\section{Introduction}\label{sec-intro}

In their seminal work, D.H. Phong and E.M. Stein \cite{Phong-Stein} study new classes of oscillatory integral operators
with singular weights, that in turn apply to the study of certain PDE problems.
Since this work appeared, there have been numerous  results
and developments of the theory, with large variety of these type of operators.
In this note we will be interested in two particular aspects of the result of Phong-Stein.
Apart from \cite{Phong-Stein}, our interest is
partially motivated by applications of oscillatory integral operators in the anaysis of boundary value problems
(see for example \cite{ASS1}-\cite{ASS3}), and recent developments in PDE problems involving oscillating boundaries (see for example \cite{KP}).
In the first part of the note we will introduce and study oscillatory integral operators
with singular kernel and linear phase, where integration is carried out on
smooth hypersurfaces. Here we will aim at obtaining precise estimates with respect to the smoothness norm of the
hypersurface. This is done in order to obtain non trivial bounds when one allows
the hypersurface to oscillate. A type of an oscillating hypersurface considered here
is technically smooth, however due to its oscillatory nature one can not rely in a straightforward manner on partial integration techniques
to bound the corresponding integral operator,
since the derivatives of the graph representing the surface blow-up.
In the second part of the paper, we will study a similar problem but with nonlinear phase function.
The type of phase function studied in this case is in part motivated by the Helmholtz equation,
and we will discuss one particular application of our results regarding decay estimates
for the eigenvalue problem corresponding to the Helmholtz equation in ${\mathbb R}^3$.

\vspace{0.2cm}
Let us start by fixing the setup and notation.
Throughout the text, by $``\cdot"$ we denote the standard scalar product in ${\mathbb R}^n$.
For two quantities $x$ and $y$ we write $x \lesssim y$ if there is an absolute constant $C$
for which $x\leq C y$. Likewise, if $x$ and $y$ depend on some parameter, say $\delta$, we may write
$x \lesssim_\delta y$ to indicate that the constant in the inequality depends on $\delta$,
and is otherwise absolute.

\vspace{0.2cm}

For a real-valued function $\psi\in C^\infty({\mathbb R}^n)$ satisfying $|| D^\alpha \psi||_{L^\infty({\mathbb R}^n)} \lesssim_\alpha 1$
for any $\alpha \in {\mathbb Z}^n_+$, let $\Gamma$ be the following hypersurface
\begin{equation}\label{Gamma-surface}
\Gamma=\{ (y,\psi(y))\in {\mathbb R}^{n+1}: \ y\in {\mathbb R}^n \}.
\end{equation}
For $f\in L^2({\mathbb R}^n)$, $\lambda>0$, and $(x,x_{n+1})\in {\mathbb R}^n\times {\mathbb R}$ define
\begin{equation}\label{def-T-lambda}
 T_\lambda  f(x, x_{n+1}) = \int\limits_{\Gamma } e^{ i\lambda x \cdot y   }
 \varphi_0( (x,x_{n+1}) ,y ) K(x-y, x_{n+1} - y_{n+1}) f(y) d \sigma (y,y_{n+1}),
\end{equation}
where  $ d\sigma $ is the induced surface measure on $\Gamma$, and

\begin{doublespace}
\begin{itemize}
 \item[(A1)]\label{(A1)} $ \varphi_0 $ is real-valued and from the class $C_0^\infty ( {\mathbb R}^{n+1} \times {\mathbb R}^n )$,
 \item[(A2)]\label{(A2)} $K\in C^\infty ( {\mathbb R}^{n+1} \setminus \{0\} )$ and for any $z\in {\mathbb R}^{n+1}\setminus \{0\}$ and any $\alpha \in {\mathbb Z}^{n+1}_+$ we have
 $$ | D^\alpha K(z) | \lesssim_\alpha \frac{|z|^m}{ |z|^{n + |\alpha|} }, $$
 where $0\leq m<n$. Here $n\geq 1$, and we do not assume that $m$ is necessarily an integer.
\end{itemize}
\end{doublespace}

\noindent \textbf{Organization.} In Section \ref{sec-est-of-T-lambda} we study
$T_\lambda$ as an operator from $L^2( {\mathbb R}^{n} )$ to $L^2({\mathbb R}^{n} )$ and prove decay estimates for its operator norm as $\lambda \to \infty$.
A special attention is paid to obtaining precise bounds with respect to the smoothness norms of the surface $\Gamma$.
We then conclude the section by discussing the behavior of the operator $T_\lambda$ under perturbations of the fixed surface $\Gamma$.
Next, in Section \ref{sec-maximal-operator} we consider a maximal operator associated with operators of the form (\ref{def-T-lambda})
when the surface is allowed to oscillate. More precisely, for a
family of surfaces $\{\Gamma_\varepsilon\}_{0<\varepsilon \leq 1}$,
we let $T_\lambda^\varepsilon$ be the operator defined by (\ref{def-T-lambda}) for the surface $\Gamma_\varepsilon$, where the parameter $0<\varepsilon\leq 1$ is meant to
model an oscillatory behavior of the given family of hypersurfaces.
Then we analyse boundedness of the following maximal operator $T_\lambda^*f (x,x_{n+1}) =\sup\limits_{0<\varepsilon\leq 1} |T_\lambda^\varepsilon f (x,x_{n+1})|$,
where $(x,x_{n+1})\in {\mathbb R}^n \times {\mathbb R}$.
Finally, Section \ref{sec-nonlinear} studies operators of type $T_\lambda$, however instead of a linear phase, there we have
a ``fractional"-type nonlinearity, namely $x\cdot y$ in the exponential is replaced by $|x-y|^\gamma$ with $\gamma\geq 1$. This change requires
a radically different approach. We discuss in subsection \ref{sub-sec-Helmholtz} how the case $\gamma=1$ applies to Helmholtz
equation. At the end of Section \ref{sec-nonlinear} we
show that the obtained upper bounds of some of the operators considered in the article are sharp.

\section{The main estimate for $T_\lambda$}\label{sec-est-of-T-lambda}

Throughout this section we will be working with operators defined on $L^2({\mathbb R}^n)$
and with values in $L^2({\mathbb R}^n)$. Thus if $T$ is an operator of this type,
by $||T|| $ we denote its operator norm.
Also, when estimating a certain quantity, we will be only concerned
with constants that depend on the surface, i.e. the function $\psi$ of (\ref{Gamma-surface}).
The following is our main estimate for the operator $T_\lambda$ defined by (\ref{def-T-lambda}).

\vspace{0.2cm}

\begin{theorem}\label{Thm-main-on-T-lambda}
For any $1\leq m< n$, and any $x_{n+1}\in {\mathbb R}$ we have
\begin{equation}
 || T_\lambda||   \lesssim \lambda^{-\frac{m}{2} \frac{n }{n+1}}  [ 1+ || \nabla  \psi||_{ L^\infty({\mathbb R}^n ) }  ] \times
 [ 1+ || \psi||_{ L^\infty({\mathbb R}^n) }   ],
\end{equation}
where the constant depends on the cut-off function $\varphi_0$, and is independent of $x_{n+1}$ and $\psi$.
\end{theorem}

\vspace{0.2cm}

We start by decomposing $T_\lambda$ as follows. Fix a parameter $0\leq \beta \leq 1$ that will be specified below,
and a smooth function $\varphi :{\mathbb R}^n \to [0,1] $ such that $ \varphi(y)=0 $ for $|y|\leq 1$
and $\varphi(y)=1$ for $|y|\geq 2$.
We then have the following decomposition
$$
T_\lambda f  = A_\lambda f  + B_\lambda  f,
$$
where $B_\lambda$ in terms of volume integral is
\begin{multline*}
B_\lambda f(x,x_{n+1}) = \int_{{\mathbb R}^n } e^{ i \lambda x \cdot y   }
\varphi_0( (x,x_{n+1}) , y ) K(x -y , x_{n+1} -   \psi(y  ) )
\times  \\  \varphi((x -y ) \lambda^\beta )  ( 1+|   \nabla  \psi( y   ) |^2  )^{1/2}   f(y ) d y .
\end{multline*}

\vspace{0.2cm}

\begin{lem}\label{Lem-A-lambda}
For any $\lambda \geq 1$, any $m>0$, and any $0 \leq  \beta \leq 1$ one has $$
|| A_\lambda||  \lesssim [ 1+ || \nabla  \psi ||_{L^\infty ({\mathbb R}^n ) } ]  \lambda^{-m \beta}.
$$
\end{lem}
\begin{proof}
Rewriting $A_\lambda $ in terms of volume integral
we get
\begin{multline*}
A_\lambda f(x, x_{n+1}) = \int_{{\mathbb R}^n } e^{ i \lambda x \cdot y   }
\varphi_0((x,x_{n+1}) ,y ) K(x -y , x_{n+1} -  \psi(y  ) )
\times  \\ [ 1-\varphi((x -y ) \lambda^\beta )] ( 1+| \nabla  \psi( y  ) |^2  )^{1/2}   f(y  ) d y .
\end{multline*}

Now, using assumption (A2) for kernel $K$, along with the definition of $\varphi$ we obtain
\begin{equation}\label{A-est1}
 | A_\lambda f(x,x_{n+1}) | \lesssim [ 1+|| \nabla  \psi ||_{L^\infty ({\mathbb R}^n) } ] \mathcal{I}_\lambda (x ),
\end{equation}
where the constant depends on the supremum norm of $\varphi_0$, and the appropriate constant involved in (A2), and we have
\begin{multline*}
\mathcal{I}_\lambda (x ): = \int\limits_{ | x  -y  |\leq 2\lambda^{-\beta} }  \frac{| f(y ) | }{  | x  - y  |^{n  - m } } dy  \leq \\
\sum\limits_{2^{-k} \leq 4\lambda^{-\beta}} \int\limits_{ 2^{-k-1} \leq |x-y| \leq 2^{-k} } \frac{| f(y) | }{  | x - y |^{n  -m } } dy  \lesssim \\
\sum_{ 2^{-k} \leq 4 \lambda^{-\beta} }  2^{ -m k } \frac{1}{2^{-k n}} \int\limits_{ 2^{-k-1} \leq |x -y | \leq 2^{-k} } | f(y ) |    dy  \lesssim \\
\mathcal{M} f(x ) \sum_{ 2^{-k} \leq 4 \lambda^{-\beta} }  2^{ -m k } \lesssim
\mathcal{M} f(x ) \lambda^{-\beta m},
\end{multline*}
where $\mathcal{M}$ is the Hardy-Littlewood maximal function.
Since $\mathcal{M}$ has strong $(2,2)$ type, the last inequality combined with (\ref{A-est1})
completes the proof of the Lemma.
\end{proof}

\noindent To study $B_\lambda f $ we rewrite it in the form
\begin{equation}\label{def-B-lambda}
B_\lambda f(x,x_{n+1}) = \int_{{\mathbb R}^n } k_\lambda \big( (x,x_{n+1}), y \big) K \big(x - y , x_{n+1} -\psi(y ) \big) g(y ) dy ,
\end{equation}
where we have set
$$
k_\lambda \big( (x,x_{n+1}), y \big) := e^{ i \lambda x  \cdot y  } \varphi_0 \big( (x,x_{n+1}), y \big) \varphi( (x -y ) \lambda^{\beta} ) ,
$$
and
\begin{equation}\label{g-in-B-lambda}
g(y ) :=   f(y ) (  1+|\nabla \psi(y )|^2 )^{1/2} , \qquad  y \in {\mathbb R}^n .
\end{equation}
For the estimate of $B_\lambda$ we start with a lemma, where the constants are allowed to depend on the norm
of derivatives of the function $\psi$ representing the surface. This dependence will be revised later on.

