Norm convolution inequalities in $L_p$

Let $1\leq p\leq\infty$, $L_p\equiv L_p(\mathbb {R})$ and let the convolution operator be given by
$$ (Af)(x)=(K*f)(x)=\int_{{\mathbb R}} K(x-y) f(y) dy, \qquad K\in L_{\text{loc}}. $$

Let $d>0$ and let
– $M_1$ be the set of all intervals of length $\leq d$;
– $M_2$ be the set of all measurable sets $e\subset[-d,d]$ such that $\operatorname{diam}(e)=\sup_{x,y\in e}|x-y|\leq d$;
– $W_1$ be the set of all finite arithmetic progressions of integer numbers;
– $W_2$ be the set of all finite sets $w\subset{\mathbb Z}$ such that $\min_{i,j\in w}|i-j|\geq 2$.
Now we define the sets $\mathfrak{L}_{d}, \mathfrak{U}_{d}, \mathfrak{V}_{d}$ as follows:
\begin{align*} \mathfrak{L}_d&=\biggl\{E=\bigcup_{k\in w}(e+kd): e\in M_1, \, w\in W_1\biggr\}, \\ \mathfrak{U}_d&= \biggl\{E= \bigcup_{k\in w}(e_k+kd): e_k\in M_2, \, w\in W_2, \, |e_k|=|e_j|, \, k,j\in w \biggr\}, \\ \mathfrak{V}_{d}&= \biggl\{E=\bigcup_{x\in e}(x+w(x)d): e\in M_2, \,w(x)\in W_2, \, |w(x)|=|w(y)|, \, x,y\in e\biggr\}, \end{align*}
where $|e|$ is the measure of a set $e\in M_i$ and $|w|$ is the number of elements of $w \in W_i$. Note that $\mathfrak{L}_d\subset\mathfrak{U}_d\cap\mathfrak{V}_d$. If $E\in\mathfrak{L}_d$, then $|E|=|e||w|$, where $e$, $w$ are the sets from the representation of $E$. Similarly, this property holds for $E\in \mathfrak{U}_d$ and $E\in\mathfrak{V}_d$.
Theorem. Let $1<p<q<\infty$. If for some $d>0$ we have either
$$ \sup\limits_{E\in \mathfrak{U}_{d}}\frac{1}{|E|^{1/p-1/q}}\int_{E}|K(x)|\,dx\leq D $$
or
$$ \sup\limits_{E\in \mathfrak{V}_{d}}\frac{1}{|E|^{1/p-1/q}}\int_{E}|K(x)|\,dx\leq D, $$
then the operator $Af=K*f$ is bounded from $L_p(\mathbb R)$ to $L_q(\mathbb R)$ and
$$ \|A\|_{L_p\rightarrow L_q}\leq C(p,q) D, $$
where $C(p,q)$ depends on $p$ and $q$.
Theorem. Let $1<p<q<\infty$, $d>0$, and the operator $Af=K*f$ be bounded from $L_p({\mathbb R})$ to $L_q({\mathbb R})$. If for any $B>0$ we have
$$ \sup_{\substack{E\in \mathfrak{L}_d\\ |E|\leq B}}\frac{1}{|E|^{1/p-1/q}}\biggl|\int_{E}K(x)\,dx\biggr|\leq C(B)<\infty, $$
then
$$ \sup_{E\in \mathfrak{L}_d}\frac{1}{|E|^{1/p-1/q}}\biggl|\int_{E}K(x)\,dx\biggr|\leq C(p,q)\|A\|_{L_p\rightarrow L_q}. $$

Corollary. Let $1<p\leq q<\infty$ and $\lambda = 1-(\frac1p-\frac1q)$. Let also
$$ \mathcal{K}(x)= \frac{e^{i|x|^a}}{|x|^b}\mspace{2mu}, $$
where $a\neq 0,$ $a \neq 1$, and $b\neq \lambda$. If
$$ \max(q, p')>\frac{a}{\lambda-b}>0, $$
then the operator $Af=\mathcal{K}*f$ is not bounded from $L_p$ to $L_q$.