This article is motivated by our papers treating homogenization of Dirichlet problem,
the classical paper by Phong and Stein, and a recent interest in PDE problems involving rough boundaries, in particular
the paper by Kenig and Prange. For a real-valued function $\psi \in C^\infty(\mathbb{R}^n)$ having bounded derivatives of all orders,
consider the hypersurface $\Gamma = \{ (y, \psi(y)) \in \mathbb{R}^{n+1}: \ y \in \mathbb{R}^n \} $.
For $f\in L^2(\mathbb{\mathbb{R}^n})$, $ \lambda>0$, and $(x, x_{n+1} ) \in \mathbb{R}^n \times \mathbb{R}$ define
$$
T_\lambda f (x, x_{n+1}) = \int_\Gamma e^{i \lambda x\cdot y} \varphi_0((x,x_{n+1}),y )
K(x-y, x_{n+1}-y_{n+1}) f(y ) d\sigma(y,y_{n+1}) ,
$$
where $d\sigma$ is the surface measure on $\Gamma$, $\varphi_0$ is a real-valued
function from the class $C_0^\infty (\mathbb{R}^{n+1} \times \mathbb{R}^n )$, and $K$ is a singular
kernel satisfying $K\in C^\infty(\mathbb{R^{n+1}} \setminus \{0\} )$ and
$| \nabla^\alpha K(z) | \lesssim_\alpha \frac{|z|^m}{|z|^{n+|\alpha|}}$ with $0\leq m < n$ for all $z\in \mathbb{R}^{n+1}\setminus \{0\}$ and
any multi-index $\alpha \in \mathbb{Z}^n_+$. Here we have $n\geq 1$ and do not assume that $m$
is necessarily an integer.
For each fixed $x_{n+1}$ 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 hypersurface $\Gamma$,
as that estimates are being used to analyse the behavior of the operator $T_\lambda$
under small perturbations of a given fixed surface $\Gamma$. The latter problem
is motivated by PDE problems with rough boundaries, where, for instance, the boundary can be technically $C^\infty$
however oscillating rapidly (and hence having large smoothness norms).
In that setting standard techniques by partial integration toward controlling operators of the form $T_\lambda$,
become inefficient, in view of the high oscillations of the surface.
The approach we propose here handles efficiently the effects coming from surface oscillations.
In a similar spirit, when we allow the surface to oscillate,
we consider a certain maximal operator associated with operators of the form $T_\lambda$
which captures the effect of the oscillation of the surface. More precisely, for a
family of hypersurfaces $\{\Gamma_\varepsilon\}_{0< \varepsilon\leq 1}$ (having a certain structure) we analyse the boundedness of the operator
$ T_\lambda^\ast 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+1} $
and $T_\lambda^\varepsilon$ is defined as above but for the surface $\Gamma_\varepsilon$.
Here the small parameter $0< \varepsilon \leq 1$ is meant to model an oscillatory behavior of the given
family of hypersurfaces.
The second part of the paper studies operators of the form $T_\lambda$ where instead of the linear phase $x\cdot y$
we consider a fractional-type nonlinearity in the form of $|x-y|^\gamma$ with real $\gamma \geq 1$.
This special choice of the phase function is partially motivated by PDE problems, in particular by the Helmholtz operator,
and requires a completely new approach. We then apply our analysis to eigenvalue problem for the Helmholtz equation
in $\mathbb{R}^3$ with some special source terms, and establish a decay estimate for solutions satisfying Sommerfeld radiation condition, as the eigenvalue tends to infinity.
The paper is essentially self-contained, and the proofs are based, among other things, on various decomposition arguments.
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