\begin{lem}\label{Lem-fixed-surface}
For $(x,x_{n+1}) \in {\mathbb R}^n \times {\mathbb R} $ set
$$
(\widetilde{B}_\lambda f)(x,x_{n+1} ) = \int_{{\mathbb R}^n } k_\lambda \big( (x,x_{n+1}) , y  \big) K \big(x  - y , x_{n+1}- \psi(y )\big) f(y ) dy .
$$
Then
$$
|| \widetilde{B}_\lambda ||  \leq C_\psi \lambda^{ n (\beta - 1/2) - m\beta },
$$
where $0\leq m < n$, $C_\psi$ is independent of $\lambda$, and $x_n$, and depends on finite number
of derivatives of $\psi$.
\end{lem}

\begin{proof}
The proof follows closely the lines of Proposition 1 on p. 134 of \cite{Phong-Stein}.
Observe that $\widetilde{B}_\lambda$ has kernel
$$
L_\lambda (x , y ) = e^{ i \lambda x  \cdot y   } \varphi_0( (x,x_{n+1}), y  ) \varphi( (x  -y ) \lambda^\beta ) K(x  - y , x_{n+1} - \psi(y ) ),
$$
and the conjugate operator $\widetilde{B}^*_\lambda$ has kernel $ \overline{L_\lambda (y  , x )} $. It follows that
$ \widetilde{B}_\lambda  \widetilde{B}^*_\lambda $ has kernel
\begin{multline*}
M_\lambda (x, y) = L_\lambda (x, z) \overline{L_\lambda ( y, z )} d z  = \\
\int_{{\mathbb R}^n } e^{ i\lambda ( x  - y  ) \cdot z  } \varphi_0( (x,x_{n+1}), z ) \varphi_0( (y, x_{n+1}) , z )
\varphi ( (x -z ) \lambda^\beta ) \varphi ( (y -z ) \lambda^\beta ) \times
\\ K( x -z , x_{n+1}- \psi(z ) ) \overline{ K ( y - z , x_{n+1}- \psi(z ) )} dz .
\end{multline*}
Since we will obtain uniform estimates in $x_{n+1}$ the dependence of the kernels on $x_{n+1}$ is dropped from the notation.
Using the estimate (A2) for $K$, for any $\alpha \in {\mathbb Z}^n_+$ we have
$$
| (D_{z }^\alpha  K ) (x  - z ,  x_{n+1} - \psi(z ) )  | \lesssim_\psi \frac{1}{ | x - z |^{n  - m + |\alpha|} }.
$$
In view of the last inequality, integrating by parts $N\geq 0$ times  in the kernel $M$ we get
\begin{equation}\label{M-estimate}
| M_\lambda (x,y) | \lesssim_\psi ( \lambda |x - y| )^{-N} \sum\limits_{ 0\leq k+l \leq N } \mathcal{I}_{k,l}(x , y ),
\end{equation}
where
$$
 \mathcal{I}_{k,l} (x, y): = \int\limits_{\substack{ \lambda^{-\beta} \leq | x  - z  | \leq C \\  \lambda^{-\beta} \leq | y  - z  | \leq C  } }
| x  -z  |^{-n+m - k}  | y - z   |^{-n+m  - l} d z .
$$
Assuming $n-2m+N>0$ and using $k+l\leq N$, it is easy to see that each $\mathcal{I}_{k,l}$ is uniformly bounded above by
$$
\int\limits_{ \lambda^{-\beta} \leq  |z  |\leq C } |z |^{ -2n+2m -N } dz
\lesssim \int\limits_{ \lambda^{-\beta} }^1 \frac{r^{n-1}}{ r^{2n-2m +N} } dr \lesssim
\lambda^{ \beta (   n-2m+N  ) }.
$$
From here, getting back to (\ref{M-estimate}) we obtain
\begin{equation}\label{M-est-final}
| M(x  , y) |  \lesssim_\psi \frac{\lambda^{ \beta(n-2m ) }  }{ ( \lambda^{1-\beta} | x  - y | )^N  } ,
\end{equation}
where $N\geq 0$ is an integer satisfying $n-2m+N>0$. By a convexity argument as in Proposition 1 of \cite{Phong-Stein} p.134, we can eliminate
the condition on $N$ being an integer, thus getting (\ref{M-est-final}) with non integer values of $N$.
We then choose $N_1< n $ and $N_2>n $ in (\ref{M-est-final}),
which is allowable since $m<n$. It follows that
\begin{multline*}
\int_{ {\mathbb R}^n } | M(x , y ) | d y  \lesssim_\psi
\int\limits_{ \lambda^{1-\beta} | z  | \leq 1  } \frac{\lambda^{ \beta(n-2m  ) }  }{ ( \lambda^{1-\beta} | z | )^{N_1}  } d z  +
\int\limits_{ \lambda^{1-\beta} | z  | > 1  } \frac{\lambda^{ \beta(n-2m  ) }  }{ ( \lambda^{1-\beta} | z | )^{N_2}  } d z  \lesssim \\
\big[ \text{substituting } u= \lambda^{1-\beta} z  \big] \ \lesssim_\psi  \lambda^{ n( 2\beta -1 ) - 2m \beta  },
\end{multline*}
uniformly in $x$. By symmetry, the same estimate holds for integration with respect to $x$ and fixed $y$.
Using the obtained mixed $L^1$-norm estimate of the kernel $M$ we can easily bound the $L^2 \mapsto L^2$ norm of the corresponding operator,
thus obtaining
$$
|| \widetilde{B}_\lambda || \lesssim_\psi \lambda^{ n (\beta - 1/2) - m\beta  },
$$
where we choose $0\leq \beta \leq 1$ satisfying
$ \beta \leq  \frac{n }{ 2(n-m ) }$ to get a bound for the norm of $\widetilde{B}_\lambda$. The proof is complete.
\end{proof}

We are now ready to prove the main estimate on $B_\lambda$ defined by (\ref{def-B-lambda}).

\begin{prop}\label{Thm-B-lambda-0}
Assume $1\leq m <n $, and for fixed $(x, x_{n+1}) \in {\mathbb R}^n\times {\mathbb R}$ let $B^{(0)}_\lambda$ be defined as follows
$$
B_\lambda^{(0)} g(x,x_{n+1}) = \int_{{\mathbb R}^n } e^{ i \lambda x  \cdot y  } \varphi_0( (x,x_{n+1}) , y )
\varphi( (x  - y ) \lambda^\beta ) K(x  - y , x_{n+1} - \psi(y ) ) g(y ) d y  .
$$
Then
$$
|| B_\lambda^{(0)} ||  \lesssim
\lambda^{ n (\beta -1/2) -m \beta  } + || \psi||_{L^\infty} \lambda^{ n (\beta -1/2) -m\beta+\beta  }.
$$
\end{prop}

\begin{remark}
Observe that if $B_\lambda$ is the operator defined by (\ref{def-B-lambda}), then
by (\ref{def-B-lambda}) and (\ref{g-in-B-lambda}) we have
$ B_\lambda f = B_\lambda^{(0)} g $ with $g(y ) = f(y )   ( 1+|\nabla  \psi(y )|^2 )^{1/2}$.
Hence Proposition \ref{Thm-B-lambda-0} implies the following bound
\begin{equation}\label{B-lambda-norm}
|| B_\lambda ||  \lesssim [ 1+|| \nabla  \psi||_{L^\infty{ ({\mathbb R}^n )} } ] \times
\left( \lambda^{ n (\beta -1/2) -m \beta   } + || \psi||_{L^\infty({\mathbb R}^n) } \lambda^{ n (\beta -1/2) -m\beta+ \beta  } \right).
\end{equation}

\end{remark}

\bigskip

\begin{proof}[Proof of Proposition \ref{Thm-B-lambda-0}]
We have
$$
K \big(x - y , x_{n+1} - \psi(y ) \big) - K(x  - y , x_{n+1}) = \int\limits_{ x_{n+1} }^{ x_{n+1} - \psi (y ) } \partial_{n+1} K(x  - y , t) dt.
$$
Recall that $ k_\lambda ( (x,x_{n+1}), y ) = e^{ i \lambda x  \cdot y  } \varphi_0( (x,x_{n+1}), y ) \varphi( (x  - y ) \lambda^\beta ) $,
we thus get
\begin{multline*}
B_\lambda^{(0)} g(x,x_{n+1} ) = \int_{{\mathbb R}^n } k_\lambda \big( (x,x_{n+1}), y \big) K \big(x  - y , x_{n+1} - \psi(y ) \big) g(y ) d y  = \\
\int_{ {\mathbb R}^n } k_\lambda \big((x,x_{n+1}), y \big) K\big(x  - y  , x_{n+1} \big) g(y ) d y   + \\
\int_{ {\mathbb R}^n } k_\lambda \big( (x,x_{n+1}), y \big) \left( \int\limits_{ x_{n+1} }^{ x_{n+1} - \psi(y) }
\partial_{n+1} K(x  - y  , t) dt  \right) g(y ) d y  =: \\ B^{(1)}_\lambda g(x,x_{n+1}) + B^{(2)}_\lambda g(x,x_{n+1}).
\end{multline*}
It follows by Lemma \ref{Lem-fixed-surface} that
\begin{equation}\label{est-on-D-lambda-fixed}
|| B^{(1)}_\lambda  ||  \lesssim \lambda^{ n( \beta - 1/2) - m \beta    }.
\end{equation}

\noindent For $B^{(2)}_\lambda$ we have the following decomposition
\begin{multline}
B^{(2)}_\lambda g(x,x_{n+1}) = - \iint\limits_{E_1} k_\lambda ( (x,x_{n+1}), y ) \partial_{n+1} K(x  - y , t) g(y ) d y  dt + \\
\iint\limits_{E_2} k_\lambda ( (x,x_{n+1}) ,y ) \partial_{n+1} K(x  - y , t) g(y ) d y  dt,
\end{multline}
where for $M=|| \psi ||_{L^\infty({\mathbb R}^n ) }$ we let
\begin{multline*}
E_1 := \{ (y , t): \ \psi(y ) >0, \ x_{n+1} - \psi(y ) < t< x_{n+1} \} = \\ \{ (y , t): \ x_{n+1}  - M <t <x_{n+1}, \ \psi(y ) >x_{n+1} -t \}
\end{multline*}
and
\begin{multline*}
E_2 := \{ (y, t): \ \psi(y ) < 0, \ x_{n+1} < t <  x_{n+1}+ \psi(y )  \} =\\ \{ (y , t): \ x_{n+1}   <t < x_{n+1} +M , \ \psi(y ) <x_{n+1} -t \}.
\end{multline*}
From here we obtain
\begin{multline}
 B_\lambda^{(2)} g(x, x_{n+1}) = \\
 - \int\limits_{x_{n+1} - M}^{x_{n+1}} \int\limits_{{\mathbb R}^n} k_\lambda ( (x,x_{n+1}), y)
 \partial_{n+1} K(x  - y , t) \mathbb{I}_{ \{ y : \ \psi(y ) > x_{n+1} -t \} } g(y ) dy dt + \\
 \int\limits_{x_{n+1} }^{x_{n+1} + M} \int\limits_{{\mathbb R}^n}
 k_\lambda ( (x,x_{n+1}), y ) \partial_{n+1} K(x  - y , t) \mathbb{I}_{ \{ y : \ \psi(y ) < x_{n+1} -t \} } g(y ) dy dt,
\end{multline}
where $\mathbb{I}$ stands for the characteristic function. Denote
$ g_{1,t} (y) : = \mathbb{I}_{ \{ y : \ \psi(y ) > x_{n+1} -t \} } g(y ) $,
$g_{2,t}(y ):= \mathbb{I}_{ \{ y : \ \psi(y ) < x_{n+1} -t \} } g(y )$, and for $h\in L^2({\mathbb R}^n)$ set
$$
F_t h(x,x_{n+1}) : = \int_{{\mathbb R}^n } k_\lambda ( (x,x_{n+1}),y ) \partial_{n+1} K(x  - y  , t) h(y ) d y , \qquad  (x ,x_{n+1}) \in {\mathbb R}^{n+1}.
$$
With this notation we have
$$
B_\lambda^{(2)} g(x,x_{n+1}) = -\int\limits_{x_{n+1} - M }^{x_{n+1}} F_t g_{1,t} (x,x_{n+1}) dt +
\int\limits_{x_{n+1}  }^{x_{n+1} + M} F_t g_{2,t} (x,x_{n+1}) dt.
$$
Observe that Lemma \ref{Lem-fixed-surface} implies $ || F_t||  \lesssim \lambda^{ n (\beta -1/2) -m\beta + \beta  } $,
with constants independent of $\psi$, hence using Minkowski's inequality for integrals we obtain
\begin{equation}\label{est-D-lambda-2}
|| B_\lambda^{(2)} ||  \lesssim M \lambda^{ n (\beta -1/2) -m\beta+ \beta  }.
\end{equation}
Finally, combining (\ref{est-on-D-lambda-fixed}) and (\ref{est-D-lambda-2}) we get
$$
||B_\lambda^{(0)}||  \lesssim \lambda^{ n (\beta -1/2) -m \beta     } +
|| \psi||_{L^\infty({\mathbb R}^n ) } \lambda^{ n (\beta -1/2) -m\beta+ \beta  },
$$
completing the proof the Proposition.
\end{proof}

\begin{proof}[Proof of Theorem \ref{Thm-main-on-T-lambda}]
Getting back to the original operator $T_\lambda$ defined by (\ref{def-T-lambda}),
in view of Lemma \ref{Lem-A-lambda} and Proposition \ref{Thm-B-lambda-0} we have
\begin{multline*}
 || T_\lambda || \leq || A_\lambda||   + || B_\lambda||  \lesssim \\ [ 1+ || \nabla \psi||_{ L^\infty({\mathbb R}^n ) }  ] \times  [ \lambda^{-m\beta} +
  \lambda^{ n(\beta - 1/2 ) - m\beta    }   + || \psi||_{ L^\infty({\mathbb R}^n)} \lambda^{ n ( \beta -  1/2 ) - m\beta +  \beta }  ],
\end{multline*}
which, if optimized in $\beta$ implies
\begin{equation}
 || T_\lambda||   \lesssim \lambda^{-\frac{m}{2} \frac{n }{n+1}}  [ 1+ || \nabla  \psi||_{ L^\infty({\mathbb R}^n ) }  ] \times
 [ 1+ || \psi||_{ L^\infty({\mathbb R}^n) }   ].
\end{equation}

The last estimate completes the proof of Theorem.
\end{proof}

\subsection{Perturbing the surface}\label{sec-perturb-surface}
Here we discuss the case when the operator $T_\lambda$ is defined on a perturbation of some fixed surface $\Gamma_0$.
Assume we have
$$
\Gamma_0 = \{ (y,\psi_0(y)): \ y\in {\mathbb R}^n  \},
$$
where $\psi_0\in C^\infty({\mathbb R}^n)$ is real-valued and has all its derivatives bounded on ${\mathbb R}^n$.
Now, for $\varepsilon>0$ consider a family of smooth and real-valued functions $\psi_\varepsilon(y)$, along with the hypersurfaces
$$
\Gamma_\varepsilon = \{ ( y,\psi_0(y) +\psi_\varepsilon(y) ) : \ y\in {\mathbb R}^n  \},
$$
that is we perturb the fixed surface $\Gamma_0$ by $\psi_\varepsilon$. Here as well, the assumption is that each $\psi_\varepsilon$
has bounded derivatives of all orders on ${\mathbb R}^n$, however we do not impose any uniform bound with respect to $\varepsilon$,
neither we assume any structural restriction on $\psi_\varepsilon$.

