):\sigma _M(y,y')=0\,\,\forall y'\in T_{(x,\xi )}\Lambda \}. \end{aligned}$$

  • We say that \(\Lambda \subset T^*M\setminus 0\) is conic if \((x,\xi )\in \Gamma \) implies that \((x,t\xi )\in \Gamma \) for all \(t>0.\)

  • Let \(\Sigma \subset X\) be a smooth submanifold of X of dimension k. Its conormal bundle in \(T^*X\) is defined by

    $$\begin{aligned} N^{*}\Sigma :=\{(x,\xi )\in T^*X: \,x\in \Sigma ,\,\,\xi (\delta )=0 \,\forall \delta \in T_{x}\Sigma \}. \end{aligned}$$
    (2.2)

The following facts characterise the Lagrangian submanifolds of \(T^*M.\)

  • Let \(\Lambda \subset T^*M\setminus 0 \) be a closed submanifold of dimension n. Then \(\Lambda \) is a conic Lagrangian manifold if and only if the 1-form \(\alpha \) in (2.1) vanishes on \(\Lambda .\)

  • Let \(\Sigma \subset X \) be a submanifold of dimension k. Then its conormal bundle \(N^*\Sigma \) is a conic Lagrangian manifold.

The Lagrangian manifolds have the following property.

  • Let \(\Lambda \subset T^{*}M\setminus 0 \) be a conic Lagrangian manifold and let

    $$\begin{aligned} d\pi _{(x,\xi )}:T_{(x,\xi )}\Lambda \rightarrow T_{x}M \end{aligned}$$
    (2.3)

    have constant rank equal to k for all \((x,\xi )\in \Lambda .\) Then each \((x,\xi )\in \Gamma \) has a conic neighborhood \(\Gamma \) such that

    1. 1.

      \(\Sigma =\pi (\Gamma \cap \Lambda )\) is a smooth manifold of dimension k.

    2. 2.

      \(\Gamma \cap \Lambda \) is an open subset of \(N^*\Sigma .\)

The Lagrangian manifolds have a local representation defined in terms of phase functions that can be defined as follows. For this, let us consider a local trivialisation \(M\times ({\mathbb {R}}^n\setminus 0), \) where we can assume that M is an open subset of \({\mathbb {R}}^n.\)

Definition 2.1

(Real-valued phase functions). Let \(\Gamma \) be a cone in \(M\times ({\mathbb {R}}^N\setminus 0). \) A smooth function \(\phi :M\times ({\mathbb {R}}^N\setminus 0)\rightarrow {\mathbb {R}}, \) \((x,\theta )\mapsto \phi (x,\theta ),\) is a real phase function if it is homogeneous of degree one in \(\theta \) and has no critical points as a function of \((x,\theta ),\) that is

$$\begin{aligned} \forall t>0,\, \phi (x,t\theta )=t\phi (x,\theta ), \text { and }d_{(x,\theta )}\phi (x,\theta )\ne 0 \quad \forall (x,\theta )\in M\times ({\mathbb {R}}^N\setminus 0).\nonumber \\ \end{aligned}$$
(2.4)

Additionally, we say that \(\phi \) is non-degenerate in \(\Gamma \) if for \((x,\theta )\in \Gamma \) such that \(d_{\theta }\phi (x,\theta )=0,\) one has that

$$\begin{aligned} d_{(x,\theta )}\frac{\partial \phi }{\partial \theta _{j}}(x,\theta ),\,\,1\le j\le N, \end{aligned}$$
(2.5)

is a system of linearly independent vectors.

The following facts describe locally a Lagrangian manifold in terms of a phase function.

  • Let \(\Gamma \) be a cone in \(M\times ({\mathbb {R}}^N\setminus 0), \) and let \(\phi \) be a non-degenerate phase function in \(\Gamma .\) Then there exists an open cone \({\tilde{\Gamma }}\) containing \(\Gamma \) such that the set

    $$\begin{aligned} U_{\phi }=\{(x,\theta )\in {\tilde{\Gamma }}:d_\theta \phi (x,\theta )=0\} \end{aligned}$$
    (2.6)

    is a smooth conic submanifold of \(M\times ({\mathbb {R}}^N\setminus 0)\) of dimension n. The mapping

    $$\begin{aligned} L_{\phi }: U_\phi \rightarrow T^*M\setminus 0,\,\, L_{\phi }(x,\theta )=(x,d_x\phi (x,\theta )), \end{aligned}$$
    (2.7)

    is an inmersion. Let us denote \(\Lambda _\phi =L_{\phi }(U_\phi ).\)

  • Let \(\Lambda \subset T^*M\setminus 0\) be a submanifold of dimension n. Then \(\Lambda \) is a conical Lagrangian manifold if and only if every \((x,\xi )\in \Lambda \) has a conic neighborhood \(\Gamma \) such that \(\Gamma \cap \Lambda =\Lambda _\phi \) for some non-degenerate phase function \(\phi .\)

Remark 2.2

The cone condition on \(\Lambda \) corresponds to the homogeneity of the phase function.

