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On the regularization of a multidimensional integral equation in Lebesgue spaces with variable exponent. (English) Zbl 1278.45003
Math. Notes 93, No. 4, 583-592 (2013); translation from Mat. Zametki 93, No. 4, 575-585 (2013).
The first-kind multidimensional integral equation with potential-type kernel of small order is considered $\mathbf{M}^{\alpha}\varphi:=\int_{\mathbb{R}^n} \frac{c(x,y)}{|x-y|^{n-\alpha}}\varphi(y)dy=f(x), \quad x\in\mathbb{R}^n, \tag{1}$ where $$0<\alpha<1$$ and the function $$c(x,y)$$ satisfies the following conditions:
$c(x,y)\in C(\mathbb{R}^n\times \mathbb{R}^n),\quad c(x,x)\in L^{\infty}(\mathbb{R}^n), \quad \inf\limits_{x\in\mathbb{R}^n}|c(x,x)|>0;$

$|c(x,y)-c(z,y)|\leq \frac{C|x-z|^{\lambda}}{(1+|x|)^{\lambda}(1+|z|)^{\lambda}}, \quad \alpha<\lambda\leq 1,$ where $$C>0$$ is independent of $$x,y,z$$.
The integral equation (1) can be reduced to an equation of the second kind by regularization of the form $$\mathbf{RM}^{\alpha}\varphi=\varphi+\mathbf{A}\varphi$$, where $$\mathbf{R}$$ is some hypersingular operator, $$\mathbf{A}$$ is an operator compact in $$L^p(\mathbb{R}^n)$$, $$1<p<n/\alpha$$.
The authors extend this result to the case of Lebesque spaces $$L^{p(\cdot)}(\mathbb{R}^n)$$ with variable exponent.
##### MSC:
 4.5e+11 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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##### References:
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