For $\varepsilon>0$ let the operator $T_\lambda^\varepsilon$ be defined as in (\ref{def-T-lambda}) where integration is over
$\Gamma_\varepsilon$. Then by Theorem \ref{Thm-main-on-T-lambda} we get
$$
||T_\lambda^\varepsilon||  \lesssim_{\psi_0} \lambda^{-\frac{m}{2} \frac{n }{n+1}}  [ 1+ || \nabla  \psi_\varepsilon  ||_{ L^\infty({\mathbb R}^n ) }  ] \times
 [ 1+ || \psi_\varepsilon ||_{ L^\infty({\mathbb R}^n) }   ],
$$
uniformly in $\varepsilon>0$ and $\lambda>0$. The point of the last estimate is that even with a rough surface,
in a sense that there is no uniform control over the derivatives of the graph representing the surface, we may still control
the norms of the corresponding operators.

\section{Maximal operator for oscillating surfaces}\label{sec-maximal-operator}

The aim of this section is to illustrate that for small oscillations of the surface,
we may as well control the maximal operator associated with the oscillations. For $\varepsilon>0$ set
\begin{equation}\label{Gamma-eps}
 \Gamma_\varepsilon = \{ ( y , \varepsilon^{\gamma} \psi( y  / \varepsilon ) ): \ y \in {\mathbb R}^n \},
\end{equation}
where $ \psi \in C^\infty ({\mathbb R}^n) $ is bounded and has bounded derivatives
of all orders, and $\gamma>0$ is a fixed parameter which will be specified below in Theorem \ref{Thm-bdd-maximal-op}.
As in (\ref{def-T-lambda}), for $\lambda>0$, $\varepsilon >0$ and $(x,x_{n+1}) \in {\mathbb R}^{n}\times {\mathbb R}$ define
\begin{equation}
 T_\lambda^\varepsilon f(x,x_{n+1}) = \int\limits_{\Gamma_\varepsilon} e^{ i\lambda x \cdot y }
 \varphi_0 \big( (x,x_{n+1}), y \big) K(x-y, x_{n+1} - y_{n+1}  ) f(y ) d \sigma_\varepsilon (y,y_{n+1}),
\end{equation}
where $f\in L^2({\mathbb R}^n)$ and $\varphi_0$ and $K$ are defined as in Section \ref{sec-intro}.
Consider the following maximal operator
\begin{equation}\label{max-oper-T}
T_\lambda^* f(x,x_{n+1}) : = \sup\limits_{ 0<\varepsilon \leq 1 } | T_\lambda^\varepsilon f(x,x_{n+1}) |, \qquad (x,x_{n+1}) \in {\mathbb R}^n\times {\mathbb R}.
\end{equation}
Similar type of maximal operators related to integral operators
for a parameterized family of smooth surfaces had been considered in \cite{Sogge-Stein}.
More precisely \cite{Sogge-Stein} deals with maximal operator corresponding to $L^p$-averaging operators defined on a family of surfaces converging to
a smooth immersed surface with Gaussian curvature nowhere vanishing of infinite order.
While the idea of taking the pointwise upper bound with respect to the family of hypersurfaces is the same as in \cite{Sogge-Stein},
the problem and the analysis discussed here  are completely different from \cite{Sogge-Stein}.
Our main result concerning (\ref{max-oper-T}) is the following $L^2({\mathbb R}^n) \mapsto L^2({\mathbb R}^n) $ bound.

\begin{theorem}\label{Thm-bdd-maximal-op}
For $\gamma> 3/2 $ and $1\leq m<n$, we have $ || T_\lambda^* ||  \lesssim \lambda^{  -\frac{ m}{2} \frac{n}{n+2}    } $.
\end{theorem}

\bigskip

As for $T_\lambda$ defined by (\ref{def-T-lambda}), start
by fixing a smooth function $\varphi :{\mathbb R}^n \to [0,1] $ such that $ \varphi(y )=0 $ for $|y |\leq 1$
and $\varphi(y )=1$ for $|y |\geq 2$. Fix also a parameter $0<\beta<1$ that will be specified in the proof of Theorem \ref{Thm-bdd-maximal-op} below.
We have the following decomposition
$$
T_\lambda^\varepsilon f = A_\lambda^\varepsilon f + B_\lambda^\varepsilon f ,
$$
where
$$
A_\lambda^\varepsilon f(x,x_{n+1}) = \int\limits_{\Gamma_\varepsilon} e^{ i\lambda x \cdot y }
\varphi_0( (x,x_{n+1}) , y ) K(x-y,x_{n+1} - y_{n+1}  ) [ 1-\varphi((x -y ) \lambda^\beta )]  f( y ) d \sigma_\varepsilon(y).
$$
Clearly we get
$$
T_\lambda^* f(x,x_{n+1}) \leq A^*_\lambda f(x,x_{n+1}) + B_\lambda^* f(x,x_{n+1}),
$$
where the maximal operators on the right-hand side are defined in analogy with (\ref{max-oper-T}).

\begin{lem}\label{Lem-A-lambda-max}
For any $\lambda \geq 1$, any $\gamma \geq 1$, and any $ \beta \geq 0 $ one has
$$
|| A_\lambda^* ||  \lesssim  [ 1+ || \nabla  \psi ||_{L^\infty({\mathbb R}^n )} ]   \lambda^{-m \beta}.
$$
\end{lem}

\begin{proof}
We use the fact that $\gamma\geq 1$ and repeat the proof of Lemma \ref{Lem-A-lambda}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{Thm-bdd-maximal-op}]
For fixed $\varepsilon>0$ and $(x,x_{n+1}) \in {\mathbb R}^n\times {\mathbb R}$ we have
\begin{equation}\label{B-integral}
B_\lambda^\varepsilon f(x,x_{n+1}) = B_\lambda^0 f(x,x_{n+1}) + \int\limits_0^\varepsilon \frac{d}{d \tau} B_\lambda^\tau f(x,x_{n+1})   d\tau,
\end{equation}
where $B_\lambda^0 f(x,x_{n+1}) = \lim\limits_{\varepsilon \to 0} B_\lambda^\varepsilon f(x,x_{n+1}) $.
Since $\gamma>1$ this limit exists, as well as the differential with respect to $\tau$ in (\ref{B-integral}).
By (\ref{B-integral}) we have
$$
|B_\lambda^* f | \leq  | B_\lambda^0 f  |  + \int\limits_0^1 \left| \frac{d}{d \varepsilon} B_\lambda^\tau f     \right| d\varepsilon,
$$
from which, using Minkowski's inequality for integrals we get
\begin{equation}\label{B-lambda-star}
 || B_\lambda^* ||  \lesssim || B_\lambda^0 ||  +
 \int\limits_0^1 \left| \left|  \frac{d}{d \varepsilon} B_\lambda^\varepsilon  \right| \right|  d \varepsilon.
\end{equation}
Rewriting $B_\lambda^\varepsilon$ as volume integral gives
\begin{multline*}
B_\lambda^\varepsilon f(x,x_{n+1}) =  \int_{{\mathbb R}^n } e^{ i \lambda x \cdot y   }
\varphi_0( (x,x_{n+1}), y ) K(x -y , x_{n+1} - \varepsilon^{\gamma} \psi(y  / \varepsilon) )
\times  \\  \varphi((x -y ) \lambda^\beta )  ( 1+| \varepsilon^{\gamma-1} (\nabla  \psi) ( y / \varepsilon ) |^2  )^{1/2}   f(y ) d y ,
\end{multline*}
from where we obtain
\begin{equation}\label{B-lambda-0}
 B_\lambda^0 f(x,x_{n+1}) = \int_{ {\mathbb R}^n  }
 e^{ i \lambda x \cdot y   } \varphi_0( (x, x_{n+1}), y ) K(x-y, x_{n+1}   ) \varphi((x -y ) \lambda^\beta ) f(y ) d y .
\end{equation}
By (\ref{B-lambda-0}) and Lemma \ref{Lem-fixed-surface} we have
\begin{equation}\label{B-lambda-0-norm-est}
 || B_\lambda^0||  \lesssim \lambda^{ n(\beta - 1/2) - m\beta   }.
\end{equation}
For the derivative of $B_\lambda^\varepsilon$ with respect to $\varepsilon$ one has
$$
\frac{d}{d \varepsilon}  (B_\lambda^\varepsilon f(x,x_{n+1}) )  =    \mathcal{A}_\lambda^\varepsilon f( x,x_{n+1} ) + \mathcal{B}_\lambda^\varepsilon f( x,x_{n+1} ),
$$
where $\mathcal{A}_\lambda^\varepsilon$ contains the
derivatives of $( 1+| \varepsilon^{\gamma-1} (\nabla  \psi) ( y/ \varepsilon ) |^2  )^{1/2}$, and in $\mathcal{B}_\lambda^\varepsilon$
we collect the differential of the kernel. We have
\begin{multline*}
\frac{d}{d \varepsilon} \left(  1+    \varepsilon^{2\gamma-2}  | (\nabla  \psi) (y/ \varepsilon) |^2    \right)^{1/2} =
\left(  1+   \varepsilon^{2\gamma-2} |(\nabla  \psi) (y  / \varepsilon) |^2    \right)^{-1/2} \times \\
 \big[ (2\gamma -2) \varepsilon^{2\gamma -3}  | (\nabla  \psi) (y  / \varepsilon) |^2  -
2 \varepsilon^{ 2\gamma -4  } (\nabla  \psi)^T (y  / \varepsilon)  (\mathrm{Hess} \psi) (y  / \varepsilon) y      \big].
\end{multline*}
Now if $\gamma>3/2$ by (\ref{B-lambda-norm}) from the last expression we obtain
\begin{equation}\label{A-lambda-eps-est}
 || \mathcal{A}_\lambda^\varepsilon   || \lesssim_\psi a(\varepsilon) \lambda^{ n(\beta -1/2) -m\beta +\beta },
\end{equation}
where $a(\varepsilon)$ is a positive and integrable function on the interval $(0,1)$, and the constant depends
on the bound of derivatives of $\psi$ up to second order.
We will assume that $\beta< \frac{n}{2(n-m+1)}$ to get a decay in the norm of $\mathcal{A}_\lambda^\varepsilon$.

Next, we proceed to the estimate of $\mathcal{B}_\lambda^\varepsilon$
which contains the derivative of the kernel $K$. Differentiating the kernel we get
\begin{multline*}
\frac{d}{d \varepsilon} K( x  - y , x_{n+1} - \varepsilon^\gamma \psi(y  / \varepsilon)  ) =
\partial_{n+1} K( x  - y , x_{n+1} - \varepsilon^\gamma \psi(y  / \varepsilon)  ) \times \\
[ -\gamma \varepsilon^{\gamma-1} \psi(y  / \varepsilon) + \varepsilon^{\gamma-2} \nabla  \psi(y  / \varepsilon) \cdot y  ].
\end{multline*}
Thus we get an operator as in (\ref{B-lambda-norm}) however the kernel here has higher singularity.
Applying the estimate (\ref{B-lambda-norm}) with $m$ replaced by $m-1$ implies
\begin{equation}\label{B-lambda-eps-est}
 || \mathcal{B}_\lambda^\varepsilon ||  \lesssim_\psi b(\varepsilon) \lambda^{ n(\beta -1/2) - m\beta + 2\beta },
\end{equation}
which gives a decay if
\begin{equation}\label{beta-in-B-lambda-eps}
0<\beta<\frac{n }{2(n-m+2)}.
\end{equation}
Here as well, $b(\varepsilon)$ is a positive and integrable function on $(0,1)$.