Remark 2.3

Although we have given a definition when a real phase function of \((x,\theta )\) is non-degenerate, the same can be defined if one considers functions of \((x,y,\theta ).\) Indeed, a real valued phase function \(\phi (x,y,\theta )\) homogeneous of order 1 at \(\theta \ne 0\) that satisfies the following two conditions

$$\begin{aligned} \text {det}\partial _x \partial _\theta (\phi (x,y,\theta ))\ne 0,\,\, \text {det}\partial _y \partial _\theta (\phi (x,y,\theta ))\ne 0,\,\,\theta \ne 0, \end{aligned}$$
(2.8)

is called non-degenerate.

2.2 Smooth factorisation condition for real phases

We can assume that X, Y are open sets in \({\mathbb {R}}^n\). One defines the class of Fourier integral operators \(T\in I^\mu _{\rho }(X, Y;\Lambda )\) by the (microlocal) formula

$$\begin{aligned} Tf(x)=\int \limits _Y\int \limits _{{\mathbb {R}}^N} e^{i\Psi (x,y,\theta )} a(x,y,\theta ) f(y)d\theta \;dy, \end{aligned}$$
(2.9)

where the symbol a is a smooth function locally in the class \(S^\mu _{\rho ,1-\rho }(X\times Y\times ({\mathbb {R}}^n\setminus 0) ),\) with \(1/2\le \rho \le 1.\) This means that a satisfies the symbol inequalities

$$\begin{aligned} |\partial _{x,y}^{\beta }\partial _\theta ^\alpha a(x,y,\theta )| \le C_{\alpha \beta }(1+|\theta |)^{\mu -\rho |\alpha |+(1-\rho )|\beta |} \end{aligned}$$

for (x, y) in any compact subset K of \(X\times Y,\) and \(\theta \in {\mathbb {R}}^N\setminus 0,\) while the real-valued phase function \(\Psi \) satisfies the following properties:

  1. 1.

    \(\Psi (x,y,\lambda \theta )= \lambda \Psi (x,y,\theta ) \) for all \(\lambda >0\);

  2. 2.

    \(d\Psi \not =0\);

  3. 3.

    \(\{d_\theta \Psi =0\}\) is smooth (i.e. \(d_\theta \Psi =0\) implies \(d_{(x,y,\theta )}\frac{\partial \Psi }{\partial \theta _j}\) are linearly independent).

Here \(\Lambda \subset T^*(X\times Y)\setminus 0\) is a Lagrangian manifold locally parametrised by the phase function \(\Psi ,\)

$$\begin{aligned} \Lambda =\Lambda _\Psi =\{(x,d_x\Psi ,y,d_y\Psi ): d_\theta \Psi =0\}. \end{aligned}$$

The canonical relation associated with T is the conic Lagrangian manifold in \(T^*(X\times Y)\backslash 0\), defined by \(\Lambda '=\{(x,\xi ,y,-\eta ): (x,\xi ,y,\eta )\in \Lambda \}.\) In view of the equivalence-of-phase-functions theorem (see e.g. [11, Theorem 1.1.3, Page 9]), the notion of Fourier integral operator becomes independent of the choice of a particular phase function associated to a Lagrangian manifold \(\Lambda .\) Because of the diffeomorphism \(\Lambda \cong \Lambda ',\) we do not distinguish between \(\Lambda \) and \(\Lambda '\) by saying also that \(\Lambda \) is the canonical relation associated with T.

Let us consider the canonical projections:

figure a

The smooth factorisation condition for \(\Psi \) can be formulated as follows (see [12] or e.g. [11, Page 45]). Suppose that there exists a number k,  with \(0\le k\le n-1,\) such that for any

$$\begin{aligned} \lambda _0=(x_0,\xi _0,y_0,\eta _0)\in \Lambda _\Psi , \end{aligned}$$

there exists a conic neighborhood \(U_{\lambda _0}\subset \Lambda _{\Psi } \) of \(\lambda _0,\) and a smooth homogeneous of order zero map

$$\begin{aligned} \pi _{\lambda _0}:U_{\lambda _0}\rightarrow \Lambda _{\Psi }, \end{aligned}$$
(2.10)

with constant rank \(\text {rank}(d\pi _{\lambda _0})=n+k,\) for which one has

$$\begin{aligned} \text {(RFC): } \boxed { \pi _{X\times Y}|_{U_{\lambda _0}}=\pi _{X\times Y}|_{\Lambda _{\Psi }}\circ \pi _{\lambda _0} }. \end{aligned}$$
(2.11)

In this case, we say that the canonical relation \(\Lambda :=\Lambda _\Psi \) satisfies the factorisation condition \(\text {(RFC)}.\)

3 Sharpness of Seeger–Sogge–Stein orders

Now, we will prove the sharpness of the Seeger–Sogge–Stein order for \(\rho =1\) in the case of elliptic Fourier integral operators.