Now combining estimates (\ref{A-lambda-eps-est}) and (\ref{B-lambda-eps-est}), along with the
estimate of $B_\lambda^0$ given by (\ref{B-lambda-0-norm-est}), from (\ref{B-lambda-star}) we obtain
\begin{equation}\label{B-star-final}
 || B_\lambda^* ||  \lesssim_\psi   \lambda^{ n(\beta -1/2) - m\beta + 2\beta  },
\end{equation}
which gives a decay if $0<\beta<1$ satisfies
\begin{equation}\label{beta-min}
 0< \beta <\frac{n}{ 2(n-m+2) }.
\end{equation}
Putting together Lemma \ref{Lem-A-lambda-max} and estimate (\ref{B-star-final}) we get
\begin{equation}\label{B-lambda-final-2}
 || T_\lambda^*||  \lesssim_\psi  \lambda^{ -m\beta  } + \lambda^{ n(\beta -1/2) - m\beta + 2 \beta   } ,
\end{equation}
which holds under the condition (\ref{beta-min}). Optimizing in $\beta$ gives $\beta= \frac{n}{2(n+2)}$ which
satisfies (\ref{beta-min}). For this choice of $\beta$ the proof of the Theorem is complete.
\end{proof}

\section{Nonlinear phase}\label{sec-nonlinear}

In this section we will consider one particular case of a nonlinear phase function with an application to Helmholtz equation.
Let $\psi_0 \in C_0^\infty({\mathbb R}^n \times {\mathbb R}^{n-1} )$ and set $K(z) = |z|^{-(n-m-1)}$, $z\in {\mathbb R}^n\setminus\{0\}$,
where $0<m<n-1$, and $n\geq 2$. For $f \in L^2({\mathbb R}^{n-1})$ consider the operator
\begin{equation}\label{def-of-T-Helmholtz}
T_\lambda f(x) = \int\limits_{{\mathbb R}^{n-1}} e^{i\lambda |x - (y',0)|^\gamma } \psi_0 (x, y') K \big( x- (y',0) \big) f(y') dy',
\end{equation}
for $x\in {\mathbb R}^n $, $\gamma\geq 1$,  and $\lambda \geq 2$. We shall study decay of the norm of $T_\lambda$
as an operator from $L^2 ({\mathbb R}^{n-1})$ to $L^2({\mathbb R}^n)$. In this section we denote this norm by $|| T_\lambda||$,
and have the following result.

\begin{theorem}\label{Thm-Helmholtz}
Set $\alpha= (n-1)/2$, and assume $\gamma>1$. Then
one has
$$
|| T_\lambda ||  \lesssim
 \begin{cases}
                                 \lambda^{-\frac{1}{\gamma} (m+1/2)}, &\text{  $m<\gamma \alpha - 1/2$}, \\
                                 \lambda^{-\alpha} \log \lambda, &\text{  $m=\gamma \alpha - 1/2$}, \\
                                 \lambda^{-\alpha}, &\text{  $m>\gamma \alpha - 1/2$} .
                            \end{cases}
$$
\end{theorem}

\bigskip

\subsection{Preliminaries}

Before proving Theorem \ref{Thm-Helmholtz} we shall make some preliminary observations.
We assume $\gamma\geq 1$ and let
$$
\Phi(x',\xi) = | \xi - (x',0) |^\gamma = ( |\xi'  - x'|^2 + \xi_n^2 )^{\gamma /2} = d^\gamma,
$$
where $d =( |\xi'  - x'|^2 + \xi_n^2 )^{1/2}$, $1/2\leq d \leq 2$,  $\xi=(\xi', \xi_n)\in {\mathbb R}^{n-1}\times {\mathbb R}$,
and $x'=(x_1,...,x_{n-1})$. We have
$$
\frac{\partial^2 \Phi}{\partial x_i \partial \xi_j} = - \gamma d^{\gamma-2} \left( \delta_{ij} +
(\gamma-2) \frac{x_i - \xi_i}{d}  \frac{x_j - \xi_j}{d}  \right),
$$
for $1\leq i,j \leq n-1$, where $\delta_{ij}$ is the Kronecker symbol. Setting
$a_i = (x_i - \xi_i)/d$ for $i=1,2,...,n-1$ we have
$$
\frac{\partial^2 \Phi}{\partial x_i \partial \xi_j} =  -\gamma d^{\gamma -2} \left( \delta_{ij} + (\gamma-2) a_i a_j \right).
$$
We set $D: = \mathrm{det} ( \delta_{ij} + (\gamma-2) a_i a_j )_{i,j=1}^{n-1}$.
The determinant here can be computed, for example, by Sylvester's determinant theorem (cf. ``matrix determinant lemma''), which states that
 $\mathrm{det} (I_k + AB )= \mathrm{det}(I_p + BA)$,
where matrices $A,B$ have dimensions respectively $k\times p$ and $p \times k$,
and $I_k$ and $I_p$ are correspondingly $k\times k$, and $p\times p$ identity matrices.
Applying this identity to $D$ we obtain
$D = 1+ (\gamma-2) \sum_{i=1}^{n-1} a_i^2$.
Thus
\begin{multline*}
D=1+(\gamma-2) \frac{|\xi' - x'|^2}{d^2} = \frac{|\xi' - x'|^2 + \xi_n^2 + (\gamma-2)|\xi' - x'|^2}{d^2} = \\
\frac{(\gamma-1) |\xi' - x'|^2 + \xi_n^2 }{d^2},
\end{multline*}
from which we conclude that
\begin{equation}\label{gamma-larger-1}
 \text{for } \gamma>1 \text{ one has } D\geq c>0 \text{ for }  \frac 12 \leq d \leq 2,
\end{equation}
and
\begin{equation}\label{gamma-is-1}
 \text{for } \gamma=1 \text{ one has } D \geq c>0 \text{ for } \frac 12 \leq d \leq 2 \text{ and } |\xi_n|\geq c_1 >0.
\end{equation}
For the proof of Theorem \ref{Thm-Helmholtz} we will use the following result.
\begin{theorem}\label{Thm-Stein}{\normalfont(see Stein \cite{Stein}, p. 377)}
Let $\psi_1 \in C_0^\infty({\mathbb R}^n \times {\mathbb R}^n)$ and $\lambda>0$ and let $\Phi$ be real-valued and smooth.
Set
$$
\mathcal{U}_\lambda f(\xi) = \int_{{\mathbb R}^n} e^{i \lambda \Phi(x, \xi)} \psi_1(x,\xi) f(x) d x , \qquad \xi \in {\mathbb R}^n,
$$
and assume that $\mathrm{det} \left(  \frac{\partial^2 \Phi(x, \xi)}{ \partial x_i \partial \xi_j } \right) \neq 0$
on the support of $\psi_1$. Then one has
$$
|| \mathcal{U}_\lambda f ||_{L^2({\mathbb R}^n) } \leq C \lambda^{-n/2} || f ||_{L^2({\mathbb R}^n)}.
$$
\end{theorem}

\noindent We next give the proof of the main result of this section.
\bigskip

\subsection{Proof of Theorem \ref{Thm-Helmholtz}}
 We start with a decomposition of the kernel $K$. Following Stein \cite{Stein} page 393,
there exists a function $\psi \in C_0^\infty ({\mathbb R}^n)$ such that
$\mathrm{supp} \psi \subset \{ x\in {\mathbb R}^n: \ \frac 12 \leq |x| \leq 2 \}$, and
$$
K(z) = \sum\limits_{k=-\infty}^\infty 2^{k (n-1-m)} \psi  (  2^k z  ).
$$
As such function one may take $\psi(z) = |z|^{-(n-1-m)} [ \eta( z ) - \eta( 2 z) ] $ where $\eta$
is a smooth function satisfying $\eta(y) =1 $ for $|y| \leq 1$ and $\eta (y) =0$ for $|y| \geq 2$.
It is clear that there is $k_0\in {\mathbb Z}$ such that
$$
K(z) = \sum\limits_{k=k_0}^\infty 2^{k (n-1-m)} \psi (  2^k z ), \text{ for } z=x-(y',0) \text{ where } (x,y')\in \mathrm{supp} \psi_0.
$$
We will assume that $k_0=0$, the proof in the general case is the same. We get
$$
T_\lambda f= \sum\limits_{k=0}^\infty T_{\lambda, k } f ,
$$
where
$$
T_{\lambda, k} f(x) = \int\limits_{{\mathbb R}^{n-1}} e^{ i \lambda | x- (y',0) |^\gamma } \psi_0(x, y') 2^{ k(n-1-m) }
\psi \big( 2^k ( x-  (y',0) ) \big) f(y') dy' .
$$
From an application of Hardy-Littlewood's maximal operator (see the proof of Lemma \ref{Lem-A-lambda})
we have $ || T_{\lambda, k}  || = || T_{\lambda, k} ||_{L^2({\mathbb R}^{n-1}) \mapsto L^2({\mathbb R}^n)}  \leq C 2^{-mk}$.
Here we shall improve this estimate.

Take $\varphi \in C_0^\infty ({\mathbb R}) $ such that $\mathrm{supp} \varphi \subset [ -\frac 12 - \frac{1}{10},  \frac 12 + \frac{1}{10} ]$
and $\sum\limits_{j=-\infty }^\infty \varphi( t-j )=1$. Set $\varphi_j (t) = \varphi (t-j)$, and for $t=(t_1,...,t_{n-1})$ define
$ \chi ( t ) := \varphi  (t_1 ) \varphi  (t_2  )\cdot ... \cdot  \varphi (t_{n-1}) $.
Then, for $j=(j_1,...,j_{n-1})\in {\mathbb Z}^{n-1}$ set
$$
\chi_j( t ) := \chi(t-j) =   \varphi_{j_1} (t_1 ) \varphi_{j_2} (t_2  )\cdot ... \cdot  \varphi_{j_{n-1}} (t_{n-1}).
$$
Clearly $\sum\limits_{j\in {\mathbb Z}^{n-1}} \chi_j(t)=1$, and
$$
1 = \sum\limits_{j\in {\mathbb Z}^{n-1}} \chi_j (2^k t) = \sum\limits_{j\in {\mathbb Z}^{n-1}} \chi ( 2^k t -j  ) =
\sum\limits_{j\in {\mathbb Z}^{n-1}} \chi \big( 2^k ( t -2^{-k} j )   \big).
$$
Setting $f_j (t): = f( t ) \chi \big( 2^k ( t-2^{-k} j ) \big) $, implies
$$
f = \sum\limits_{ j\in {\mathbb Z}^{n-1} } f_j.
$$
We get the following decomposition
$$
T_{\lambda, k} f (x) = \sum\limits_{ j\in {\mathbb Z}^{n-1} } T_{\lambda, k} f_j (x),
$$
and for every $x$ the sum has only a bounded number of non-vanishing terms. Therefore by Cauchy-Schwarz we obtain
\begin{equation}\label{Cauchy-Schwarz}
 | T_{\lambda,k} f(x)  |^2 \leq C \sum\limits_{ j\in {\mathbb Z}^{n-1} } | T_{\lambda, k} f_j (x) |^2,
\end{equation}
where the constant $C$ is independent of $x$, $\lambda$, and $k$. We have
\begin{multline*}
T_{ \lambda, k } f_j (x) = \int\limits_{ {\mathbb R}^{n-1} } e^{ i\lambda |x - (y',0) |^\gamma  } \psi_0 (x, y') 2^{ k(n-1-m) }
\psi \big( 2^k ( x- (y', 0) ) \big) f_j (y') dy' = \\ \big[\text{with } y'=2^{-k} z' \big] \
2^{-m k } \int\limits_{ {\mathbb R}^{n-1} } e^{ i \lambda | x- 2^{-k} (z', 0) |^\gamma } \psi_0 (x, 2^{-k} z') \psi \big( 2^k x- (z',0) \big) f_j (2^{-k} z') dz',
\end{multline*}
and
$$
f_j( 2^{-k} z' ) = f(  2^{-k} z') \chi \big(  2^k ( 2^{-k} z' - 2^{-k} j ) \big) = f(  2^{-k} z') \chi ( z' - j ).
$$
Hence
\begin{multline*}
 T_{ \lambda, k } f_j (x) = 2^{- m k} \int\limits_{ {\mathbb R}^{n-1} } e^{ i \lambda 2^{-k \gamma} | 2^k x - (z',0) |^\gamma }
 \psi_0 (x, 2^{-k} z' ) \psi \big( 2^k x - (z',0) \big) f(2^{-k} z' ) \chi( z' -j ) dz' = \\
 \big[ \text{with } y'= z' -j \big]  \
 2^{-m k} \int\limits_{ {\mathbb R}^{n-1} } e^{ i\lambda 2^{-k \gamma} | 2^k x - (y' +j,0) |^\gamma  } \psi_0 \big( x, 2^{-k} (y'+j) \big)
 \psi \big( 2^k x- (y'+j,0) \big) \times \\ f \big( 2^{-k} (y'+j) \big) \chi(y') dy' =  2^{-m k }
\int\limits_{ {\mathbb R}^{n-1} } e^{ i \lambda 2^{-k \gamma} \left| 2^k ( x- (2^{-k}j,0) ) - (y',0) \right|^\gamma  } \psi_0 (x, 2^{-k } j+ 2^{-k} y')
\times \\
 \psi \big(2^k (x- (2^{-k} j,0) ) -(y',0) \big) f(  2^{-k} j + 2^{-k} y' ) \chi(y') d y'.
\end{multline*}
We also have
\begin{multline*}
 \int\limits_{ {\mathbb R}^n } | T_{\lambda,k} f_j (x)  |^2 dx  = \big[ \text{with } x= u+(2^{-k} j, 0) \big] \\
 \int\limits_{ {\mathbb R}^n } \left| T_{\lambda,k} f_j \big( u+ (2^{-k} j, 0 ) \big) \right|^2 d u =  \big[ \text{with } \xi=  2^k u \big] \\
 2^{-k n} \int\limits_{{\mathbb R}^n }  \left| T_{\lambda,k} f_j \big( 2^{-k} \xi + (2^{-k}j, 0) \big)  \right|^2 d \xi .
\end{multline*}
Now let $\widetilde{\chi} \in C_0^\infty ({\mathbb R}^{n-1})$ be so that $\widetilde{\chi} =1$ on $\mathrm{supp} \chi$
and $\mathrm{supp} \widetilde{\chi} \subset [-1,1]^{n-1}$. We then have
\begin{multline*}
 T_{\lambda, k } f_j \big( 2^{-k} \xi + ( 2^{-k}j,0 )  \big) =
 2^{- m k} \int\limits_{ {\mathbb R}^{n-1} } e^{ i \lambda 2^{-k \gamma} | \xi - (y',0) |^\gamma } \psi_0( 2^{-k} \xi +(2^{-k}j, 0 ), 2^{-k} j + 2^{-k} y' ) \times
 \\ \psi \big( \xi - (y',0) \big)  f( 2^{-k} j + 2^{-k} y' ) \chi(y') \widetilde{\chi}(y') dy' =
 2^{-m k } \int\limits_{ {\mathbb R}^{n-1} } e^{ i\lambda 2^{-k \gamma} \Phi (y', \xi) } \psi_1 (y', \xi) g(y') dy' := \\ 2^{- m k} \mathcal{U}_{ \lambda 2^{-k \gamma} } g(\xi),
\end{multline*}
where
\begin{equation}\label{Phi-in-thm}
\Phi (y' ,\xi) = | \xi - (y', 0) |^\gamma = \big( |  \xi' - y'|^2 + \xi_n^2 \big)^{\gamma/2},
\end{equation}
$$
\psi_1(y', \xi)  = \psi \big(\xi - (y',0) \big) \psi_0 \big(  2^{-k} \xi + (2^{-k} j, 0), 2^{-k}  j + 2^{-k} y' \big) \widetilde{\chi}(y'),
$$
and
$$
g(y') = f( 2^{-k} j + 2^{-k} y') \chi(y').
$$
It is clear that for $R>0$ large enough, independently of $j$ and $k$, one has
$$
\mathrm{supp} \psi_1 \subset B(0,R) \times B(0,R),
$$
and that the derivatives of $\psi_1$ can be bounded uniformly in $k$ and $j$.
We have
$$
\left( \int_{{\mathbb R}^n} |T_{\lambda, k} f_j(x)|^2  dx \right)^{1/2}  = 2^{-kn/2} 2^{-mk} \left( \int_{{\mathbb R}^n} | \mathcal{U}_{\lambda 2^{-k \gamma}} g(\xi) |^2 d \xi \right)^{1/2}
$$
and we shall now use Theorem \ref{Thm-Stein}.
It follows from (\ref{gamma-larger-1}) that the determinant condition in Theorem \ref{Thm-Stein}
is satisfied and hence we obtain
$$
\left( \int_{{\mathbb R}^{n-1}} | \mathcal{U}_{\lambda 2^{-k \gamma} } g( \xi', \xi_n ) |^2 d \xi' \right)^{1/2} \leq C (\lambda 2^{-k \gamma})^{-\alpha}
\left( \int_{{\mathbb R}^{n-1}} | g(x') |^2 dx'  \right)^{1/2},
$$
where $\alpha = (n-1)/2$. Integration in $\xi_n$ demonstrates
$$
|| \mathcal{U}_{\lambda 2^{-k \gamma}} g ||_{L^2({\mathbb R}^n)} \leq C \lambda^{-\alpha} 2^{k \gamma \alpha}  || g ||_{L^2({\mathbb R}^{n-1})}.
$$
We also have
\begin{multline*}
|| g ||_{L^2({\mathbb R}^{n-1})}^2 = \int\limits_{ |y'|\leq \sqrt{n} }  | f( 2^{-k} j + 2^{-k} y' ) |^2 dy' = \ [ \text{with } z' = 2^{-k} y' ] \\
2^{ k(n-1) } \int\limits_{ |z'| \leq 2^{-k} \sqrt{n} } | f( 2^{-k} j + z'  ) |^2 dz' .
\end{multline*}
Here $ g= g_{j,k} $ and it is easy to see that
$$
\sum\limits_{j\in {\mathbb Z}^{n-1}} \int_{{\mathbb R}^{n-1}} | g_{j,k}  (y') |^2 d y'  \leq C 2^{ k(n-1) } || f ||_{L^2({\mathbb R}^{n-1})}^2.
$$
Invoking (\ref{Cauchy-Schwarz}) we obtain
\begin{multline*}
\int_{{\mathbb R}^n} | T_{\lambda,k} f (x) | ^2 dx  \leq C \sum\limits_{j \in {\mathbb Z}^{n-1}} \int_{{\mathbb R}^n} | T_{\lambda, k} f_j  |^2 dx \leq \\
C 2^{-k n } 2^{-2 m k} ( \lambda^{-\alpha} 2^{k \gamma \alpha} )^2 \sum\limits_{j } \int_{{\mathbb R}^{n-1}} |g_{j,k}|^2 dy' \leq \\
C 2^{-k n } 2^{-2 m k} ( \lambda^{-\alpha} 2^{k \gamma \alpha} )^2 2^{ k(n-1) } \int_{{\mathbb R}^{n-1}} |f |^2 dy'
\end{multline*}
and hence
\begin{equation}\label{T-lambda-k-est}
|| T_{\lambda, k} ||  \leq C 2^{-k/2} 2^{- mk} \lambda^{-\alpha} 2^{k \gamma \alpha}.
\end{equation}
In this estimate $\lambda^{-\alpha} 2^{k \gamma \alpha}$ can be replaced by 1, since we can make a trivial
estimate instead of using Theorem \ref{Thm-Stein}. Thus we also have
\begin{equation}\label{T-lambda-k-est-trivial}
|| T_{\lambda, k} ||  \leq C 2^{-k/2} 2^{- mk} .
\end{equation}
It follows that
$$
|| T_\lambda || \leq \sum\limits_{k=0}^\infty || T_{\lambda, k} || \lesssim \lambda^{-\alpha}
\sum_{2^k \leq \lambda^{1/\gamma}} 2^{ (\gamma \alpha - m - 1/2 ) k}  + \sum_{ 2^k  \geq \lambda^{1/\gamma}  } 2^{ - (m + 1/2) k }.
$$
For $m< \gamma \alpha -1/2$ we have $\gamma \alpha - m - 1/2>0$ and
$$
|| T_\lambda || \lesssim \lambda^{-\alpha} \lambda^{( \gamma \alpha - m - 1/2 ) / \gamma} + \lambda^{-(m + 1/2 ) / \gamma} \leq C \lambda^{-(m + 1/2 ) / \gamma}.
$$
If $m = \gamma \alpha -1/2$ one has $\gamma \alpha - m - 1/2 =0$ and
$$
|| T_\lambda || \lesssim \lambda^{-\alpha} \log \lambda + \lambda^{-\alpha} \leq C \lambda^{-\alpha} \log \lambda .
$$
Finally, for $m > \gamma \alpha -1/2$ we have $ \gamma \alpha -m -1/2<0$ and
$$
|| T_\lambda || \lesssim  \lambda^{-\alpha} + \lambda^{-(m+1/2)/\gamma}  \leq C \lambda^{-\alpha},
$$
and the proof of the Theorem is complete. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad $\square$

\subsubsection{\normalfont{\textbf{The case of $\gamma=1$, and the Helmholtz equation.}}}\label{sub-sec-Helmholtz}
Here we set $K(z) = |z|^{-(n-1-m)} \omega(z)$,
where $\omega \in C^\infty({\mathbb R}^n \setminus \{0\})$, is homogeneous of degree 0,
and $\omega(z) = 0 $ for all $|z|=1$ satisfying $|z_n| \leq \varepsilon$, for some given $\varepsilon>0$.
The method of the proof of Theorem \ref{Thm-Helmholtz} combined with
(\ref{gamma-is-1}) gives
\begin{equation}\label{Helmholtz-est-gamma-1}
|| T_\lambda ||  \lesssim
 \begin{cases}
                                 \lambda^{-  (m+1/2)}, &\text{  $m<  n/2 - 1$}, \\
                                 \lambda^{-\alpha} \log \lambda, &\text{  $m=n/2 -1$}, \\
                                 \lambda^{-\alpha}, &\text{  $m> n/2 - 1 $} ,
                            \end{cases}
\end{equation}
with $\alpha=(n-1)/2$.

We now discuss how the analysis can be applied to Helmholtz equation.
In ${\mathbb R}^3$ consider a smooth hypersurface $\Gamma$ given by (\ref{Gamma-surface}),
and fix some $\varphi_0\in C_0^\infty({\mathbb R}^3)$. Let $\sigma_\Gamma$ be the surface measure
of $\Gamma$, and for a measure $\mu_\Gamma  = \varphi_0(y) d \sigma_\Gamma(y)$ let
$u_\lambda$ be a solution to the following inhomogeneous Helmholtz equation
\begin{equation}\label{eq-Helmholtz}
\Delta u + \lambda^2 u =  -\mu_\Gamma   \ \ \text{ in } {\mathbb R}^3.
\end{equation}
We denote by $G_\lambda (\cdot, \cdot)$ the fundamental solution (Green's function) of (\ref{eq-Helmholtz}). It is well known that
$G_\lambda (x,y)=e^{i\lambda |x-y|}/ (4\pi |x-y|)$ where $x,y\in {\mathbb R}^3$, $x\neq y$. Now let $u_\lambda$ be the solution
to (\ref{eq-Helmholtz}) given in terms of the Green's function, namely
\begin{equation}\label{Helm-sol-Green}
u_\lambda (x) = \int_{{\mathbb R}^3} G_\lambda(x,y) d\mu_\Gamma (y).
\end{equation}
This is precisely the solution to (\ref{eq-Helmholtz}) satisfying Sommerfeld radiation condition, which in dimension three reads
$$
 \lim\limits_{r \to \infty } r \left( \frac{\partial}{\partial r } -  i \lambda \right) u_\lambda (r \nu) =0,
$$
uniformly for all directions $\nu \in \mathbb{S}^2$, where $i$ is the imaginary unit.
From (\ref{Helm-sol-Green}) we have
\begin{equation}\label{Helm-3d-on-Gamma}
u_\lambda (x) = \int_{\Gamma}  \frac{e^{i \lambda |x-y|}}{4\pi |x-y|}  \varphi_0(y) d\sigma_\Gamma(y), \qquad x\in {\mathbb R}^3.
\end{equation}
Let us show here how to obtain decay estimates on $u_\lambda$, as $\lambda \to \infty$, in the case when $\Gamma$ is a plane.
Assume $ \nu \in {\mathbb R}^3$ is the unit normal to $\Gamma$, thus $\Gamma= \{ y \in {\mathbb R}^3: y\cdot \nu =0 \}$.
Fix any $3 \times 3$ orthogonal matrix $M$ such that $M e_3 = \nu$, where $e_3=(0,0,1)$.
Next, make a change of variables in (\ref{Helm-3d-on-Gamma}) by the formula $y=M z$, and $x=M w$, where $w, z \in {\mathbb R}^3$. We get
\begin{multline}\label{Helm-reduced-to-0-plane}
u_\lambda (x) =u_\lambda (Mw) =  \int_{ z_3=0 }  \frac{e^{i \lambda |Mw-Mz|}}{4\pi |M w- Mz|}  \varphi_0(M z) d\sigma (z) = \\ \int_{{\mathbb R}^2}
\frac{e^{i \lambda | w- (z',0)  |}}{4\pi | w- (z',0)|}  \varphi_0(M (z',0)) d z', \qquad x,w \in {\mathbb R}^3.
\end{multline}

Thus the case of an arbitrary plane, by a rotation, is easily reduced to the case of $y_3=0$ in ${\mathbb R}^3$.
Now, if we have a bounded domain $D\subset {\mathbb R}^3$
which stays within a positive distance from $\Gamma$, then the determinant condition (\ref{gamma-is-1}) will be satisfied for all $x\in D$.
Next, using the argument of Theorem \ref{Thm-Helmholtz} as we do for (\ref{Helmholtz-est-gamma-1}),
with $n=3$ and $m=1$ one may show from (\ref{Helm-reduced-to-0-plane}) that
\begin{equation}
|| u_\lambda ||_{L^2(D)} \lesssim_D \lambda^{- 1} |D|^{1/2}  ,
\end{equation}
where the constant in the inequality depends on the distance
of $D$ and the plane $\Gamma$.
Thus, in this particular case
we get a quantitative decay estimate of the $L^2$-norm of solutions $u_\lambda$ as the eigenvalue tends to infinity.