Proof of Theorem 1.1. Let \(\mu >-k/2.\) Let us follow the argument in [11, Page 42] (see also [10]) and to analyse the weak (1,1) estimate at the end of the proof. Using again the equivalence-of-phase-function theorem, it is enough to consider elliptic operators T on \({\mathbb {R}}^n\) with kernels, locally defined by

$$\begin{aligned} K(x,y)=\int \limits _{{\mathbb {R}}^n}e^{i\Phi (x,y,\xi )}b(x,y,\xi )d\xi ,\,\, \Phi (x,y,\xi ):= x\cdot \xi -\phi (y,\xi ), \end{aligned}$$
(3.1)

with symbols \(b(x,y,\xi )\) compactly supported in (x, y). That \(\Lambda \) satisfies the local graph condition means that the real-value phase function \(\phi \) satisfies

$$\begin{aligned} \text {det}\partial _{y}\partial _{\xi }\phi (y,\xi )\ne 0 \end{aligned}$$
(3.2)

on the support of the symbol b,  and \(\xi \ne 0.\) Let us observe that

$$\begin{aligned} \Lambda _0:=\{\lambda \in \Lambda : \text {rank }d\pi _{X\times Y}|_{\Lambda }(\lambda )=n+k\} \end{aligned}$$
(3.3)

is not empty and is open in \(\Lambda .\) Fix a point \(\lambda _0\in \Lambda _0.\) Let \(\Delta :=\sum _{j=1}^n\partial _{x_j}^2\) be the standard Laplacian on \({\mathbb {R}}^n,\) and define the distribution

$$\begin{aligned} \varkappa (y):=(1-\Delta )^{-s/2}\delta _{y_0} \end{aligned}$$
(3.4)

for a fixed point \(y_0\in Y,\) where \(s>0\) is small enough in such a way that

$$\begin{aligned} -\frac{k}{2}+s<\mu . \end{aligned}$$
(3.5)

Since \((1-\Delta )^{-s/2}\) is an elliptic pseudo-differential operator of order \(-s\), its Schwartz kernel \(K_{s}\) is of Calderón-Zygmund type, and it satisfies the inequality \(|K_s(y,y_0)|\le C |y-y_0|^{-n+s},\) in some local coordinate system. So, the fact that \(s>0\) implies that

$$\begin{aligned} \varkappa (y)=\int \limits _{{\mathbb {R}}^n}K_{s}(y,z)\delta _{y_0}(z)dz=K_{s}(y,y_0)\in L^{1}_{\text {loc}}. \end{aligned}$$
(3.6)

Now, we are going to introduce the geometric construction in [10] (see also [11, Page 42]) in order to obtain a useful parametrisation of the phase function \(\Phi \). Let \(\Sigma =\pi _{X\times Y}(C\cap U),\) where \(U\subset \Lambda _0\) is a neighborhood of \(\lambda _0.\) Taking into account that \(\text {rank }d\pi _{X\times Y}|_{U}=n+k,\) that is the rank of \(d\pi _{X\times Y}\) is constant in U,  \(\Sigma \) is a k-dimensional submanifold defined by the equations

$$\begin{aligned} h_{j}(x,y)=0,\,1\le j\le n-k, \end{aligned}$$
(3.7)

in a neighborhood of \(y_0,\) with the system \(\{\nabla h_j:1\le j\le n-k\}\) being linearly independent on \(\Sigma .\) Then \(\Lambda \) is the conormal bundle of \(\Sigma ,\) and the phase function of T takes the representation

$$\begin{aligned} \Phi (x,y,\lambda )=\sum _{j=1}^{n-k}\lambda _jh_j(x,y). \end{aligned}$$
(3.8)