When we allow $D$ to cross the plane $\Gamma$, the determinant condition in (\ref{gamma-is-1}) becomes
invalid. However, this scenario can be handled by splitting the integral in (\ref{Helm-reduced-to-0-plane})
by means of a smooth cut-off into two regions, where one of them stays away from $\Gamma$, and the other one is in a small neighbourhood of $\Gamma$.
The former will be estimated as above, relying on (\ref{gamma-is-1}), while the latter will be handled using the smallness of the support
of the cut-off and uniform boundedness of $u_\lambda$. Thus, one may obtain decay estimates on $u_\lambda$ in this setting
as well relying on the methods discussed in the paper, however, the rate of decay will be worse than $\lambda^{-1}$.
In a similar vein we may handle the case when $\Gamma$ is a union of some finite number of planes in ${\mathbb R}^3$,
by applying the analysis on each flat piece of $\Gamma$ separately.
The details are left as an exercise for an interested reader.

It should be noted that at this stage our analysis does not extend to the case of a general hypersurface
and more interestingly to the case of oscillating (rough) boundaries considered in subsection \ref{sec-perturb-surface} and Section \ref{sec-maximal-operator}.
Understanding the properties of the operator (\ref{def-of-T-Helmholtz}) in this generality seems to be a very interesting problem.
Another interesting problem here is to understand if the operators
$T_\lambda$ with non integer $\gamma$ can be applied to the study of fractional order Helmholtz operators.
It was to our surprise that there were virtually no results in the literature concerning fundamental solutions
of fractional order Helmholtz equations.

\subsubsection{\normalfont{\textbf{The case of $\gamma=2$.}}} We have $\gamma \alpha -1/2 = n-1-1/2 = n-3/2$. For $m<n-3/2$
we have proved that
\begin{equation}\label{T-lambda-est-gamma-2}
 || T_\lambda ||_{L^2({\mathbb R}^{n-1}) \mapsto L^2({\mathbb R}^n)} \leq C \lambda^{-(m/2 + 1/4)}.
\end{equation}
Now set
$$
S_\lambda f(x) = \int_{{\mathbb R}^{n-1}} e^{i \lambda x' \cdot y'} \psi_0(x, y') K( x-(y',0) )  f(y') dy'.
$$
We have
$$
2 x' \cdot y' = 2 x \cdot (y',0) = |x|^2 + |y'|^2  - |x-(y',0)|^2
$$
and
$$
e^{i 2 \lambda x' \cdot y'} = e^{i \lambda |x|^2}  e^{i \lambda |y'|^2} e^{i \lambda |x-(y',0)|^2}.
$$
Combination of this inequality and (\ref{T-lambda-est-gamma-2}) gives
\begin{equation}\label{S-lambda-est-gamma-2-n}
 ||S_\lambda ||_{L^2({\mathbb R}^{n-1}) \mapsto L^2({\mathbb R}^n)} \leq C \lambda^{-(m/2+1/4)}.
\end{equation}
On the other hand taking $\beta=1/2$ in Lemmas \ref{Lem-A-lambda} and \ref{Lem-fixed-surface} we obtain
\begin{equation}\label{S-lambda-est-gamma-2-n-1}
 ||S_\lambda ||_{L^2({\mathbb R}^{n-1}) \mapsto L^2({\mathbb R}^{n-1})} \leq C \lambda^{-m/2 }.
\end{equation}
It is interesting to observe that we have a better decay in (\ref{S-lambda-est-gamma-2-n}) than in (\ref{S-lambda-est-gamma-2-n-1}).

\subsection{Lower bounds.}\label{subsec-lower-bounds}

Here we construct examples for which the upper bounds
(\ref{S-lambda-est-gamma-2-n}), (\ref{S-lambda-est-gamma-2-n-1}), and the first estimate of Theorem \ref{Thm-Helmholtz}
are attained, showing that these estimates are best possible.

We start with an example showing that $m/2$ in (\ref{S-lambda-est-gamma-2-n-1}) can not be replaced by a larger number.
Take $x_0' \in {\mathbb R}^{n-1}$  and set $F = \{ x' \in {\mathbb R}^{n-1}: \ |x' - x_0'| \leq c_0 \lambda^{-1/2} \}$
where $c_0$ is a small constant, which will be specified in due course. Then choose $y_0' \in {\mathbb R}^{n-1}$ such that $|x_0' - y_0' | = 100 c_0 \lambda^{-1/2}$
and set $E= \{ y' \in {\mathbb R}^{n-1}: \ |y' - y'_0| \leq c_0 \lambda^{-1/2} \}$. We have
$$
x' \cdot y' = (x' - x_0') \cdot (y' - y_0') + (x' - x_0')\cdot y_0'+ x_0' \cdot y'
$$
and
$$
e^{ i \lambda x' \cdot y' } = e^{i \lambda (x'- x_0') \cdot (y'-y_0')} e^{i \lambda (x'- x_0') \cdot y_0'} e^{i \lambda x_0' \cdot y'}.
$$
Setting $f(y') = e^{-i \lambda x_0' \cdot y'} \mathbb{I}_E (y')$ we obtain
\begin{align*}
S_\lambda f(x) = e^{i \lambda (x' - x_0')\cdot y_0'} \int_{{\mathbb R}^{n-1}} e^{i \lambda (x' - x_0') \cdot (y'-y_0')} \psi_0 (x,y')
e^{i\lambda x_0' \cdot y'} K(x-(y',0)) f(y') dy' &= \\
e^{i \lambda (x' - x_0')\cdot y_0'} \int_{{\mathbb R}^{n-1}} \psi_0(x,y') e^{i \lambda (x'-x_0') \cdot (y' - y_0')} K(x-(y',0)) \mathbb{I}_E(y') dy' &= \\
e^{i \lambda (x' - x_0')\cdot y_0'}  \int_E \psi_0(x,y') K(x-(y',0)) dy' &+ \\
e^{i \lambda (x' - x_0')\cdot y_0'} \int_E \psi_0(x,y') \big( e^{i\lambda (x' - x_0')\cdot (y'-y_0')} - 1 \big) K(x-(y',0)) dy' &:= \\ P(x) + R(x),
\end{align*}
where we have
\begin{multline*}
 |R(x)| \leq C \int_E \lambda |x'- x_0'| |y'-y'_0| K(x-(y',0)) dy' \leq \\ C \int_E \lambda c_0 \lambda^{-1/2} c_0 \lambda^{-1/2} K(x-(y',0)) dy'=
C c_0^2 \int_E K(x-(y',0)) dy',
\end{multline*}
for $x'\in F$. We now take $x=(x',0)$ and assume that $\psi_0( (x', 0),y') = 1$ for $x' \in F$ and
$y' \in E$. We then have
$$
|P(x)| = \int_E K(x'-y',0) dy',
$$
and
$$
|R(x)| \leq C c_0^2 \int_E K(x' - y',0) dy', \qquad \text{ for } x' \in F.
$$
Choosing $c_0$ small, for $x' \in F$ we obtain
$$
|S_\lambda f(x',0) | \geq \frac 12 \int_E \frac{1}{|x'-y'|^{n-1-m} } dy' \geq c (\lambda^{-1/2}  )^{n-1} \frac{1}{\lambda^{-(n-1-m)/2}} = c \lambda^{-m/2}.
$$
It follows that
$$
\left( \int_{R^{n-1}}  |S_\lambda f(x',0) |^2 d x' \right)^{1/2} \geq c \lambda^{-m/2} |F|^{1/2}.
$$
But $||f||_2 = |E|^{1/2} = |F|^{1/2}$ and hence
$$
||S_\lambda||_{L^2({\mathbb R}^{n-1}) \to L^2({\mathbb R}^{n-1}) } \geq c \lambda^{-m/2}.
$$
Thus we have proved that (\ref{S-lambda-est-gamma-2-n-1}) is sharp.

\bigskip
We shall then prove that (\ref{S-lambda-est-gamma-2-n}) is sharp as well, for which we will
use the construction described above. Define $F_1 : = F \times [0, c_0 \lambda^{-1/2}]$
and assume that $\psi_0 (x, y') = 1$ for $x \in F_1$ and $y' \in E$. Now for $x \in F_1$
we obtain as above that $S_\lambda f(x) = P(x) + R(x)$ where
$$
|P(x)| =\int_E K(x' - y', x_n) dy'.
$$
It follows that for $x \in F_1$ we have
$$
|S_\lambda f(x) | \geq \frac 12 \int_E \frac{1}{\lambda^{-(n-1-m)/2}} dy' \geq c \lambda^{-m/2},
$$
hence
$$
\left(  \int_{{\mathbb R}^n} |S_\lambda f(x)|^2 dx  \right)^{1/2} \geq c \lambda^{-m/2} \lambda^{-n/4},
$$
since $|F_1| = c \lambda^{-n/2}$. But
$$
|| f ||_2 = |E|^{1/2} = c \lambda^{-(n-1)/4}= c \lambda^{-n/4 + 1/4},
$$
and consequently
$$
|| S_\lambda ||_2 \geq c ||f||_2 \lambda^{-m/2 - 1/4},
$$
which shows that
$$
|| S_\lambda ||_{L^2 ({\mathbb R}^{n-1}) \mapsto L^2({\mathbb R}^n) }  \geq c \lambda^{-(m/2+1/4)}.
$$
Thus we have proved that (\ref{S-lambda-est-gamma-2-n}) is sharp.

\bigskip

Finally we turn to the first estimate of Theorem \ref{Thm-Helmholtz}.
For $\gamma>1$ we have
$$
T_\lambda f(x) = \int_{{\mathbb R}^{n-1}} e^{ i \lambda | x- (y',0) |^\gamma } \psi_0 (x,y') K(x- (y',0)) f(y') dy'
$$
and we shall estimate the norm of $T_\lambda$ from below. Set $F := B(x_0', c_0 \lambda^{-\beta} )$,
$F_1: = F \times [0, c_0 \lambda^{-\beta}]$, and $E:=  B(y_0', c_0 \lambda^{-\beta})$, where
$|x_0' - y_0'| = 100 c_0 \lambda^{-\beta}$. Here $B(x,R)$ denotes a ball with a center $x$
and radius $R$. We assume that $\psi_0 (x,y') =1 $ for $x\in F_1$ and $y' \in E$.
Letting $x\in F_1$ and $f=\mathbb{I}_E$ we have
$$
T_\lambda f(x) = \int_E K(x-(y',0)) dy' + \int_E \big(  e^{i \lambda |x-(y',0)|^\gamma} -1 \big) K( x-(y',0) ) dy' := P(x) + R(x).
$$
Choosing $\beta=1/\gamma$ for $x \in F_1$ and $y' \in E$ implies
$$
|e^{i \lambda |x-(y',0)|^\gamma} -1 | \leq \lambda  |x-(y',0)|^\gamma  \leq C c_0 \lambda \lambda^{-\beta \gamma} \leq C c_0.
$$
It follows that
$$
|R(x)| \leq C c_0 \int_E K( x-(y',0) ) dy'.
$$
Now taking $c_0$ small we obtain
$$
| T_\lambda f(x) | \geq c \int_E K(x-(y',0)) dy' \geq c \int_E \frac{d y'}{\lambda^{-\beta(n-1-m)}} = C \lambda^{-\beta m}
$$
and
$$
\left( \int_{F_1} | T_\lambda f  |^2 dx \right)^{1/2} \geq c \lambda^{-\beta m} \lambda^{-\beta n/2},
$$
since $|F_1| \geq c \lambda^{-\beta n}$. But $ || f ||_2 = c \lambda^{-\beta(n-1)/2}  = c \lambda^{-\beta n/2}  \lambda^{\beta /2}$
and hence
$$
|| T_\lambda|| \geq c \lambda^{-\beta m} \lambda^{-\beta /2} = c \lambda^{-\beta (m+1/2)}= c \lambda^{-( m+1/2) /\gamma}.
$$
In Theorem \ref{Thm-Helmholtz} we proved that
$$
|| T_\lambda|| \leq C \lambda^{-(m+1/2)/\gamma},
$$
for $m<\gamma \alpha - 1/2= \gamma n/2 - \gamma/2 - 1/2$
and thus we have proved that this estimate is sharp.

\section*{Acknowledgements}
Part of the paper has been written while all authors met at Institut Mittag-Leffler
during the program ``Homogenization and Random Phenomenon". H.A. thanks The Institue for its support. H.S. was partially supported by the Swedish Research Council.


\section{Introduction}
Support Vector Machine (SVM) is a classification technique based on the idea of Structural Risk Minimization (SRM). It is a kernel-based classifier which was first introduced in 1995 by Vapnik and his colleagues, at AT\&T Bell Laboratories \cite{cortes1995support}. The algorithm has been used in many classification tasks due to its success in recognizing handwritten characters in which it outperformed precisely trained neural networks. Some of these tasks are: text classification \cite{joachims1998text}, image classification \cite{chatterjee2013vision}, and bioinformatics  \cite{guyon2002gene,sartakhti2012hepatitis}.

\noindent
One of the newest versions of SVM is Least Squares Twin Support Vector Machine (LSTSVM) introduced in 2009 \cite{arun2009least}. The algorithm combines the idea of Least Squares SVM (LSSVM) \cite{suykens1999least} and Twin SVM (TSVM) \cite{khemchandani2007twin}. Although, in some classification tasks LSTSVM provides high accuracies \cite{arun2009least} it still suffers from two main drawbacks. (I) In real-world applications, the data points may not be entirely assigned to a class, while LSTSVM strictly assigns each data point to a class, (II) Although in many classification tasks data points have different importance; LSTSVM considers the data points to have same priorities.

\noindent
Many real-world applications require different values of importance for input data. In such cases, the main concern is how to determine the final classes by assigning different importance degrees to training data. Moreover, the classifier should be designed in a way that it can separate the noises from data. A good approach to cope with these challenges is to use the concept of fuzzy functions.

\noindent
The fuzzy theory is very useful for analyzing complex processes using standard quantitative methods or when the available information is interpreted uncertainly. A fuzzy function can represent uncertainty in data structures using fuzzy parameters. In the literature, the concepts of fuzzy function and fuzzy operations are introduced by different researchers \cite{negoita1975applications,zadeh1975concept,dubois1978operations,yager1979solving,dubois1980fuzzy}. A fuzzy function offers an efficient way of capturing the inexact nature of real-world problems.

\noindent
In this paper we incorporate the concept of fuzzy set theory into the LSTSVM model. Unlike the standard LSTSVM, in the training phase the proposed fuzzy LSTSVM treats training data points according to their importance degrees. In the literature, several approaches of applying fuzzy sets in SVM have been proposed \cite{inoue2001fuzzy,lin2002fuzzy,chaves2005fuzzy,wang2014interval,hao2008fuzzy}. The key feature of the proposed fuzzy LSTSVM is that it assigns fuzzy membership values to data points based on their importance degrees. In addition, we use fuzzy numbers to set the parameters of the fuzzy LSTSVM model such as the weight vector and the bias term. Using these two features, we proposed two models for fuzzy LSTSVM.

\noindent
The rest of this paper is organized as follows. A brief review of basic concepts including the SVM, TSVM, and LSTSVM is presented in Section \ref{SecII}. The proposed models for fuzzy LSTSVM are introduced in Section \ref{SecIII}. In Section \ref{SecIV} we evaluate the proposed models, and finally, Section \ref{SecV} concludes the paper.

\section{Basic Concepts}\label{SecII}
\noindent
In this section a quick review of different versions of SVM is presented, namely the standard SVM, TSVM, and LSTSVM.
\subsection{Support Vector Machine}
\noindent
The main idea behind SVM is to minimize the classification error while preserving the maximum possible margin between classes. Suppose we are given a set of training data points $x_{i}\in\mathbf{R}^{\mathbf{d}}, i=1, \cdots, \mathbf{n}$ with labels $y_{i}\in\lbrace-1,+1\rbrace$. SVM seeks for a hyperplane with equation $w.x+b=0$ with the following constraints:
\begin{flalign}\label{eq1}
&y_{i}(w.x_{i}+b) \geq1,\qquad \forall i.&
\end{flalign}
\noindent
where $w$ is the weight vector. Such a hyperplane could be obtained by solving Eq. (\ref{eq2}):
\begin{flalign}\label{eq2}
&Minimize\quad f(x)=\frac{\Vert w\Vert^{2}}{2}& \\\nonumber
&subject\:to\quad	y_{i}(w.x_{i}+b)-1\geq 0&
\end{flalign}
\noindent
The geometric interpretation of this formulation is depicted in Fig. \ref{SVM} for a toy example.
\begin{figure}[h]
\begin{center}
\includegraphics[width=2 in]{SVM.jpg}
\caption{Geometric interpretation of SVM}
\label{SVM}
\end{center}
\end{figure}
\subsection{Twin Support Vector Machine}\label{SecII.2}
\noindent
In SVM only one hyperplane does the task of partitioning the samples into two groups of positive and negative classes. For the first time in 2007, Jayadeva et al. \cite{khemchandani2007twin} proposed TSVM with the idea of using two hyperplanes in which samples are assigned to a class according to their distance from the hyperplanes. The main equations of TSVM are as follows:
\begin{flalign}\label{eq3}
&x_{i}w^{(1)}+b^{(1)}=0& \\\nonumber
&x_{i}w^{(2)}+b^{(2)}=0&
\end{flalign}
\noindent
where $w^{(i)}$ and $b^{(i)}$ are the weight vector and bias term of the $i^{th}$ hyperplane, respectively. Each hyperplane represents the samples of its class. This concept is geometrically depicted in Fig. \ref{TSVM} for a toy example. In TSVM, the two hyperplanes are non-parallel. Each of them is closest to the samples of its own class and farthest from the samples of the opposite class \cite{zhang2014sparse,Ding2014an}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=1.7 in]{TSVM.jpg}
\caption{The Geometric interpretations of Twin SVM}
\label{TSVM}
\end{center}
\end{figure}
\noindent
Let us assume that $A$ and $B$ indicate two data points of class  $+1$ and class $-1$, respectively. The two hyperplanes are obtained by solving Eq. (4) and Eq. (5).
\begin{flalign}\label{eq4}
\includegraphics[width=3 in]{4.pdf}
\end{flalign}
\begin{flalign}\label{eq5}
\includegraphics[width=3 in]{5.pdf}
\end{flalign}
\noindent
In these equations  $\xi$ represents the slack variables, $e_{i} \; (i \in \{1,2\})$ is a column vector of ones with desirable length, and $p_{1}$ and $p_{2}$ are penalty parameters.
\subsection{Least Squares Twin Support Vector Machine}\label{SecII.3}
\noindent
LSTSVM \cite{arun2009least}, \cite{shao2012least} is a binary classifier which combines the idea of LSSVM and TSVM. In other words, LSTSVM employs least squares of errors to modify the inequality constraints in TSVM to equality constraints and solves a set of linear equations rather than two Quadratic Programming Problems (QPPs). Experiments have shown that LSTSVM can considerably reduce the training time, while preserving competitive classification accuracy \cite{suykens1999least,gao20111}. Furthermore, since the time complexity of SVM is of order $m^{3}$, where $m$ is the number of constraints, theoretically when there are an equal number of positive and negative samples, the speed of the algorithm increases by the factor of four compared to the standard SVM.\\
\noindent
LSTSVM finds its hyperplanes by minimizing Eq. (6) and Eq. (7) which are linearly solvable. By solving Eq. (6) and Eq. (7), the values of $w$ and $b$ for each hyperplane are obtained according to Eq. (8) and Eq. (9).
\begin{flalign}\label{eq6}
\includegraphics[width=3 in]{6.pdf}
\end{flalign}
\begin{flalign}\label{eq7}
\includegraphics[width=3 in]{7.pdf}
\end{flalign}
\begin{flalign}\label{eq8}
&\begin{bmatrix}
	w^{(1)} \\
	b^{(1)}
\end{bmatrix}
=-(F^{T}F + \frac{1}{p_{1}} E^{T}E)^{-1}F^{T}e&
\end{flalign}
\begin{flalign}\label{eq9}
&\begin{bmatrix}
	w^{(2)} \\
	b^{(2)}
\end{bmatrix}
=(E^{T}E + \frac{1}{p_{2}} F^{T}F)^{-1}E^{T}e&
\end{flalign}
\noindent
where $E=\begin{bmatrix}
A & e
\end{bmatrix}$ and $F= \begin{bmatrix}
B & e
\end{bmatrix}$ and $A$, $B$, $e$ and $\xi$ are already introduced in Section \ref{SecII.2}.
\section{Fuzzy Least Squares Twin Support Vector Machine}\label{SecIII}
\noindent
In this section, first we explain the importance of fuzzy classification and then we introduce two approaches for improving LSTSVM using the fuzzy sets theory. Basic notations used in this section are as follows: samples of the positive and negative classes are represented by matrices $A$ and $B$, respectively. $A$ contains $m_{1}$ positive samples and $B$ contains $m_{2}$ negative samples. Membership degrees are represented by $\mu$ and slack variables are represented by vector  $\xi$. All equations will be presented in matrix form where for each matrix $M$, its transpose is represented by  $M^{T}$. $e$ is a vector with arbitrary size and all its elements are equal to $1$.
\subsection{Fuzzy Classification}
\noindent
In many real-world applications a sample in the training data does not belong exactly to a single class. Furthermore, in some applications it would be desirable for the new training samples to have higher importance than older ones. Given the uncertainty of assigning such importance values, the fuzzy sets provide an elegant way to cope with this problem. We can define a fuzzy membership degree $\mu _{i}$ for each sample in the training data. A membership degree is a number between $0$ and $1$ which can be considered as a measure of influence of the sample on the final class. Therefore, a training sample with membership degree of $\mu _{i}$ influences class $+1$ by $\mu _{i}$ and influences class $-1$ by $(1-\mu_{i})$. In addition, using fuzzy membership functions, it is possible to assign a membership degree to each sample based on its entry time. Sequential learning \cite{clegg1998sequence} is another application which induces applying fuzzy concepts in classification algorithms such as SVM.\\
\noindent
In 2008 Pei-Yi Hao introduced fuzzy SVM \cite{hao2008fuzzy}. In his paper, he introduced two approaches, $M_{1}$ and $M_{2}$for applying fuzzy sets in SVM. In the first model, $M_{1}$, he constructed a crisp hyperplane, and he also assigned a fuzzy membership to each data point. In the second model, $M_{2}$, he constructed a fuzzy hyperplane to discriminate classes. In the following sections, we integrated the fuzzy set theory into the LSTSVM algorithm in accordance with \cite{hao2008fuzzy}.
\subsection{Fuzzy LSTSVM: Model $M_{1}$}
\noindent
In this model, fuzzy memberships values are assigned to data points such that noises and outliers get small membership values. Our goal is to construct two crisp hyperplanes to distinguish target classes. In order to use this model in LSTSVM algorithm, we rewrote Eq. (6) and Eq. (7) in the form of Eq. (10) and Eq. (11):
\begin{flalign}\label{eq10}
\includegraphics[width=2.9 in]{10.pdf}
\end{flalign}

\begin{flalign}\label{eq11}
\includegraphics[width=2.9 in]{11.pdf}
\end{flalign}
\noindent
Eq. (10) and Eq. (11) represent equations of the positive and the negative class hyperplanes, respectively. In these two equations the membership degree $\mu$ appears only as error coefficient.\\

\noindent
By obtaining $\xi$ and substituting it in Eq. (10) and Eq. (11), the two equations are reformulated as Eq. (12) and Eq. (13).
\begin{flalign}\label{eq12}
\includegraphics[width=2.9 in]{12.pdf}
\end{flalign}
\begin{flalign}\label{eq13}
\includegraphics[width=2.9 in]{13.pdf}
\end{flalign}
\noindent
By differentiating Eq. (12) and Eq. (13) with respect to $w$ and $b$, we have:
\begin{flalign*}
\includegraphics[width=3 in]{135.pdf}
\end{flalign*}

\noindent
By solving the above equations using some matrix algebra, we would have Eq. (14) and Eq. (15) as the equations of the hyperplanes $J_{1}$ and $J_{2}$, respectively.
\begin{flalign} \label{eq14}
\includegraphics[width=2.9 in]{14.pdf}
\end{flalign}

\begin{flalign}\label{eq15}
\includegraphics[width=2.9 in]{15.pdf}
\end{flalign}

\noindent
These two equations can be represented as Eq. (16) and Eq. (17), respectively.
\begin{flalign}\label{eq16}
\includegraphics[width=2.9 in]{16.pdf}
\end{flalign}

\begin{flalign}\label{eq17}
\includegraphics[width=2.9 in]{17.pdf}
\end{flalign}

\noindent
Once the values of $w^{(1)}$, $b^{(1)}$, $w^{(2)}$ and $b^{(2)}$ are obtained, a new data point is assigned to a class based on its distance from the hyperplane of the corresponding class.