Because compositions of Fourier integral operators with pseudo-differential operators leaves invariant the canonical relation \(\Lambda \), we have that \(T\circ (1-\Delta )^{-s/2}\in I^{\mu -s}_1(X,Y,\Lambda ).\) So, in local coordinates, we have

$$\begin{aligned} T\varkappa (x)&=T\circ (1-\Delta )^{-s/2}\delta _{y_0}=\int \limits _{{\mathbb {R}}^n}\int \limits _{{\mathbb {R}}^{n-k}}e^{i\left( \sum _{j=1}^{n-k}\lambda _jh_j(x,y)\right) }a(x,{\overline{\lambda }})\delta _{y_0}(y)d{\overline{\lambda }}dy\\&=\int \limits _{{\mathbb {R}}^{n-k}}e^{i{\overline{\lambda }}\cdot {\overline{h}}(x,y_0)}a(x,{\overline{\lambda }})d{\overline{\lambda }}=(2\pi )^{n-k}({\mathscr {F}}^{-1}a)(x,{\overline{h}}(x,y_0)), \end{aligned}$$

where \({\overline{\lambda }}\) and \({\overline{h}}\) are vectors with components \(\lambda _j\) and \(h_j,\) respectively, and \({\mathscr {F}}\) denotes the Fourier transform. The symbol \(a\in S^{\mu -s+\frac{k}{2}}_{1,0}({\mathbb {R}}^{n-k})\) is obtained from the symbol of the operator \(T\circ (1-\Delta )^{-s/2}\) by using the stationary method phase, where we have eliminated k-variables. Computing the second argument of \((2\pi )^{n-k}{\mathscr {F}}^{-1}a,\) one has

$$\begin{aligned} (2\pi )^{n-k}{\mathscr {F}}^{-1}a(x,\zeta )=\int \limits _{{\mathbb {R}}^{n-k}}e^{i\lambda \cdot \zeta }a(x,\lambda ){\mathscr {F}}({\delta }_0)(\lambda )d\lambda =P\delta _{0}(\zeta ), \end{aligned}$$
(3.9)

where P is a pseudo-differential operator in \({\mathbb {R}}^{n-k}\) of order \(m=n-s+\frac{k}{2}.\) Denoting the Schwartz kernel of P by \(K_P,\) we have that \(P\delta _0(\zeta )=K_{P}(\zeta ,0),\) and then one has

$$\begin{aligned} |K_{P}(\zeta ,0)|\sim |\zeta |^{-(n-k)-m}, \,\,m=n-s+\frac{k}{2}. \end{aligned}$$
(3.10)

Define the set

$$\begin{aligned} \Sigma _{y_0}:=\{x:(x,y_0)\in \Sigma \}. \end{aligned}$$

We have that \(\text {distance}(x,\Sigma _{y_0})\asymp |{\overline{h}}(x,y_0)|.\) Consequently,

$$\begin{aligned} |(2\pi )^{n-k}{\mathscr {F}}^{-1}a(x,\zeta )|\asymp \text {distance}(x,\Sigma _{y_0})^{-(n-k)-(\mu -s+\frac{k}{2})}, \end{aligned}$$

locally uniformly in x. The identity (3.9) implies that \(T\varkappa \) is smooth on \(\Sigma _{y_0}.\) Now, we apply the geometric construction above to test the operator T on the distribution \(\varkappa \) to see that

$$\begin{aligned} | T\varkappa (x)|&=|K_{P}({\overline{h}}(x,y_0),0)| \asymp |{\overline{h}}(x,y_0)| \asymp \text {distance}(x,\Sigma _{y_0})^{-(n-k)-(\mu -s+\frac{k}{2})}, \end{aligned}$$

and that the singularities of \(T\varkappa \) can appear only in transversal directions to \(\Sigma .\) Now, to finish the proof, let \(\Omega =\overline{B(y_0,r)}\) be the compact neighborhood of \(y_0,\) with radius \(r>0.\) Observing that for any \(x\in \Omega ,\) \({\overline{h}}(x,y_0)\in {\mathbb {R}}^{n-k},\) \( T\varkappa \in L^{1}(\Omega ),\) if and only if

$$\begin{aligned} (n-k)+(\mu -s+\frac{k}{2})< n-k, \end{aligned}$$

and that \(T\varkappa \in L^{1,\infty }(\Omega )\setminus L^{1}(\Omega )\) if and only if

$$\begin{aligned} (n-k)+(\mu -s+\frac{k}{2})= n-k, \end{aligned}$$

it follows that \( T\varkappa \notin L^{1,\infty }(\Omega )\) if and only if

$$\begin{aligned} (n-k)+(\mu -s+\frac{k}{2})>n-k, \end{aligned}$$
(3.11)

or equivalently \(\mu >s-\frac{k}{2}\) as in (3.5). In conclusion, we have that \(\varkappa \) is locally in \(L^1\) and that \(T\varkappa \notin L^{1,\infty }(\Omega )\) showing that T is not locally of weak (1,1) type. \(\square \)