\subsection{Fuzzy LSTSVM: Model $M_{2}$}

\noindent
In this model, we construct fuzzy hyperplanes to discriminate the classes. In $M_{2}$, all parameters of the model, even the components of weight vector $w$, are fuzzy numbers. For computational simplicity all parameters used in this work are restricted to a class of "triangular" symmetric membership functions. For a symmetric triangular fuzzy number $X=(o,r)$, $o$ is the center and $r$ is the width of the corresponding membership function.\\

\noindent
Let us assume $W$ and $B$ are the fuzzy weight vector and fuzzy bias term, respectively, where each component of $W$ is shown by $W_{i}=(w_{i},c_{i})$ and $B=(b,d)$. Then the equation of a fuzzy hyperplane is defined as follows:
\begin{flalign}\label{eq18}
\includegraphics[width=2.9 in]{18.pdf}
\end{flalign}

\noindent
To find the fuzzy hyperplane for class $+1$ of our fuzzy LSTSVM, we rewrite Eq. (6) as:
\begin{flalign}\label{eq19}
\includegraphics[width=2.9 in]{19.pdf}
\end{flalign}
\noindent
In this equation, $\frac{1}{2}\Vert c^{(1)} \Vert ^{2}+d^{(1)}$ measures the vagueness of the model. As the vagueness of the model increases, the results would be more inexact. In Eq. (19) the parameter $M$ is a control parameter chosen by the user. Also $\frac{p_{1}}{2}\mu \xi^{T}\xi$ determines the amount of least squares error, where $\mu$ is the membership degree of the positive sample and the vector $\xi$ is the slack variable vector. $p_{1}$ is a trade-off parameter which controls the effect of the least squares error on the hyperplane.\\
\noindent
Eq. (\ref{eq19}) can be rewritten as Eq. (\ref{eq21})
\begin{flalign}\label{eq21}
\includegraphics[width=2.9 in]{21.pdf}
\end{flalign}
\noindent
Setting the derivation of Eq. (\ref{eq21}) with respect to $w^{(1)}$, $b^{(1)}$, $c^{(1)}$ and $d^{(1)}$ equal to zero and solving the above equations, the below system would appear:
\begin{flalign}\label{eq23}
\includegraphics[width=3 in]{23.pdf}
\end{flalign}
\noindent
Up to now we have found all the necessary parameters of the first fuzzy hyperplane. By substituting values of these parameters in Eq. (\ref{eq18}), we can obtain the equation of the first fuzzy hyperplane.\\
\noindent
For the second hyperplane, the equations are as follows:
\begin{flalign}\label{eq24}
\includegraphics[width=3 in]{24.pdf}
\end{flalign}
That can be rewritten as:
\begin{flalign}\label{eq26}
\includegraphics[width=3 in]{26.pdf}
\end{flalign}
Hence we have:
\begin{flalign}\label{eq28}
\includegraphics[width=3 in]{28.pdf}
\end{flalign}
\noindent
By solving Eq. (\ref{eq28}) and finding the values of the parameters $w^{(2)}$, $b^{(2)}$, $c^{(2)}$ and $d^{(2)}$, the equation of the second fuzzy hyperplane can be obtained using Eq. (\ref{eq18}). After finding the equations of the two fuzzy hyperplanes, the fuzzy distance between a given test data point and the fuzzy hyperplanes should be calculated. Definition 1 shows how find the fuzzy distance between a data point and a fuzzy hyperplane.\\
\noindent
\textbf{Definition 1}: $\Delta=(\delta,\gamma)$ is the fuzzy distance between a data point $x_{0}=(x_{01},\cdots,x_{0n})$ and the fuzzy hyperplane $W.x+B$, where:\\
$\delta$$=\frac{\lvert w_{1}x_{01}+\cdots+w_{n}x_{0n}+b\rvert}{\lVert W\rVert}$\\
and\\
$\gamma$$=\frac{\lvert (w_{1}+c_{1})x_{01}+\cdots+(w_{n}+c_{n})x_{0n}\rvert}{\lVert W\rVert}$.\\
\noindent
By finding fuzzy distances between the data point and the fuzzy hyperplanes, it is necessary to define a fuzzy membership function which determines the membership degree of the data point in each fuzzy hyperplane. Let us assume that $\Delta_{1}=(\delta_{1},\gamma_{1})$ and $\Delta_{2}=(\delta_{2},\gamma_{2})$ are fuzzy distances between a data point and the two hyperplanes $H_{1}$ and $H_{2}$, respectively. Then for an input data $x_{0}$, the degree that $x_{0}$ belongs to hyperplane $H_{1}$ is defined by the following membership function (by finding membership degrees for $H_{1}$, membership degrees for $H_{2}$ are easily obtainable):
\begin{flalign}\label{eq29}
&\mu_{1}(x_{0})=\begin{cases}
	1-\frac{\delta_{1}+\gamma_{1}}{\delta_{1}+\gamma_{1}+\delta_{2}+\gamma_{2}} & \delta_{1} \geq \gamma_{1}, \delta_{2}\geq \gamma_{2},\\
	1-\frac{\delta_{1}}{\delta_{1}+\delta_{2}+\gamma_{2}} & \delta_{1} < \gamma_{1}, \delta_{2}\geq \gamma_{2},\\
	1-\frac{\delta_{1}+\gamma_{1}}{\delta_{1}+\gamma_{1}+\delta_{2}} & \delta_{1} \geq \gamma_{1}, \delta_{2} < \gamma_{2}\\
	1-\frac{\delta_{1}}{\delta_{1}+\delta_{2}} & \delta_{1} < \gamma_{1}, \delta_{2} < \gamma_{2},\\
\end{cases}&
\end{flalign}
\section{Numerical Experiments}\label{SecIV}
To evaluate the performance of our proposed algorithm, we investigate its classification accuracy on both artificial and benchmark datasets. All experiments are carried out in Matlab 7.9 (R2009b) environment on a PC with Intel processor (2.9 GHz) with 2 GB RAM. The \textbf{Accuracy} used to evaluate a classifier is defined as: $(TP+TN)/(TP+FP+TN+FN)$, where $TP$, $TN$, $FP$ and $FN$ are the number of true positive, true negative, false positive and false negative, respectively. Also the accuracies are measured by the standard 10-fold cross-validation methodology \cite{duda2012pattern}. In our implementation, we focus on comparison between SVM, LSTSVM and FLSTSVM with model $M_{2}$.
\subsection{Experiment on artificial dataset}
We first consider a two-dimensional \enquote{Xor} dataset, which is a very common dataset for evaluating the effectiveness of SVM-based algorithms, shown in Fig. 3. This hand-made dataset consists of 121 records belonging to two classes. Each record has two features: a class and a value which determines how much the record belongs to the class. The stars denote the data points of positive class, while the triangular belong to the negative class.
\begin{figure}[h]
\begin{center}
\includegraphics[width=2 in]{Toydata.png}
\caption{The synthetic dataset}
\label{Toydata}
\end{center}
\end{figure}
\noindent
Table \ref{TI} shows the results of applying SVM, LSTSVM and FLSTSVM algorithms to the dataset. It should be noted that in this paper, only the linear version of all the three algorithms are studied.
\begin{table}[h]
	\centering
	\caption{A comparison of classification accuracy}
	\begin{tabular}{c c}
		\hline
		\textbf{Algorithms} & \textbf{Accuracy(\%)}\\
		\hline
		SVM & 53.0\\
		LSTSVM & 65.0\\
		FLSTSVM & 73.0\\
		\hline
	\end{tabular}
	\label{TI}
\end{table}
\noindent
The obtained values for accuracies of all three algorithms are fully justifiable. In SVM there is only one hyperplane responsible of classifying data and because the dataset is defined as a two-dimensional space, this hyperplane would be a line which is shown in figure \ref{Fig4.a}. In LSTSVM algorithm we have two lines for classification. As mentioned in Section \ref{SecII.3}, these lines should be the nearest to their corresponding class records and farthest from the opposite class records. Figure \ref{Fig4.b} shows these lines for LSTSVM. Because the data points overlap and they don't exactly lie on a line, the LSTSVM algorithm has still a large amount of error although it has higher accuracy compared to the SVM.\\
FLSTSVM also has two lines responsible for classifying data with the difference that these two lines are not crisp. Figure \ref{Fig4.c} shows the fuzzy lines of FLSTSVM. To show the fuzzy nature of each line, we have used multiple lines. As shown in the figure, these fuzzy lines discriminate the data points better than SVM and LSTSVM. Therefore FLSTSVM provides higher accuracy compared to the other two algorithms.
\begin{figure}[h!]
	\centering
	\begin{subfigure}[b]{0.28\textwidth}
		\includegraphics[width=\textwidth]{SVMonToydata.png}
		\caption{}
		\label{Fig4.a}
	\end{subfigure}%
	\hspace{0.03\textwidth}%
	\begin{subfigure}[b]{0.28\textwidth}
		\includegraphics[width=\textwidth]{LSTSVMonToydata.png}
		\caption{}
		\label{Fig4.b}
	\end{subfigure}
	\hspace{0.03\textwidth}
	\begin{subfigure}[b]{0.28\textwidth}
		\includegraphics[width=\textwidth]{FLSTSVMonToydata.png}
		\caption{}
		\label{Fig4.c}
	\end{subfigure}
	\caption{Decision lines obtained by (a). SVM, (b). LSTSVM and (c). FLSTSVM}
	\label{Fig4}
\end{figure}
\subsection{Experiments on benchmark datasets}
We also performed experiments on a collection of four benchmark datasets form UCI machine learning repository \cite{blake1998uci}. These datasets are Heart-Statlog, Australian Credit Approval, Liver Disorder and Breast Cancer Wisconsin. These datasets represent a wide range of size (from 198 to 690) and features (from 7 to 34). Details of the four datasets are listed in Table \ref{DataSets}. Also, Table \ref{Table3} lists the results of each algorithm. As shown in the table, FLSTSVM has higher accuracies compared to the other two algorithms. It should be noted once more that, in these experiments only the linear version of the FLSTSVM is considered (and so for the other two algorithms). We claim that the non-linear version of the proposed algorithm would outperform the non-linear version of SVM and LSTSVM with more meaningful differences.
\begin{table}[h!]
	\caption{Datasets}
	\includegraphics[width=3.2 in]{table2.pdf}
	\label{DataSets}
\end{table}

\begin{table}[h!]
	\centering
	\caption{Experimental results of SVM, LSTSVM, and FLSTSVM}
	\label{Table3}
	\begin{tabular}{m{2.5cm} | m{1cm} | m{1.2cm} | m{1.5cm}}
		\hline
		\multirow{2}{*}{\textbf{Dataset}} &
		\multicolumn{3}{c}{\textbf{Accuracy(\%)}} \\
		\cline{2-4}
		& SVM & LSTSVM & FLSTSVM \\
		\hline
		Heart-Statlog & 84.1 & 85.5 & 87.9 \\
		\hline
		Austrailian Credit Approval & 85.5 & 86.6 & 90.7 \\
		\hline
		Liver Disorder & 58.3 & 70.9 & 73.2 \\
		\hline
		Breast Cancer & 79.9 & 83.8 & 98.2 \\
		\hline
	\end{tabular}
\end{table}
\section{Conclusion}\label{SecV}
In this paper, we enriched LSTSVM classifier by incorporating the theory of fuzzy sets. We proposed two novel models for fuzzy LSTSVM. In the first model, $M_{1}$, a fuzzy membership was assigned to each input point and the hyperplanes were optimized based on fuzzy importance degrees of samples. In the second model, $M_{2}$, all parameters to be identified in LSTSVM are considered to be fuzzy. Also to discriminate the target class in $M_{2}$, we construct two fuzzy hyperplanes. We carried out a series of experiments to analyze our classifier against SVM and LSTSVM. The results demonstrate that FLSTSVM obtains better accuracies that the other two algorithms.As our future work, we plan to focus on non-linear version of the fuzzy LSTSVM algorithm.\\
Our method can be employed in the development of intelligent systems for decision making, pattern recognition, optimization, and control. It permits for the inclusion of vague assessments in classification problems. FLSTSVM can have several various applications in real life. Disease detection, image analysis, image denoising, weather forecasting systems, intrusion detection are some instances of its application. In addition to binary classification, it can applied on multi-class classification problems. FLSTSVM also can be employed in regression problem.

\noindent
Compliance with Ethical Standards:\\
Javad Salimi Sartakhti declares that he has no conflict of interest.
Nasser Ghadiri declares that he has no conflict of interest.
Homayun Afrabandpey declares that he has no conflict of interest.
Narges Yousefnezhad declares that she has no conflict of interest.\\

\noindent
Ethical approval:\\
This article does not contain any studies with human participants performed by any of the authors.

\bibliographystyle{apalike}